L IBOR M ARKET M ODELS - T HEORY AND A PPLICATIONS

B Y D AVID G LAVIND S KOVMAND

A dissertation submitted to

T H E F ACULTYOF S OCIAL S CIENCES

In partial fulfillment of the requirements for the degree of

Doctor of Philosophy in Economics and Management

U NIVERSITY OF A ARHUS D ENMARK

Table of Contents Preface v

Summary vii

Dansk Resumé (Danish Summary) ix

Chapter I 1 The Valuation of Callable CMS-spread bonds with floored coupons

Chapter II 53 Fast and Accurate Option Pricing in a Jump-Diffusion Libor Market Model

Chapter III 107 Alternative Specifications for the Lévy Libor Market Model: An Empirical Investigation

Preface

This thesis was written in the period October 2004 to January 2008, during my studies at the School of Economics and Management at the University of Aarhus. In this period I have spent an academic year at the Haas School of Business, University of California, Berkeley. The last 4 months have been spent at my new employer, the Aarhus School of Business, University of Aarhus. A number of people have contributed to the making of this thesis. First and foremost I thank my main advisor Professor Bent Jesper Christensen, who drafted me to the PhD program and sparked my interest in research. Secondly my co-advisor Professor Niels Haldrup I also thank for solid guidance along the way. A large thanks goes to my informal advisor, cooperator, and friend Professor Peter Løchte Jørgensen. Without our discussions and your advice on matters aca- demic, as well as real-world, I would never have gotten this far. From September 2006 to May 2007 I visited the Haas School of Business at University of California, Berkeley. My stay was extremely pleasant and I would like to thank Professor Greg Duffee for inviting me and Michael Verhofen for good collaboration and very fruitful discussions. I am also grateful to my new co-workers in the Finance Research Group at the Aarhus School of Business, especially Elisa Nicolato and Thomas Kokholm, for helping and encouraging me in the final stages of completing the thesis. The good people at Scanrate Financial Systems I thank for providing me with advice, data and friendship over the years. I look forward to continuing our collab- oration. At the University of Aarhus I thank my fellow PhD students for providing a friendly work environment as well as excuses to occasionally step away from the computer. The administrative staff I thank for assistance with practical matters concerning teaching and traveling. Søren Staunsager and his staff should also be mentioned in helping me numerous times with computer issues. Finally I thank my family and friends for putting up with me and for general encouragement, love and support.

David Glavind Skovmand, Arhus,˚ January 2008

v

Updated Preface

The pre-defense took place on April 1, 2008. I am grateful to the members of the assessment committee, Klaus Sandmann, Claus Munk and Peter Ove Christensen, for their careful reading of the dissertation and their many useful comments and suggestions. Many of the suggestions have been incorporated in the present version of the thesis while others remain for future work on the chapters.

David Glavind Skovmand, Arhus,˚ April 2008

vii

Introduction

This thesis consists of three self contained chapters with a common theme of pric- ing of interest rate derivatives using the Libor Market Model of Miltersen, Sand- mann, and Sondermann (1997), Brace, Gatarek, and Musiela (1997) and Jamshidian (1997). This model has become a favorite among banks and other financial institu- tions for pricing derivatives and managing risk. The original papers have spawned a new branch of academic literature that deals with the many theoretical as well as practical issues surrounding the Libor Market model and its extensions. This thesis can be considered an attempt at contributing to that paradigm. Summary of Chapters

Chapter I: The Valuation of Callable Bonds with Floored CMS-spread coupons

(With Peter Løchte Jørgensen) Published: Wilmott Magazine, December 2007 In this paper we investigate a new type of structured bond that has recently been introduced with enormous success - primarily among private investors - in many countries in Europe. The bonds are medium term and with fixed and very high initial coupons. The remaining coupons are determined as a constant multiplier times the spread between a long and a short swap interest rate. These coupons are floored at or near zero, and the bond investment can thus be seen as a bet on the steepening of future term structure curves. However, if the term structure becomes too steep, the bonds may be called by the issuer. The paper studies the pricing and the optimal call strategy of these highly exotic bonds in a stochastic interest rate framework. We implement two versions of the LIBOR Market Model as well as a Gaussian two-factor short rate model. We show how to adapt the Least-Squares Monte Carlo procedure to handle the callability of the product in a numerically efficient manner. We also calculate lower bounds for the product as well as delta and vega ratios. Chapter II: Fast and Accurate Option Pricing in a Jump-Diffusion Libor Market Model

Submitted to the Journal of Computational Finance This paper extends, improves and analyzes cap and swaption approximation formu- lae for the jump-diffusion Libor Market Model derived in Glasserman and Merener (2003).

ix More specifically, the case where the Libor rate follows a log-normal diffusion pro- cess mixed with a compound Poisson process under the spot measure is investigated. The paper presents an extension that allows for an arbitrary parametric specification of the log-jump size distribution, as opposed to the previously studied log-normal case. Furthermore an improvement of the existing swaption pricing formulae in the log-normal case is derived. Extensions of the model are also proposed, including displaced jump-diffusion and stochastic volatility. The formulae presented are based on inversion of the Fourier transform which is approximated using the method of cumulant expansion. The accuracy of the ap- proximations is tested by thorough Monte Carlo experiments and the errors are found to be at acceptable levels. Chapter III: Alternative Specifications for the L´evy Libor Market Model: An Empirical Investigation This paper introduces and analyzes specifications of the L´evy Market Model origi- nally proposed by Eberlein and Ozkan (2005). An investigation of the term structure of option implied moments shows that the Brownian motion and homogeneous L´evy processes are not suitable as modelling devices and consequently a variety of more appropriate models is proposed. Besides a diffusive component the models have jump structures with low or high frequency combined with constant or stochastic volatility. The models are subjected to an empirical analysis using a time series of data for Euribor caps. The results of the estimation show that pricing performances are improved when a high frequency jump component is incorporated. Specifically, excellent results are achieved with the 4 parameter Self-Similar Variance Gamma model which is able to fit an entire surface of caps with an average absolute per- centage pricing error of less than 3%.

x Dansk Resume(Danish Summary)

Denne afhandling indeholder tre selvstændige kapitler som alle omhandler prisfast- sættelse af rentederivater ved brug af Libor Market Modellen oprindeligt beskrevet i Miltersen, Sandmann, and Sondermann (1997), Brace, Gatarek, and Musiela (1997) og Jamshidian (1997). Denne model er blevet utrolig populær blandt banker og andre finansielle institutioner til derivatprisfastsættelse og risikostyring. De tre originale artikler har skabt en ny gren af akademisk litteratur som omhandler de mange teoretiske s˚avel som praktiske aspekter ved Libor Market Modellen og dens udvidelser. Denne afhandling kan betragtes som et forsøg p˚aat bidrage til dette paradigme.

xi References

Brace, A., D. Gatarek, and M. Musiela (1997): “The Market Model of Interest Rate Dynamics,” Mathematical Finance, 7(2), 127–154.

Eberlein, E., and F. Ozkan (2005): “The L´evy Libor Model,” Finance and Stochastics, 9, 327348.

Glasserman, P., and N. Merener (2003): “Cap and swaption approximations in Libor market models with jumps,” Journal of Computational Finance, 7(1), 1–36.

Jamshidian, F. (1997): “Libor and Swap Market Models and Measures,” Finance and Stochastics, 1(4), 261–291.

Miltersen, K. R., K. Sandmann, and D. Sondermann (1997): “Closed form solutions for term structure derivatives with log-normal interest rates,” Journal of Finance, 52(2), 409–430.

xii Chapter I

The Valuation of Callable Bonds with Floored CMS-spread coupons ∗

David Skovmand,† University of Aarhus and CREATES

Peter Løchte Jørgensen‡ Aarhus School of Business May 1, 2008

Abstract A new type of structured bond has recently been introduced with enormous success - primarily among private investors - in many countries in Europe. The bonds are medium term and with fixed and very high initial coupons. The remaining coupons are determined as a constant multiplier times the spread between a long and a short swap interest rate. These coupons are floored at or near zero, and the bond investment can thus be seen as a bet on the steepening of future term structure curves. However, if the term structure becomes too steep, the bonds may be called by the issuer. The paper studies the pricing and the optimal call strategy of these highly exotic bonds in a stochastic interest rate framework. We implement two versions of the LIBOR Market Model as well as a Gaussian two-factor short rate model. We show how to adapt the Least-Squares Monte Carlo procedure to handle the callability of the product in a numerically efficient manner. We also calculate lower bounds for the product as well as delta and vega ratios.

∗Financial support from the Danish Mathematical Finance Network is gratefully acknowledged. The authors are grateful for helpful comments from Mark Joshi, Svend Jakobsen, B.J. Christensen, and Malene Shin Jensen as well as from participants at the March 2007 World Congress on Com- putational Finance in London, the conference on Quantitative Methods in Finance in Sydney, December 2005, and at a seminar at Cass Business School in London. Any remaining errors are the authors’ responsibility. †Current affiliation: Aarhus School of Business and the Center for Research in Econometric Analysis of Time Series (CREATES), www.creates.au.dk. Corresponding address: Aarhus School of Business, Department of Business Studies, Fuglesangs All´e4, DK-8210 Aarhus V, Denmark, e-mail: [email protected] ‡Address: Aarhus School of Business, Department of Business Studies, Fuglesangs All´e 4, DK- 8210 Aarhus V, Denmark, e-mail: [email protected] 1 Introduction

The market for complex interest rate dependent products has been growing rapidly in the past two years. Fueled by the low interest rate environment investors have been seeking higher returns on their investments which has caused banks to come up with more and more complicated products to sell. This paper investigates one of these products called the callable Constant Matu- rity Swap (CMS)-steepener. This type of product is typically sold as a medium term bond. The bond provides the investor with a high fixed coupon (typically 6-10%) for an initial period but after this initial period the bond pays a floating coupon determined as some multiplum of the spread between a long and short constant maturity swap rate determined in arrears. The coupons are floored at or near zero to avoid negative payments. The bond can also be called at par on coupon dates after the initial period making it a Bermudan type product. From the investor’s point of view this type of product is essentially a bet on the steepening of the term structure. Other than in the unlikely event of issuer default, the worst case scenario for the investor is being locked into a below market coupon until maturity of the bond. The high initial coupons are meant to compensate for this risk. The issuer has a limited downside since he can call the bond at par if the CMS spread coupons he has to pay become too high. CMS spread related products (callable and non-callable) were extremely popular with investors in 2005 with an estimated $50 billion sold on a Worldwide basis (see Jeffery (2006)). However, the recent flattening of the term structure of interest rates has caused a significant drop in the secondary market value of the products. This has led to a wide criticism of issuers from the investor community where many feel they have purchased something they did not understand. Indeed the largest pool of buyers have been unsophisticated retail investors. Adding to the controversy banks have also been reluctant to disclose the profits made from CMS Steepeners. According to Sawyer (2005) there have even been suggestions that the banks themselves have had difficulties with both pricing and managing risk. In the Nordic region sales of CMS Steepeners have been particularly high. The market was primarily triggered by the huge success of Forstædernes Bank (Forbank) who – in cooperation with Dexia Bank – issued a DKK 2.4 Billion ($400 Million) callable CMS Steepener in the beginning of 2005. We use this particular issue as a case of study in our investigation of the products. The academic literature has dealt with these products only on a very abstract level where the most notable is Piterbarg (2004b). In a sense any callable bond can be viewed as a non-callable bond subtracted the value of the issuer’s embedded option to call the bond at par. Since the bond can be called only at discrete times

2 this embedded option is a Bermudan. Therefore Piterbarg (2004b) uses the well- known method of Least-Squares Monte Carlo (LSMC) (see Longstaff and Schwartz (2001), Carri`ere (1996), Tsitsiklis and Roy (1999) and Stentoft (2004)) to price the option part independent from the product itself. In this paper we propose a simpler and more efficient way of pricing these bonds that is based on pricing the entire bond instead of splitting it into a non-callable and an option part. We present the techniques in a framework that is general enough to be used for any issuer callable bond. A well-known issue in pricing American or Bermudan options using the LSMC procedure is finding the proper exercise strategies. This has been widely studied in the case of simple fixed for floating Bermuda swaptions (Andersen (2000) and Sven- strup (2005)). In the case of CMS spread callables this problem is more complex, and we therefore perform a numerical investigation to evaluate the quality of differ- ent exercise strategies. The technology to perform this quality check is developed in the papers of Andersen and Broadie (2004) and Haugh and Kogan (2004). The intuition behind these papers is basic: The price of any option is equal to the value of the hedge portfolio, but to hedge a Bermudan option we would need an optimal stopping rule. The LSMC procedure only provides us with an estimated stopping rule. The present value loss or duality gap from following a suboptimal stopping rule determines the quality of the stopping rule approximation. We adapt the procedure in Andersen and Broadie (2004) and show both theoretically and numerically how the duality gap can be calculated in the case of issuer callable bonds. We would not expect the pricing of these products to be independent of the underlying assumption of the distribution of interest rates. We therefore implement three different models of the term structure: The simple Gaussian short rate G2++ model of Brigo and Mercurio (2006) as well as the log-normal (see Miltersen, Sand- mann, and Sondermann (1997), Brace, Gatarek, and Musiela (1997), and Jamshid- ian (1997)) and the constant elasticity of variance (CEV) extended version of the LIBOR market model (see Andersen and Andreasen (2000) and Hull and White (2001)). We discuss the different properties of these models as well as the specifics of their calibration. Perhaps even more important than pricing are the sensitivities of the product – the deltas and vegas. Calculating these in the LIBOR market model is a non- trivial exercise. We show how the pathwise approach to sensitivities in Glasserman and Zhao (1999), Piterbarg (2004a), and Piterbarg (2004b) can be adapted to our specific case and our results are consistent with intuition. The paper is arranged as follows. First, we fix notation and describe the CMS Steepener in greater detail. Secondly, we describe the different models of the term structure. Third, we outline the LSMC procedure as well as a theoretical derivation of the duality gap of a generic suboptimal strategy. Fourth, we show how hedge

3 ratios are calculated and outline the numerical procedure which ends the theoretical part of our paper. Next we deal with the calibration and numerical implementation of the different models of the term structure. Finally, we present the numerical results and discuss the implications. We end with a conclusion.

2 Basic Setup

In this section we introduce some basic notation and briefly recapitulate some central valuation expressions regarding simple interest rate derivatives. These are used later in the paper mainly in relation to market calibration of our various dynamic models. The present section also describes the CMS Steepener and its valuation on an abstract level. We assume a standard probability space (Ω, , P) with sigma-algebra filtration ∞ F t t=0 and real world probability measure P. We also make the standard assump- tions{F } of no arbitrage and frictionless markets. For a tenor structure T0 <.....

P (t, Tk−1) P (t, Tk) Fk(t)= F (t, Tk−1, Tk)= − , (1) τkP (t, Tk)

where P (t, Tk) denotes the time t price of a zero coupon bond which matures at Tk. + A call option on the forward rate giving the payoff τk(Fk(Tk−1) K¯ ) at Tk is referred to as a caplet. The time t value of the caplet can be expressed− as

F Cpl(t)= P (t, T )τ E k [(F (T ) K¯ )+], (2) k k t k k−1 − where Fk refers to the forward measure with corresponding numeraire P (t, Tk) (see e.g. Bj¨ork (2004)). As is well-known, the forward rate Fk(t) must be a martingale under this measure. A Tα (Tβ Tα) payer swap is a contract that entitles the holder to a series of floating× LIBOR− payments in exchange for paying a series of fixed payments at times Tα+1 <.....

P (t, Tα) P (t, Tβ) Sα,β(t)= β − . (3) i=α+1 τiP (t, Ti) P 4 An equivalent representation of the forward swap rate (see e.g. Rebonato (2002)) is

β

Sα,β(t)= wi(t)Fi(t), i=α+1 X with

τiP (t, Ti) wi(t)= β . (4) j=α+1 τjP (t, Tj)

This expression shows that the forwardP swap rate can be seen as a weighted average of forward LIBOR rates. Note, however, that the weights will be changing in a stochastic manner over time. It is often more convenient to express the spot swap rate in terms of the length of the swap tenor, l:

S (T ) S (T ), (5) l i ≡ i,β(Ti+l) i where β(t)= i if t

PS(t)= C (t)Eα,β[(S (T ) K¯ )+], α,β t α,β α − α,β where Et [ ] denotes expectation formed under the swaption measure Cα,β. The · β associated numeraire is defined as Cα,β(t)= i=α+1 τiP (t,ti). We define a callable CMS Steepener as a bond with a principal of 1 and payments P CFi at t1 <... ti ...c≤  i l2 i−1 − l1 i−1 where R is the fixed initial coupon, and Sl2 and Sl1 are the swap rates set in arrears as defined in (5) with l2 > l1. These rates are referred to as Constant Maturity Swap

5 (CMS) rates because as time passes the rates have a fixed time to maturity.1 m is a constant multiplier (e.g. 3) and f is a small positive constant (e.g. 5 bps) which effectively floors the coupon payments just above zero (for tax reasons). Let us now consider the general valuation at time t, t

N N CF δ F (t ) C = N E i − i i i−1 . (7) t t t N "i=η+1 ti # X The entire bond value can therefore be written as

N CF V = N EN N −1 + i C , (8) t t t tN N − t " i=1 ti # X where the first term on the right-hand side of course corresponds to the value of an otherwise identical non-callable CMS Steepener bond. The LSMC algorithm can be implemented on both (6) and (8). We shall argue below that it will be computationally more efficient to work with the single expression (6) than to work with the decomposition in (8) as is often done in similar contexts (see e.g. Piterbarg (2004b)).

3 Model Choice

In order to evaluate the model risk in pricing the CMS Steepener we will value this structured bond using three distinct models for the evolution of the term structure

1A CMS rate is not the fixed rate that sets the price of a Constant Maturity Swap equal to zero. A Constant Maturity Swap is more general expression than a standard swap in that the floating LIBOR rates are normally replaced with a longer term fixed maturity rate for example the 20 year swap rate. Pricing a Constant Maturity swap is a non-trivial exercise (see e.g. Hunt and Kennedy (2004)). 2The choice of valuation date before the first coupon date is made purely for notational sim- plicity. Valuation at later dates entails no further complications.

6 of interest rates. We start out with the 2-factor classical Gaussian model termed the G2++ model and proposed by Brigo and Mercurio (2006). This model’s main advantage is tractability. Since the model has Gaussian state variables a closed form expression for their transition densities can be derived. This makes the evaluation of complex floating payoffs particularly easy. But the tractability comes at a price and the model is not able to simultaneously price a large segment of the vanilla instruments used for calibration. An entirely different approach is the so-called market models pioneered by Mil- tersen, Sandmann, and Sondermann (1997), Brace, Gatarek, and Musiela (1997), and Jamshidian (1997). These models differ from the classical interest rate models in that they model directly the evolution of discretely compounded market rates such as the LIBOR or the swap rate. As we will see, the main advantage of these models is the ease at which they can be calibrated to price a large segment of vanilla products such as caps and swaptions. Many different market models exist and in this paper will use the standard log-normal forward LIBOR market model as well as the CEV-extension proposed by Andersen and Andreasen (2000).

3.1 The G2++ Model

In the G2++ model the instantaneous short rate under the risk-neutral measure, Qc, is given by

r(t)= x(t)+ y(t)+ ϕ(t), r(0) = r0, (9) with stochastic factor processes, x( ) and y( ), described by the Ornstein-Uhlenbeck system · ·

dx(t)= ax(t) dt + σdW (t), x(0) = 0, − 1 dy(t)= bx(t) dt + ξ dW (t), y(0) = 0, − 2 with

dW1(t) dW2(t)= ρdt.

In this dynamic system W1(t) and W2(t) are correlated Brownian motions, a, σ, b and ξ are positive constants, and 1 ρ 1. In (9), ϕ( ) is a deterministic function used to match the initial observed− term≤ ≤ structure of interest· rates. The model is affine and the zero coupon bond price can thus be written as

P (t, T )= A(t, T ) exp B(a,t,T )x(t) B(b, t, T )y(t) , (10) {− − } 7 where P M (0, T ) 1 A(t, T )= exp [V (t, T ) V (0, T )+ V (0,t)] , P M (0,t) 2 −   1 e−z(T −t) B(z,t,T )= − , z σ2 2 1 3 V (t, T )= T t + e−a(T −t) e−2a(T −t) a2 − a − 2a − 2a   ξ2 2 1 3 + T t + e−b(T −t) e−2b(T −t) b2 − b − 2b − 2b   σξ e−a(T −t) 1 e−b(T −t) 1 e−(a+b)(T −t) 1 + 2ρ T t + − + − − , ab − a b − a + b   where P M (0, ) denotes the market observed discount function. The forward LIBOR and swap rates· can now be generated in closed form through formula (1) and (3)-(4). A pricing formula for swaptions is described in the Appendix. For further details we refer to Brigo and Mercurio (2006).

3.2 LIBOR Market Models The log-normal LIBOR market model (LMM) was derived with the intent of having a model that was consistent with the standard market practice of using the Black- 76 formula to value caplets. Consistence with the Black-76 formula requires the discretely compounded or simple forward rate to be a log-normal martingale under the forward measure, and short rate models are not able to achieve this. The HJM- model by Heath, Jarrow, and Morton (1992) is similar to LMM in spirit but models the instantaneous forward rate instead of the simple forward rate. HJM was also the starting point for one derivation of the market model of Brace, Gatarek, and Musiela (1997), and indeed Hunt and Kennedy (2004) show that the market models can be seen as a subset of the HJM model class. A PDE approach to market models can be found in Miltersen, Sandmann, and Sondermann (1997) but perhaps a more natural derivation which is also the one used by most textbooks is the change-of- numeraire technique advocated by Jamshidian (1997). We refer to these papers or the textbooks by Brigo and Mercurio (2006) and Rebonato (2002) for an extensive treatment. The log-normal assumption of the forward rate distribution was known to be at odds with reality long before LIBOR Market Models were invented. Therefore several extensions have been proposed some of which still nest the log-normal version. The goal of the extensions was to have a model that realistically replicates the so-called volatility skew observed in market prices of interest rate options. The

8 volatility skew refers to the decreasing curve of Black-76 volatilities as a function of strike observed in market prices. The simplest extension is the CEV model investigated in Andersen and Andreasen (2000) and Hull and White (2001). This model uses a different distribution for the forward rate which allows for replication of simple versions of the skew. In reality the skew is more like a hockey-stick shape and if one wants to replicate this pattern exactly one would need stochastic volatility or jumps in the LIBOR rate process. This causes problems with calibration as well as inherent difficulties with hedging as these type of models describe incomplete markets. We therefore stick to the CEV-extension of Andersen and Andreasen (2000) as this retains completeness. The general version of the LIBOR market model we work with is the following:

dF (t)= σ (t)φ(F (t))dZ (t), t T , k = 1,...,M, k k k k ≤ k−1 ∀ where Zk(t) is a Brownian motion under the forward measure with numeraire P (t, Tk). Note that forward rates are martingales under their respective measures as they should be. We also assume constant correlations, i.e.

dZ (t)dZ (t)= ρ dt, i, j = 1,...,M. i j i,j ∀ The choice of φ-function determines the distribution of the forward rates. We look at two different specifications,

φ(x)= x, φ(x)= xp , 0

The first of these yields a log-normal distribution of the forward rates and the second implies a non-central χ2-distribution (see Andersen and Andreasen (2000)). The two most important products we need to price are caplets and swaptions. If we were to price a caplet the current formulation of the model would suffice since a caplet depends on one forward rate only. Recall from (2) that the time t price of a caplet on F ( ) and expiring at time T is given as k · k F Cpl(t)= P (t, T )τ E k [(F (T ) K¯ )+]. k k t k k−1 − With φ = x this immediately leads to the well-known Black-76 pricing formula, cf. Appendix B.2. When φ(x) = xp the distribution of the forward rate is skewed to the left and the Black-76 formula no longer applies. This skewing of the distribution causes in-the-money caplets to be priced higher and out-of-the-money caplets to be priced lower implying a volatility skew. A closed-form formula can also be derived and is shown in Appendix B.2. For swaptions no exact pricing formula exists in our reference LIBOR market models and one must resort to approximations or simulation to calculate prices.

9 In such a simulation we would need realizations from the joint distribution of the forward rates comprising the underlying swap, and we therefore need the dynamics of all relevant forward rates under a single measure. As it turns out a forward rate Fk(t) is only driftless under its ”own” measure with numeraire P (t, Tk). The no arbitrage conditions determine the drift under other numeraires. The usual risk neutral measure (Qc cf. earlier) takes the continuously accrued bank account as the numeraire, but since we are working with discrete rates it is more convenient to work under the discrete money market measure which we denote by Qd and which has the discrete bank account as numeraire. The discrete bank account is defined as follows:

B0 = 1, B = B [1 + τ F (T )], 1 k

dFk(t)=σk(t)φ(Fk(t))(µkdt + dZk(t)), (11) k ρ τ σ (t)φ(F (t)) µ = 1 k,j j j j , k {t

4 Pricing Methodology

Standard Monte Carlo simulation techniques cannot be used for pricing callable products like the CMS Steepener since these methods do not identify the stopping

10 rule (call strategy) necessary for characterizing the value of the security. One way to estimate such a stopping rule goes via the Least-Squares Monte Carlo procedure normally attributed to Tsitsiklis and Roy (1999), Carri`ere (1996), and Longstaff and Schwartz (2001). This procedure has countless extensions and applications where Piterbarg (2004b) is the most notable in our context since he focuses exclusively on the LIBOR Market Model. The idea of the LSMC procedure as used in Piterbarg (2004b) is described in the following. To price any American or Bermudan product we essentially need two things ev- erywhere in the state space: The exercise value and the continuation value of the product. In general neither of these are readily available so they have to be esti- mated. Consider, for example, the option embedded in the callable CMS Steepener whose value was defined in (7). Once this option is exercised the holder ”receives” a series of payments determined by CFi δiFi(ti−1). The value of these payments cannot be calculated in closed form for− the LIBOR Market Models or the G2++ model. We must therefore either approximate or simulate the value. Piterbarg (2004b) shows in a quite general setting how this value can be approximated by regression. However, if instead of focusing on the option to call the bond we look at the bond in its entirety, then the exercise value for the issuer is simply the principal and therefore readily observable. Although this simple insight avoids an entire regression step in the LSMC procedure it is rarely used in the literature. Notable exceptions are Joshi (2006) and Amin (2003) which both use this ”trick” in different contexts. To the authors’ knowledge it has never been applied in the case of issuer callable bonds, and we will therefore more carefully describe the procedure in a setting appropriate for these types of bonds. We assume an economy with a time t vector of state variables Xt. The objective of our analysis is to price a bond with maturity tN and a pricipal of 1. The bond is issued at time t0 = 0, has payments CFi on payment dates = t1,...,ti,...,tN , and is callable at par on all payment dates. The valuation ofT this{ bond at issuance} is an optimal stopping problem of type η P N −1 −1 V0 /N0 = infE0 [Nt + Nt CFi]. η η i i=1 X The superscript on V indicates that this is the solution to what we shall later refer to as the Primal problem. As before N is a martingale measure with corresponding numeraire Nti at time ti. Note also that the index of exercise η is unknown and therefore a stopping time in a mathematical sense. To price the product we need an estimate of the stopping time, or a stopping rule. To obtain such an exercise strategy we proceed as follows: At each exercise time the issuer can either pay back the principal or wait until the next exercise time. If we assume that the coupons

11 are reinvested in numeraire bonds the (accumulated) value at an exercise time is therefore i −1 N −1 Vti = min 1+ Nti Ntk CFk , Nti Eti Nti+1 Vti+1 . (12) ! Xk=0 h i A more convenient way of writing this is to define

i Q = N EN N −1 V N N −1CF ti ti ti ti+1 ti+1 − ti tk k h i Xk=0 = N EN N −1 CF + min(1, Q ) , ti ti ti+1 { i+1 ti+1 } h i and then rewrite (12) in terms of Qti as follows,

i −1 Vti = Nti Ntk CFk + min (1 , Qti ) . Xk=0

Vti cannot be directly evaluated since it depends on the exercise behavior of the issuer at ti+1. The LSMC procedure is essentially an algorithm to approximate

Vti by simulation. But instead of approximating Vti directly we approximate the

Qti ’s instead. The two approaches are essentially equivalent as one can always calculate one from the other using the above formula. But approximating Qti means that we avoid approximating coupons that have already been determined. for the r×n+1 approximation of Qi we choose for each time a polynomial function f( , ) :R Rr R where n is the order of the polynomial · · × + → Q f(β ; R(X )=β(0) + β(1)R (X )+ β(2)R (X )2 + ...β(n)R (X )n (13) ti ≈ i ti) i i 1 ti i 1 ti i 1 ti . . (14) (r×n−n+1) (r×n−n+2) 2 (r×n) n +βi Rr(Xti )+ βi Rr(Xti ) + βi Rr(Xti ) ··· (15)

Xti is a M dimensional vector of state variables and R(X)=(R1(X),..., IRr(X)) M where the Ri’s are vector functions R+ R+ that each map the state variables into a key rate determined a priori to be important→ for the continuation value of the bond. The important thing here is that the rates used must be measurable with respect

to the filtration ti , as the issuers decision to call can only be based on information known at the callF time. Choosing these rates is not an exact science, and it is very specific the particular problem at hand. In general one should include rates that determine the level and the slope of the term structure at a given point in time.

12 Piterbarg (2004b) also advocates the use of a so-called core swap rate. That is the swap rate that sets on the day of exercise and expires on the day the entire bond matures. For the case of CMS Steepeners we have found that using the short forward rate, the core swap rate, and the two swap rates that determine the coupons give excellent results.

As the functional form for f( ; R(Xti )) we use polynomial functions which seems to be the industry standard. Choosing· f to be linear in β is a smart choice from a computational perspective since it allows us to estimate β with standard ordinary least squares (OLS). This choice does not appear to be overly restrictive (see Piter- barg (2004b)) and we will therefore use the linear model throughout the paper. The LSMC procedure goes as follows:

N 1. Simulate K paths of the state variables Xti (ωj) i=1 and calculate the nu- N { N } meraires Nti (ωj) i=1 and coupons CFi(ωj) i=1 for each path j = 0,...,K, ω Ω. { } { } j ∈ 2. At the terminal date t set Q (ω ) = 0 j = 0,...,K, ω Ω. N tN j ∀ j ∈

3. At the last exercise time tN−1 set −1 QtN−1 (ωj)= NtN−1 (ωj)NtN (ωj) (1 + CFN (ωj)) K R − (X − (ω ))Q (ω ) ˆ j=1 N 1 N 1 j tN−1 j and calculate the OLS estimator βN−1 = K ′ . j=1PRN−1(XN−1(ωj )) RN−1(XN−1(ωj )) P 4. For the next exercise time tN−2 set Q (ω )= N (ω )N −1 (ω ) CF (ω )+min(1, f(βˆ ; R (X (ω )) , tN−2 j tN−2 j tN−1 j { N−1 j N−1 N−1 tN−1 j } and calculate the OLS estimator βˆN−2 in the same manner as in step 3.

5. Repeat previous step until the first the exercise time t1.

We now have an exercise strategy through the estimated parameters of the fi- functions. We can define the indicator function for exercise as ˆ 1 if 1 f(βi; Ri(Xti (ωj)) or ti = tN Ii(ωj)= ≤ . 0 if 1 > f(βˆ ; R (X (ω )))  i i ti j The stopping time index for each path is

η(ω )= min(i [1,...,N] : I (ω )=1). j ∈ i j Finally, the MC price estimate of the bond price can the be calculated as follows

K η(ωj ) 1 −1 −1 Vˆ0 = N0 Nt (ωj) + N (ωj)CFi(ωj) . (16) K  η(ωj ) ti  j=1 i=1 X X   13 4.1 Lower bounds

The method described in the previous section does not guarantee that the chosen fi- functions are optimal. This means that exercise might occur at sub-optimal points in time. If this is the case the estimate in (16) will not converge to the true value of the bond as K is increased. Exercise is done by the issuer who seeks to minimize the value of the bond. Therefore if exercise is done sub-optimally the value of the bond will be higher than in the optimal case. One can therefore interpret the price in (16) as an upper bound on the true bond price. The method does not give any measure on how far one is from the true price and the estimate (16) could thus in principle be severely upward biased. In this section we therefore show how to calculate a corresponding lower bound estimate to complement the upper bound. The difference between the lower and upper bound is called the duality gap and can be seen as a measure of the quality of the estimated exercise strategy. The problem of calculating a lower bound is similar to calculating an upper bound for American options. In the case of straight American options it is the holder of the option that exercises, and the challenge in that case is therefore to establish an upper bound since a suboptimal strategy provides a downward biased estimate. A method of finding proper upper bounds for American options using simulation has been recently developed in three seminal papers by Rogers (2002), Haugh and Kogan (2004) and Andersen and Broadie (2004) (for other methods see Glasserman (2004)). A less mathematical approach is that of Joshi (2006) that gives a very good economic interpretation of the upper bound. In all the cited cases the tool to calculate the upper bound is the concept of duality which is well-known in, for example, the operations research literature. For more details on this subject we refer to the above cited papers and the references therein. Below we show how a lower bound in relation to issuer callable bonds can be developed using the ideas from the American options literature. The main points can easily be lost in the mathematics and we therefore end with an economic argument based on the intuition provided in Joshi (2006). Recall that the problem of pricing an issuer callable bond is that of computing

η P N −1 −1 V0 /N0 = inf E0 [Nt + Nt CFi)], (17) η∈I η i i=0 X for a series of exercise times = t1,...,tN , with indices I = 1,...,N . We have earlier termed this the primalT problem.{ Now,} for an arbitrary {finite submartingale}

Mtn – abbreviated Mn – we can define a so-called dual function F (n, Mn) as follows,

η N −1 −1 F (n, M)= Mn + Et min Nt + Nt CFi Mη) . n η∈[n,N]∩I η i − " i=1 !# X 14 The dual problem consists of finding the maximum submartingale,

η V D/N = sup F (0, M) = sup M + EN min(N −1 + N −1CF M ) . 0 0 0 0 I tη ti i η M M η∈ − " i=1 #! X (18)

Recall from the previous section that we defined Vtn as the value of the bond newly issued at time tn plus the coupons up until that time invested in numeraire bonds:

n −1 N −1 Vtn = min 1+ Ntn Nti CFi , Ntn Etn Ntn+1 Vtn+1 . (19) i=0 ! X   We can now show the following theorem:

Theorem 1: Given the primal and dual problems in (17) and (18), the follow- ing statements are true

P D 1. The duality relation holds i.e V0 = V0 .

∗ 2. An optimal solution Mn for the dual problem is the discounted bond price

process Vtn /Ntn . Proof: See Appendix A. 2

The dual problem can now be used to construct a lower bound for the bond price. We know the optimal submartingale, but solving this process is equivalent to solving the primal problem. But if we where to use an arbitrary submartingale instead we get the inequality,

η ∗ N −1 −1 ∗ V0/N0 =M0 + E0 min(Nt + Nt CFi Mt ) η∈I η i − " i=1 # Xη N −1 −1 M0 + E0 min(Nt + Nt CFi Mt) . (20) ≥ η∈I η i − " i=1 # X Hence any submartingale will give a lower bound. Suppose we use the suboptimal ˜ strategy estimated in the previous section and let Vti be defined as Vt in (19) when ˜ Vt1 following this estimated suboptimal strategy instead. Set M0 = V˜0/N0 and M1 = Nt1 and

V˜ V˜ V˜ n M = M + tn+1 tn + I EN tn+1 N −1 + N −1CF . n+1 n N − N n tn N − tn ti i tn+1 tn " tn+1 i=1 !# X 15 We first notice that this is actually a martingale. Assume that the suboptimal strategy says ”continue” at time tn, that is In = 0. In this case we have:

˜ ˜ N N Vtn+1 Vtn Etn [Mn+1]=Mn + Etn = "Ntn+1 − Ntn # ˜ ˜ N Vtn+1 N Vtn+1 Mn + Etn Etn = Mn. "Ntn+1 − "Ntn+1 ##

This means that Mn is a martingale in the continuation region. If we assume the strategy says exercise at tn so In = 1, then we have

V˜ n EN [M ]=M + EN tn+1 N −1 + N −1CF tn n+1 n tn N − tn ti i " tn+1 i=1 !# X V˜ n EN tn+1 N −1 + N −1CF = M . − tn N − tn ti i n " tn+1 i=1 !# X

So Mt is also a martingale in the exercise region. Martingales are also submartingales so we can insert this in equation (20) to obtain a lower bound. Hence, a lower bound for the bond can be written at time 0:

η L ˜ N −1 −1 V0 = V0 + E0 min(Nt + Nt CFi Mη) . (21) η∈I η i − " i=1 # X The first term is the upper bound calculated from the LSMC algorithm described in the previous section. At the first exercise time (or at maturity if the bond is never called) the exercise value is always equal to M. This means that the second term will always be less than or equal to zero. L V0 can be interpreted as the price the buyer of the bond will be willing to pay, whereas the price in (16) can be seen as the sellers price. The argument goes as follows. The seller of the bond decides when he wants to call the bond. This may or may not be at an optimal time. The buyer wants to be hedged even if the bond is called optimally, but without knowledge of the optimal strategy, exact replication is not possible. But the buyer can form a subreplicating strategy based on selling the bond himself and following a suboptimal exercise strategy. The initial value of the L hedge will be V0 . When following this strategy 4 situations can occur at a given point in time:

1. The buyer and seller exercise. In this case their prices will agree and the hedge will be perfect.

16 2. The buyer and seller do not exercise. Again the prices will agree. 3. The seller exercises and the buyer does not. If the time is optimal the buyer will take a loss on his hedging position which is financed by selling numeraire bonds. 4. The buyer exercises and the seller does not. The buyer can resell the product with 1 less exercise date to continue the hedge. If the time was not optimal then losses from reselling are financed by selling numeraire bonds. In all 4 cases the buyer’s price will be equal to or below that of the seller’s. All the small losses are discounted back to present time. The losses are equal to the second term in (21) which is called the duality gap. We define this term as follows: η L ˜ N −1 −1 ∆0 = V0 V0 = E0 min(Nt + Nt CFi Mη) . (22) − η∈I η i − " i=1 # X This term is calculated through simulation. The algorithm is as follows 1. Estimate an exercise strategy using the technique from the previous section. N 2. Simulate K paths of the state variables Xti (ωj) i=1 and calculate the nu- N { N } meraires Nti (ωj) i=1 and coupons CFi(ωj) i=1 for each path j = 0,...,K, ω Ω. { } { } j ∈ 3. For the first path, ω1, do the following: Start from t1 and calculate an estimate, Mˆ 1, using a sub-simulation with K1 subpaths. The subpaths are evolved from time t1 in path ω1. Repeat this for all possible exercise times creating a series of estimates Mˆ 1,..., Mˆ N−1. −1 n −1 ˆ Now calculate Ntn (ω1) + i=1 Nti (ω1) CFi(ω1) Mn n = 1,...,N and find the minimum. − ∀ P 4. Repeat the previous step for all paths ωj and average the result to get an estimate of duality gap ∆ˆ 0.

With N exercise times, K paths, and K1 subpaths the computational time for calculating the lower bound is approximately of order K N K1 and therefore extremely slow. × ×

5 Hedging and Sensitivities

The hedging of derivatives is in general at least as important as pricing, and we therefore dedicate this section to some considerations on the determination of the two most important hedge ratios – the delta and the vega. This analysis focuses mainly on the LIBOR market models.

17 5.1 Deltas

Delta denotes the sensitivity of the price with respect to shifts in the initial vector of forward rates

F¯(0) = (F0(0),F1(0), . . . . ., FM (0)). (23)

The mth individual delta ratio is defined as follows

∂V0 ∆m = . (24) ∂Fm(0)

One can easily calculate a first order approximation for this number by shifting the initial term structure and calculating the corresponding shift in the price using the methodology provided in the previous sections. If we denote the shifted price V¯0 and rate F¯m(0) then

∂V V¯ V 0 0 − 0 . (25) ∂F (0) ≈ F¯ (0) F (0) m m − m In simple cases the payoff will be a smooth function of the initial curve making this “bump-and-revalue” approach stable enough for practical purposes. However, in our case the price of a callable CMS Steepener is not a smooth function of the initial term structure. The reason is the callable nature of the product. The decision to call the bond at a point in time is binary and a small shift in the initial rates could therefore change this decision. The corresponding price change could be quite large since the amount of payments on the relevant path will have changed. Another reason to avoid using (25) is the computational cost since a new price would have to be calculated for each perturbation. A superior method that both alleviates the instability problem as well the com- putational cost is the pathwise approach described in Glasserman and Zhao (1999) and refined in Piterbarg (2004a) and Piterbarg (2004b). The idea is as follows. Recall the forward rate dynamics

dF (t)=σ (t)φ(F (t))(µ dt + dZ (t)), t T , k = 1,...,M, k k k k k ≤ k−1 ∀ k ρ τ σ (t)φ(F (t)) µ = 1 k,j j j j . k {t

18 with respect to rates, ′ d(∆mFk(t)) =σk(t)φ (Fk(t))∆mFk(t)(µk(t, Fk(t))dt + dZk(t)) M ∂µ (t, F¯(t)) +σ (t)φ(F (t)) k ∆ F (t) k k ∂x m j j=0 j ! X t T , k = 1,...,M, ≤ k−1 ∀ where ′ ∂µk (φ (Fj(t))(1 + τjFj(t)) φ(Fj(t))τj) τjρk,jσj(t) = − 2 , ∂xj (1 + τjFj(t)) xj =Fj (t)

φ(x)=xp,

φ′(x)=pxp−1. The starting value of this SDE is 1 for m = k ∆ F (0) = . m k 0 for m = k  6 An approximation that saves computational time is fixing the drifts at the time zero value of the forward rates. This avoids having to recalculate the drift for each time step. This results in the following SDE, ′ d(∆mFk(t)) =σk(t)φ (Fk(t))∆mFk(t)(µk(t, Fk(0))dt + dZk(t)) M ∂µ (t, F¯(0)) +σ (t)φ(F (t)) k ∆ F (t) , (26) k k ∂x m j j=0 j ! X t T , k = 1,...,M. ≤ k−1 ∀ Glasserman and Zhao (1999) show that the errors resulting from this approximation are insignificant. We can now calculate the delta of the product. Given an estimate of the optimal exercise time η we have seen that the initial bond price can be represented as η ˜ N −1 −1 V0 = E0 Btη + Bti CFi . " i=1 # X The deltas can now be calculated as follows η ˜ N −1 −1 ∆mV0 =E0 ∆m(Btη + Bti CFi) " i=1 # X η η N −1 N −1 N −1 =E0 ∆mBtη + E0 ∆m(Bti )CFi) + E0 Bti ∆m(CFi) . (27) " i=1 # " i=1 # h i X X 19 Notice that when differentiating through we have ignored that the optimal time η also depends on the initial forward rates. Piterbarg (2004a) gives a formal argument why this is reasonable based on the smooth pasting conditions of the optimization problem. The first two terms are calculated as follows with t [t ,t ], ∈ n n+1 n τ ∆ (B−1)= B−1 i ∆ F (t ) m t − t 1+ τ F (t ) m i i−1 i=1 i i i−1 X t t B−1 − n ∆ F (t ). − t 1+(t t)F (t )(1 + τ F (t )) m n+1 n n+1 − n+1 n n n+1 n For details see Piterbarg (2004a). The last term in (27) is more difficult. For l>s we have τ max((S (t ) S (t )) m, f) for η i > T CF = i l i−1 s i−1 lockout , i τ R − ∗ for 0 ≥ i T  i ≤ ≤ lockout where R is the fixed rate in the lockout period. A standard way to approximate the swap rates is the following formula due to Rebonato (2002),

β(ti−1+l) β(ti−1+l) S (t )= w (t )F (t ) w (0)F (t ) l i j i j i ≈ i j i j=i−1 j=i−1 X X β(ti−1+s) β(ti−1+s) S (t )= w (t )F (t ) w (0)F (t ) s i j i j i ≈ i j i j=i−1 j=i−1 X X β(t)=m if t

β(ti−1+l)

∆m(Sl(ti−1)) = (∆mwj(0)Fj(ti)+ wj(0)∆mFj(ti)) . j=i−1 X From simple calculations we get that

1 1 ∆ w (0) = τ w (0) . m j m j β(t − +l) k − 1 τmFm(0) 1+ i 1 1  − k=m j=i−1 1+τj Fj (0)   This term is very small especially for large l andP can inQ practice be ignored but we include it since it comes at little computational cost as it can be calculated

20 before any simulation is performed. We can now evaluate all the relevant terms and calculate the deltas in equation (26) by Monte Carlo. Note that the approach described above is not applicable to G2++ model since this is not a model of forward rates. In this case we therefore use (25). The standard error of the delta estimate depends on the size of the rate shift or perturbation F¯m(0) Fm(0). Because of the discontinuity this might not be at the smallest possible− level. We have experimented with several shifts and found that a 1 basis point perturbation gives decent results.

5.2 Vegas

Vega denotes the price sensitivity with respect to the volatility in the market. In practice traders not only perform delta hedging but also vega hedging in order to be hedged against moves in volatility. One can form a vega hedging portfolio in several ways by using volatility dependent products such as cap, floors and swaptions. We have initially chosen to calibrate our models to a large segment of swaptions. How- ever, finding the hedge portfolio of swaptions is difficult since swaption volatilies are not direct inputs to the versions of the LIBOR market model used in this paper. One could proceed with the naive bump-and-revalue approach by shifting a market price and then recalibrating the model. In our case this does not give meaningful results since the perturbation will be obscured by calibration error. One way to get meaningful swaption vegas would therefore be to start over and apply a local calibration procedure (see for example Brigo and Mercurio (2006) or Pietersz and Pelsser (2006)) instead of our more global approach. This procedure fits the model to a small subset of swaptions without error. This subset could for example be the most liquid or the core swaptions. The downside of the local approach is of course that you take the chance of fitting the model perfectly to prices that might reflect noise such as illiquidity and non-synchronous trading.3 This avenue is beyond the scope of our paper and we therefore proceed with calculating the sensitivities of the volatilities of the individual rates which amounts to finding a vega hedge portfolio of caplets. This approach also has the advantage of having a more direct interpretation with respect to forward rates than the swaption approach. We proceed in a manner similar to the previous section. We calculate the price sensitivity of a unit shock to the volatility curve for each rate and we denote each vega: ∂V0 . This partial derivative can be calculated using the pathwise approach ∂σm in a similar manner as we did for the deltas in the previous section. Differentiating

3An alternative to this is the indirect approach described in Piterbarg (2004b). We have tried this approach but where not able to get meaningful results.

21 through and fixing the drifts we now get the following SDE: ∂ ∂ d( F (t)) = φ(F (t))1 + σ (t)φ′(F (t)) F (t) (µ (t, F (0))dt + dZ (t)) ∂σ k k {m=k} k k ∂σ k k k k m  m  M ∂µ (t, F¯(0)) ∂ +σ (t)φ(F (t)) k F (t) , k k ∂x ∂σ j j=0 j m ! X t T , k = 1,...,M. ≤ k−1 ∀ ∂ Note that Fk(0) = 0 m and ∂σm ∀ ′ ∂µk (φ (Fj(t))(1 + τjFj(t)) φ(Fj(t))τj) τjρk,jσj(t) = − 2 ∂xj (1 + τjFj(t)) xj =σj

φ(Fj(t))τjρk,j + 1{j=k} . (1 + τjFj(t)) Besides the extra indicator function term this is the same as in the previous section. We can now calculate the vegas of the CMS Steepener bond as follows: η ∂ ∂ V˜ =EN (B−1 + B−1CF ) ∂σ 0 0 ∂σ tη ti i m " m i=1 # X η η ∂ ∂ ∂ =EN B−1 + EN (B−1)CF + EN B−1 CF . 0 ∂σ tη 0 ∂σ ti i 0 ti ∂σ i  m  " i=1 m # " i=1 m # X X (28) The three terms are calculated with Monte Carlo as in the previous section but with ∆ replaced by ∂ . m ∂σm Since prices are quoted in terms of Black-76 implied volatility, it is necessary to express the vegas in this metric. By using the chain rule of partial differentiation we have ∂V˜ ∂σ ∂V˜ = BS76 ∂σm ∂σm ∂σBS76 ˜ ∂V˜ ∂V ∂σm = ∂σ (29) ⇒ ∂σBS76 BS76 ∂σm The numerator is calculated in 28. The denominator cannot be calculated in closed form for the CEV model but it can easily be found by numerical differentiation. For the standard LMM case we of course have σm = σBS76 so the vegas are given in the proper metric from 28 directly In the G2++ model there is, in our setup, no proper way to calculate vegas. A pathwise approach as well as a bump-and-revalue approach will be too noisy since all volatility parameters in the G2++ model have a global influence.

22 6 Case Study: The Dexia Dannevirke 2005/2016 CMS Steepener

In this section we further investigate the particular issue of ”Dexia Dannevirke”. This was the largest structured bond issue ever seen in Denmark with DKK 2.4 billion (about EUR 320 million or USD 425 million) sold. The success of this particular issue incited the larger share of all Nordic banks to issue a string of similar products. The bond was issued on January 25, 2005 which we denote time zero. The initial fixed coupon was set at 9.55% and the floating coupons were determined as 3 times the floored (at 5bps) difference between the 20 and the 2 year EURIBOR swap rate determined in arrears at the previous coupon time. The following payments CFi were specified, δ 9.55% if i 2 CF = i i δ max [3[S (t ) S (t )], 0.05%] if 23≤ i> 2,  i 20 i−1 − 2 i−1 ≥ where δi is determined from a 30/360 day-count fraction. The payments are made on the dates listed in Table 1. The bond can be called by the issuer at time t2 and onwards. If the bond is called the coupon on the call date is paid as well as the principal which we for simplicity assume is 1.

7 Model Calibration

The concept of calibration is to fit the parameters in the model so they are able to reproduce the relevant prices observed in the market with a certain amount of ac- curacy. There is naturally a tradeoff between the number of calibration instruments used and the fitting quality. We therefore make a selection of calibration instru- ments that we deem most relevant for the the callable CMS Steepener described in the previous section. From Jyske Bank we have obtained market quotes on Euro denominated EU- RIBOR caps and swaptions. All market quotes are in Black-76 implied volatility. The swaptions in the European market are settled annually and the caps are settled either quarterly or semiannually. Both the LMMs and the G2++ model are marked to market with the zero curve available from Datastream. All rates and quotes are from January 25, 2005 – the day the bond was issued.

7.1 G2++ The G2++ model has a limited number of parameters which again limits the amount of calibration instruments. Since the bond we are aiming to price depends mainly on

23 swap rates we choose to calibrate to swaptions only. A subsection of the swaption matrix of implied volatilities can be seen in Table 2. The bond price depends among other things on the twenty year swap rate that sets in 11 years. This means that swaptions with a timespan that stretches further than 31 years are of little relevance to the bond price. Excluding this portion of the swaption matrix leaves us with 106 datapoints to fit. The G2++ model has a closed form solution for the swaption price. This formula is shown in Appendix B.1. Following the market convention we translate the model prices into volatilities by inverting the Black-76 formula. We then minimize the distance between model volatilities and market volatilities to get the optimal set of parameters, Θ˜ = (˜a, ˜b, σ,˜ η,˜ ρ˜), using the following Root-Mean-Square distance metric,4

2 1 σi,j σi,j (a, b, σ, ξ, ρ) Θ˜ = arg min market − model . (a,b,σ,η,ρ)v106 i,j u M T σmarket ! u 1≤i≤10,1≤j≤10,ti +tj ≤31 u X t i,j M Here σmarket is the implied volatility of a payer swaption with maturity date ti and T i,j swap length (tenor) equal to tj , and σmodel( ) is the corresponding model implied volatility from inverting formula (31) in the Appendix.· Minimization is done in the Ox programming language using the Simulated Annealing optimization procedure to get starting values and the standard BFGS optimization routine to refine the solution. As Table 3 shows the errors are tolerable for smaller tenors but increase up to 13% for larger tenors and swaption maturities. This can be expected from a model with only five parameters and 106 datapoints to fit. Note from the table that we estimate a correlation between the factors equal to 0.891 similar to the case investigated in Brigo and Mercurio (2006). Since the fact− ors are theoretical constructs this number does not have a direct interpretation. But with a value different from 1 or -1 it does mean that the model is able to pick up some of the information on the correlation between rates available in the swaption matrix.

7.2 LIBOR Market Models The LIBOR Market Model in its currently stated form does not readily lend itself to calibration. First recall the general market model under the forward numeraire. With tenor structure T < < T we have 1 ··· M dF (t)= σ (t)φ(F (t))dZ (t), t T , k = 1,...,M, k k k k ≤ k−1 ∀ 4We also tried minimizing prices instead and achieved almost identical results.

24 where

dZi(t)dZj(t)= ρi,jdt.

Since we have semiannual payments in the bond we are currently analyzing, we choose forward rates with equidistant maturities of 6 months, meaning Tk Tk−1 = 0.5. The furthest maturity we need to deal with in pricing is 31 years. This− means that TM = 31 and the number of rates is 62. In the log-normal case, ie. when φ(x) = x, we have only the volatilities σk(t) and the correlations ρi,j to determine. As recommended by Rebonato (2002) and Brigo and Mercurio (2006) we use the following parametric form for the volatilities,

σ (t) = Φ [a(T t)+ d] e−b(Tk−1−t) + c . k k k−1 − This specification is very tractable. It allows the volatilities to
exhibit the typical humped shape observed in the market. It is also time-inhomogeneous since the term Φk is forward rate dependent which allows a better initial fit to the market. We do however set the Φ’s pairwise equal:

Φ = Φ , k odd . k k+1 ∀ This eases calibration as it reduces the number of parameters to estimate. In un- constrained calibration a common problem is that the parameters oscillate between forward rates yielding an unrealistic evolution of the volatility (see Brigo and Mer- curio (2006)). To avoid this we impose a regularity constraint,

Φ Φ < 0.1, k. | k+1 − k| ∀

−b(Tk−1−t) The second term, [a(Tk−1 t)+ d] e + c, is homogeneous through time and depends only on time to− maturity of the forward rate. This term determines the shape of the volatility structure. The correlations are parametrized through the well-known approach justified in Schoenmakers and Coffey (2000), j i ρ = exp[ | − | (ln(ρ ) i,j −M 1 − ∞ − i2 + j2 + ij 3Mi 3Mj + 3i + 3j + 2M 2 M 4 + η − − − − 1 (M 2)(M 3) − − i2 + j2 + ij Mi Mj 3i 3j + 3M + 2 η − − − − )]. − 2 (M 2)(M 3) − − This particular approach yields a correlation matrix that is flexible enough to have the particular characteristics observed in several empirical studies, see e.g. the survey

25 in Rebonato (2002)). In order to facilitate direct comparison with the G2++ model we choose to calibrate to the same swaption data as in the previous section.5 A practical issue often ignored is the fact that the LMM models semiannual rates whereas the EURIBOR swaptions we calibrate to are settled annually. Throughout the paper we therefore use the following adjusted swap rate for a Tα (Tβ Tα)- swap. × −

β P (t, Tα) P (t, Tβ) Sα,β(t)= (α−β)/2 − = wi(ti)Fi(t), i=1 P (t, Tα+2i) i=α+1 X P P (t, Ti) wi(t)= (β−α)/2 . i=1 2τiP (t, Tα+2i) This swap rate is describedP in Schoenmakers (2002). As before we minimize the distance between the market quoted volatilities and the model implied volatilities using a root mean square distance metric, yielding the optimal parameters:

Θ=(˜˜ a, ˜b, c,˜ d,˜ Φ˜ , η˜ , η˜ , ρ˜ )= { k}k≥1 1 2 ∞ 2 1 σi,j σi,j (a, b, c, d, Φ ,η ,η ,ρ ) arg min market − model { k} 1 2 ∞ . v106 i,j u M T σmarket ! u 1≤i≤10,1≤j≤10,ti +tj ≤31 u X t i,j Here σmarket is the Black-76 implied volatility of a payer swaption with maturity date M T i,j ti and swap length (tenor) equal to tj , and σmodel( ) is the corresponding model Black-76 volatility from formula (32) in the Appendix.· We use the constrained optimization algorithm of Lawrence and Tits (2001) to obtain the minimum distance parameters. In Table 4 we report the corresponding pricing errors and in Table 5 the estimated parameters. We obtain an average error of 1.7% which might seem high when using 39 parameters to fit 106 datapoints. But 32 of the parameters (the Φ’s) are heavily constrained and have only a local influence. If we look at the correlation parameters we see that the second parameter η2 has reverted to zero as in the calibration experiments performed in Schoenmakers and Coffey (2000). We also observe that ρ∞ = 0.41. This parameter is interpreted as the limiting correlation between rates when distance between maturity is increased. Looking at the 3d-plot of the correlation matrix in Figure 1 we can observe two important features. The first is decorrelation, i.e. the fast reduction in dependency when the distance between maturity is increased. The second is the increase in

5The LMM with the current parametrization can also be calibrated to both swaptions and caps but as described in Rebonato (2002) there are inherent differences between the swaption and the cap market that complicate simultaneous calibration to both markets.

26 interdependency between equidistant rates as maturity is increased. Graphically this means that the steepness of the surface when moving away from the diagonal is decreasing.

7.3 The CEV LMM We also perform a market calibration of the CEV LIBOR market model which has φ = xp. As in the previous section we assume the same parametric form for volatilities and correlation. The difference is that now we have an extra parameter p to determine. This parameter is highly dependent on the skew of the implied volatilies of interest rate options. Ideally we would like to determine p from OTM- Swaptions, since swap rates are our main concern. But data for these products are not readily available as the market has very low liquidity. We therefore follow Hull and White (2001) and use OTM-Caplets instead.6 We have access to prices quoted in Black-76 volatilities for 13 different strikes ranging from 1.5% to 10%. The procedure is in two steps. First we calibrate p and (a, b, c, d, Φk k≥1,η1,η2,ρ∞) from OTM-Caplet prices using formula (33) in the Appendix.{ Sec}ondly we fix p at the value estimated in step 1 and recalibrate (a, b, c, d, Φk k≥1,η1,η2,ρ∞) to swaption implied volatilities by using formula (35) using the{ values} from step 1 as starting values in the optimization procedure.7 Again optimization is done using the standard root mean square distance metric. The errors we get in Table 6 are tolerable and comparable to the LMM case.

8 Numerical Results

8.1 Simulation Details We use an Euler discretization of the forward LIBOR dynamics under the discrete money market measure given in the SDE in (11). We choose stepsizes so each coupon payment date/exercise time t1,...,t23 are hit. We use 4 Euler steps between each coupon date totaling 92 steps per path. In the the pre-simulation step we use 25000 antithetic paths (total 50000) to estimate the parameters of the exercise strategy. In the pricing step we use another 25000 (50000) paths. The sub-simulations involved

6Caps are settled quarterly for maturities below 2 years and semiannually thereafter. We therefore translate quarterly volatilities into semiannual volatilies using the procedure in Brigo and Mercurio (2006). Note also that caplet volatilites are stripped from cap volatilities using the standard “bootstrap” procedure also described in Brigo and Mercurio (2006). 7Note that we do not have a closed formula for the Black-76 volatility in the CEV case but only a pricing formula. We therefore plug in the price from formula (35) and invert the Black-76 formula.

27 in calculating the lower bound are extremely slow and we therefore estimate this with only 25 (50) antithetic paths for 200 (400) sub-simulations. To further increase speed we have also employed a so-called Runge-Kutta drift approximation in the sub-simulations only. This significantly increases the speed as we avoid having to continuously calculate the state dependent drift for each Euler step. The errors from this approximation are in general extremely small (see for example Rebonato (2002) and Glasserman and Zhao (1999)). Zero coupon bond prices setting on non- tenor dates are calculated using the constant interpolation technique described in Piterbarg (2004a). The short rate dynamics of the G2++ model is simulated in all cases without discretization error under the terminal forward measure with numeraire P (t,t23). We use the same amount of paths as in the LIBOR Market Models. For details on this we refer to Brigo and Mercurio (2006).

8.2 Prices In Table 8 we have calculated the theoretical price of the Dexia Dannevirke bond with a 10000 bp principal. The value of the call option was calculated by finding the difference in values to the price of a similar non-callable bond valued using the same random numbers which can be done at almost zero computational expense. We used a second order polynomial of the core swap rate, the 6m LIBOR rate and the 20 and 2 year swap rate with all rates setting on the respective exercise dates to calculate the exercise strategy. We have experimented with many other rates such as medium term swap rates and other LIBOR rates as well as higher order polynomial functions. In all of these cases the duality gap was larger, in some cases higher than 100 bp. The results of one of these experiments are in Table 9 in the Appendix where we have used a 4th order polynomial function as well as included more rates. From this table we can see that the duality gaps have increased with an order of almost 8, and the option prices have been approximately halved. These results corroborate the existing notion in the literature (see for example Piterbarg (2004b)) that basis functions should be kept simple. We have also supplied a 95% confidence interval which is calculated as follows:

2 2 sV s∆ sV V˜0 ∆˜ 0 z1−α/2 + ; V˜0 + z1−α/2 ,  − − s K K1 √K    where sV and s∆ are the standard deviations for the upper bound and the duality gap respectively. In Table 8 we see that the prices for the different models are statistically different from each other. The fact that the log-normal model gives higher prices than the CEV model is expected since the non-central chi-square distribution of the

28 CEV model gives more weight to the lower end of the distribution of forward rates. The normal distribution of the G2++ model implies even more weight to the lower end than the CEV model and therefore yields the smallest price of the three. In Table 10 we have tabulated the probability of exercise at each exercise time point. The overall distribution is very similar for the 3 models having at most 28.09% percent chance of early exercise. This is consistent with papers such as Andersen and Andreasen (2001) and Svenstrup (2005) which investigate the exercise strategies from lower factor models in Bermudan swaptions. The only thing that stands out is the large discrepancy at the final exercise time t22, where the G2++ model has more than 5 times the higher probability of exercise. One reason for this discrepancy is that there is less than one month between the last exercise time t22 and maturity at t23. The small day-count fraction makes the coupon payment at t23 very small and therefore the exercise decision is more idiosyncratic and prone to error. Perhaps one of the most interesting things to observe from this section is the implied estimate of the profit the issuing bank must have made from selling these bonds. The bonds where sold at par (10,000bps) and DKK 2.4 billion worth of bonds where sold. Looking at the lower end of the confidence interval for the G2++ model and the upper end for the log normal LMM in Table 8 suggests that the profit margin for Forbank ranges between 6.8% to 9.07%. In monetary terms this translates to a profit ranging from DKK 163.2 million to DKK 217.6 million. Con- sidering that this profit is riskless if a proper hedge portfolio is formed, the amounts are considerable. If we make the somewhat unfair and unscientific extrapolation of this result to a USD 50 billion market of CMS spread related products (see Jeffery (2006)) then these type of products have made considerable profits to the issuers.

8.3 Deltas In figure 2 we show the delta profile for the product for each forward rate in the term structure. All deltas are in basis points. The LIBOR market model deltas are calculated using the pathwise approach and the G2++ model deltas are calculated by the standard bump-and-revalue approach with a 1bp perturbation. The effect of shifting the initial term structure has two opposite effects. The downwards effect is due to two reasons. One is simple and due to higher discounting of the coupons; the other is due to the fact that higher initial interest rates make the short CMS rate higher causing smaller coupons. We can see that the effect begins to shift around rates setting in 13 years or more which is natural since these rates only affect the long CMS rate and not the short one. Overall the delta ratios are very close to each other for the different models. This would suggest that all models are appropriate for delta hedging.

29 8.4 Vegas The vegas in Figure 3 are calculated with the pathwise approach for the LMM and CEV-LMM for the volatility of all forward rates. We notice a somewhat complex vega profile with large differences in the two models. The specific complex pattern is mainly due to the negative vega component in the product. As the volatility of the 2 year CMS rate increases the price of the coupons decreases. However, the volatility of the 2 year CMS rate cannot increase without also affecting the 20 year swap, and hence the effects will offset each other at some points. Looking closer at Figure 3 we see that rates of low maturity have close to zero vega but as maturity increases the vegas begin to increase as the rates affect more and more coupon payments. This effect is offset by the negative vega component in coupon payments which causes the dip around rates of 7-9 years maturity. The vegas increase again around 11 years which is the maturity date of the bond, but drop as maturity increases further due to the decreasing effect they have on coupons.

9 Conclusions

This paper has presented a methodology to price and risk manage the so-called callable CMS Steepener structured bond. We have used the simple G2++ model for the short rate as well as two different LIBOR market models. We have adapted the Least-Squares Monte Carlo procedure and shown how looking at the bond in its entirety can avoid a regression step. We have also adapted the procedure of Andersen and Broadie (2004) to give a downward biased price estimate of issuer callable bonds. This is an important supplement to the upward biased estimate of the LSMC procedure as the gap between the two determines the quality of the exercise strategy. We then proceeded to show how hedge ratios can be calculated. We have applied the theoretical insights on a particular callable CMS spread bond. We have found that the different models of the term structure yield signifi- cantly different prices when calibrating to the same data and using the same exercise strategy. We have also shown that our specific exercise strategy gives duality gaps of a tolerable size. An interesting bi-product of our analysis is an estimate of the sizeable profits the banks must have made from selling this particular bond. To the authors’ knowledge this paper is the first about these particular products and we have therefore opted to use relatively simple models of the term structure of interest. However, the use of stochastic volatility and/or jumps in the forward rates are becoming more and more normal in both academia and practice (see Hagan, Kumar, Lesniewski, and Woodward (2002) and Glasserman and Kou (2003)). Our results on pricing could easily be extended to these cases. However, in the area of hedging there are still many unanswered questions in the more advanced model

30 setups. Future work on CMS spread callables could be to evaluate the hedging per- formance of different models over time. At lot of work has been done recently on simpler products such as European and Bermudan swaptions (see Pietersz and Pelsser (2006) and the references therein) and the conclusions have mostly been that properly calibrated simple models perform very well against more complicated alternatives. Our results for delta hedging also point in that direction but further analysis are still needed.

31 A Proof of Theorem 1

For any finite submartingale we have η P N −1 −1 V0 /N0 = inf E0 [Nt + Nt CFi)] η∈I η i i=0 Xη N −1 −1 = inf E0 [Nt + Nt CFi) Mη + Mη] η∈I n i − i=0 Xη N −1 −1 inf E0 [Nt + Nt CFi) Mη]+ M0 ≥η∈I η i − i=0 X η N −1 −1 E0 min Nt + Nt CFi Mη + M0. (30) ≥ η∈I η i − " i=0 !# X The first inequality follows from from the optional sampling theorem for submartin- gales (see for example Hoffmann-Jorgensen (1994)). Let us now set Mn = Vtn /Ntn P where M0 = V0 /N0. We first observe Mn is a submartingale as V n M = tn = min N −1 + N −1CF ,EN N −1V n N tn ti i ti ti ti+1 tn i=0 ! N XN   E N −1 V = E [M ] . ≤ tn tn+1 tn+1 tn n+1 Plugging this into the dual problem gives us

η D P N −1 −1 Vtη V0 V0 + N0E0 min(Nη + Nt CFi ) . ≥ η∈I i − N " i=1 tη # X The second term in the above equation is positive since the exercise value of the bond will always be larger than or equal to the continuation value when following D P an optimal strategy. We therefore have V0 V0 . Taking the infimum of (30) we D P ≥ D P get V0 V0 which proves the duality relation V0 = V0 attained at the optimal ≤ ∗ submartingale Mn = Vtn /Ntn .

B Calibration formulas

B.1 G2++ Model The formula for a payer swaption is rather involved and it is stated here for com- pleteness. For the proof we refer to Brigo and Mercurio (2006).

32 Consider an interest rate swap with nominal value of 1 and strike K¯ and payments at t <.....

− 2 ∞ −1/2 x µx n e ( σx ) PS(0, K¯ )= Φ( h (x)) λ (x)eκi(x)Φ( h (x)) dx, (31) √ − 1 − i − 2 −∞ σx 2π " i=0 # Z X where

2 y¯ µy ρxy(x µx) h1(x)= − − σ 1 ρ2 − σ 1 ρ y − xy x − xy h (x)=h (px)+ B(b, t ,t )pσ 1 ρ2 2 1 0 i y − xy −B(a,t0,ti)x q λi(x)=ciA(t0,ti)e x µ κ (x)= B(b, t ,t ) µ 1/2(1 ρ2 )σ2B(b, t ,t )+ ρ σ − x , i − 0 i y − − xy y 0 i xy y σ  x  and σ2 σξ σ2 ρσξ µ = + ρ 1 e−at0 + 1 e−2at0 + 1 e−t0(a+b) x − a2 ab − 2a2 − b(a + b) −   ξ2 σξ   ξ2   ρσξ   µ = + ρ 1 e−bt0 + 1 e−2bt0 + 1 e−t0(a+b) y − b2 ab − 2b2 − a(a + b) −   1 e−2at0       σx =σ − r 2a 1 e−2bt0 σy =ξ − r 2b ρσξ −(a+b)t0 ρxy = 1 e . (a + b)σxσy −   B.2 LIBOR Market Models Caplet prices in the log-normal LIBOR market model with tenor structure = T < < T coincide with the Black(76) formula. The price of a caplet T 1 ··· M paying at time Ti the difference between the strike K¯ and the forward rate setting at Ti−1 is

Cpl(0, T , T , K,v¯ )=P (0, T )τ Ei(F (T K¯ )+) i−1 i i i i i−1 − =P (0, T )τ F (0)Φ(d ) K¯ Φ(d2) i i i 1 −   33 where log(F (0)/K¯ )+ v2/2 d = i 1 v log(F (0)/K¯ ) v2/2 d = i − 2 v Ti−1 2 v = σi (t)dt. Z0 Swaption prices in the log-normal LMM cannot be derived in closed form. However, a well established approximation due to Rebonato (2002) can be used. Consider again an interest rate swap with nominal value of 1 and strike K¯ and payments at the tenor structure ˜ = T˜ <.....< T˜ and denoteτ ˜ = T˜ T˜ . The price of T α+1 β i i+1 − i a payer swaption at time t = 0 which gives the right to enter the swap at T˜α is

PS( ,K)(0) = C (t) S (0)Φ(d ) K¯ Φ(d ) , T α,β α,β 1 − 2 where   ¯ 2 log(Sα,β(0)/K)+ vα,β/2 d1 = vα,β ¯ 2 log(Sα,β(0)/K) vα,β/2 d2 = − vα,β β ˜ w (0)w (0)F (0)F (0)ρ Tα v2 = i j i j i,j σ (t)σ (t)dt, (32) α,β S (0) i j i,j=1 α,β 0 X Z β ˜ where Cα,β(0) = i=α+1 τiP (0, Ti) is the present value of a series of basis points also referred to as the swaption numeraire. The weights wi follow from the fact that the swap rate can beP written as a weighted sum of forward rates,

β

Sα,β(t)= wi(t)Fi(t). i=α+1 X In this paper the tenor structure of the LIBOR Market Model is semiannual and this does not coincide with the tenor structure of the swaptions in the European market that are annually settled. The weights in the above formula is therefore determined as follows P (t, T˜ ) w (t)= i . i (β−α)/2 ˜ i=1 2τiP (t, Tα+2i) P 34 For further details we refer to Schoenmakers (2002). In the extended LMM cap prices can be derived in closed form. With a CEV parameter 0

Cpl(0, T , T , K¯ )=P (0, T )τ Ei((F (T ) K¯ )+) (33) i−1 i i i i i−1 − =P (0, T )τ F (0)(1 χ2(a, b + 2,c)) Kχ¯ 2(c, b, a) , (34) i i i − − where   K¯ 2(1−p) a = (1 p)2v − 1 b = 1 p − F (0)2(1−p) c = i (1 p)2v − Ti−1 2 v = σi(t) dt, Z0 where χ2( , x, y) is the distribution function for a non-central χ2-distributed ran- dom variable· with non centrality parameter value x and y degrees of freedom. For standard approximations of this function we refer to classical textbook of Johnson and Kotz (1973). Again we use an approximation due to Andersen and Andreasen (2000) to find the swaption value:

PS( ,K)= C (0) S (0)(1 χ2(a, b + 2,c)) Kχ¯ 2(c, b, a) , (35) T α,β α,β − − where   K¯ 2(1−p) a = (1 p)2v − 1 b = 1 p − S (0)2(1−p) c = α,β (1 p)2v − β p p T˜α 2 wi(0)wj(0)Fi(0) Fj (0)ρi,j vα,β = 2p σi(t)σj(t)dt. i,j=1 Sα,β 0 X Z

35 References

Amin, A. (2003): “Multi-Factor Cross Currency LIBOR Market Mod- els: Implementation, Calibration and Examples,” preprint, available from http://www.geocities.com/anan2999/.

Andersen, L. (2000): “A Simple Approach to the Pricing of Bermudan Swaptions in the Multifactor LIBOR Market Model,” Journal of Computational Finance, 3(2), 5–32.

Andersen, L., and J. Andreasen (2000): “Volatility Skews and Extensions of the Libor Market Model,” Applied Mathematical Finance, 7, 1–32.

(2001): “Factor Dependence of Bermudan Swaptions: Fact or Fiction,” Journal of Financial Economics, 62(1), 3–37.

Andersen, L., and M. Broadie (2004): “Primal-Dual Simulation Algorithm for Pricing Multidimensional American Options,” Management Science, 50(9), 1222– 1234.

Bjork,¨ T. (2004): Arbitrage Theory in Continuous Time, Second Edition. Oxford University Press.

Brace, A., T. Dun, and G. Barton (2001): Towards a Central Interest Rate Model. Cambridge University Press.

Brace, A., D. Gatarek, and M. Musiela (1997): “The Market Model of Interest Rate Dynamics,” Mathematical Finance, 7(2), 127–154.

Brigo, D., and F. Mercurio (2006): Interest Rate Models, Theory and Practice, 2nd Edition. Springer Finance.

Carriere,` P. (1996): “Valuation of Early-Exercise Price of Options using Simu- lations and Nonparametric Regression,” Insurance: Mathematics and Economics, 19(1), 19–30.

Glasserman, P. (2004): Monte Carlo Methods in Financial Engineering,. Springer Verlag.

Glasserman, P., and S. Kou (2003): “The Term Structure of Simple Forward Rates with Jump Risk,” Mathematical Finance, 13(3), 383–410.

Glasserman, P., and X. Zhao (1999): “Fast Greeks by Simulation in Forward LIBOR models,” Journal of Computational Finance, 3(1), 5–39.

36 Hagan, P. S., D. Kumar, A. S. Lesniewski, and D. E. Woodward (2002): “Managing Smile Risk,” Wilmott Magazine, pp. 84–108.

Haugh, M. B., and L. Kogan (2004): “Pricing American Options: A Duality Approach,” Operations Research, 52(2), 258–270.

Heath, D., R. Jarrow, and A. Morton (1992): “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valua- tion,” Econometrica, 60(1), 77–105. Hoffmann-Jorgensen, J. (1994): Probability with a View toward Statistics. Chapman & Hall.

Hull, J., and A. White (2001): “Forward Rate Volatilities, Swap Rate Volatili- ties, and Implementation of the Libor Market Model,” Journal of Fixed Income, 10(2), 46–63.

Hunt, P. J., and J. E. Kennedy (2004): Financial Derivatives in Theory and Practice. Wiley. Jamshidian, F. (1997): “Libor and Swap Market Models and Measures,” Finance and Stochastics, 1(4), 261–291. Jeffery, C. (2006): “Getting Flattened,” Risk Magazine, 19(2).

Johnson, N. L., and S. Kotz (1973): Continuous Univariate Distributions, vol. 2. Houghton-Mifflin Company Boston. Joshi, M. (2006): “Monte Carlo Bounds for Callable Products with Non-Analytic Break Costs,” Working Paper available www.markjoshi.com.

Lawrence, C., and A. Tits (2001): “A Computationally Efficient Feasible Se- quential Quadratic Programming Algorithm,” SIAM Journal of Optimization, 11(4), 1092–1118.

Longstaff, F., and E. Schwartz (2001): “Valuing American Options by Sim- ulation: A Simple Least-Square Approach,” Review of Financial Studies, 14(1), 113–147.

Miltersen, K. R., K. Sandmann, and D. Sondermann (1997): “Closed form solutions for term structure derivatives with log-normal interest rates,” Journal of Finance, 52(2), 409–430.

Pietersz, R., and A. Pelsser (2006): “A Comparison of Single Factor Markov- Functional and Multi Factor Market Models,” Unpublished Working Paper.

37 Piterbarg, V. (2004a): “Computing Deltas of Callable Libor Exotics in Forward Libor Models,” Journal of Computational Finance, 7(2), 107144.

(2004b): “Pricing and Hedging Callable Libor Exotics in Forward Libor Models,” Journal of Computational Finance, 8(2), 65–119.

Rebonato, R. (2002): Modern Pricing of Interest Rate Derivatives, The Libor Market Model and Beyond. Princeton University press.

Rogers, L. (2002): “Monte Carlo valuation of American options,” Mathematical Finance, 12(1), 271–286.

Sawyer, N. (2005): “A Difference of Opinion,” Risk Magazine, 18(10), 20–22.

Schoenmakers, J. (2002): “Calibration of LIBOR models to Caps and Swaptions: A Way round Intrinsic Instabilities via Parsimonious Structures and a Collateral Market Criterion,” Working Paper, Weierstrass Institute.

Schoenmakers, J., and B. Coffey (2000): “Stable Implied Calibration of a Multi-factor LIBOR Model via a semi-parametric correlation structure,” Working Paper, Weierstrass Institute.

Stentoft, L. (2004): “Convergence of the Least Squares Monte Carlo Approach to American Option Valuation,” Management Science, 50(9), 1193–1203.

Svenstrup, M. (2005): “On the Suboptimality of Single-factor Exercise Strategies for Bermudan Swaptions,” Journal of Financial Economics, 78(3), 651–684.

Tsitsiklis, J., and B. V. Roy (1999): “Optimal Stopping of Markov Processes: Hilbert Space Theory, Approximation Algorithms, and an Application to Pric- ing High Dimensional Financial Derivatives,” IEEE Transactions on Automatic Control, 44, 1840–1851.

38 Tables and Graphs

Table 1: Payment dates

t1 20-06-2005 t13 20-06-2011 t2 20-12-2005 t14 20-12-2011 t3 20-06-2006 t15 20-06-2012 t4 20-12-2006 t16 20-12-2012 t5 20-06-2007 t17 20-06-2013 t6 20-12-2007 t18 20-12-2013 t7 20-06-2008 t19 20-06-2014 t8 20-12-2008 t20 20-12-2014 t9 20-06-2009 t21 20-06-2015 t10 20-12-2009 t22 20-12-2015 t11 20-06-2010 t23 25-01-2016 t12 20-12-2010

39 Table 2: Swaption Matrix (of implied volatilities in percent)

M T Maturity ti / Tenor tj 1234567 1 22.3 21.8 21.1 20.5 19.7 18.6 17.8 2 22.2 21.3 20.2 19.1 18.1 17.4 16.7 3 20.6 20.0 19.1 17.9 17.1 16.4 16.0 4 19.5 18.9 17.9 16.8 16.2 15.7 15.4 5 18.3 17.7 16.8 15.9 15.4 15.0 14.8 10 15.1 14.5 13.9 13.3 13.0 12.9 12.8 15 13.0 12.5 12.0 12.0 11.9 11.8 11.8 20 11.9 11.6 11.6 11.6 11.6 11.6 11.7 25 11.4 11.4 11.4 11.5 11.5 11.5 - 30 11.0 ------M T Maturity ti / Tenor tj 8 9 10 15 20 25 30 1 17.1 16.5 16.1 14.5 13.3 12.9 12.6 2 16.2 15.8 15.4 14.2 13.1 12.7 - 3 15.6 15.3 15.1 13.9 13.0 12.6 - 4 15.1 14.8 14.6 13.8 12.9 12.5 - 5 14.6 14.4 14.3 13.4 12.7 12.4 - 10 12.7 12.6 12.5 13.0 11.4 - - 15 11.8 11.8 11.8 11.3 - - - 20 11.7 11.7 11.7 - - - - 25 ------30 ------

40 Table 3: Calibration errors (in percent) in G2++

M T Maturity ti / Tenor tj 123 45 67 1 -0.41 -0.37 -2.28 -1.92 -1.79 -3.16 -3.35 2 -0.64 -1.54 -3.47 -4.81 -5.69 -5.32 -5.49 3 -2.72 -1.72 -2.29 -4.43 -4.85 -5.22 -4.25 4 -1.10 -0.01 -1.42 -3.92 -3.96 -3.89 -2.85 5 0.11 0.74 -1.08 -3.37 -3.62 -3.51 -2.19 10 2.25 0.36 -1.79 -4.01 -3.87 -2.30 -0.78 15 0.42 -1.40 -3.78 -2.04 -1.09 -0.26 1.35 20 -1.66 -2.75 -1.40 -0.24 1.15 2.66 4.77 25 -2.76 -1.95 -0.65 1.63 3.11 4.64 - 30 -3.91 ------M T Maturity ti / Tenor tj 8 9 10 15 20 25 30 1 -3.46 -3.52 -2.73 0.36 1.84 5.77 8.46 2 -4.99 -4.26 -3.81 0.92 2.25 5.63 - 3 -3.67 -2.68 -1.24 2.39 4.32 7.17 - 4 -2.02 -1.38 -0.21 5.13 6.32 8.76 - 5 -1.00 0.07 1.77 5.48 7.38 10.20 - 10 0.54 1.69 2.74 13.26 6.82 - - 15 2.86 4.18 5.49 7.39 - - - 20 6.10 7.40 8.66 - - - - 25 ------30 ------Parameter values are a = 3.47,b = 0.0461 ,σ = 0.0537, ξ = 0.00842 ρ = 0.891 . With data calibrated to the Swaption implied volatilities on January 28, 2005. −

41 Table 4: Calibration errors (in percent) in the log-normal LMM

M T Maturity ti / Tenor tj 1234567 1 -0.78 -0.58 0.64 2.07 2.26 0.73 0.65 2 -0.27 1.06 0.56 -0.54 -1.46 -0.83 -0.69 3 -0.64 0.98 0.88 -1.00 -0.81 -0.60 -0.59 4 -0.60 0.93 0.37 -0.87 0.09 -0.74 -0.45 5 -1.01 0.87 1.16 0.50 -0.70 -1.30 -0.35 10 0.42 0.36 -0.39 -0.99 -2.50 -1.88 -0.84 15 0.27 0.17 -0.29 -0.66 -0.72 -0.87 0.12 20 0.17 0.05 -0.14 -0.37 -1.44 -0.69 0.57 25 0.12 -0.09 -0.24 -0.29 0.08 0.80 - 30 -0.10 ------M T Maturity ti / Tenor tj 8 9 10 15 20 25 30 1 0.70 -0.26 -0.29 -0.26 -2.06 -1.20 -1.10 2 -1.13 -1.22 -1.35 0.33 -1.57 -1.31 - 3 -0.80 -0.35 0.68 1.40 -0.12 -0.34 - 4 -0.06 0.26 1.37 3.18 1.08 0.40 - 5 0.59 1.70 2.37 2.82 1.24 1.13 - 10 0.39 0.35 0.55 7.20 -3.80 - - 15 0.50 0.72 0.49 -2.52 - - - 20 0.82 0.69 0.96 - - - - 25 ------30 ------Average Error=1.74% Max error=7.19%

42 Table 5: Parameters in the log-normal LMM

a 0.033807 η1 0.881421 b 0.904988 η2 0 c 0.141077 ρ∞ 0.414192 d 0.000169

i = Θi i = Θi 1, 2 1.374 33, 34 0.849 3, 4 1.473 35, 35 0.795 5, 6 1.473 37, 38 0.884 7, 8 1.388 39, 40 0.840 9, 10 1.326 41, 42 0.833 11, 12 1.260 43, 44 0.798 13, 14 1.177 45, 46 0.830 15, 16 1.077 47, 48 0.838 17, 18 1.007 49, 50 0.883 19, 20 1.084 51, 52 0.799 21, 22 1.045 53, 54 0.807 23, 24 0.985 55, 56 0.812 25, 26 0.937 57, 58 0.843 27, 28 0.864 59, 60 0.801 29, 30 0.957 61, 62 0.774 31, 32 0.906

43 Table 6: Calibration errors (in percent) in the Extended/CEV LMM

M T Maturity ti / Tenor tj 1234567 1 -0.88 -0.45 0.85 2.19 2.27 0.62 0.56 2 -0.38 1.15 0.66 -0.52 -1.57 -0.90 -0.77 3 -0.69 1.04 0.93 -1.08 -0.82 -0.63 -0.62 4 -0.64 0.97 0.33 -0.80 0.13 -0.69 -0.46 5 -1.02 0.82 1.33 0.64 -0.56 -1.23 -0.34 10 0.47 0.37 -0.36 -0.98 -2.51 -1.93 -0.92 15 0.32 0.19 -0.28 -0.62 -0.70 -0.90 0.07 20 0.20 0.08 -0.14 -0.38 -1.40 -0.70 0.53 25 0.11 -0.07 -0.24 -0.26 0.12 0.79 - 30 -0.08 ------M T Maturity ti / Tenor tj 8 9 10 15 20 25 30 1 0.60 -0.34 -0.39 -0.30 -2.03 -1.11 -0.97 2 -1.20 -1.31 -1.45 0.30 -1.53 -1.22 - 3 -0.86 -0.43 0.61 1.38 -0.08 -0.25 - 4 -0.11 0.22 1.33 3.18 1.13 0.49 - 5 0.58 1.69 2.36 2.85 1.32 1.23 - 10 0.30 0.28 0.48 7.17 -3.80 - - 15 0.43 0.65 0.45 -2.56 - - - 20 0.77 0.65 0.93 - - - - 25 ------30 ------Average Error=1.76% Max error=7.20%

44 Table 7: Parameters in the extended/CVV LMM

a 0.047104 η1 0.922098 b 1.410679 η2 0 c 0.092664 ρ∞ 0.395819 d -0.01779 p 0.665

i = Θi i = Θi 1, 2 0.610 33, 34 0.466 3, 4 0.691 35, 35 0.452 5, 6 0.722 37, 38 0.469 7, 8 0.698 39, 40 0.468 9, 10 0.683 41, 42 0.459 11, 12 0.663 43, 44 0.435 13, 14 0.628 45, 46 0.450 15, 16 0.580 47, 48 0.451 17, 18 0.570 49, 50 0.479 19, 20 0.557 51, 52 0.432 21, 22 0.570 53, 54 0.430 23, 24 0.534 55, 56 0.435 25, 26 0.522 57, 58 0.431 27, 28 0.469 59, 60 0.446 29, 30 0.520 61, 62 0.408 31, 32 0.498

45 Table 8: Prices

Model Price Call option Duality gap 95% CL Log-normal LMM 9315.45(2.0) 91.02(1.6) 12.74(1.6) [9297.68 ; 9319.37] CEV-LMM 9292.54(1.77) 112.29(1.8) 13.79(1.8) [9273.80 ; 9296.01] G2++ 9091.80(0.8) 127.42(1.3) 17.09(2.1) [9070.30 ; 9093.36] All prices in basis points. Standard errors in parentheses. All prices are calculated with the same basis function which is a second order polynomial of four rates. The 6m forward LIBOR, core swap rate and 2 and 20 year swap rate. All rates setting on each exercise date.

46 Table 9: Prices with an alternative stopping strategy

Model Price Call option Duality gap 95% CL Log-normal LMM 9382.49(2.2) 24.02(1.1) 96.88(8.5) [9268.40 ; 9386.81] CEV-LMM 9344.51(1.9) 46.83(1.4) 91.05(8.1) [9237.15 ; 9348.23] G2++ 9158.98(0.9) 60.24(1.3) 106.09(6.1) [9040.80 ; 9160.74] All prices are calculated using a basis function consisting of a 4th order polynomial of the 6m forward LIBOR, the core swap rate, the 2 and 20 year swap rates, and the 20 year forward LIBOR. All rates setting on exercise dates.

47 Table 10: Call time distribution

Times LMM CEV-LMM G2++ Times LMM CEV-LMM G2++ t2 0.68% 0.64% 0.54% t13 0.75% 0.84% 0.88% t3 1.13% 1.10% 1.26% t14 0.67% 0.81% 0.99% t4 1.14% 1.12% 1.30% t15 0.84% 0.88% 1.02% t5 1.24% 1.25% 1.32% t16 1.01% 0.99% 1.00% t6 1.08% 1.17% 1.14% t17 1.13% 1.14% 1.04% t7 1.06% 1.14% 1.19% t18 1.38% 1.32% 1.16% t8 0.75% 0.83% 1.09% t19 1.16% 1.38% 1.16% t9 0.74% 0.88% 0.96% t20 1.43% 2.02% 1.46% t10 0.71% 0.79% 1.06% t21 1.88% 2.30% 2.56% t11 0.74% 0.80% 0.98% t22 0.65% 1.00% 5.09% t12 0.67% 0.76% 0.87% t23 79.16% 76.87% 71.91%

48 Figure 1: Implied Correlation 1.0 0.8

ρ i,j 0.6

30 20 30 25 20 T 10 15 i 10 5 T j

49 Figure 2: Deltas

1

0,5

0

5 5 5 5 5 5 5 5 5 5 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,5 11,5 12,5 13,5 14,5 15,5 16,5 17,5 18,5 19,5 20,5 21,5 22,5 23,5 24,5 25,5 26,5 27,5 28,5 29,5 30,5 31,5

G2 -0,5 CEV-LMM LMM

-1

-1,5

-2

50 Figure 3: Vegas

2,00%

1,50%

1,00%

CEV-LMM 0,50% LMM

0,00%

,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 ,5 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3

-0,50%

-1,00%

51 Chapter II Fast and Accurate Option Pricing in a Jump-Diffusion Libor Market Model ∗

David Skovmand,† University of Aarhus and CREATES May 1, 2008

Abstract

This paper extends, improves and analyzes cap and swaption approxima- tion formulae for the jump-diffusion Libor Market Model derived in Glasser- man and Merener (2003a). More specifically, the case where the Libor rate follows a log-normal diffusion process mixed with a compound Poisson process under the spot measure is investigated. The paper presents an extension that allows for an arbitrary parametric specification of the log-jump size distribution, as opposed to the previously studied log-normal case. Furthermore an improvement of the exist- ing swaption pricing formulae in the log-normal case is derived. Extensions of the model are also proposed, including displaced jump-diffusion and stochas- tic volatility. The formulae presented are based on inversion of the Fourier transform which is approximated using the method of cumulant expansion. The accuracy of the approximations is tested by Monte Carlo experiments and the errors are found to be at acceptable levels.

∗The author would like to thank Elisa Nicolato for helpful comments †Current affiliation: Aarhus School of Business and the Center for Research in Econometric Analysis of Time Series (CREATES), www.creates.au.dk. Corresponding address: Aarhus School of Business, Department of Business Studies, Fuglesangs All´e4, DK-8210 Aarhus V, Denmark, e-mail: [email protected] 1 Introduction

The term Libor Market Model (LMM) is coined with reference to the models de- scribed in the three seminal papers Brace, Gatarek, and Musiela (1997), Miltersen, Sandmann, and Sondermann (1997) and Jamshidian (1997). The novelty of these three papers is that they present a unifying no-arbitrage approach to modeling the term structure of discretely compounded rates such as Libor rates and swap rates. The main assumption in the papers is log-normality of the underlying Libor rate, which stands in opposition to stylized facts observed in financial data such as the implied volatility smile in interest rate options. Extensions of the log-normal LMM have since been proposed in order to remedy this deficiency, mirroring a similar development in the equity options literature. One popular approach has been to model the Libor rates via a jump-diffusion as proposed in Glasserman and Kou (2003), Glasserman and Merener (2003a), Glasserman and Merener (2003b), and Jarrow, Li, and Zhao (2007). All these papers have focused mainly on log-normal jump sizes as originally proposed by Merton (1976). However, the equity option pricing literature has recently seen several deviations from this assumption. Most notable examples are the log-uniformly distributed jumps in Yan and Hanson (2006), Pareto-Beta jump sizes in Ramezani and Zeng (2007), and double exponential dis- tribution investigated in Kou (2002), Kou and Wang (2004), and Kou and Wang (2003). These alternative specifications are motivated by a need for a higher degree of flexibility, such as asymmetry in the number of positive and negative jumps. In the double exponential case the main motivation is the ability to easily price certain exotic options. These motivations also apply to the Libor modelling case, but so far very little work has been done in an LMM context. If the Libor rates are modeled as affine jump-diffusions (AJD, see Duffie, Pan, and Singleton (2000)) under the rates corresponding forward measures, the extension to different jump sizes in the LMM would be completely analogous to the equity case. Indeed the price of a call option on the Libor (a caplet) can be reduced to a single numerical integration. This paper investigates the less simple case when rates are specified as a diffusion plus a compound Poisson process, under the discrete risk-neutral measure (spot measure), defined by a discretely updated bank account numeraire. Modelling rates under the spot measure has the advantage that simulation from the model is simpler and results in a smaller discretization bias as well as variance than the alternative forward measure specification, as demonstrated in Glasserman and Zhao (2000). The spot measure specification also allows for an easy handling of the dimension- ality of the model, which is not the case for the forward measure specification that requires as many jump processes as Libor rates in order for the forward rates to be AJDs under their respective measures. As the number of rates are not uncom-

54 mon to range in the 60’s this can seriously complicate the use the forward measure specification. The drawback of the spot measure specification used in this paper, is that Libor rates are no longer AJDs under their respective forward measures. This means that one has to resort to approximation techniques to price caplets as done in Glasserman and Merener (2003a). This paper derives approximative expressions for caplets and swaptions using differ- ent methods than Glasserman and Merener (2003a). The novelty of the alternative route taken in this paper, is that it allows for an arbitrary parametric specification of the jump size distribution. The method employed, is based on approximating the characteristic function describing the jump sizes using a cumulant expansion tech- nique. The derived characteristic function can subsequently be used in the standard Fourier inversion pricing methodology introduced by Carr and Madan (1999). However the classic Merton-76 style jump-diffusion is unable to capture implied volatility smiles across both maturity and strike. This problem is well known in the equity option pricing literature, and has recently been documented in a LMM context by Jarrow, Li, and Zhao (2007) and Skovmand (2008). Specifically, if short maturity smiles are too be matched then the smile flattens too quickly as a function of maturity due to the fast decline of skewness and kurtosis generated by the jumps. One solution to this problem is to displace the jump-diffusion with a constant, which results in skewness even for long maturity rates. In addition kurtosis can be gener- ated in the long maturity rates, by including stochastic volatility. Both extensions are included and examined in this paper, and their consequences in terms of the derived approximations are discussed. All the proposed approximation formulas are subjected to a number of tests of ac- curacy using Monte Carlo simulations as a benchmark for the true price. Assuming log-normal jump sizes the accuracy is also compared with the approximations pro- posed in Glasserman and Merener (2003a), and it is found that the proposed formu- las in this paper perform better when pricing swaptions, whereas in the caplet case the differences are negligible. The case where jump sizes are log-double exponential is also investigated with and without a displacement factor. In the log-normal as well as the log-double exponential case the approximation works very well for shorter maturities. For long maturities comparatively larger errors in prices are observed, but in terms of percentages they are only above 1% in very few cases.

2 General Setup

Let P (t, Tk) be the time t zero coupon bond price with maturity at Tk. Consider a tenor structure T <.....

55 are defined as

P (t, Tk) P (t, Tk+1) Fk(t)= F (t, Tk, Tk+1)= − . δP (t, Tk+1)

2.1 Arbitrage Free Dynamics The discrete bank account is defined as follows

B0 = 1, B = B [1 + δF (T )], 1 k

dF (t) k = [γ′ µ α ] dt + γ′ dW Q(t)+ dJ (t), (1) F (t ) k k − k k k k − k δγjFj(t ) µk(t)= − , 1+ δFj(t ) j=Xβ(t) − k x 1+ δFj(t ) αk(t)= (e 1) − x λf(x)dx, R − 1+ δFj(t )e Z j=Yβ(t)  −  N(t) J (t)= (eXℓ 1), k − Xℓ=1 0 t T , k = 1,...,K, ≤ ≤ k Q where W (t) is an D-dimensional Brownian motion under Q. γk is a D dimensional constant vector. N(t) is a Poisson process independent of W Q(t) with intensity λ jumping at the random times τ1 τn < t. The Xℓ’s are IID stochastic vari- ables normally referred to as marks≤···≤ with a distribution given by the density function f(x) and the characteristic function φX (z). β(t) is the index of the first forward Libor rate that has not expired at t. This specification is nested in the general semi-martingale specification in Jamshid- ian (1999), but this specific model has to the authors knowledge only been studied in Glasserman and Merener (2003a). A more convenient way to write the jump process Jk(t) is as an integral over a ran- dom measure µ(dx,dt), also referred to as the jump measure, that assigns a mass of

56 one for each pair (Xℓ,τℓ)

N(t) t (eXℓ 1) = (ex 1)µ(dx,dt). − 0 R − Xℓ=1 Z Z with a corresponding compensator νQ(dx,dt) which has the ability of making

N(t) t Q h(Xℓ) h(x)ν (dx,dt), − 0 R Xℓ=1 Z Z a martingale for any function h satisfying standard regularity conditions. In the simple compound Poisson case above the compensator is deterministic and given as

νQ(dx,dt)= λf(x)dxdt.

3 Caplet Pricing

A caplet with strike K is a claim that pays δ(F (T ) K)+ at time T . The price k k − k+1 under the forward measure Fk with P (0, Tk+1) as numeraire is known from standard textbooks to be: F Cpl = δP (0, T )E k (F (T ) K)+ . k k+1 k k − In order to evaluate this expectation the first step is to switch to the forward measure

Proposition 1. Assume the dynamics are given by equation (1) under the discrete bank account measure Q. Changing measures to Fk with P (t, Tk+1) as numeraire yields the following arbitrage free dynamics:

dFk(t) ′ Fk x Fk =γkdW (t) (e 1)ν (dx,dt)+ dJk(t), t Tk, (2) F (t ) − R − ∀ ≤ k − Z N(t) where J (t)= (eXℓ 1), with compensator k ℓ=1 − P k Fk 1+ δFj(t ) ν (dx,t)= − x λf(x)dx. (3) 1+ δFj(t )e ) j=Yβ(t)  −  Proof. In Glasserman and Kou (2003) it is proved that the dynamics in (2) are free of arbitrage. (3) follows from Theorem 7 in Jamshidian (1999) or Lemma 4.1 in Glasserman and Merener (2003a) .

57 This proposition shows that changing from the spot measure to the forward measure complicates the dynamics of the forward rates. From the compensator in (3) we see that the jump process is no longer compound Poisson as it has state dependency in the jump arrival rate as well as the jump size. Unfortunately this means that an explicit caplet price cannot be derived and one has to resort to approximative methods. Following Glasserman and Merener (2003a) I therefore seek to approximate the dynamics in equation (2) with the following martingale dynamics:

ˆ Nˆ(t) dFk(t) x Xˆℓ =ˆγkdW (t) (e 1)λˆ(t)fˆ(x, t)dxdt + d (e 1) (4) Fˆk(t ) − R −  −  − Z Xℓ=1   where W (t) is a one dimensional Brownian motion,γ ˆk is a scalar, Nˆ(t) is a Poisson process with time dependent deterministic intensity λˆ(t) and the Xˆℓ’s are realiza- tions from a random variable Xˆ, with a distribution described by a time dependent deterministic density function fˆ(x, t). In order to determine λˆ(t) and fˆ(x, t), the state dependent compensator from (3) is investigated. Taylor expanding around the time zero term structure gives us.

νF(x, t)= λf(x) k × k x 1+ δFj(0) δ(1 e ) x + − x 2 [Fj(t) Fj(0)] + O([Fj(t) Fj(0)]) dx. 1+ δFj(0)e (1 + δFj(0)e ) − − j=Yβ(t)   (5)

1+δFj (0) The term x is clearly close to 1, whereas the higher order terms are of 1+δFj (0)e lower orders of magnitude. Ignoring these higher order terms yields the following approximate deterministic expression for the compensator

k F 1+ δFj(0) νk (x, t) νˆk(dx,t)= x λf(x)dx. ≈ 1+ δFj(0)e j=Yβ(t)   As shown in the above definition the approximation is akin to the classical trick of freezing the rates at time zero used very frequently in the LMM literature (see for example Brigo and Mercurio (2006)). Fixing time and integrating the approximate compensator gives us the approximate

58 arrival rate of the jumps:

λˆ(t)= νˆk(dx,t) R Z k 1+ δFj(0) = x λf(x)dx, R 1+ δFj(0)e Z j=Yβ(t)   and similarly the approximate density

k ˆ νˆk(x, t) 1+ δFj(0) λ f(x, t)= = x f(x). R νˆk(dx,t)) 1+ δFj(0)e λˆ(t) j=Yβ(t)   Finally the diffusion parameterR is reduced to a scalar by setting

′ γˆk = [γkγk]. q Note that this last step is not approximative. In the absence of jumps (4) is in fact equivalent in distribution to the dynamics in (2). Using the dynamics in (4) and the above defined approximations gives us the closed form option price Proposition 2. Assume the Libor rate follows 1. Then the approximate price of a Tk-caplet with strike K is given by δ ∞ φ(z (α + 1)i) Cpl = P (0, T ) e−(iz−α) log(K) − dz, (6) k k+1 π α2 + α z2 + i(2α + 1)z Z0 − where α> 0 is a tuning parameter satisfying φ(z (α + 1)i) < , z R . | − | ∞ ∀ ∈ + ˆ The characteristic function φ(z)= E[eiz log(Fk(Tk))] is given by

φ(z)= φcont(z)φjump(z), with components

1 2 iz log(Fˆk(0))− iz(z+i)ˆγ φcont(z)= e 2 k , (7) k izx φjump(z) = exp izαˆk + δ λˆ(Tj) e 1 fˆ(x, Tj)dx , (8) − R − Z j=1 ! X
where αˆ = δ k (ex 1)fˆ(x, T )λˆ(T )dx k R j=1 − j j (PROOF)R SeeP appendix

59 3.1 Cumulant Expansion for φjump(z) The expression in equation (8) is complicated in the sense that its evaluation requires solving a numerical integral. Evaluating the outer integral in (6) with another integral nested in the integrand is slow, and this effectively means the formula cannot be used for calibration purposes. Fortunately one can approximate the φjump(z) function very accurately. This can be done by first observing that (8) can be written as

k

φ (z) = exp izαˆ + δ λˆ(T )(φ ˆ (z) 1) , jump − k j X − j=1 ! X izx ˆ ˆ where φXˆ (z)= R e f(x, t) is the characteristic function of the random variable X with density fˆ(x, t). The objective is therefore to approximate φ (z), which is done R Xˆ by writing it as an expansion in terms of the characteristic function for the mark distribution under the spot measure. Let X be the random variable with distribution given by the density f(x) that defines the mark distribution under the spot measure in (1). Its corresponding characteristic function φX (z) can be written as a cumulant expansion:

∞ (iz)n φ (z)= E[eizX ] = exp κ , (9) X n n! n=1 ! X where κn are the cumulants of the distribution which are implicitly defined as the coefficients{ } of a Taylor expansion of the logarithm of the characteristic function (see for example Abramovitz and Stegun (1972))

∞ (iz)n log(φ (z)) = κ . (10) X n n! n=0 X In a similar manner φXˆ (z) can be expressed as a cumulant expansion:

∞ (iz)n φ (z) = exp κˆ (t) . (11) Xˆ n n! n=1 ! X Putting equation (9) and (11) together yields

∞ n izXˆ (iz) φ ˆ (z)= E[e ] = exp (ˆκ (t) κ ) φ (z). X n − n n! X n=1 ! X If X is Gaussian then the above formula is normally referred to as an Edgeworth or Gram-Charlier expansion. The above series is a well known method in statistics used

60 to approximate one distribution with another. The underlying statistical theory can be in Kolassa (2006). f(x) has known cumulants κn , whereas the cumulants of Xˆ are easily calculated via the recursive formula1 { } n−1 n 1 κˆ (t) =m ˆ (t) − κˆ (t)m ˆ (t), (12) n n − k 1 k n−k Xk=1  −  wherem ˆ n(t) is the nth moment calculated numerically as

n mˆ n(t)= x fˆ(x, t)dx. R Z Summarizing the results and assuming a truncation point P , the jump component in the characteristic function in Proposition 2 can be approximated by

φjump(z) k P (iz)n exp izαˆ + δ λˆ(T ) exp (ˆκ (T ) κ ) φ (z) 1 . (13) ≈ − k j n j − n n! X − j=1 n=1 ! !! X X This approximation significantly increases the speed of the caplet pricing formula. In all cases studied in this paper P is set to 4 which provides more than sufficient accuracy.

4 Swaption Pricing

A Ta (Tb Ta) payer swaption of strike K is a contract that gives the right but not the obligation× − to, enter into a T (T T ) payer swap at time T . This means that a × b − a b if the swaption is exercised the holder will pay the fixed payments of δK and receive the floating payments of δFi−1(Ti−1) for i = a + 1, . . . , b. The payoff at expiration can easily be shown to be (see for example Hunt and Kennedy (2004)) C (T )(S (T ) K)+, a,b a a,b a − where b−1

Ca,b(t)= δP (t, Ti+1), i=a X P (t, Ta) P (t, Tb) Sa,b(t)= b−1 − , i=a δP (t, Ti+1)

n 1 ∂ log(φX (z)) The formula is a simple consequence ofP linking the two relations, κn = ∂zn and z=0 n n ∂ φX (z) mn = ( i) ∂zn − z=0

61 where is called the Sa,b the swap rate. Ca,b is normally referred to as the Present Value of a Basis Point(PVBP) and it is used as the numeraire which defines the swaption measure Sa,b. The swap rate can also be represented as a weighted sum of forward rates.

b−1

Sa,b(t)= wi(t)Fi+1(t), i=a X P (t, Ti) wi(t)= b−1 . j=a δP (t, Tj)

Note that the weights themselves areP dependent of the forward rates through their relation to the zero coupon bonds, making the swap rate a complicated non-linear function of the underlying forward rates. A perhaps less misleading expression is

1 b−1 1 j=a 1+δFj (t) Sa,b(t)= − δ b−1 i 1 i=Qa j=a 1+δFj (t) P Q It then follows that the value of a payer swaption under the swaption measure S is given by (see again Hunt and Kennedy (2004))

S PS(t)= C (t)E a,b [(S (T ) K)+]. a,b t a,b a − To evaluate this expression the next step is to change the numeraire.

Proposition 3. Assume the dynamics are given by equation (1). Changing measures b−1 from the spot measure Q to the swaption measure Sa,b with Ca,b(t)= i=a δP (t, Ti+1) as the numeraire yields the following arbitrage free dynamics for the forward rates (ignoring drifts)2: P

dF (t) k =γ′ dW a,b(t) [. . . ]dt + dJ (t), (14) F (t ) k − k k − a,b where W (t) is D dimensional Brownian motion under Sa,b and

N(t) J (t)= (eXℓ 1), k − Xℓ=1 2The drift in (14) is complicated and in this case irrelevant hence it is left it out. In the case without jumps it is given in Brigo and Mercurio (2006) Proposition 6.8.2

62 with state dependent compensator νa,b(dx,t):

b−1 k 1+ δFj(t ) νa,b(dx,t)= wk(t) − x λf(x)dx, 1+ δFj(t )e Xk=a j=Yβ(t)  − 

P (t,Tk+1) where wk(t)= b−1 . i=a δP (t,Ti+1) Proof. FollowsP from Lemma 4.2 in Glasserman and Merener (2003a).

Ignoring the drifts the swap rate dynamics follow using Itˆo’s lemma for jump- diffusions (see for example Cont and Tankov (2004)):

b−1 ∂S (t ) S dS (t)= a,b − F (t )γ′dW (t) a,b ∂F (t ) i − i i=a i X − x + [Sa,b(t ,F (t )(e )) Sa,b(t, F (t ))] µ(dx,dt) R − − − − Z + [. . . ]dt, (15) where the swap rate is written as a vector function R RK R of the Libor term + × → structure Sa,b(t, F (t)) where F (t)=(F1(t),...,FK (t)). Furthermore it follows from Andersen and Andreasen (2001):

∂S (t) S (t)δ P (t, T ) b δP (t, T ) a,b = a,b b + k=i k . (16) ∂F (t) 1+ δF (t) P (t, T ) P (t, T ) C (t) i i " a − b P a,b # Due to the complicated nature of (15) and (16) the swap rate dynamics are highly intractable. This means that to price a swaption we again have to resort to approx- imative techniques. Using the logic of the previous section the following martingale dynamics are posed as an approximation to the true dynamics in (15):

dSˆa,b(t) =ˆγa,bdWˆ (t) Ha,b(x)λˆa,b(t)fˆa,b(x, t)dt Sˆ (t ) − R a,b − Z Nˆ(t) + d H (Xˆ ) ,  a,b ℓ  Xℓ=1   where Wˆ (t) is a univariate Brownian motion, Nˆ(t) is a Poisson process independent of Wˆ (t) with deterministic time dependent intensity λˆa,b(t) and Xˆ1, Xˆ2,... are IID stochastic variables drawn from a distribution with time dependent density fˆa,b(x, t).

63 The diffusion parameter is defined following Andersen and Andreasen (2000)

′ b−1 ∂S (0) F (0) b−1 ∂S (0) F (0) γˆ = a,b j γ a,b j γ . a,b v ∂F (0) S (0) j ∂F (0) S (0) j u j=a j a,b ! j=a j a,b ! u X X t To derive the jump specific terms λˆa,b(t) and fˆa,b(x, t) the rates are again frozen at time zero to get an approximate compensator

b−1 k 1+ δFj(0) νa,b(x, t) νˆa,b(x, t)= wj(0) x λf(x). ≈ 1+ δFj(0)e Xk=a j=Yβ(t)   As before the approximate intensity is found by integrating the above expression

b−1 k ˆ 1+ δFj(0) λa,b(t)= wj(0) x λf(x)dx, R 1+ δFj(0)e Z Xk=a j=Yβ(t)   and the density is found in a similar manner

νˆa,b(x, t) fˆa,b(x, t)= . R νˆa,b(y,t))dy

For the function Ha,b(x) that translatesR marks into jump sizes I propose using the following x Sa,b 0,F (0)e Sa,b(0,F (0)) Ha,b(x)= − . (17) Sa,b(0
,F (0)) For a flat term structure the swap rate is well known to be close to a linear function of the forward rates (see for example Rebonato (2002)) causing the function in (17) to resemble ex 1. The formula in Section 5.4 in Glasserman and Merener (2003a) assumes exactly− that H (x)= ex 1, in a slightly different setup. The choice in (17) a,b − captures the initial non-linearity in the swap rate vector function Sa,b(t, F (t)) and can therefore be expected to be more accurate than the Glasserman and Merener (2003a) alternative. Using the approximative swap rate dynamics, swaption pricing is largely equivalent to the caplet as seen in the following proposition. Proposition 4. Swaption Pricing Assume the Libor rate dynamics are given by equation (1). The approximate price of a T (T T ) payer swaption with strike K is given by a × b − a 1 ∞ φ(z (α + 1)i) PS(T , T )= C (0) e(−iz−α) log(K) − dz, a b a,b π α2 + α z2 + i(2α + 1)z Z0 −

64 where α> 0 is a tuning parameter satisfying φ(z (α + 1)i) < . | − | ∞ ˆ The characteristic function φ(z)= E[eiz log(Sa,b(Ta))] is given by

φ(z)= φcont(z)φjump(z), with components

1 2 iz log(Sˆa,b(0))− iz(z+i)γ φcont(z)= e 2 a,b , a iz log(Ha,b(x)+1) φjump(z) = exp izαˆa,b + λˆ(Tj)δ(e 1)fˆa,b(x, Tj)dx , − R − j=0 ! Z X a ˆ ˆ where αˆa,b = R j=1 Ha,b(x)λa,b(Tj)fa,b(x, Tj)dx R Proof. SubstitutingP Fˆ (t) with Sˆ (t), δP (0, T ) with C (0) and (ex 1) with k a,b k+1 a,b − Ha,b(x) the proof is identical to Proposition 2

Equivalent to the previous section a cumulant expansion is used in the jump com- ponent of the characteristic function to speed up the evaluation. The derivation is equivalent to the caplet case except it is done in terms of the transformed stochastic variable log(Ha,b(Xˆ) + 1) where Xˆ has density fˆa,b(x, t). Assuming a truncation point P this gives us the following approximation

φjump(z) a P (iz)n exp izαˆ + δ λˆ (T ) exp (ˆκ (T ) κ ) φ (z) 1 , ≈ − a,b a,b j n j − n n! X − j=1 n=1 ! !! X X where moments and cumulants are given by

n mˆ n(t)= log(Ha,b(x)+1) fˆa,b(x, t)dx (18) R Z n−1 n 1
κˆ (t) =m ˆ (t) − κˆ (t)m ˆ (t). n n − k 1 k n−k Xk=1  −  5 Displacing the Jump-Diffusion

A simple and popular way generate to implied volatility skews in the Libor rate options has been to model the displaced rate Fk(t)+ d as a log-normal martingale

65 under the forward measure.3 Specifically with d> 0 the volatility skew, declines in terms of moneyness and it is retained for long maturity options, making it a very powerful extension despite its simplicity. Furthermore mixing a displaced diffusion with a compound Poisson process has the interesting feature of being able to generate an asymmetric smile in the short end an a skew in the long end of the maturity spectrum. The displacement can easily be applied in this setup, since the displaced model is nested in the general semi-martingale model of Jamshidian (1999). It follows that the arbitrage free spot measure dynamics in (1) can be modified to4

dF (t) k = [γ′ µ α ] dt + γ′ dW Q(t)+ dJ (t), (19) F (t )+ d k k − k k k k − k δγj(Fj(t )+ d) µk(t)= − , 1+ δFj(t ) j=Xβ(t) − k x 1+ δFj(t ) αk(t)= (e 1) x − x λf(x)dx, R − 1+ δFj(t )e + d(e 1) Z j=Yβ(t)  − −  N(t) J (t)= (eXℓ 1). k − Xℓ=1

Jk(t) is a compound Poisson process with intensity λ and mark distribution f(x). Despite its immediate appeal the model has the problematic feature that for d> 0 the Libor rates are no longer guaranteed to be positive. Whether this poses any significant arguments against the model is unclear5.

5.1 Caplet Pricing Caplets are priced by first noticing that

F Cpl = δP (0, T )E k (F (T ) K)+ k k+1 k k − F = δP (0, T )E k (F˜ (T )(d) K˜(d))+ , k+1 k k − h i 3In equity option pricing this approach was pioneered by Rubinstein (1983) and extensively studied for example in Rebonato (2004). In an LMM context both Rebonato (2002) and Brigo and Mercurio (2006) study it in detail 4 Specifically from Jamshidian (1999) Theorem 7 with βi = Φi = Li + d where Li is the Libor rate in that particular setup 5A similar case of non-positivity has frequently been made, to no apparent avail, against the classic Vasicek (1977)-model for the short interest rate

66 (d) (d) with F˜k(Tk) = Fk(Tk)+ d and K˜ = K + d. The price can then be seen as a call ˜(d) ˜ (d) ˜ (d) option on the displaced rate Fk (Tk) with strike K . The dynamics of Fk(Tk) can then be approximated in the exact same manner as in Section 3. The resulting formula is stated in the appendix.

5.2 Swaption Pricing Following Section 4 gives us the following the swap rate rate dynamics(ignoring drifts) under the swaption measure

b−1 ∂S (t ) S dS (t)= a,b − (F (t )+ d)γ′dW (t) a,b ∂F (t ) i − i i=a i X − x + [Sa,b(t , (F (t )+ d)(e 1)) Sa,b(t, F (t ))] µ(dx,dt) R − − − − − Z + [. . . ]dt, where µ(dx,dt) is the jump measure with compensator

b−1 k d 1+ δFj(t ) νa,b(dx,t)= wk(t) x − x λf(x)dx. (20) 1+ δFj(t )e + d(e 1) Xk=s j=Yβ(t)  − −  This SDE is approximated by the martingale dynamics ˆ dSa,b(t) d ˆ d ˆd ˆd =ˆγa,bdW (t) Ha,b(x)λa,b(t)fa,b(x, t)dt Sˆ (t)+ d − R a,b Z Nˆ(t) + d Hd (Xˆ ) ,  a,b ℓ  Xℓ=1   ˆ ˆd where N(t) is a Poisson process with intensity λa,b(t)= R νˆa,b(x, t)dx whereν ˆa,b(dx,t) is an approximation to the compensator in (20) defined as R b−1 k 1+ δFj(0) νˆa,b(dx,t)= wk(0) x x λf(x)dx. 1+ δFj(0)e + d(e 1) Xk=s j=Yβ(t)  −  ˆ ˆ d νˆa,b(x,t) Furthermore X1, X2,... are IID random variables with density fa,b(x, t)= ˆd . λa,b(t) Following the non-displaced case the function translating the marks into jumps is defined as x d Sa,b 0, (F (0) + d)(e 1) Sa,b(0,F (0)) Ha,b(x)= − − . Sa,b(0,F (0))
+ d

67 Finally following Andersen and Brotherton-Ratcliffe (2005) the diffusion parameter is approximated by

′ b−1 ∂S (0) F (0) + d b−1 ∂S (0) F (0) + d γˆd = a,b i γ a,b i γ a,b v ∂F (0) S (0) + d j ∂F (0) S (0) + d j u j=a j a,b ! j=a j a,b ! u X X t As with caplets, the pricing formula is largely the same and given in the appendix.

6 Stochastic Volatility

The displacement factor adds limited flexibility since it generates solely monotonic skews in implied volatility. This is an obvious drawback since non-monotonic smiles are observed in the long maturity spectrum (see Skovmand (2008)). The most popular solution to this problem, in equities as well as interest rate models, has been to add stochastic volatility in the diffusion component. Following Andersen and Brotherton-Ratcliffe (2005) who study the pure diffusion case we can extend the dynamics in (1) to include stochastic volatility as follows

dF (t) k = [V (t)γ′ µ α ] dt + γ′ V (t)dW Q(t)+ dJ (t), F (t ) k k − k k k k − k p δγjFj(t ) µk(t)= − , 1+ δFj(t ) j=Xβ(t) − k x 1+ δFj(t ) αk(t)= (e 1) − x λf(x)dx, R − 1+ δFj(t )e Z j=Yβ(t)  −  N(t) J (t)= (eXj 1), k − j=1 X 0 t T , k = 1,...,K, ≤ ≤ k where V (t) denotes the stochastic volatility process which is assumed to follow the mean reverting dynamics posed in Cox, Ingersoll, and Ross (1985)

dV (t)= η(κ V (t))dt + ǫ V (t)dZ(t). − Z(t) is a Brownian motion independent of W Q(pt). Fortunately adding stochastic volatility does not introduce further error in the caplet approximation since the V (t) process is unaffected by a change to the forward measure. The same cannot be said when pricing swaptions, but the resulting error is well studied and quite small

68 as demonstrated in Andersen and Brotherton-Ratcliffe (2005). Since the extension only affects the diffusion component it follows immediately from Andersen and Brotherton-Ratcliffe (2005) Lemma 2 that the resulting prices for caplets and swaptions are found by replacing φcont(z) in Proposition 2 and 4 with log(Fk(0)) log(Sa,b(0)) e I(Tk, z) and e I(Ta, z) respectively, where

I(t, z) = eA(t,z)+B(t,z)V (0), with κ D(z) 1 e−D(z)t B(t, z)= − − , ǫ2 1 k−D(z) e−D(z)t − k+D(z) k−D(z) e−D(z)t 1 −2 k+D(z) − A(t, z)= κθǫ (κ D(z))t 2 log κ−D(z) . − − 1 !! κ+D(z) − D(z) = 2 κ2 + z(z + i)ǫ2

Finally it can also bep noted that the stochastic volatility component is without a ”leverage effect” i.e a correlation between Z(t) and W Q(t). Introducing leverage adds another approximation step since the V (t) process becomes state dependent under the forward measure as shown in Wu and Zhang (2006) who derive cap and swaption approximations in the pure diffusion case. However, the asymmetric smile normally generated from the leverage effect (see for example Heston (1993)) is also obtainable simply by displacing the dynamics with a constant as done in the previous section. This approach has the advantage that it does not induce a state dependent volatility term and hence requires no further approximation. Caplets and swaptions can then be priced by the applying the same adjustments to the displaced jump- diffusion formulas given in the appendix.

7 Numerical Testing

In this section the formulas are tested using two different examples of jump size distribution; the normal and the double exponential distribution. Furthermore the displaced diffusion case is investigated, butthe stochastic volatility case is omitted since it relates purely to the diffusion component, which has already been studied in Andersen and Brotherton-Ratcliffe (2005). The diffusion component is set to be 2-dimensional, and since the jump component is the point of interest in the paper the diffusion volatility is chosen to be low with ′ γk = (0.0007, 0.0008) . The initial term structure of the Libor rates is taken from the EURIBOR market

69 recorded from Datastream on May 5th 2006. The the term structure is plotted in Figure 1 revealing a humped term structure increasing until around the 15 year maturity and decreasing for longer maturities. There are two approximative steps in the procedure in the paper. The first is fixing the rates at zero in the state dependent compensator in equation (5), and the second is the cumulant expansion approximation in equation (13) and (18). Errors coming from the latter approximation can be controlled by adding more terms in the expansion. Furthermore an analysis shows that the maximum percentage pricing errors from this approximation with P = 4 are around 0.01% for swaptions, and significantly smaller for caplets. As this section will show the total approximation errors are generally at least one order of magnitude bigger, therefore P is kept at 4 and the error terms are ignored in the remainder. Another issue with using the Fourier inversion formulas in propositions 2 and 4, is the choice of the tuning parameter α. The optimal choice of α minimizes the total variation in the integrand, which causes the numerical integration scheme to require a minimum number of function evaluations for a desired level of accuracy. Optimality of α is beyond the scope of this paper therefore the tuning parameter is somewhat arbitrarly set at α = 0.75. This allows for the entire caplet price surface with 77 prices to be calculated in about 0.5 seconds using Matlab. A number of papers exist on this topic for example Lee (2004) and Lord and Kahl (2007) and these can be referred to if further speed is required.

7.1 Normal Mark Distribution Recall the the normal distribution has density

(x−µ )2 − J 1 σ2 f(x)= e 2 J , 2 2πσJ with characteristic function p

2 2 µJ iz−z σ /2 φX (z)= e .

2 The cumulants of the normal distribution are κ1 = µJ ,κ2 = σJ and κ3 = κ4 = 0. The model is tested with two different sets of parameters. ···

Parameter Set ANORM: µ = 0.02, σ = 0.08 and λ = 5. J − J Parameter Set BNORM: µ = 0.025, σ = 0.15 and λ = 1.5. J − J The first specification has jumps that occur on average 5 times a year, with a jump

70 size negatively biased, and the second specification has jumps that are rarer events occurring only 1.5 times a year on average, but with a bigger magnitude. Both have the same ATM implied caplet volatility at around 18%. Caplet and swaption prices are found by generating monte carlo simulations under the spot measure with a standard log-Euler discretization scheme (see Glasserman (2004)) applied to the SDE in equation (1). The jumps are simulated independently using algorithm 6.2 in Cont and Tankov (2004). 500000 paths are simulated using antithetic sampling for the Brownian motion, and the same seeds are used in both parameter sets in order to facilitate comparison. In general the simulation is partic- ularly slow because the drift features an integral that has to be reevaluated at each time point. Perhaps some of the many approximative techniques already existing for the Libor Market Model (see Rebonato (2002)) could be applied to speed up the computations without loss of accuracy, but this is left for future research. A fixed grid is used in the discretization of equation (1) with a stepsize equal to 0.125. Due to the state-dependent drift, discretization error cannot be avoided. But the magnitude can be evaluated following Glasserman and Zhao (2000), who argue that a good measure of discretization error is the percentage differences between bond prices from the simulation, and bond prices calculated from the Libor term structure. In the simulations performed in this paper the percentage differences are of the order 0.001% for the 20 year maturity, and lower in absolute terms for shorter maturities hence discretization biases can be safely ignored. The caplet prices are shown in the Table 1 and 2. The table shows Monte Carlo prices as well as the price from Proposition 2(CE), found using the cumulant ex- pansion approximation, and the price calculated using the Glasserman and Merener (2003a) formula(G&M). Except for a few cases in Table 2 the caplet prices reveal no noticeable difference between the CE formula, and the G&M-formula. A closer analysis shows that differences are at the level of 10−5 Bps and therefore without any financial relevance. Except for a few cases the differences between the approximative price and the MC price are not small enough to be explained by Monte Carlo variation, which is per- haps a bit troubling. The approximation appears to deteriorate further as maturity increases. To evaluate these errors in relative terms, the percentage pricing errors along with the half-widths of the 95% confidence limit divided by the MC price is plotted in Figures 2 and 3. While the errors are outside the of the MC confidence limit, they are still quite small and only outside the 1% for deep OTM prices. As a final check on the approximation errors, implied volatilies are plotted in Figures 4 and 5. For both parameter sets it shows a pronounced volatility smile for the 1 and 3 year maturity, and an almost flat volatility curve for the longer maturities. The deteriation of the approximation is markedly clearer when looking at implied vol. But even for the longest maturitues the errors levels are well within normal bid-ask

71 spreads typically around 0.005.

The results for Swaption prices are shown in Tables 3 and 4, for the two different parameter sets. In this case Monte Carlo confidence limits are larger and the errors fluctuate in and out the 95% interval across maturity and moneyness. The approx- imation errors grow as maturity increases similar to the caplet case. It can also be observed that the two approximations are indistinguishable for short tenors which can be expected since the 1 year tenor swaption behaves essentially like a caplet. But for longer tenors the CE formula is consistently better than the G&M formula in both parameter sets. The differences in the two approximations are investigated further in terms of percentage pricing errors plotted in Figures 8 and 9. Here it can be observed that in general percentage errors increase heavily as tenor increases. This was also found in the numerical experiments performed by Glasserman and Merener (2003a). The figures show that the CE formula generally performs better than the G&M formula in the long tenor case. Finally, implied volatilities are plotted figures 6 and 7. It can be seen that some of the larger percentage price errors translate into very small errors in implied volatil- ity. The errors increase when maturity increases, but as in the caplet case they are not critical around the at-the-money level.

7.2 Double Exponential Mark Distribution A more flexible jump size distribution can be introduced, by choosing marks follow- ing a double exponential distribution. This distribution has been used with some success in the equity and currency option price literature in Kou (2002), Kou and Wang (2003), and Kou and Wang (2004). It was also suggested for the use as a mark distribution in the Libor Market Model in Glasserman and Kou (2003), but they did not further investigate its use. The double exponential distribution has the following density function

f(x)= pη e−η1x1 + (1 p)η eη2x1 , 1 {x≥0} − 2 x<0 here p denotes the probability of an upward jump and (1 p) the corresponding − probability of a downward jump. The two parameters η1 and η2 control the tail decay of positive and negative axis. Hence the distribution allows for both positive and negative skewness. For use in our pricing formulas we need the characteristic function given in Kou and Wang (2003) Lemma 2.1 as

pη (1 p)η φ (z)= 1 + − 2 , X η iz η + iz 1 − 2 72 This function can be extended to the complex domain with a strip of regularity given by η2 < Im(z) < η1. This means the tuning parameter α has to satisfy η2 < − n − (1 + α) < η . Recall the nth moment of the distribution is m = ∂ φX (z) (0) − 1 n ∂zn yielding p 1 p m1 = − , η1 − η2 p 1 p m2 =2 2 + 2 −2 , η1 η2 p 1 p m3 =6 3 6 −3 , η1 − η2 p 1 p m4 =24 4 + 24 −4 , η1 η2 The corresponding cumulants may then be calculated from equation (12). The re- sulting expressions are very lengthy, and is available from the author by request. The formulas are tested using the same diffusion component as used in the normal distribution case, and with the following parameter set.

Parameter Set DE: p = 0.5 η1 = 1/0.05 and η2 = 1/0.07, with jump intensity λ = 5.

This corresponds to a mean jump size in log differences 5% and 7% for positive and positive and negative jumps respectively. This parameter set is chosen to approxi- mately match the corresponding level of ATM implied volatility used when testing normally distributed jump sizes. The caplet and swaption prices can be found in tables 5 and 6 and what can be seen is that error levels are similar to the normal case, and the accuracy goes from being very good in the short maturity spectrum, to poorer in long maturities. The percentage errors and implied vol plotted in figures 12, 13 reveal the same pattern.

7.3 Displaced Diffusion Here the dynamics in (19) are assumed. The dynamics are chosen to have double exponential marks and the same diffusion component and term structure of interest rates as in the previous sections.

Parameter Set DEDD: d = 0.04, p = 0.5, η1 = 1/0.03, η2 = 1/0.03 and λ = 5.

Note that these parameters no longer have interpretation of positive and negative

73 mean jump size in log changes because of the displacement factor. But they are again chosen to have the same level of ATM implied vol as the previous sections around 18%. The caplet and swaption prices are given in tables 7 and 8 and the errors show that the approximation fares equally well compared to the previous cases. The implied volatilities are graphed in figures 15 and 17 and here the effect of displacing the diffusion is noticeable since the short maturities are practically indistinguishable from the previous cases, but in the long run the skew is retained.

8 Summary

This paper derives approximate expressions for caplets and swaptions under the as- sumption that the Libor rates follow a diffusion with compound Poisson jumps under the spot measure. The model extends the existing pricing methods by allowing for an arbitrary parametric distribution driving the jump sizes. Furthermore, the accu- racy of the derived formulas in the classic log-normal jump size case are compared to existing approximations by Glasserman and Merener (2003a). The results from a Monte Carlo analysis show that the derived approximations are superior in pricing swaptions with longer tenors. Finally, the paper also extends the framework by displacing the jump-diffusion, and allowing for stochastic volatility in the diffusion component.

74 A Proof of Proposition 2

Proof. Defining f˜(t) = log(Fˆk(t)) and using Ito’s lemma for jump-diffusions (see for example Cont and Tankov (2004)) we get

˜ x ˆ ˆ 1 2 ˆ df(t)= (e 1)λ(t)f(x, t)dx γˆk dt +γ ˆkdW (t) − R − − 2  Z  x + log Fˆk(t )+ Fˆk(t )(e 1) log(Fˆk(t )) µˆ(dx,dt) R − − − − − Z  h i  x ˆ ˆ 1 2 ˆ = (e 1)λ(t)f(x, t)dx γˆk dt +γ ˆkdW (t)+ xµˆ(dx,dt), − R − − 2 R  Z  Z whereµ ˆ(dx,dt) is the jump measure corresponding to the compound Poisson process with compensatorν ˆk(x, t)= λˆ(t)fˆ(x, t). The terminal state can then be written as

Tk Tk ˜ ˜ 1 2 ˆ f(Tk)=f(0) αˆk γˆkTk + γˆkdW (t)+ xµˆ(dx,dt), (21) − − 2 R Z0 Z0 Z whereα ˆ = δ k (ex 1)fˆ(x, T )λˆ(T )dx. Since the underlying jump density is k R j=1 − j j the same between tenor points we can write R P

Nˆ(Tj ) Tk k ˆ j xµˆ(dx,dt)= δ Xn, R 0 j=0 ˆ Z Z X n=NX(Tj−1)+1

ˆ j ˆ j where X1 ,..., Xn are IID random marks drawn from the distribution with density fˆ(x, T ). Defining ∆Nˆ(j) = Nˆ(T ) Nˆ(T ) 1 and observing that ∆Nˆ(j) j j − j−1 − ∼ po(λˆ(Tj)δ), as well as conditioning on the number of jumps in each interval and

75 using their independence yields

ˆ k N(Tj ) j Tk iz Xn iz xµˆ(dx,dt) n Nˆ T E[e 0 R ]= E[e = ( j−1)+1 ] P R R j=0 Y k ∞ j = P (∆N(j)= n)E[eizXn)]n j=1 n=0 Y X k ∞ ˆ n n −λˆ(Tj )δ (λ(Tj)δ) izx = e e fˆ(x, Tj)dx n! R j=1 n=0 Y X Z  k izx = exp λˆ(Tj)δ e fˆ(x, Tj)dx 1 R − j=1 Y  Z  k izx = exp δ λˆ(Tj) e 1 fˆ(x, Tj)dx . R − Z j=1 ! X
From (21) and the above result, it follows that

izf˜(Tk) E[e ]= φcont(z)φjump(z), where

1 2 iz log(Fˆk(0))− iz(z+i)ˆγ φcont(z) = e 2 k k izx φjump(z) = exp izαˆk + δ λˆ(Tj) e 1 fˆ(x, Tj)dx . − R − Z j=1 ! X
The formula in (6) then follows from equation (6) in Carr and Madan (1999)

76 B Pricing Caplets with a Displacement

Following the exact same approximation steps as Section 3 gives us the approximate dynamics

ˆ ˆ Nd(t) dFk(t) x Xˆℓ =ˆγkdW (t) (e 1)λˆd(t)fˆd(x, t)dx + d (e 1) , Fˆk(t )+ d − R −  −  − Z Xℓ=1 0 t T , k = 1,...,K,   ≤ ≤ k where the Nˆd(t) is a Poisson process with intensity

k ˆ 1+ Fj(0) λ(t)= x x λf(x)dx, (22) R 1+ δFj(0)e + d(e 1) Z j=Yβ(t)  −  and the Xˆℓ’s are distributed with density

k ˆ 1+ Fj(0) fd(x, t)= x x λf(x)dx. (23) R 1+ δFj(0)e + d(e 1) Z j=Yβ(t)  −  This yields the following proposition Proposition 5. Assume the Libor rate follows the dynamics 19. Then the approx- imate price of a Tk-caplet with strike K is given by δ ∞ φ(z (α + 1)i) Cpl = P (0, T ) e−(iz−α) log(K+d) − dz, k k+1 π α2 + α z2 + i(2α + 1)z Z0 − where α is a predetermined dampening constant. ˆ The characteristic function φ(z)= E[eiz log(Fk(Tk)+d)] is given by by

φ(z)= φcont(z)φjump(z). The components of the characteristic function are

1 2 iz log(Fˆk(0)+d)− iz(z+i)ˆγ φcont(z)= e 2 k k izx φjump(z) = exp izαˆk + δ λˆ(Tj) e 1 fˆ(x, Tj)dx , − R − Z j=1 ! X
k x ˆ ˆ ˆ ˆ where αˆk = δ R j=1(e 1)f(x, Tj)λ(Tj)dx with fd(x, Tj) and λd(Tj) defined in (23) and (22) respectively − R P the cumulant expansion is the same as equation (13) with fˆ(x, t) and λˆ(t) replaced by fˆd(x, t) and λˆd(t)

77 C Pricing Swaptions with a Displacement

Proposition 6. Swaption Pricing Assume the Libor rate dynamics are given by equation (19). The price of a Ta (T T ) payer swaption with strike K is given by × b − a 1 ∞ φ(z (α + 1)i) PS(T , T )= C (0) e(−iz−α) log(K+d) − dz, a b a,b π α2 + α z2 + i(2α + 1)z Z0 − ˆ where α> 0.5 is a tuning parameter. The characteristic function φ(z)= E[eiz log(Sa,b(Ta))+d] is given by

φ(z)= φcont(z)φjump(z),

where the two components of the characteristic function are

1 2 iz log(Sˆa,b(0)+d)− iz(z+i)γ φcont(z)= e 2 a,b a d ˆ iz log(Ha,b(x)+1) ˆd φjump(z) = exp izαˆa,b + λ(Tj)δ(e 1)fa,b(x, Tj)dx , − R − j=0 ! Z X a d ˆd ˆd with αˆa,b = R j=1 Ha,b(x)λa,b(Tj)fa,b(x, Tj)dx R P The cumulant expansion can be carried out as in equation (18) with Ha,b(x), fˆa,b(x, t) ˆ d ˆd ˆd and λa,b(t) replaced by Ha,b fa,b(x, t) and λa,b(t).

78 References

Abramovitz, M., and I. Stegun (1972): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York Dover.

Andersen, L., and J. Andreasen (2000): “Volatility Skews and Extensions of the Libor Market Model,” Applied Mathematical Finance, 7, 1–32.

(2001): “Factor Dependence of Bermudan Swaptions: Fact or Fiction,” Journal of Financial Economics, 62(1), 3–37.

Andersen, L., and R. Brotherton-Ratcliffe (2005): “Extended Libor Mar- ket Models with Stochastic Volatility,” Journal of Computational Finance, 9(1), 1–40.

Brace, A., D. Gatarek, and M. Musiela (1997): “The Market Model of Interest Rate Dynamics,” Mathematical Finance, 7(2), 127–154.

Brigo, D., and F. Mercurio (2006): Interest Rate Models, Theory and Practice, 2nd Edition. Springer Finance.

Carr, P., and D. Madan (1999): “Option Pricing and the Fast Fourier Trans- form,” Journal of Computational Finance, 2(4), 61–73.

Cont, R., and P. Tankov (2004): Financial Modelling with Jump Processes. Chapman & Hall/CRC Financial Mathematics Series.

Cox, J., J. Ingersoll, and S. Ross (1985): “A Theory of the Term Structure of Interest Rates,” Econometrica, 53, 385–407.

Duffie, D., J. Pan, and K. Singleton (2000): “Transform Analysis and Asset Pricing for Affine Jump Diffusions,” Econometrica, 68(6), 1343–1376.

Glasserman, P. (2004): Monte Carlo Methods in Financial Engineering,. Springer Verlag.

Glasserman, P., and S. Kou (2003): “The Term Structure of Simple Forward Rates with Jump Risk,” Mathematical Finance, 13(3), 383–410.

Glasserman, P., and N. Merener (2003a): “Cap and swaption approximations in Libor market models with jumps,” Journal of Computational Finance, 7(1), 1–36.

(2003b): “Numerical Solution of Jump-Diffusion LIBOR Market Models,” Finance and Stochastics, 7, 1–27.

79 Glasserman, P., and X. Zhao (2000): “Arbitrage-free discretization of lognor- mal forward Libor and swap rate models,” Finance and Stochastics, 4, 35–68.

Heston, S. L. (1993): “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options,” Review of Financial Studies, 6(2), 327–343.

Hunt, P. J., and J. E. Kennedy (2004): Financial Derivatives in Theory and Practice. Wiley.

Jamshidian, F. (1997): “Libor and Swap Market Models and Measures,” Finance and Stochastics, 1(4), 261–291.

(1999): “Libor Market Model with Semimartingales,” Working Paper, NetAnalytic Ltd., London.

Jarrow, R., H. Li, and F. Zhao (2007): “Interest Rate Caps ’Smile’ Too! But Can the LIBOR Market Models Capture the Smile?,” Journal of Finance, 62, 345–382.

Kolassa, J. (2006): Series Approximation Methods in Statistics, 3rd Edition. Lec- ture Notes in Statistics, New York: Springer Verlag.

Kou, S. G. (2002): “Option Pricing Under a Double Exponential Jump Diffusion Model,” Management Science, 48(8), 10861101.

Kou, S. G., and H. Wang (2003): “First Passage Times of a Jump Diffusion Model,” Advances in Applied Probability, 35, 504–531.

(2004): “Option Pricing Under a Double Exponential Jump Diffusion Model,” Management Science, 50, 1178–1192.

Lee, R. (2004): “Option Pricing by Transform Methods: Extensions, Unification, and Error Control,” Journal of Computational Finance, 7(3), 51–86.

Lord, R., and C. Kahl (2007): “Optimal Fourier Inversion in Semi-Analytical Option Pricing,” SSRN eLibrary.

Merton, R. C. (1976): “Option Pricing When Underlying Stock Returns Are Discontinuous,” Journal of Financial Economics, 3(1-2), 125–144.

Miltersen, K. R., K. Sandmann, and D. Sondermann (1997): “Closed form solutions for term structure derivatives with log-normal interest rates,” Journal of Finance, 52(2), 409–430.

80 Ramezani, C. A., and Y. Zeng (2007): “Maximum likelihood estimation of the double exponential jump-diffusion process,” Annals of Finance, 3, 487–507.

Rebonato, R. (2002): Modern Pricing of Interest Rate Derivatives, The Libor Market Model and Beyond. Princeton University press.

(2004): Volatility and Correlation: The Perfect Hedger and the Fox. Wiley.

Rubinstein, M. (1983): “Displaced Diffusion Option Pricing,” Journal of Finance, 38(1), 213–217.

Skovmand, D. (2008): “Alternative Specifications for the L´evy Libor Market Model: An Empirical Investigation,” Working Paper.

Vasicek, O. (1977): “An Equilibrium Characterization of the Term Structure,” Journal of Financial Economics, 5(2), 177–188.

Wu, L., and F. Zhang (2006): “Libor Market Model With Stochastic Volatility,” Journal of Industrial and Management Optimization, 2(2), 199227.

Yan, G., and F. Hanson (2006): “Option Pricing for a Stochastic-Volatility Jump-Diffusion Model with Log-Uniform Jump-Amplitudes,” Proceedings of American Control Conference, pp. 2989–2994.

81 Table 1: Caplet Prices for the Normal distribution, Parameter Set ANORM 0.5 0.8 1 1.2 1.5 T=1(MC) 90.86 (0.00082) 38.28 (0.00057) 12.88 (0.00032) 2.61 (0.00015) 0.18 (4e-005) T=1(CE) 90.85 38.27 12.88 2.61 0.18 T=1(G&M) 90.85 38.27 12.88 2.61 0.18 T=3(MC) 90.12 (0.0055) 43.29 (0.0039) 22.37 (0.0028) 10.36 (0.0019) 2.89 (0.0011) T=3(CE) 90.09 43.25 22.35 10.35 2.89 T=3(G&M) 90.09 43.25 22.35 10.35 2.89 T=7(MC) 84.51 (0.026) 47.88 (0.021) 31.41 (0.017) 20.24 (0.014) 10.33 (0.01) T=7(CE) 84.42 47.79 31.33 20.18 10.30 T=7(G&M) 84.42 47.79 31.33 20.18 10.30

T=10(MC) 78.27 (0.048) 47.89 (0.04) 34.00 (0.034) 24.07 (0.03) 14.41 (0.024) 82 T=10(CE) 78.09 47.72 33.85 23.95 14.32 T=10(G&M) 78.09 47.72 33.85 23.95 14.32 T=15(MC) 65.68 (0.081) 43.84 (0.071) 33.60 (0.064) 25.92 (0.057) 17.81 (0.049) T=15(CE) 65.45 43.59 33.37 25.70 17.63 T=15(G&M) 65.45 43.59 33.37 25.70 17.63 T=20(MC) 53.14 (0.098) 37.62 (0.088) 30.21 (0.08) 24.49 (0.074) 18.17 (0.065) T=20(CE) 52.89 37.35 29.94 24.23 17.93 T=20(G&M) 52.89 37.35 29.94 24.23 17.93

This table shows the caplet price for different maturities across moneyness, X/FK (0). MC denotes the Monte Carlo price with the halfwidth of the 95% confidence limit in paranthesis. CE denotes the price calculated using the approximation in Proposition 2, and G&M is the caplet formula derived in Section 5.3 in Glasserman and Merener (2003a). The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. . Table 2: Caplet Prices for the Normal distribution, Parameter Set BNORM 0.5 0.8 1 1.2 1.5 T=1(MC) 90.86 (0.0011) 38.45 (0.00079) 11.72 (0.0005) 3.01 (0.00029) 0.42 (0.00013) T=1(CE) 90.87 38.47 11.74 3.01 0.43 T=1(G&M) 90.87 38.47 11.74 3.01 0.43 T=3(MC) 90.09 (0.0066) 43.20 (0.005) 21.98 (0.0038) 10.23 (0.0028) 3.22 (0.0018) T=3(CE) 90.18 43.26 22.03 10.26 3.24 T=3(G&M) 90.18 43.26 22.03 10.26 3.24 T=7(MC) 84.38 (0.03) 47.70 (0.025) 31.20 (0.021) 20.09 (0.018) 10.37 (0.014) T=7(CE) 84.54 47.82 31.28 20.16 10.43 T=7(G&M) 84.54 47.82 31.28 20.16 10.43 83 T=10(MC) 78.05 (0.054) 47.68 (0.046) 33.79 (0.041) 23.91 (0.036) 14.36 (0.029) T=10(CE) 78.22 47.78 33.87 23.98 14.41 T=10(G&M) 78.22 47.79 33.87 23.98 14.42 T=15(MC) 65.43 (0.088) 43.62 (0.077) 33.40 (0.07) 25.75 (0.064) 17.68 (0.055) T=15(CE) 65.56 43.68 33.43 25.76 17.69 T=15(G&M) 65.56 43.68 33.43 25.76 17.70 T=20(MC) 52.97 (0.1) 37.48 (0.091) 30.08 (0.084) 24.37 (0.077) 18.07 (0.068) T=20(CE) 52.99 37.45 30.02 24.30 17.99 T=20(G&M) 52.99 37.45 30.03 24.30 17.99

This table shows the caplet price for different maturities across moneyness, X/FK (0). MC denotes the Monte Carlo price with the halfwidth of the 95% confidence limit in paranthesis. CE denotes the price calculated using the approximation in Proposition 2, and G&M is the caplet formula derived in Section 5.3 in Glasserman and Merener (2003a). The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. . Table 3: Swaption Prices for the Normal distribution, Parameter Set ANORM 0.5 1 1.5 1 X 1(MC) 181.69 (0.0021) 25.77 (0.0009) 0.35 (0.00012) 1 X 1 181.67(CE) 181.67(G&M) 25.75(CE) 25.76(G&M) 0.35(CE) 0.35(G&M) 1 X 5(MC) 889.81 (0.029) 126.14 (0.016) 1.66 (0.0019) 1 X 5 889.71(CE) 889.71(G&M) 126.09(CE) 126.37(G&M) 1.69(CE) 1.72(G&M) 1 X 10(MC) 1680.92 (0.096) 237.72 (0.054) 2.95 (0.0055) 1 X 10 1680.71(CE) 1680.72(G&M) 237.68(CE) 239.30(G&M) 3.06(CE) 3.21(G&M) 10 X 1(MC) 155.05 (0.097) 67.37 (0.071) 28.56 (0.049) 10 X 1 154.71(CE) 154.71(G&M) 67.07(CE) 67.07(G&M) 28.39(CE) 28.39(G&M) 10 X 5(MC) 714.90 (0.55) 310.94 (0.42) 131.86 (0.3)

10 X 5 713.29(CE) 713.33(G&M) 309.61(CE) 309.73(G&M) 131.22(CE) 131.33(G&M) 84 10 X 10(MC) 1283.37 (1.2) 558.39 (0.89) 236.32 (0.63) 10 X 10 1280.70(CE) 1280.75(G&M) 557.03(CE) 557.18(G&M) 236.62(CE) 236.77(G&M) 20 X 1(MC) 104.95 (0.19) 59.67 (0.16) 35.90 (0.13) 20 X 1 104.45(CE) 104.45(G&M) 59.15(CE) 59.15(G&M) 35.42(CE) 35.42(G&M) 20 X 5(MC) 474.85 (0.9) 270.18 (0.74) 162.56 (0.59) 20 X 5 472.90(CE) 472.84(G&M) 268.26(CE) 268.13(G&M) 160.92(CE) 160.77(G&M) 20 X 10(MC) 837.69 (1.6) 476.74 (1.3) 286.25 (1) 20 X 10 835.44(CE) 834.71(G&M) 475.56(CE) 474.11(G&M) 286.33(CE) 284.75(G&M)

This table shows swaption prices for different maturities across moneyness, X/FK (0). MC denotes the Monte Carlo price generated using the simulation scheme in the Appendix with the halfwidth of the 95% confidence limit in paranthesis. CE denotes the price calculated using the approximation in Proposition 4 with P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. G&M is the caplet formula derived in Section 5.4 in Glasserman and Merener (2003a). Table 4: Swaption Prices for the Normal distribution, Parameter Set BNORM 0.5 1 1.5 1 X 1(MC) 181.68 (0.0027) 23.44 (0.0013) 0.84 (0.00035) 1 X 1 181.71(CE) 181.71(G&M) 23.49(CE) 23.50(G&M) 0.86(CE) 0.86(G&M) 1 X 5(MC) 889.73 (0.038) 115.07 (0.021) 3.92 (0.0049) 1 X 5 889.89(CE) 889.90(G&M) 115.30(CE) 115.55(G&M) 4.04(CE) 4.10(G&M) 1 X 10(MC) 1680.75 (0.12) 217.67 (0.071) 6.88 (0.014) 1 X 10 1681.08(CE) 1681.11(G&M) 218.03(CE) 219.46(G&M) 7.20(CE) 7.50(G&M) 10 X 1(MC) 154.63 (0.11) 66.96 (0.083) 28.44 (0.061) 10 X 1 154.95(CE) 154.95(G&M) 67.11(CE) 67.11(G&M) 28.55(CE) 28.56(G&M) 10 X 5(MC) 713.44 (0.61) 309.22 (0.47) 130.95 (0.35) 85 10 X 5 714.46(CE) 714.51(G&M) 309.60(CE) 309.73(G&M) 131.52(CE) 131.65(G&M) 10 X 10(MC) 1281.77 (1.2) 555.82 (0.98) 234.22 (0.71) 10 X 10 1283.00(CE) 1283.07(G&M) 556.79(CE) 556.97(G&M) 236.38(CE) 236.57(G&M) 20 X 1(MC) 104.63 (0.2) 59.43 (0.17) 35.72 (0.13) 20 X 1 104.65(CE) 104.66(G&M) 59.30(CE) 59.31(G&M) 35.53(CE) 35.53(G&M) 20 X 5(MC) 474.20 (0.92) 269.74 (0.76) 162.12 (0.61) 20 X 5 473.89(CE) 473.84(G&M) 268.92(CE) 268.81(G&M) 161.26(CE) 161.14(G&M) 20 X 10(MC) 838.00 (1.6) 477.13 (1.3) 286.29 (1) 20 X 10 837.34(CE) 836.66(G&M) 476.75(CE) 475.35(G&M) 286.76(CE) 285.23(G&M)

This table shows swaption prices for different maturities across moneyness, X/FK (0). MC denotes the Monte Carlo price generated using the simulation scheme in the Appendix with the halfwidth of the 95% confidence limit in paranthesis. CE denotes the price calculated using the approximation in Proposition 4 with P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. G&M is the caplet formula derived in Section 5.4 in Glasserman and Merener (2003a). Table 5: Caplet Prices, Double Exponential Marks, Parameter Set DE 0.5 0.8 1 1.2 1.5 T=1(MC) 90.87 (0.00094) 38.62 (0.00067) 12.96 (0.0004) 2.85 (0.00021) 0.29 (7.7e-005) T=1(CEFourier) 90.87 38.62 12.96 2.85 0.29 T=3(MC) 90.25 (0.0061) 43.88 (0.0045) 22.98 (0.0033) 10.87 (0.0024) 3.25 (0.0015) T=3(CEFourier) 90.24 43.87 22.97 10.86 3.25 T=7(MC) 84.88 (0.029) 48.78 (0.024) 32.42 (0.02) 21.21 (0.017) 11.13 (0.013) T=7(CEFourier) 84.84 48.72 32.37 21.17 11.10

T=10(MC) 78.72 (0.052) 48.87 (0.044) 35.11 (0.039) 25.19 (0.034) 15.40 (0.028) 86 T=10(CEFourier) 78.64 48.77 35.02 25.11 15.34 T=15(MC) 66.24 (0.088) 44.83 (0.077) 34.73 (0.07) 27.09 (0.064) 18.93 (0.055) T=15(CEFourier) 66.08 44.66 34.56 26.92 18.78 T=20(MC) 53.63 (0.1) 38.49 (0.094) 31.21 (0.087) 25.54 (0.08) 19.23 (0.071) T=20(CEFourier) 53.51 38.34 31.04 25.37 19.05

This table shows the caplet price for different maturities across moneyness, X/FK (0). MC denotes the Monte Carlo price with the halfwidth of the 95% confidence limit in paranthesis. CE denotes the price calculated using the approximation in Proposition 2, and G&M is the caplet formula derived in Section 5.3 in Glasserman and Merener (2003a). The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. . Table 6: Swaption Prices, Double Exponential Marks, Parameter Set DE 0.5 1 1.5 1 X 1(MC) 181.72 (0.0024) 25.92 (0.0011) 0.58 (0.00021) 1 X 1(CE) 181.72 25.92 0.58 1 X 5(MC) 889.96 (0.033) 126.97 (0.019) 2.72 (0.003) 1 X 5(CE) 889.92 127.02 2.77 1 X 10(MC) 1681.28 (0.11) 239.49 (0.064) 4.81 (0.0088) 1 X 10(CE) 1681.12 239.69 5.00 10 X 1(MC) 156.00 (0.1) 69.61 (0.077) 30.56 (0.055) 10 X 1(CE) 155.79 69.39 30.41 10 X 5(MC) 719.44 (0.59) 321.46 (0.45) 141.15 (0.33) 87 10 X 5(CE) 718.44 320.50 140.63 10 X 10(MC) 1291.95 (1.2) 577.75 (0.96) 253.15 (0.7) 10 X 10(CE) 1290.30 577.08 253.81 20 X 1(MC) 105.94 (0.21) 61.67 (0.17) 38.02 (0.14) 20 X 1(CE) 105.69 61.32 37.64 20 X 5(MC) 479.62 (0.95) 279.54 (0.79) 172.41 (0.64) 20 X 5(CE) 478.69 278.33 171.19 20 X 10(MC) 846.80 (1.7) 493.88 (1.4) 304.08 (1.1) 20 X 10(CE) 846.02 493.86 304.97

This table shows swaption prices for different maturities across moneyness, X/FK (0). MC denotes the Monte Carlo price generated using the simulation scheme in the Appendix with the halfwidth of the 95% confidence limit in paranthesis. CE denotes the price calculated using the approximation in Proposition 4 with P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. G&M is the caplet formula derived in Section 5.4 in Glasserman and Merener (2003a). Table 7: Caplet Prices, Double Exponential Marks, Displaced Diffusion, Parameter Set DEDD 0.5 0.8 1 1.2 1.5 T=1(MC) 90.87 (0.00094) 38.62 (0.00067) 12.96 (0.0004) 2.85 (0.00021) 0.29 (7.7e-005) T=1(CEFourier) 90.87 38.62 12.96 2.85 0.29 T=3(MC) 90.25 (0.0061) 43.88 (0.0045) 22.98 (0.0033) 10.87 (0.0024) 3.25 (0.0015) T=3(CEFourier) 90.24 43.87 22.97 10.86 3.25 T=7(MC) 84.88 (0.029) 48.78 (0.024) 32.42 (0.02) 21.21 (0.017) 11.13 (0.013) T=7(CEFourier) 84.84 48.72 32.37 21.17 11.10

T=10(MC) 78.72 (0.052) 48.87 (0.044) 35.11 (0.039) 25.19 (0.034) 15.40 (0.028) 88 T=10(CEFourier) 78.64 48.77 35.02 25.11 15.34 T=15(MC) 66.24 (0.088) 44.83 (0.077) 34.73 (0.07) 27.09 (0.064) 18.93 (0.055) T=15(CEFourier) 66.08 44.66 34.56 26.92 18.78 T=20(MC) 53.63 (0.1) 38.49 (0.094) 31.21 (0.087) 25.54 (0.08) 19.23 (0.071) T=20(CEFourier) 53.51 38.34 31.04 25.37 19.05

This table shows the caplet price for different maturities across moneyness, X/FK (0). MC denotes the Monte Carlo price with the halfwidth of the 95% confidence limit in paranthesis. CE denotes the price calculated using the approximation in Proposition 2, and G&M is the caplet formula derived in Section 5.3 in Glasserman and Merener (2003a). The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. . Table 8: Swaption Prices, Double Exponential Marks, Displaced Diffusion, Parameter Set DEDD 0.5 1 1.5 1 X 1(MC) 181.78 (0.0027) 26.45 (0.0012) 0.69 (0.00018) 1 X 1(CE) 181.79 26.42 0.68 1 X 5(MC) 890.09 (0.036) 125.37 (0.019) 2.83 (0.0041) 1 X 5(CE) 890.13 125.29 2.80 1 X 10(MC) 1681.32 (0.11) 230.38 (0.062) 4.49 (0.011) 1 X 10(CE) 1681.34 230.39 4.50 10 X 1(MC) 158.44 (0.079) 65.16 (0.051) 22.27 (0.03) 10 X 1(CE) 158.37 65.11 22.29 10 X 5(MC) 729.62 (0.44) 298.68 (0.29) 101.12 (0.17) 89 10 X 5(CE) 729.25 298.63 101.43 10 X 10(MC) 1309.30 (0.92) 534.71 (0.63) 179.69 (0.38) 10 X 10(CE) 1308.88 535.57 181.35 20 X 1(MC) 109.40 (0.14) 57.85 (0.1) 29.08 (0.072) 20 X 1(CE) 109.27 57.77 29.06 20 X 5(MC) 496.09 (0.68) 262.95 (0.5) 132.41 (0.35) 20 X 5(CE) 495.57 262.84 132.68 20 X 10(MC) 878.82 (1.2) 468.11 (0.92) 236.65 (0.65) 20 X 10(CE) 878.75 469.55 239.06

This table shows swaption prices for different maturities across moneyness, X/FK (0). MC denotes the Monte Carlo price generated using the simulation scheme in the Appendix with the halfwidth of the 95% confidence limit in paranthesis. CE denotes the price calculated using the approximation in Proposition 4 with P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. G&M is the caplet formula derived in Section 5.4 in Glasserman and Merener (2003a). 0.048

0.046

0.044

0.042 Rate

0.04

0.038

0.036 0 5 10 15 20 25 30 Maturity

Figure 1: EURIBOR Term Structure This figure shows the EURIBOR forward rates with 6 month tenor taken from Datastream the 6th of May 2006. This term structure is used in all numerical experiments.

90 T=1 T=3 T=7 0.15 0.15 0.5

0.1 0.4

0.1 0.05 0.3

0 0.2 0.05 −0.05 0.1 Pct. Error Pct. Error Pct. Error

−0.1 0 0

−0.15 −0.1

−0.2 −0.05 −0.2 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

T=10 T=15 T=20 0.6 1.2 1.5 CE 0.5 1 95% CL 91

0.4 0.8 1

0.3 0.6

0.2 0.4 0.5 Pct. Error Pct. Error 0.1 Pct. Error 0.2

0 0 0

−0.1 −0.2

−0.2 −0.4 −0.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

Figure 2: Caplet Price Percentage errors, Normal Marks, Parameter Set ANORM This figure shows the caplet price percentage error defined as 100 times the simulated price minus the CE price divided by the simulated price. The errors are plotted for the different maturities across moneynessX/FK (0). MC prices are generated using the simulation scheme in the Appendix along with the 95% confidence limit. CE denotes the price calculated using the approximation in Proposition 2. The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. T=1 T=3 T=7 0.5 0.1 0.2

0 0.1 0 0 −0.1 −0.5 −0.1 −0.2 −1 −0.2 −0.3 Pct. Error Pct. Error Pct. Error −0.3 −1.5 −0.4 −0.4 −2 −0.5 −0.5

−2.5 −0.6 −0.6 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

T=10 T=15 T=20 0.3 0.4 0.6 CEFourier 0.2 0.3 95% CL 0.4

0.2 92 0.1 0.1 0.2 0 0 −0.1 Pct. Error Pct. Error Pct. Error 0 −0.1 −0.2 −0.2 −0.2 −0.3 −0.3

−0.4 −0.4 −0.4 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

Figure 3: Caplet Price Percentage errors, Normal Marks, Parameter Set BNORM This figure shows the caplet price percentage error defined as 100 times the simulated price minus the CE price divided by the simulated price. The errors are plotted for the different maturities across moneynessX/FK (0). MC prices are generated using the simulation scheme in the Appendix along with the 95% confidence limit. CE denotes the price calculated using the approximation in Proposition 2. The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. T=1 T=3 T=7 0.25 0.2 0.194

0.24 0.192 0.195 0.23 0.19

0.22 0.188 0.19

0.21 0.186

0.185 Implied Vol. Implied Vol. Implied Vol. 0.2 0.184

0.19 0.182 0.18 0.18 0.18

0.17 0.175 0.178 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 MC Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) CEFourier 95% CL T=10 T=15 T=20 0.192 0.192 0.194 93 0.19 0.19 0.192

0.188 0.188 0.19

0.186 0.186 0.188 Implied Vol. Implied Vol. Implied Vol. 0.184 0.184 0.186

0.182 0.182 0.184

0.18 0.18 0.182 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

Figure 4: Caplet Implied Volatility , Normal Marks, Parameter Set ANORM This figure shows the caplet implied volatility. MC prices are generated using the simulation scheme in the Appendix along with the 95% confidence limit. CE denotes the price calculated using the approximation in Proposition 2. The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. T=1 T=3 T=7 0.26 0.205 0.194

0.192 0.2 0.24 0.19 0.195 0.188 0.22

0.19 0.186

0.2 Implied Vol. Implied Vol. Implied Vol. 0.184 0.185 0.182 0.18 0.18 0.18

0.16 0.175 0.178 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

T=10 T=15 T=20 0.192 0.189 0.189 MC 0.188 CEFourier 0.19 0.188

95% CL 94 0.187 0.188 0.187 0.186

0.186 0.185 0.186 Implied Vol. Implied Vol. Implied Vol. 0.184 0.184 0.185 0.183 0.182 0.184 0.182

0.18 0.181 0.183 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

Figure 5: Caplet Implied Volatility, Normal Marks, Parameter Set BNORM This figure shows the caplet implied volatility. MC prices are generated using the simulation scheme in the Appendix along with the 95% confidence limit. CE denotes the price calculated using the approximation in Proposition 2. The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. 1 X 1 1 X 5 1 X 7 1 X 10

0.24 0.24 0.24 0.24 0.22 0.22 0.22 0.22 0.2 0.2 0.2 0.2 0.18 0.18 0.18 0.18 Implied Vol. Implied Vol. Implied Vol. Implied Vol. 0.16 0.16 0.16 0.16 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 5 X 1 5 X 5 5 X 7 5 X 10 0.2 0.2 0.2 0.2

0.19 0.19 0.19 0.19

0.18 0.18 0.18 0.18 Implied Vol. Implied Vol. Implied Vol. Implied Vol.

MC 0.17 0.17 0.17 0.17 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 CE Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) G&M 15 X 1 15 X 5 15 X 7 15 X 10 95% CL 0.195 0.2 0.2 0.2

95 0.19 0.19 0.19 0.19 0.185 Implied Vol. Implied Vol. Implied Vol. Implied Vol.

0.18 0.18 0.18 0.18 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 20 X 1 20 X 5 20 X 7 20 X 10 0.2 0.2 0.2 0.2

0.19 0.19 0.19 0.19 Implied Vol. Implied Vol. Implied Vol. Implied Vol.

0.18 0.18 0.18 0.18 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

Figure 6: Swaption Implied Volatility , Normal Marks, Parameter Set ANORM This figure shows the swaption implied volatility. The title for each subgraph denotes Maturity × Tenor. MC prices are generated using the simulation scheme in the Appendix along with the 95% confidence limit. CE denotes the price calculated using the approximation in Proposition 4. The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. 1 X 1 1 X 5 1 X 7 1 X 10 0.3 0.3 0.3 0.3

0.2 0.2 0.2 0.2 Implied Vol. Implied Vol. Implied Vol. Implied Vol.

0.1 0.1 0.1 0.1 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 5 X 1 5 X 5 5 X 7 5 X 10 0.2 0.2 0.2 0.2

0.19 0.19 0.19 0.19

0.18 0.18 0.18 0.18 Implied Vol. Implied Vol. Implied Vol. Implied Vol.

MC 0.17 0.17 0.17 0.17 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 CE Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) G&M 15 X 1 15 X 5 15 X 7 15 X 10 95% CL 0.195 0.195 0.195 0.195

0.19 0.19 0.19 0.19 96 0.185 0.185 0.185 0.185 Implied Vol. Implied Vol. Implied Vol. Implied Vol.

0.18 0.18 0.18 0.18 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 20 X 1 20 X 5 20 X 7 20 X 10 0.195 0.195 0.195 0.2

0.19 0.19 0.19 0.19 0.185 0.185 0.185 Implied Vol. Implied Vol. Implied Vol. Implied Vol.

0.18 0.18 0.18 0.18 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

Figure 7: Swaption Implied Volatility, Normal Marks, Parameter Set BNORM This figure shows the swaption implied volatility. The title for each subgraph denotes Maturity × Tenor. MC prices are generated using the simulation scheme in the Appendix along with the 95% confidence limit. CE denotes the price calculated using the approximation in Proposition 4. The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. 1 X 1 1 X 5 1 X 7 1 X 10 0.5 2 5 5

0 0 0 0

Pct. Err Pct. Err −2 Pct. Err −5 Pct. Err −5

−0.5 −4 −10 −10 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 5 X 1 5 X 5 5 X 7 5 X 10 0.5 0.5 0.5 1

0 0 0 0

Pct. Err Pct. Err Pct. Err −0.5 Pct. Err −1

−0.5 −0.5 −1 −2 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 CE Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) G&M 15 X 1 15 X 5 15 X 7 15 X 10 95%CL 2 1 1 0.5

97 1 0.5 0.5 0 Pct. Err Pct. Err Pct. Err 0 Pct. Err 0 0

−1 −0.5 −0.5 −0.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 20 X 1 20 X 5 20 X 7 20 X 10 2 2 1 1

1 1 0.5 0.5 Pct. Err Pct. Err Pct. Err 0 Pct. Err 0 0 0

−1 −1 −0.5 −0.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

Figure 8: Swaption Price Percentage errors, Normal Marks, Parameter Set ANORM This figure shows the swaption price percentage error defined as 100 times the simulated price minus CE price divided by the simulated price. The errors are plotted for Maturity × Tenor across moneynessX/FK (0). MC prices are generated using the simulation scheme in the Appendix along with the 95% confidence limit. CE denotes the price calculated using the approximation in Proposition 4. The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. 1 X 1 1 X 5 1 X 7 1 X 10 2 5 5 5

0 0 0 0 Pct. Err Pct. Err Pct. Err −2 Pct. Err −5 −5

−4 −5 −10 −10 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 5 X 1 5 X 5 5 X 7 5 X 10 0.5 1 1 2

0 0 0 0 Pct. Err Pct. Err Pct. Err −0.5 Pct. Err −1 −1 −2

−1 −2 −2 −4 CE 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 G&M Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 95%CL 15 X 1 15 X 5 15 X 7 15 X 10 0.5 0.5 0.5 0.5

0 0 0 0 98 Pct. Err Pct. Err Pct. Err Pct. Err

−0.5 −0.5 −0.5 −0.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 20 X 1 20 X 5 20 X 7 20 X 10 1 1 1 0.5

0.5 0.5 0.5 0 Pct. Err Pct. Err Pct. Err 0 Pct. Err 0 0

−0.5 −0.5 −0.5 −0.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

Figure 9: Swaption Price Percentage errors, Normal Marks, Parameter Set BNORM This figure shows the swaption price percentage error defined as 100 times the simulated price minus CE price divided by the simulated price. The errors are plotted for Maturity × Tenor across moneynessX/FK (0). MC prices are generated using the simulation scheme in the Appendix along with the 95% confidence limit. CE denotes the price calculated using the approximation in Proposition 4. The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. −3 x 10 T=1 T=3 T=7 3 0.015 0.06

2 0.01 0.04

1 0.005 0.02 Bps Error Bps Error Bps Error 0 0 0

−1 −0.005 −0.02

−2 −0.01 −0.04 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) CEFourier 95% CL T=10 T=15 T=20 0.1 0.25 0.2

0.08

99 0.2 0.15

0.06 0.15 0.1 0.04 0.1 0.05 0.02 0.05 0 Bps Error Bps Error 0 Bps Error 0 −0.05 −0.02

−0.04 −0.05 −0.1

−0.06 −0.1 −0.15 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

Figure 10: Caplet Price Percentage errors, Double Exponential Distribution, Parameter Set ANORM This figure shows the caplet price percentage error defined as 100 times the simulated price minus the CE price divided by the simulated price. The errors are plotted for the different maturities across moneynessX/FK (0). MC prices are generated using the simulation scheme in the Appendix along with the 95% confidence limit. CE denotes the price calculated using the approximation in Proposition 2. The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. T=1 T=3 T=7 0.26 0.21 0.2

0.198 0.205 0.24 0.196 0.2 0.194 0.22

0.195 0.192

0.2 Implied Vol. Implied Vol. Implied Vol. 0.19 0.19 0.188 0.18 0.185 0.186

0.16 0.18 0.184 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) MC CEFourier 95% CL T=10 T=15 T=20 0.2 0.2 0.198

0.198 0.198 0.196 100 0.196 0.196 0.194 0.194 0.194 0.192 0.192 Implied Vol. Implied Vol. Implied Vol. 0.192 0.19 0.19 0.19 0.188

0.186 0.188 0.188 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

Figure 11: Caplet Implied Volatility, Double Exponential Distribution, Parameter Set ANORM This figure shows the caplet implied volatility. MC prices are generated using the simulation scheme in the Appendix along with the 95% confidence limit. CE denotes the price calculated using the approximation in Proposition 2. The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. 1 X 1 1 X 5 1 X 7 1 X 10 0.5 2 2 5

0 0 0 0

Pct. Err Pct. Err −2 Pct. Err −2 Pct. Err

−0.5 −4 −4 −5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 5 X 1 5 X 5 5 X 7 5 X 10 0.2 0.2 0.5 0.5

0 0 0 0

Pct. Err Pct. Err Pct. Err Pct. Err −0.5

−0.2 −0.2 −0.5 −1 CE 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 95%CL 15 X 1 15 X 5 15 X 7 15 X 10 1 1 0.5 0.5

0.5 0.5 101 0 0 Pct. Err Pct. Err 0 Pct. Err 0 Pct. Err

−0.5 −0.5 −0.5 −0.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 20 X 1 20 X 5 20 X 7 20 X 10 1 1 0.5 0.5

0.5 0.5 0 0 Pct. Err Pct. Err 0 Pct. Err 0 Pct. Err

−0.5 −0.5 −0.5 −0.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

Figure 12: Swaption Price Percentage errors, Double Exponential Distribution, Parameter Set ANORM This figure shows the swaption price percentage error defined as 100 times the simulated price minus CE price divided by the simulated price. The errors are plotted for Maturity × Tenor across moneynessX/FK (0). MC prices are generated using the simulation scheme in the Appendix along with the 95% confidence limit. CE denotes the price calculated using the approximation in Proposition 4. The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. 1 X 1 1 X 5 1 X 7 1 X 10 0.3 0.3 0.3 0.3

0.2 0.2 0.2 0.2 Implied Vol. Implied Vol. Implied Vol. Implied Vol.

0.1 0.1 0.1 0.1 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 5 X 1 5 X 5 5 X 7 5 X 10 0.21 0.21 0.21 0.21

0.2 0.2 0.2 0.2

0.19 0.19 0.19 0.19 Implied Vol. Implied Vol. Implied Vol. Implied Vol.

0.18 0.18 0.18 0.18 MC 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 CE Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 95% CL 15 X 1 15 X 5 15 X 7 15 X 10 0.2 0.2 0.2 0.2

0.19 0.19 0.19 0.19 102 Implied Vol. Implied Vol. Implied Vol. Implied Vol.

0.18 0.18 0.18 0.18 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 20 X 1 20 X 5 20 X 7 20 X 10 0.2 0.205 0.205 0.205

0.2 0.2 0.2 0.19 0.195 0.195 0.195 Implied Vol. Implied Vol. Implied Vol. Implied Vol.

0.18 0.19 0.19 0.19 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

Figure 13: Swaption Implied Volatility , Double Exponential Distribution, Parameter Set ANORM This figure shows the swaption implied volatility. The title for each subgraph denotes Maturity × Tenor. MC prices are generated using the simulation scheme in the Appendix along with the 95% confidence limit. CE denotes the price calculated using the approximation in Proposition 4. The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. T=1 T=3 T=7 0.1 0.1 0.2

0 0 0.1

−0.1 −0.1 0 −0.2 −0.2 −0.1 −0.3 −0.3 −0.2 Pct. Error −0.4 Pct. Error −0.4 Pct. Error −0.3 −0.5 −0.5

−0.6 −0.6 −0.4

−0.7 −0.7 −0.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

T=10 T=15 T=20 0.2 0.2 0.3 CE 0.1 0.2 95% CL 103 0.1 0 0.1

0 −0.1 0

−0.2 −0.1 Pct. Error Pct. Error Pct. Error −0.1

−0.3 −0.2 −0.2 −0.4 −0.3

−0.5 −0.3 −0.4 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

Figure 14: Caplet Price Percentage errors, Double Exponential Distribution, Displaced Diffusion, Parameter Set DEDD This figure shows the caplet price percentage error defined as 100 times the simulated price minus CE price divided by the simulated price. The errors are plotted for the different maturities across moneynessX/FK (0). MC prices are generated using the simulation scheme in the Appendix along with the 95% confidence limit. CE denotes the price calculated using the approximation in Proposition 5 in the appendix. The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. T=1 T=3 T=7 0.3 0.24 0.22

0.28 0.23 0.21

0.22 0.26 0.2 0.21 0.24 0.19 0.2 Implied Vol. Implied Vol. Implied Vol. 0.22 0.18 0.19

0.2 0.18 0.17

0.18 0.17 0.16 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) MC CE 95% CL T=10 T=15 T=20 0.22 0.22 0.22

0.21 0.21 0.21 104

0.2 0.2 0.2

0.19 0.19 0.19 Implied Vol. Implied Vol. Implied Vol. 0.18 0.18 0.18

0.17 0.17 0.17

0.16 0.16 0.16 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

Figure 15: Caplet Implied Volatility, Double Exponential Distribution, Displaced Diffusion, Parame- ter Set DEDD This figure shows the caplet implied volatility. MC prices are generated using the simulation scheme in the Appendix along with the 95% confidence limit. CE denotes the price calculated using the approximation in Proposition 2. The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. 1 X 1 1 X 5 1 X 7 1 X 10 2 1 1 2

1 0 0 0 Pct. Err Pct. Err Pct. Err Pct. Err 0 −1 −1 −2

−1 −2 −2 −4 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 5 X 1 5 X 5 5 X 7 5 X 10 0.2 0.5 0.5 0.5

0 0 0 0

Pct. Err Pct. Err Pct. Err −0.5 Pct. Err −0.5

−0.2 −0.5 −1 −1 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 CE Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 95%CL 15 X 1 15 X 5 15 X 7 15 X 10 0.2 0.5 0.5 0.5

0 105 0 0 0 Pct. Err Pct. Err Pct. Err Pct. Err −0.5

−0.2 −0.5 −0.5 −1 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 20 X 1 20 X 5 20 X 7 20 X 10 0.5 0.5 0.5 1

0 0 0 0 Pct. Err Pct. Err Pct. Err Pct. Err −1

−0.5 −0.5 −0.5 −2 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

Figure 16: Swaption Price Percentage errors, Double Exponential Distribution, Displaced Diffusion, Parameter Set DEDD This figure shows the swaption price percentage error defined as 100 times the simulated price minus CE price divided by the simulated price. The errors are plotted for Maturity × Tenor across moneynessX/FK (0). MC prices are generated using the simulation scheme in the Appendix along with the 95% confidence limit. CE denotes the price calculated using the approximation in Proposition 6 in the appendix. The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. 1 X 1 1 X 5 1 X 7 1 X 10 0.3 0.3 0.3 0.3

0.2 0.2 0.2 0.2 Implied Vol. Implied Vol. Implied Vol. Implied Vol.

0.1 0.1 0.1 0.1 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 5 X 1 5 X 5 5 X 7 5 X 10 0.22 0.22 0.22 0.24

0.22 0.2 0.2 0.2 0.2 0.18 0.18 0.18 0.18 Implied Vol. Implied Vol. Implied Vol. Implied Vol. 0.16 0.16 0.16 0.16 MC 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 CE Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 95% CL 15 X 1 15 X 5 15 X 7 15 X 10 0.22 0.22 0.22 0.22

0.2 0.2 0.2 0.2 106 0.18 0.18 0.18 0.18 Implied Vol. Implied Vol. Implied Vol. Implied Vol.

0.16 0.16 0.16 0.16 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) 20 X 1 20 X 5 20 X 7 20 X 10 0.22 0.22 0.24 0.24

0.2 0.2 0.22 0.22 0.2 0.2 0.18 0.18 0.18 0.18 Implied Vol. Implied Vol. Implied Vol. Implied Vol. 0.16 0.16 0.16 0.16 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 Moneynes(X/F) Moneynes(X/F) Moneynes(X/F) Moneynes(X/F)

Figure 17: Swaption Implied Volatility, Double Exponential Distribution, Displaced Diffusion, Pa- rameter Set DEDD This figure shows the swaption implied volatility. The title for each subgraph denotes Maturity × Tenor. MC prices are generated using the simulation scheme in the Appendix along with the 95% confidence limit. CE denotes the price calculated using the approximation in Proposition 6 in the appendix. The cumulant expansion is cut off at P = 4 and the tuning parameter for the numerical integration is set at α = 0.75. Chapter III Alternative Specifications for the L´evy Libor Market Model: An Empirical Investigation∗

David Skovmand† University of Aarhus and CREATES May 1, 2008

Abstract This paper introduces and analyzes specifications of the L´evy Market Model originally proposed by Eberlein and Ozkan¨ (2005). An investigation of the term structure of option implied moments shows that the Brownian motion and homogeneous L´evy processes are not suitable as modelling de- vices, and consequently a variety of more appropriate models is proposed. Besides a diffusive component the models have jump structures with low or high frequency combined with constant or stochastic volatility. The models are subjected to an empirical analysis using a time series of data for Euri- bor caps. The results of the estimation show that pricing performances are improved when a high frequency jump component is incorporated. Specifi- cally, excellent results are achieved with the 4 parameter Self-Similar Variance Gamma model, which is able to fit an entire surface of caps with an average absolute percentage pricing error of less than 3%.

∗The author would like to thank Elisa Nicolato, Thomas Kokholm and Peter Løchte Jørgensen, for useful comments †Current affiliation: Aarhus School of Business and the Center for Research in Econometric Analysis of Time Series (CREATES), www.creates.au.dk. Corresponding address: Aarhus School of Business, Department of Business Studies, Fuglesangs All´e4, DK-8210 Aarhus V, Denmark, e-mail: [email protected] 1 Introduction

The Libor Market Model formulated in the seminal papers by Miltersen, Sand- mann, and Sondermann (1997), Brace, Gatarek, and Musiela (1997), and Jamshid- ian (1997) was a breakthrough in the world of academics as well as practitioners. It provided a theoretical framework for the market practice of using the classic Black and Scholes (1973) formula for pricing options on Libor rates, such as caps and swaptions. However, the model was almost immediately extended due to its in- ability to fit the volatility smile already known to be present in options on Libor rates. The extensions have mirrored the development in the equity option pricing literature and they include Libor market models with stochastic volatility (Andersen and Brotherton-Ratcliffe (2005), Hagan, Kumar, Lesniewski, and Woodward (2002), and Wu and Zhang (2006)) and/or jumps generated by a compound Poisson process (Jarrow, Li, and Zhao (2007) and Glasserman and Kou (2003) ). In spite of these advances it is still an open question how to optimally price liquid interest rate options in the cross-section of both maturity and strike. The strike dimension alone is well fitted by for example the SABR model of Hagan, Kumar, Lesniewski, and Woodward (2002), which has gained popularity to the point where traders have begun to quote smiles in terms of its parameters. But the failure of the SABR model is manifest since practitioners must use maturity dependent parame- ters in order to fit the entire surface. More advanced models have also been developed such as the L´evy process driven Li- bor Market Model of Eberlein and Ozkan¨ (2005), Eberlein and Kluge (2007), Kluge (2005), and the more general semi-martingale model of Jamshidian (1999). These papers generalize previous approaches by allowing high frequency jump processes, as opposed to the classic compound Poisson jump-diffusion framework, originally proposed in Merton (1976). High frequency jump models have already been tremen- dously successful in pricing equity options, as shown in Carr, Geman, Madan, and Yor (2002), Carr, Geman, Madan, and Yor (2003), and Carr, Geman, Madan, and Yor (2007). But whether the high frequency jump components are relevant as mod- eling devices in interest rate models has, to the authors knowledge, not yet been thoroughly investigated from an empirical point of view. This paper analyzes different specifications of jump processes coupled with stochas- tic volatility in a Libor Market Model, and sheds light on their comparative ability to fit a time series of Euribor cap prices in both the maturity and strike dimensions. The different models are formulated in a unified framework which can be seen as an extension of Eberlein and Ozkan¨ (2005). The framework also has the advantage of allowing for a fast and accurate pricing of caps through Fourier inversion techniques in the spirit of Carr and Madan (1999). The specific choice of driving processes is motivated by an investigation of the term

108 structure of higher moments such as variance, skewness and kurtosis implied from the data. The analysis is inspired by Konikov and Madan (2002) who examine equity options and find that kurtosis and negative skewness are increasing as a function of maturity. The cap market investigated in this paper displays similar patterns in the term structure of implied moments and therefore the driving processes are selected according to their capability of capturing these stylized features. The chosen Libor market models are divided in three categories. 1) The Classic Models , where the building block is a displaced diffusion with stochastic volatil- ity and/or compound Poisson jumps. This class can be seen as fusing the approach taken in Andersen and Brotherton-Ratcliffe (2005) and Glasserman and Kou (2003). A similar specification was also analyzed in a slightly different setup in Skovmand (2008). 2) The Self-Similar Additive Models , where the driving process is an in- homogeneous L´evy process with the added feature of self-similarity. This process class has been introduced and thoroughly studied in Sato (1991) and Sato (1999). The utility of self-similar additive processes in the context of equity option pricing has been convincingly demonstrated in Carr, Geman, Madan, and Yor (2007) and Galloway (2006). The concrete specification used in this paper to describe Libor rates is a self-similar additive extension of the well known Variance Gamma model of Madan, Carr, and Chang (1998). 3) Time-Changed L´evy Models , where the driv- ing process is a homogeneous L´evy process, specifically a Variance Gamma process, subordinated, in order to incorporate stochastic volatility, to an integrated Cox, Ingersoll, and Ross (1985) process. This modelling framework was first studied in Carr, Geman, Madan, and Yor (2003) and further analyzed for example in Carr and Wu (2004), although still in the context of equity options. Each model class is characterized by a different jump structure. In the classic mod- els, jumps are big and infrequent whereas the last two model classes have an infinity of small jumps, resulting in a behavior similar to a diffusion process. Whether an actual diffusion component is still relevant in this setup, is explored by mixing the infinite activity models with a standard Brownian motion, as well as a Brownian motion with stochastic volatility. For each different model a daily recalibration is performed and the pricing errors are scrutinized in the maturity and strike dimension, as well as across the sample period. The models are ranked in terms of their average absolute percentage pricing errors (APE), both in- and out-of-sample. In order to carry out formal tests, a sample-wide estimation based on the General- ized Method of Moments is implemented, and the results are used to perform the Diebold and Mariano (1995) test for superior performance. The empirical investigation first documents the smile in the Euribor cap market and provides a more general description of the data. The models are then taken to the data and the results show that the Time-Changed L´evy Models and Self-Similar

109 Additive Models outperform the classic models, with average absolute percentage errors of 2.5%, 2.8% compared to 4-4.5% for the classic specifications. Mixing a stochastic volatility diffusion component with either a self-similar additive or time-changed L´evy jump component has a small ( 0.1 0.2%) but statistically significant effect on performances compared to the pure≈ jump− case. On the other hand including a constant volatility diffusion component is only significant for the time-changed L´evy models. Finally, perhaps the most striking result is the impres- sive performance of the Self-Similar Additive Model with no diffusion component. This is the most parsimoneous model studied in the paper, with only 4 process spe- cific parameters. Despite its simplicity it is able to capture an entire price surface consisting of 97 prices, with a performance comparable to the best model specifi- cation, which is a 9 parameter time-changed L´evy process mixed with a stochastic volatility diffusion component. The paper is structured in the following manner. First, the Generalized Libor Mar- ket Model framework is presented. Then the dataset is explored and the smile and other stylized features are documented in the Euribor cap dataset. Subsequently, an analysis of the term structure of moments is performed and the specific processes are motivated, and further analyzed in detail. Finally, the empirical methodology is de- scribed, and the results are presented, followed by a discussion of their implications. The paper ends with a conclusion.

2 The General Libor Market Model

Let P (t, Tk) be the time t zero coupon bond price with maturity at Tk. For a tenor structure T <.....

P (t, Tk) P (t, Tk+1) Fk(t)= F (t, Tk, Tk+1)= − (1) δP (t, Tk+1) 0 t T , k = 1,...,K. ≤ ≤ k From (1) it follows that the Libor rate Fk(t) can be seen as a price process scaled by the numeraire P (t, Tk+1). Therefore absence of arbitrage implies that for each rate Fk(t) there exists an associated probability measure Fk+1, termed the forward measure, under which Fk(t) is a martingale. Of particular relevance is the final forward measure or terminal measure denoted by ∗ F := FK+1. In what follows the entire Libor market model will be specified along the lines of Eberlein and Ozkan¨ (2005) by defining the evolution of the terminal rate under F∗

110 and subsequently using a backward induction procedure to derive the dynamics of the remaining rates in the term structure. More precisely the explicit construction can be carried using the following inputs. The initial term structure Fk(0) for k = 1,...,K; a series of rate specific constants λ1,...,λK allowing for flexibility in modelling the volatilities across maturity; finally the dynamics for the terminal rate ∗ FK (t) under the terminal measure F which are given as follows t t F∗ FK (t)= FK (0) exp bK (s)ds + λK dLs . (2) Z0 Z0  F∗ The driving source of randomness is the stochastic process Lt defined as t F∗ F∗ ∗ Lt = c(s)dWs + Jt , (3) Z0 F∗ where Wt is a 1-dimensional Brownian motion and c(t) is a positive process inde- F∗ pendent of Wt satisfying standard regularity conditions and T ∗ c(s)2ds < F∗ a.s. (4) ∞ Z0 ∗ The process Jt is a purely discontinuous martingale obtained by a time change as follows ∗ Jt = X(Y (t)), where X(t) is an additive martingale process1, and Y (t) is the integrated process t Y (t)= y(s)ds, (5) 0 Z ∗ with y(t) being a strictly positive process independent of X(t) and W F . ∗ t Denoting with µ(dt,dx) and νF (dt,dx) the random measure of jumps and its com- ∗ pensator respectively, Jt can be rewritten as t ∗ F∗ Jt = x(µ ν )(ds,dx), 0 R − ∗ Z Z νF (dt,dx)= y(t)k(t, x)dtdx t T , ∀ ≤ K where k(t, x) is the deterministic L´evy system associated with the additive process X(t). For simplicity it is also assumed that T ∗ (1 x )k(t, x)dxdt < , R ∧ | | ∞ Z0 Z 1An additive process has independent but possibly non-stationary increments

111 ∗ implying that the jump process Jt has finite variation. However most of the concrete specifications in this paper will display infinite activity i.e

k(t, x)dx = , t TK . R ∞ ∀ ≤ Z ∗ In other words the process Jt jumps an infinite number of times on any finite time interval. Another technical assumption that ensures the martingale property of FK (t) F∗ ∗ E [exp(uJ ∗ )ds] < (6) T ∞ for u < (1 + ǫ)M where ǫ> 0 and M are constants such that K λ < M. | | j=1 j P To complete the description of the terminal rate FK (t) the drift term bK (t) is defined as

∗ 1 2 λK x F bK (t)= c(t) e 1 λK x ν (t,dx). (7) −2 − R − − Z Under the regularity conditions in (6) and (4) the above
definition only ensures that FK (t) is a martingale if the time-change Y (t) integrated process is deterministic. For a stochastic Y (t) defined in (5), the X(Y (t)) process lacks independent increments when conditioning on the filtration including Y (t) implying that the FK (t) process is in general not a martingale. This means the associated market model is exposed to the possibility of dynamic arbitrage strategies as noted in Carr, Geman, Madan, and Yor (2003). One can argue whether the ability to create arbitrage strategies based on continuous observation of the Y (t) process is realistic in practice, as Y (t) is far from being directly observable. Moreover, following the same arguments as Carr, Geman, Madan, and Yor (2003) one can show that the the terminal rate FK (t) possesses the so called martingale marginals property i.e there exists a process, which is a martingale and exhibits the same marginal distributions as FK (t) in (2), albeit defined on a different filtration.

We can now proceed to specify the remaining rates Fk(t) for k = 1,...,K 1 under their own forward measures Fk+1. The connection between the terminal measure− and the remaining forward measures can be established through the backward induction procedure described in Eberlein and Ozkan¨ (2005). Assuming that the forward measures Fj+1 for j = k + 1,...,K and correspondingly the martingale forward rates Fj(t) have been constructed, the (k + 1)-th forward measure is then defined via the Radon-Nikodym derivative

K ∗ K dFk+1 1+ δFj(Tk+1) P (0, T ) ∗ = = (1 + δFj(Tk+1)). dF 1+ δFj(0) P (0, Tk+1) j=Yk+1 j=Yk+1

112 k+1 The dynamics of the Libor rate Fk(t) under F are then defined as t t Fk+1 Fk(t)= Fk(0) exp bk(s)ds + λk dLs , (8) Z0 Z0  Fk+1 with the process Lt given by t t Fk+1 Fk+1 Fk+1 Lt = c(s)dWs + x(µ ν )(ds,dx), R − Z0 Z0 Z Fk+1 k+1 Fk+1 k+1 where Ws is an F -Brownian motion and ν denotes the F -compensator of the random measure of jumps µ. Using Girsanov’s theorem for general semi- martingales (see for example Jacod and Shiryaev (1987)) it follows that the connec- tion between the Brownian motions is K F k+1 ∗ δc(t)Fj(t ) dWt = dWt − dt. (9) − 1+ δFj(t ) j=Xk+1 − while the relation between the compensators is given by2

K λj x F δ(e 1)Fj(s) F∗ ν k+1 (dt,dx)= 1+ − ν (dt,dx). (10) 1+ δFj(s) j=Yk+1   Finally the martingale condition (with the discussed caveats) is secured by setting the drift as

k+1 1 2 λkx F bk(t)= c(t) e 1 λkx ν (t,dx). (11) −2 − R − − Z For completeness the dynamics of k-th Libor rate under
the terminal measure F∗ can be expressed using (9) and (10) as follows

t t F (t)= F (0) exp b∗(s)ds + λ dL∗ k = 1,...,K, (12) k k k k s ∀ Z0 Z0  with K ∗ δc(t)Fj(t ) bk(t)= − − 1+ δFj(t ) j=Xk+1 − K λj x ∗ 1 2 λkx δ(e 1)Fj(t ) F c(t) λkx (e 1) 1+ − − ν (t,dx). − 2 − R − − 1+ δFj(t ) ! Z j=Yk+1  −  (13)

2This result is also derived in a different setting in Jamshidian (1999)

113 The derivation of expression (13) is given in the appendix.

The model can be extended to nest the displaced diffusion framework studied rig- orously in Rebonato (2002) and Brigo and Mercurio (2006) for the continuous case. Carrying out the steps outlined above one may then obtain the following dynamics for the Libor rates under the terminal measure t t F (t)+ α =(F (0) + α) exp b∗(s)ds + λ dL∗ k = 1,...,K, (14) k k k k s ∀ Z0 Z0  with

K ∗ δc(t)(Fj(t )+ α) bk(t)= − − 1+ δFj(t ) j=Xk+1 − K λj x ∗ 1 2 λkx δ(e 1)(Fj(t )+ α) F c(t) λkx (e 1) 1+ − − ν (t,dx). − 2 − R − − 1+ δFj(t ) ! Z j=Yk+1  −  (15)

From (12) and (14) it can be observed that the driving process for all rates under ∗ the terminal measure is the Lt process defined in (3). The overall result is a 1 factor model for the entire term structure implying that all Libor rates are perfectly correlated. This obviously unrealistic implication is immaterial to this study since only caps are investigated and these are unaffected by correlation.3

2.1 Caplet Pricing

A cap is a portfolio of call options on Libor rates. A Tk+1-cap with strike K pays (Fi(Ti) K) at Ti+1 for i = 1,...,k. The total time t < T1 value of the cap is therefore−

k F Cap(t, T ,K)= δP (t, T )E i+1 [(F (T ) K)+]. k+1 i+1 i i − i=1 X The individual payments, referred to as caplets, have value:

F δP (t, T )E i+1 [(F (T ) K)+]. i+1 i i − 3Correlation, can be easily included by using a multidimensional Brownian motion, and this has been the topic of a large part of the LMM literature (see Brigo and Mercurio (2006) and Rebonato (2002) for an overview in the pure diffusion case. Dependence structure with jumps is a more delicate issue has to the authors knowledge only been considered in LMM context in Belomestny and Schoenmakers (2006))

114 By investigating the dynamics of Fk(t) under its own forward measure given in (8), it is easily concluded that a closed form expression for the above expectation is not possible to derive in general since the compensator is state dependent via its relationship in (10). Fortunately an accurate approximation is derived in Kluge (2005).4 Proposition 1. Given a Libor Market Model described by (14) the approximated price of a Caplet with strike K and maturity Tk+1 is given by

Cpl(0,Tk+1,K)= K + α ∞ K + α R+iu 1 δP (0, Tk+1) Re π 0 Fk(0) + α (R + iu)(R +1+ iu) Z h  R+iu φ if +(iR u)λ , T φ if iλ , T × − k − k k × − k − k k −(R+1+iu) φ ifk, Tk du,

(16) − where


K δ(Fk(0) + α) fk := λj, 1+ δFk(0) j=Xk+1 ∗ and φ(u, t) is the characteristic function of the driving process Lt

F∗ ∗ φ(u, t)= E [exp(uiLt )] .

R is a properly chosen constant such R< 1 and φ if +(iR u)λ , T < − − k − k k ∞ The caplet pricing formula in the above proposition is very efficient since
it only requires the calculation of a one-dimensional numerical integral.

3 Exploring the Data

The primary dataset is on Euribor caps and it is retrieved through Reuters; the term structure information is collected from Datastream. The Euribor is the Euro- equivalent of the standard Libor rate, but in order to avoid confusion both will be referred to as Libor. The data is recorded daily in the sample but for computational reasons the analysis is restricted to each Wednesday in the sample (If missing, Thursday). The dataset covers the period from May 7th, 2003, to November 10th, 2004, resulting in 80 days

4Kluge (2005) derives the formula without a displacement nevertheless the proof can be trivially extended to encompass this slightly more general case

115 of data. Euribor caps are given for a range of maturities 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, and 20 years. It is also given for a large range of strikes, specifically 1.75, 2.0, 2.25, 3.0, 3.5, 4.0, 4.5, 5.0, 6.0, 7.0, 8.0, and 10.0 percent, and on June 20th, 2003, the 1.5 strike is added and 4.5 strike is removed. Ideally, one would like to investigate prices for caplets instead of caps since these provide more fundamental information on the distributional properties of the indi- vidual rates. Therefore a bootstrapping algorithm is implemented as described in Piza (2005), to extract caplet prices from cap prices. Unfortunately, Euribor caps for the 1 and 2 year maturity are struck on rates with 3 month to expiry, whereas the remaining maturities are struck on the 6 month rate. This complicates the use of a bootstrapping algorithm, so for simplicity it is assumed that the implied volatilities for the 3 month rates are the same as the 6 month. This is not as simplifying as it might seem. In markets where caps on both underlying 6 month and 3 month rates are observed (such as the Danish, Norwegian and Swiss markets) the volatilities are close to identical and differences are normally well within bid-ask spreads. Since the rates change every day so does the moneyness (K/Fk(0)) of the option. For computational reasons it is convenient to have constant moneyness through time therefore caplet prices are interpolated between strikes, for each day in the sample. Following Jarrow, Li, and Zhao (2007) and Li and Zhao (2006) the interpolation is done using local cubic polynomials to retain the structure of the prices. Moneyness levels are restricted to 0.6, 0.7, 0.8, 0.9, 1, 1.1, 1.2, 1.3 except for the 1 year caplets where due to liquidity reasons only 0.9, 1 and 1.1 are included and for the 2 year 1.3 and 0.6 are excluded. This leaves 7760 option prices for the entire data with 97 prices each Wednesday. The average price level in basis points is drawn in Figure 1 and it shows a fairly smooth and consistent surface with prices increasing in moneyness but a clear non- monotonic behavior in the maturity dimension. The corresponding average implied volatilities are found by inverting the Black- Scholes formula, and the results are drawn in Figure 2. The surface shows a slight hump in the volatility surface peaking at 2 years and declining for all other ma- turities. This is consistent with other studies, such as Rebonato (2002) who per- forms a thorough investigation and provides some economic explanations of this phenomenon. Figure 3 is a 2-dimensional version of Figure 2. Here we clearly observe a smile with in-the-money (ITM) and out-of-the money (OTM) options having higher implied volatility than the at-the-money level. But the smile is more like a smirk since it is tilted to the right, with ITM being bigger than the corresponding OTM level. Figure 4 shows the term structure of volatility as it evolves through time for three different levels of moneyness. The long maturity volatility appears close to being constant whereas the short maturities show an upward increasing trend, as well as

116 being significantly more erratic. Again this is true for the three different levels of moneyness. Finally, in Figure 5 the term structure of 6 month Libor rates throughout the sample is plotted. The figure shows a persistent increasing term structure with a decreas- ing overall level and slope. It also shows long rates being somewhat volatile in the beginning of the sample.

4 Implied Features of the market

In light of the model described in Section (2) the main question is : What kind of ∗ features should the distribution of Lt have? Essentially the objective is to generate a realistic implied volatility/price surface. Looking again at Figure 2 there are several effects that have to be accounted for. The first is the hump in implied volatility around the 2 year maturity. This effect can easily be generated with a maturity dependent scaling (the λk’s), as shown in the next section. Another more delicate issue is the smile and its behavior across maturity. In order to generate the smile the distribution of the underlying has to exhibit positive excess kurtosis meaning that extreme moves have to occur more often than what is predicted by a normal distribution. Furthermore there is asymmetry the smile or smirk prevalent in Figure 3 which implies that a negatively skewed distribution is needed. Non-Gaussian models are of course well established in the literature (see various papers cited in the introduction) but the majority of them suffer from the fact that they cannot price options in both the maturity and strike dimension simultaneously. One of the simplest examples of a non-Gaussian model with this ”flaw” is the Vari- ance Gamma (VG) model of Madan, Carr, and Chang (1998). The Variance Gamma process is constructed by time changing a Brownian motion with drift with respect to a gamma process. So the law of the VG process is given by

ν ν XV G(t)= θGt + σW (Gt ),

ν where Gt is a gamma process with mean rate 1 and variance ν and W (t) is a Brownian motion. The process has variance, skewness and excess kurtosis equal to

2 2 (variance) µ2 = t(θ ν + σ ), 1 (2θ3ν2 + 3σ2θν) (skewness) γ1 = , √t (θ2ν + σ2)3/2 1 3σ2ν + 12σ2θ2ν2 + 6θ4ν3 (excess kurtosis) γ = . 2 t (θ2ν + σ2)2

117 Skewness is controlled by the θ parameter and kurtosis is partly controlled by the ν parameter. What can also be observed is that variance increases linearly as a function of time, and skewness and excess kurtosis decreases with a rate of 1/√t and 1/t respectively. This means that for long maturities the distribution generated by the model is essentially Gaussian. Furthermore Konikov and Madan (2002) show that all homogeneous L´evy processes converge to Gaussianity in this manner. This stands in contrast to Figure 3 that shows a smile with an almost constant slope across maturity. In order to asses the scope of this problem an analysis similar to Konikov and Madan (2002) of the term structure of moments is performed. A separate Variance Gamma ∗ Libor market model with Lt = XV G(t) is calibrated to each option maturity in the sample. For each maturity the variance, skewness, and kurtosis is calculated using the above formulas and the results are averaged within four different periods of the sample. The three series are then plotted against maturity in Figure 6-8. Looking first at the variance in Figure 6 we see a clear concave function of variance in time; a behavior different from the linear scaling implied by a L´evy process. The skewness in Figure 7 starts at a level close to zero but is increasing in absolute terms; the complete opposite of a L´evy process. The excess kurtosis is plotted in Figure 8 and it shows the same behavior as the skewness but with opposite sign. What can be inferred from these graphs is that a model based on a single homogeneous L´evy process is in direct conflict with the data and therefore not appropriate as a modeling device. A minimum criterium for a ”good” model is therefore different than linear scaling of variance as well as the the ability to retain skewness and excess kurtosis in time i.e be able delay the convergence to Gaussianity. The next section presents 3 classes of models with these features.

5 Specifying the Driving Process

In this section three classes of models all nested in the general specification in Section 2 are studied. The driving stochastic processes are defined and their characteristic functions are given, along with their domain of existence which is essential for a correct numerical implementation of the pricing formula in Proposition 1. ∗ Besides setting the Lt process, there is also considerable freedom in specifying the constants λk for k = 1,...,K which as described in Section 2 determines the volatil- ity structure. For simplicity these are chosen as λk = 1 for k = 2,...,K and allowing only λ1 to be determined by the data. This allows for the investigated models to generate the observed hump across maturity in the implied volatility surface. This choice is primarily motivated by simplicity and could easily be extended to more

118 realistic, but also less parsimonious structures (see Brigo and Mercurio (2006) for an overview in the log-normal case).

5.1 Classic models In this setup a standard log-normal process is perturbed by 1) a displacement factor α, 2) stochastic volatility and 3) a compound Poisson jump component with normally distributed jump sizes.

5.1.1 DDSV The first model studied is characterized by a displaced diffusion and stochastic volatility hence the name DDSV. The Libor rates have dynamics given under the terminal measure F∗ by

t t ∗ ∗ Fk(t)+ α =(Fk(0) + α) exp bk(s)ds + λk dLs , (17) Z0 Z0  ∗ ∗ where bk(s) is given in expression (15) and the driving process Lt is defined as

t ∗ ∗ Lt = y(s)dWs , 0 Z p where y(t) follows the classic mean reverting diffusion

dy(t)= κ(η y(t))dt + ǫ y(t)dZ(t). (18) − p ∗ where Z(t) is a Brownian motion independent of Wt . The properties of the square- root process y(t) are studied in Cox, Ingersoll, and Ross (1985) and Heston (1993). In particular it is well known that if the mean reversion speed κ is positive then y(t) ∗ is stationary and ergodic implying that the process Lt will converge to Gaussianity. ∗ However, it is shown in the appendix that Lt displays positive excess kurtosis given by5

t 3V ar( 0 y(s)ds) γ2 = 2 . (19) ηt +(y(0) η) 1 (1 e−κt) −R κ − allowing for more flexibility in the term structure of moments,
in particular permit- ting a lower decay rate of convergence to Gaussianity compared to the homogeneous L´evy process.

5An explicit, but less intuitive, expression for the kurtosis is derived in Das and Sundaram (1999)

119 ∗ ∗ The assumption that Wt is independent of Z(t) means that the Lt process is sym- metric due to the symmetry of the Brownian motion. This is also referred to as absence of the popularly coined ”leverage” effect meaning that negative moves of the interest rate are followed by an increase in the volatility. There is empirical evidence suggesting that this assumption is reasonable under the physical measure. For example Chen and Scott (2004) find that the correlation between short rates and volatility is very small but this may or may not apply to the martingale measure distribution studied in this paper.6 In any case, from an option pricing perspective a leverage effect is important only because it generates skewness in the martingale measure distribution of the underlying which results in an asymmetric smile. In the DDSV model in (17) skewness is added directly in the rates through the dis- placement factor α with positive values corresponding to negative skewness. The ∗ skewness in the rates will in fact persist as Lt converges to Gaussianity, retaining an implied volatility skew – even for long maturities as shown in Rebonato (2002). The downside of this model is that the Libor rates are not guaranteed to be positive but whether this poses any other than purely theoretical problems is questionable. ∗ The characteristic function φSV (u, t) of Lt is given by (see for example in Cox, Ingersoll, and Ross (1985))

∗ φSV (u, t) := E[exp(iuLt )] = A exp(y(0)B), (20) iu2 x = , 2 D = √κ2 2ǫ2ix, − ηκ2t exp ǫ2 A = 2ηκ , κ  ǫ2 cosh(Dt/2) + D sinh(Dt/2) 2ix B = .
κ + D coth(Dt/2)

5.1.2 Strip of regularity for φSV In order to calculate the price in Proposition 1 it is necessary to evaluate the char- acteristic function in the complex domain. In doing this we must ensure that the characteristic function is only evaluated in its strip of regularity z C φ (z,t) < . { ∈ | | SV | ∞} Unfortunately the strip of regularity cannot be derived explicitly, but instead one can set up parameter restrictions.

6In fact Jarrow, Li, and Zhao (2007) find that there is evidence that points toward correlation between rates and volatility

120 From Andersen and Piterbarg (2005) Proposition 3.1 it follows that for each z = z1 + iz2 there exists a critical time T˜ when φSV (z, T˜) = . with E =(κ2 ǫ2(z + z2)) the explosion times| are given| ∞ as − 2 2 1. If E 0 or z [0, 1] then ≥ 2 ∈ T˜ = . ∞ 2. If E < 0 then

1 √E T˜ = 4 π + arctan 0.5 . √E − κ !!

This means that for each z2 the parameters have to satisfy

1 √E E 0 or T ∗ < 4 π + arctan 0.5 . ≥ √E − κ !! These restrictions ensure that the characteristic function is well defined up to the longest maturity T ∗ = 20 years in the sample of caplet prices.

5.1.3 DDSVJ The DDSVJ model extends the previous DDSV model by adding a jump component described by a compound Poisson process. This extension allows the model to generate more kurtosis and therefore steeper smiles in the short end of the maturity spectrum. Since a compound Poisson process is a L´evy process it will normally converge to Gaussianity faster than the stochastic volatility component, resulting in only secondary effects on the shape of the smile for longer maturities. As before the rates are specified as

t t ∗ ∗ Fk(t)+ α =(Fk(0) + α) exp bk(s)ds + λk dLs , Z0 Z0  and jumps are introduced by setting

t N(t) ∗ ∗ Lt = y(s)dWs + Xj, 0 j=1 Z p X where N(t) is a Poisson process with intensity λCP and the Xj’s are IID random 2 variables drawn from a normal distribution with mean µCP and variance σCP .

121 ∗ The Lt process can be rewritten in the random measure notation used in Section 2 as

t t ∗ ∗ Lt = y(s)dWs + x(µ(ds,dx) kCP (x)dsdx), 0 0 R − Z p Z Z where the L´evy density kCP is given by λ (x µ )2 k (x)= CP exp − − CP . CP √2πσ σ2 CP  CP  Using standard calculations (see for example Cont and Tankov (2004)) the charac- ∗ teristic function for Lt is given by

∗ E[exp(iuLt )] = φDDSVJ (u, t) = φ (u, t) exp tλ exp( σ2 u2/2+ iuµ ) 1 ) . SV × CP − CP CP − The second exponential term is non-explosive u C so the parameter restrictions
 are µ R, σ > 0 and those in Section 5.1.2.∀ ∈ CP ∈ CP 5.2 Self-Similar Additive Models A process is called self-similar if it has marginal laws that obey

L(ta) =d aγL(t) a> 0, ∀ where γ is termed the exponent of self-similarity and =d denotes equality in law. The Brownian motion is a trivial example of a self-similar process with γ = 0.5. Processes that are both self-similar and additive have been studied in Sato (1991) and are therefore often referred to as Sato processes. These processes are very much related to the concept of self-decomposability. A random variable X is self- decomposable if for any constant c [0, 1] there exists an independent random variable X(c) such that X can be decomposed∈ as

X =d cX + X(c).

h(x) X is also self-decomposable if the corresponding L´evy density has the form x where h(x), is increasing for negative x and decreasing for positive x. h(x) is referred to as the self-decomposability characteristic for the random variable X. Sato (1991) showed that a law is self-decomposable if and only if it is the law at unit time of an additive and self-similar process. This means that a Sato process can be constructed from any self-decomposable law X.

122 A very desirable feature of these processes is that skewness and kurtosis are constant through time and variance scales with t2γ as discussed in Carr, Geman, Madan, and Yor (2007). In terms of matching the implied variance in Figure 6 a scaling of t2γ is indeed promising since the concave behavior observed in the figure resembles the case with γ < 0.5. For skewness and kurtosis the constant implication is far from the increasing patterns seen in Figure 7 and 8, but it is nevertheless closer to the observed behavior than the decreasing patterns of the homogeneous L´evy process. Motivated by its superior performance and flexibility demonstrated in Carr, Geman, Madan, and Yor (2007) this paper will focus on the Sato process associated with the Variance Gamma (VG) law. Recall that the VG distribution is obtained as the unit time law of the VG process previously introduced in Section 4, i.e

XV G = XV G(1). Carr, Geman, Madan, and Yor (2002) show that the L´evy density of the VG law is C exp(Gx) C exp( Mx) k (x) = 1 + 1 − , (21) V G x<0 x x>0 x | | | | with 1 C = , ν −1 θ2ν2 σ2ν θν G = + , r 4 2 − 2 ! −1 θ2ν2 σ2ν θν M = + + , r 4 2 2 ! and characteristic function given by GM C φ (u)= . (22) V G GM +(M G)iu + u2  −  From (21) one can notice that the VG law is self-decomposable with self-decomposability characteristic given by h(x) = 1 C exp(Gx) + 1 C exp( Mx). x<0 x>0 − Therefore for a given exponent of self-similarity γ a Variance Gamma Self-Similar Self-Decomposable (VGSSD) process XV GSSD(t) satisfying

d γ XV GSSD(t) = t XV G

123 can be constructed. Its characteristic function φV GSSD(u, t) is given by GM C φ (u, t)= φ (utγ)= . (23) V GSSD V G GM +(M G)iutγ + u2t2γ  −  For z R we see that φ (iz,t) is well defined only when ∈ V GSSD GM (M G)ztγ z2t2γ > 0 − − − G M < z < , ⇔ −tγ − tγ Again z can be considered fixed and the parameters can be restricted according to the above expression. The process XV GSSD(t) can also be written in terms of random measures as

t XV GSSD(t)= x(µ(dx,dt) kV GSSD(t, x)dxdt), R − Z0 Z where kV GSSD(t, x) is the L´evy system x 1 x 1 k (t, x) = 1 h′ 1 h′( ) V GSSD x<0 tγ t1+1γ − x<0 tγ t1+1γ   as derived in Theorem 1 in Carr, Geman, Madan, and Yor (2007). Three different models are now constructed using the VGSSD process with the Libor rate dynamics given as in Section 2 by

t t ∗ ∗ Fk(t)= Fk(0) exp bk(s)ds + λk dLs . Z0 Z0  5.2.1 VGSSD The first self-similar additive model has no diffusion component and is simply defined as

∗ Lt = XV GSSD(t) with characteristic function φV GSSD(u, t) given in (23)

5.2.2 VGSSDC This model is driven by the VGSSD process and a Brownian motion with a constant diffusion parameter. It is defined as

t ∗ ∗ Lt = cdW¯ s + XV GSSD(t). Z0

124 It follows from standard calculations that the above process has characteristic func- tion

φ (u, t) = exp( c¯2tu2/2)φ (u, t). V GSSDC − V GSSD 5.2.3 VGSSDSV Here stochastic volatility is added in the diffusion component yielding

t ∗ ∗ Lt = y(s)dWs + XV GSSD(t), 0 Z p where y(t) is a CIR process defined in (18). The above process has characteristic function

φV GSSDSV (u, t)= φSV (u, t)φV GSSD(u, t), with φSV (u, t) defined in (20) subject to regularity conditions described Section 5.1.2.

5.3 Time-Changed L´evy processes As discussed earlier, the standard L´evy processes suffer from the fact that they converge to Gaussianity when maturity increases. One way to postpone this central- limit-theorem effect is to time-change the L´evy process with respect to an increasing process. The time-change is also referred to as adding stochastic volatility to a L´evy process (see Carr, Geman, Madan, and Yor (2003)). Taking XV G(t) from Section 4 and a rate of time change y(s) given by a CIR process independent of XV G(t), we construct the new process XV GSV as follows

XV GSV (t)= XV G(Y (t)).

t where Y (t)= 0 y(s)ds. The characteristic function follows from standard calculations (see for exampe Carr, R Geman, Madan, and Yor (2003))

φ (u, t)= φ ( i log(φ (u))) , V GSV SV − V G with φSV and φV G defined in (20) and (22) respectively.

The strip of regularity follows from the arguments in Section 5.1.2:

125 For z = z + iz we have φ (z) < if the following two conditions are met 1 2 | V GSV | ∞ 1. G< z < M. − 2 2. For w = Im( i log(φ (z + iz , 1)) and E =(κ2 ǫ2(w + w2)) − V G 1 2 − 1 √E E 0 or T ∗ < 4 π + arctan 0.5 . ≥ √E − κ !!

The VGSV process can also be defined using the random jump measure µ as

t XV GSV (t)= x(µ(dx,dt) kV G(x)y(t)dxdt), R − Z0 Z

where kV G(x) is the L´evy density for the Variance Gamma law defined in (21). Several models can now be built using the VGSV process.

5.3.1 VGSV The first model is a pure jump process simply defined as

∗ Lt = XV GSV (t)

where the above process has characteristic function φV GSV (u, t).

5.3.2 VGSVC Adding a Brownian motion with a constant diffusion parameter yields

t ∗ ∗ Lt = cdW¯ s + XV GSV (t), Z0 where the above process has characteristic function

φ (u, t) = exp( c¯2tu2/2)φ (u, t). VGSVC − V GSV 5.3.3 VGSVD In this model a stochastic volatility process for the diffusion component is also included. Specifically

t ∗ ∗ Lt = y˜(s)dWs + XV GSV (t), (24) 0 Z p 126 wherey ˜(t) is again described by a CIR process

dy˜(t) =κ ˜(˜η y˜(t))dt +ǫ ˜ y(t)dZ˜ . − t ∗ p y˜(t) is assumed independent of Wt and XV GSV (t). The characteristic function of ∗ Lt in (24) is then given by

φVGSVD(u, t)= φ˜SV (u, t)φV GSV (u, t), where φ˜SV is equal to φSV with κ,η,ǫ and y(0) replaced byκ, ˜ η,˜ ǫ˜ andy ˜(0).

6 Estimation Methodology

The problem of estimating an option pricing model is that we have observations of the underlying interest rate process as well as the prices of options on the underlying. The underlying is observed under the physical measure but caplets are priced under the forward measure. Reconciling the differences between the physical measure and a martingale measure is a technically daunting task (see for example Jones (2003) for the equity case) and requires many assumptions on the structure of the risk premiums that arise when moving from one measure to another. In this paper the focus is exclusively on modeling the martingale measure, and therefore the relation to the physical measure is ignored. This means that the underlying Libor rate term structure will only appear as an input to the option pricing formula. This approach is similar to well known studies of the risk-neutral distribution such as Bakshi, Cao, and Chen (1997) and Huang and Wu (2004) who study S&P 500 index options and equity options respectively. This section is divided into two different subsections. In Section 6.1 the single day calibration approach is described. The purpose of this avenue is to investigate pa- rameter stability and pricing performance, on single day data alone. Throughout the paper the parameters are estimated by minimizing the sum of squared percentage pricing errors (SSE), whether on a daily or sample-wide ba- sis. For each day t = 1,...,T = 80 we observe moneyness mi and maturity τi for i = 1,...,M = 97. The percentage pricing error is defined as

C(t,mi,τi) Cˆ(t,mi,τj, Θ) ui,t = − . (25) C(t,mi,τi) The sum of squared percentage errors for time t is defined as

2 M C(t,m ,τ ) Cˆ(t,m ,τ , Θ) SSE = u′ u = i i − i j , t t t C(t,m ,τ ) i=1 i i ! X 127 where C() and Cˆ() denote model and market price respectively. Θ is the vector of parameters in the model. The SSE is appealing as an objective function for two reasons. First, taking percent- ages means that the procedure weighs pricing errors evenly across moneyness and maturity. Second, the quadratic nature of the SSE penalizes larger errors thereby creating less variability in the percentage error matrix. This approach is also prefer- able to an objective function based on absolute deviations since that would favor the more expensive long maturity options. Naturally, there are other metrics that would achieve the same objectives, for example an SSE using implied volatilities instead of prices. This would perhaps be more appropriate due to the market practice of quot- ing prices and bid-ask spreads in terms of their implied volatilities. Unfortunately this approach is not computationally feasible in a larger dataset such as this one due to implied volatility not having a closed form representation, in the generalized L´evy models studied in this paper.

6.1 Daily Recalibration For each day and for each model, the following optimization problem is solved

′ Θˆ = argmin ut(Θ) ut(Θ), t = 1,...,T.

The starting value of the volatility process y(0) is normalized to 1 for identification reasons. The optimization procedure used is the Nelder and Mead (1964)-algorithm with starting values from the previous day. For the first day in the sample, parameter values are found using the randomized global search algorithm CMAES by Hansen, Muller, and Koumoutsakos (2003).

6.2 Sample Wide Analysis A different way of evaluating the model performance is by a sample-wide estimation which tests the models ability to simultaneously fit the entire time-series of data. Specifically this procedure allows us to perform a more formal test of superior per- formance. The estimation is performed by solving the optimization problem:

1 T Θˆ = argmin SSE (Θ). T t t=1 X Again the starting value for the y(0) process is normalized to 1. Alternatively, the volatility process y(t) could have been treated as a latent variable and estimated using a filtering approach (see for example Pan (2002)). Filtering out the volatility

128 essentially creates a new parameter to estimate for each day in the sample, giv- ing much more flexibility to the models where the stochastic volatility component accounts for most of the randomness. This effectively biases any performance com- parison toward these models hence this procedure is avoided, and the starting value of the volatility process(y(0)) is kept fixed. The covariance matrix of Θˆ can be estimated using the classical robust estimator of White (1980) which is trivially extended to the present nonlinear setting (see for example in Mittelhammer, J., and Miller (2000) Section 15.4.1)

−1 T ∂u (Θ, yˆ ) ∂u (Θ, yˆ ) ′ cov(Θ)ˆ = t t t t ∂Θ ∂Θ Θ=Θˆ " t=1 # X   T ′ ∂ut(Θ, yˆt) [ ∂ut(Θ, yˆ t) SSEt × " ∂Θ ∂Θ Θ=Θˆ # t=1   X −1 T ∂u (Θ, yˆ ) ∂u (Θ, yˆ ) ′ t t t t . (26) × " ∂Θ ∂Θ Θ=Θˆ # t=1   X The gradient vectors in the above expression can be derived in closed form for all the models studied in this paper. This is done by inserting the pricing formula from Proposition (1) in the expression for ut in (25) and differentiating with respect to the parameters to calculate the gradient vectors. These results are rather lengthy and available from the author upon request. To compare the performance of two different models, a t-test is performed based on i j the sample differences of the sum of squared errors. Defining dt = SSEt SSEt as ¯ 1 T − the difference in errors for the ith and jth model and setting d = T t=1 dt we get a test statistic : P d¯ S = . stdev(d¯)

Diebold and Mariano (1995) show that S is approximately standard normal under the null hypothesis of equal mean squared percentage errors. The denominator is adjusted using the Newey and West (1987) correction for autocorrelation in errors:

q T v T stdev(d¯)= dˆ2 + 2 1 (dˆ dˆ ), v t − q + 1 t t−v u i=1 v=1 t=v+1 uX X   X t where q is the number of lags determined through the optimal selection procedure of Andrews (1991)

129 7 Results

7.1 Results: Daily recalibration The overall pricing performance for each model is reported as the average absolute percentage pricing error(APE) across both time, maturity and strike.

1 T M C(t,m ,τ ) Cˆ(t,m ,τ , Θ) AP E = i i − i j . T M C(t,mi,τi) t=1 i=1 X X

As a benchmark case the pure log-normal model of Brace, Gatarek , and Musiela (1997), Miltersen, Sandmann, and Sondermann (1997) and Jamshidian (1997) with ∗ t ∗ Lt = 0 cdW¯ s , is estimated for each day. As expected the log-normal model has a rather high APE of 20.02% (s.e 0.25%) withc ¯ = 0.134. R The results for all the models specified in the previous section can be found in Tables 1 to 3. The first thing to note is a considerably better pricing performance for all the different specifications. The lowest APE is found with the VGSVD model with an APE of 2.457%. This is not surprising as it is by far the most flexible model with a whopping 9 parameters. However the simpler time-changed L´evy and self-similar additive models have only a slightly worse performance so it appears that little is gained when increasing the complexity. In terms of performance the classic models, DDSV and DDSVJ are ranked the lowest with APE’s of 4.65% 3.91% respectively. These levels are, however, still lower than the 5% mark often considered to be an upper threshold of a tolerable APE. Figure 9 plots the daily APE across time, and here it can be seen that moving from DDSV to DDSVJ has a clear significant change in overall performance. These changes are not at all apparent when moving between specifications in the self- similar additive or time-changed L´evy models. In fact adding a constant or stochas- tic volatility diffusion component seems to increase the performance very little. Turning to the parameters and starting with the classic models in Table 1 it is first noticed that α is of a considerable size in the DDSV case. This confirms empirical results in section (4), that the interest rate distribution is heavily skewed to the left. α falls slightly when adding the compound Poisson jumps as this process also adds to the left-skewness by having a mean jump µCP of negative size. The average intensity λCP = 1.5 corresponds to an infrequent jump approximately every 8 months. The volatility of volatility appears very low at 0.005. However one must be careful not to interpret the values in log-normal terms. Ignoring stochastic volatility and jumps, Rebonato (2004) show that as α increases the process converges to an arithmetic Brownian motion. Gaining intuition from this limit case, we would indeed expect the parameters to be of a lower magnitude since as α increases the parameters begin to determine variance in the level of rates as opposed to the variance of log differ-

130 ences. Moving to the self-similar additive models in Table 2 it is again seen that consider- able skewness is needed to capture the caplet prices with an average M being more than twice the size of the average G. A constant volatility diffusion component appears to be unnecessary asc ¯ is very small whereas adding stochastic volatility has a slightly bigger impact. The stochastic volatility component has a volatility of volatility ǫ of 0.09 meaning that it accounts for a significant part of the randomness. Finally the γ parameter is fairly constant, both across models and time, with an average value around 0.09-0.11 corresponding to a ”smaller than linear” scaling of variance with time which was also observed in Figure 6. Finally, moving to the time-changed L´evy models in Table 3 it can first be seen in VGSVC that the constant volatility diffusion component is of slightly higher mag- nitude than in VGSSDC. It also seems to have a stabilizing effect as the standard errors have decreased compared to VGSV. Adding stochastic volatility in the diffu- sion component in the VGSVD model has an impact mainly in the short run since the mean reversion speed is very fast.

7.1.1 Error Analysis

In order to investigate how the error is distributed across time, the daily APE is plotted as a function of time in Figure 9. Here we see that the self-similar additive models and time-changed L´evy models are very similar within their respective model classes. This is not the case for the classic models since going from DDSV to DDSVJ has a noticeable effect. Moving along the time dimension the errors appear fairly stationary without any particular critical days. In Tables 4 to 6 the average percentage errors are given across moneyness and maturity, but I refrain from taking absolute values in order to detect any systematic over- or underpricing. Starting with the classic models in Table 4 we see that if we move from low to high moneyness there is a tendency to heavily underprice ITM caplets, slightly overprice ATM and then again underprice deep OTM options. This pattern indicates the classic model’s overall inability to create enough curvature in the smile. Looking further at the self-similar additive models in Table 5, we see an overall improvement especially for out-of-the money pricing errors which have decreased to almost a third of the level in the classic models. The same effect is observed for the time-changed L´evy models in Table 6 but with a slightly lower overall absolute error level.

131 7.1.2 Out-Of Sample Performance

As a robustness check an out-of-sample analysis is performed. Since an interest rate model of the kind investigated in this paper is mainly used for hedging purposes it is not only relevant how the model replicates todays prices but also future prices. For each day in the sample the calibrated parameters are used to calculate the pricing error today, 1 week, 1 month, 3 months, and 6 months ahead. The results are averaged and reported in Table 7. The out-of-sample pricing performance ranking of the models is almost the same as its in-sample counterparts, with the only change being the VGSSD model faring slightly better than the VGSSDC model. The errors are very similar across models for the 6 month horizon, but what is perhaps the most relevant to option traders is the 1 week and 1 month ahead pricing errors since these correspond to the most often used rebalancing frequencies of a hedge portfolio. In these two horizons the gain from using the time-changed L´evy or self-similar additive models over the classic models is clear.

7.2 Results: Sample-wide estimation

The results of the sample-wide estimation procedure is shown for the three different classes of models in Tables 8-10. Overall the parameters are also largely of the same magnitude as in the previous section, and one can repeat the interpretation therein. One slight difference however is in DDSVJ model which for the sample wide analysis has a significantly higher frequency of jumps with λCP = 15.97 compared to an average of 1.5 in the daily recalibration case. The higher frequency causes the jump component to generate higher skewness but the effect is offset by a lower displacement coefficient α. This could indicate that the two sources of skewness in the DDSVJ model are not fully identified by the data. The benefit of doing a sample wide analysis is that it allows us to perform the Diebold-Mariano test for superior pricing performance. The results can be found in Table 11. A negative (i, j)’th value means that model i is superior to model j and critical values, for the null hypothesis of equal performance, are values smaller than -1.645 and larger than 1.645. We can see that both the self-similar additive models and the time-changed L´evy models outperform the classic models in a statistically significant manner. Comparing the self-similar additive models and time-changed L´evy models we see that the latter outperforms the former in all cases. One can also note that adding a constant volatility diffusion component has a significant impact on performance for the time-changed L´evy models but not for the self-similar additive models.

132 8 Conclusion

In this paper a theoretical and empirical analysis is performed concerning the pric- ing of caplets. By investigating the term structure of moments the paper provides an actual desideratum for the behavior of the driving process in an extended L´evy Libor Market Model. The models presented in this paper can be considered a first attempt at achieving the posed goals. The empirical analysis in the paper show that high frequency jump models outperform traditional diffusion or finite activity jump-diffusion based models, in pricing caplets. It also shows that adding stochastic volatility diffusion component to an infinite activity jump model improves the per- formance at a statistically significant but economically negligible level. Furthermore the Self-Similar Additive Variance Gamma model is shown to have an impressive performance despite its parsimony. High frequency jump models no doubt have the potential of becoming as popular in interest rate modeling as they have recently become in stock price modeling. How- ever, the popularity is contingent on the ease of which the model can price not just caps but also swaptions. Clearly the perfect correlation implication of the 1 factor approach taken in this paper is insufficient as swaptions are correlation sensitive products. Several factors, with a well specified dependence structure, would have to be added to the existing framework in order to get realistic prices. Needless to say this is a non-trivial task left for future research.

133 Appendix A: Deriving the drift under the Terminal Measure

Proof. Inserting (9) and (10) in (7) and (3) you get

K λj x ∗ 1 2 λkx δ(e 1)Fj(t ) F bk(t)= c(t) e 1 λkx 1+ − − ν (t,dx), − 2 − R − − 1+ δFj(t ) Z j=k+1  − 
Y and

t K t t Fk+1 δc(s)Fj(s) F∗ Fk+1 Lt = ds + c(s)dWs + x(µ ν )(ds,dx). − 0 1+ δFj(s) 0 0 R − Z j=Xk+1 Z Z Z t Fk+1 t F∗ F∗ Fk+1 Using (10) and 0 R x(µ ν )(ds,dx) = 0 R x(µ ν + ν ν )(ds,dx) the above equation can be− written as − − R R R R t K Fk+1 δc(s)Fj(s) Lt = ds − 0 1+ δFj(s) Z j=Xk+1 K t λj x δ(e 1)Fj(s) F∗ ∗ + x 1 1+ − ν (ds,dx)+ Lt . 0 R − 1+ δFj(s) ! Z Z j=Yk+1   Note that the second term is finite because of the assumption of finite variation in the jump process. A new drift can then be defined as

K ∗ δc(t)Fj(t ) bk(t)= − − 1+ δFj(t ) j=Xk+1 − K λj x ∗ 1 2 λkx δ(e 1)Fj(t ) F c(t) λkx (e 1) 1+ − − ν (t,dx), − 2 − R − − 1+ δFj(t ) ! Z j=Yk+1  −  and the Libor rate process under the Terminal measure can be written as

t t F (t)= F (0) exp b∗(s)ds + λ dL∗ k = 1,...,K k k k k s ∀ Z0 Z0 

134 Appendix B: Excess Kurtosis of a Brownian motion with CIR Stochastic Volatility

Defining

t Z = y(s)dWs. 0 Z p The excess kurtosis is defined as E[(Z E[Z])4] γ = − 3 (27) 2 E[(Z E[Z])2]2 − − Since the Brownian motion has mean zero we get

E[Z] = 0, (28) and from the Itˆoisometry we get

t t 2 2 E[Z ]= E[( y(s)dWs) ]= E[y(s)]ds. 0 0 Z p Z Using Cox, Ingersoll, and Ross (1985) equation 19 we have that E[y(s)] = η+(y(0) − η)e−κs. Inserting this in the above gives us 1 E[Z2]= ηt +(y(0) η) (1 e−κt). (29) − κ − Since the fourth moment of the normal distribution is 3 it follows that

t t E[Z4]= E[( y(s)ds)23] = 3V ar( y(s)ds) (30) Z0 Z0 Inserting (28) (29) and (30) in (27) we get the result in equation (19)

135 References

Andersen, L., and R. Brotherton-Ratcliffe (2005): “Extended Libor Mar- ket Models with Stochastic Volatility,” Journal of Computational Finance, 9(1), 1–40.

Andersen, L., and V. Piterbarg (2005): “Moment Explosions in Stochastic Volatility Models,” Working Paper.

Andrews, D. W. K. (1991): “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation,” Econometrica, 59(3), 817–58.

Bakshi, G., C. Cao, and Z. Chen (1997): “Empirical Performance of Alternative Option Pricing Models,” The Journal of Finance, 52(5), 2003–2049.

Belomestny, D., and J. Schoenmakers (2006): “A jump-diffusion Libor model and its robust calibration,” SFB 649 Discussion Paper 2006-037, Weierstrass In- stitute for Applied Analysis and Stochastics, Berlin, Germany.

Black, F., and M. Scholes (1973): “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, 81, 637–654.

Brace, A., D. Gatarek, and M. Musiela (1997): “The Market Model of Interest Rate Dynamics,” Mathematical Finance, 7(2), 127–154.

Brigo, D., and F. Mercurio (2006): Interest Rate Models, Theory and Practice, 2nd Edition. Springer Finance.

Carr, P., H. Geman, D. Madan, and M. Yor (2002): “The fine structure of asset returns: An empirical investigation,” Journal of Business, 75(2), 305–332.

(2003): “Stochastic Volatility for L´evy Processes,” Mathematical Finance, 13(3), 345–382.

(2007): “Self-Decomposability and Option Pricing,” Mathematical Finance, 17(1), 31–73.

Carr, P., and D. Madan (1999): “Option Pricing and the Fast Fourier Trans- form,” Journal of Computational Finance, 2(4), 61–73.

Carr, P., and L. Wu (2004): “Time-changed L´evy processes and option pricing,” Journal of Financial Economics, 71, 113–141.

Chen, R.-R., and L. Scott (2004): “Stochastic Volatility and Jumps in Interest Rates: An International Analysis,” SSRN eLibrary.

136 Cont, R., and P. Tankov (2004): Financial Modelling with Jump Processes. Chapman & Hall/CRC Financial Mathematics Series.

Cox, J., J. Ingersoll, and S. Ross (1985): “A Theory of the Term Structure of Interest Rates,” Econometrica, 53, 385–407.

Das, S., and R. Sundaram (1999): “Of Smiles and Smirks: A Term Structure Perspective,” Journal of Financial and Quantitative Analysis, 34(2), 211–239.

Diebold, F., and R. S. Mariano (1995): “Comparing Predictive Accuracy,” Journal of Business and Economic Statistics, 13, 253–265.

Eberlein, E., and W. Kluge (2007): “Calibration of L´evy Term Structure Mod- els,” Advances in Mathematical Finance: In Honor of D. Madan, M. Fu, R.A. Jarrow, J.-Y Yen and R.J. Elliot(Eds.), pp. 147–172.

Eberlein, E., and F. Ozkan¨ (2005): “The L´evy Libor Model,” Finance and Stochastics, 9, 327348.

Galloway, M. (2006): “Option Pricing with Selfsimilar Additive Processes,” PhD Thesis, University of Florida.

Glasserman, P., and S. Kou (2003): “The Term Structure of Simple Forward Rates with Jump Risk,” Mathematical Finance, 13(3), 383–410.

Hagan, P. S., D. Kumar, A. S. Lesniewski, and D. E. Woodward (2002): “Managing Smile Risk,” Wilmott Magazine, pp. 84–108.

Hansen, N., S. Muller, and P. Koumoutsakos (2003): “Reducing the time complexity of the derandomized evolution strategy with covariance matrix adap- tation (CMA-ES).,” Evolutionary Computation, 11(1), 1–18.

Heston, S. L. (1993): “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options,” Review of Financial Studies, 6(2), 327–343.

Huang, J., and L. Wu (2004): “Specification Analysis of Option Pricing Models Based on Time-Changed L´evy Processes,” Journal of Finance, 59(3), 1405–1440.

Jacod, J., and A. Shiryaev (1987): Limit Theorems for Stochastic Processes. Berlin Heidelberg New York: Springer.

Jamshidian, F. (1997): “Libor and Swap Market Models and Measures,” Finance and Stochastics, 1(4), 261–291.

137 (1999): “Libor Market Model with Semimartingales,” Working Paper, NetAnalytic Ltd., London.

Jarrow, R., H. Li, and F. Zhao (2007): “Interest Rate Caps ’Smile’ Too! But Can the LIBOR Market Models Capture the Smile?,” Journal of Finance, 62, 345–382.

Jones, C. S. (2003): “The dynamics of stochastic volatility: Evidence from under- lying and options markets,” Journal of Econometrics, 116(1-2), 181–224.

Kluge, W. (2005): “Time-inhomogeneous L´evy processes in interest rate and credit risk models,” PhD Thesis, University of Freiburg.

Konikov, M., and D. Madan (2002): “Option Pricing Using Variance Gamma Markov Chains,” Review of Derivatives Research, 5, 81–115.

Li, H., and F. Zhao (2006): “Unspanned Stochastic Volatility: Evidence from Hedging Interest Rate Derivatives,” The Journal of Finance, 61(1), 341–378.

Madan, D., P. Carr, and E. Chang (1998): “The Variance Gamma Process and Option Pricing,” European Finance Review, 2, 79–105.

Merton, R. C. (1976): “Option Pricing When Underlying Stock Returns Are Discontinuous,” Journal of Financial Economics, 3(1-2), 125–144.

Miltersen, K. R., K. Sandmann, and D. Sondermann (1997): “Closed form solutions for term structure derivatives with log-normal interest rates,” Journal of Finance, 52(2), 409–430.

Mittelhammer, R., J. G. J., and D. J. Miller (2000): Econometric Founda- tions. Cambridge University Press.

Nelder, J., and R. Mead (1964): “A simplex method for function minimization,” The Computer Journal, pp. 308–313.

Newey, W., and K. West (1987): “A Simple Positive Semi-Definite Het- eroscedasticity and Autocorrelation Consistent Covariance Matrix,” Economet- rica, 53, 703–708.

Pan, J. (2002): “The Jump-Risk Premia Implicit in Options: Evidence from an Integrated Time-Series study,” Journal of Financial Economics, 63, 3–50.

Piza, M. (2005): “Cap Stripping Method for RACL,” Bloomberg Documentation.

138 Rebonato, R. (2002): Modern Pricing of Interest Rate Derivatives, The Libor Market Model and Beyond. Princeton University press.

(2004): Volatility and Correlation: The Perfect Hedger and the Fox. Wiley.

Sato, K. (1991): “Self-similar Processes with Independent Increments,” Probability and Related Fields, 89(3), 285–300.

(1999): L´evy Processes and Infinitely Divisible Distributions. Cambridge University Press.

Skovmand, D. (2008): “Fast and Accurate Option Pricing in a Jump-Diffusion Libor Market Model,” Working Paper.

White, H. (1980): “A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity,” Econometrica, 48(4), 817–838.

Wu, L., and F. Zhang (2006): “Libor Market Model With Stochastic Volatility,” Journal of Industrial and Management Optimization, 2(2), 199227.

139 Table 1: Classic Models: Single day calibration Θ DDSV DDSVJ α 0.5956 0.4362 (0.01384) (0.01831) κ 0.9813 2.636 (0.01878) (0.6734) η 0.000115 1.09e-005 (1.413e-005) (3.587e-006) ǫ 0.005712 0.004863 (0.0004473) (0.0003668) λ(1) 0.9275 0.5836 (0.01079) (0.01474) λCP - 1.544 (0.4126) µCP - -0.002502 (0.006779) σCP - 0.01048 (0.003175) AP E 0.04656 0.03908 (0.00036653) (0.00040048) This table shows the average parameter values for the classic models. Parameters are estimated by minimizing the SSE for each Wednesday from May 7th 2003 to November 10th 2004 and then averaged. The APE denotes the average absolute percentage caplet pricing error. Standard errors are denoted in parenthesis

140 Table 2: Self-similar Additive Models: Single day calibration Θ VGSSD VGSSDC VGSSDSV C 1.315 1.304 1.243 (0.01666) (0.01886) (0.02253) G 2.848 2.833 2.556 (0.06433) (0.06728) (0.07757) M 6.173 6.139 6.819 (0.06266) (0.06798) (0.161) γ 0.07221 0.07022 0.05995 (0.006818) (0.007003) (0.007732) λ1 0.504 0.5031 0.517 (0.004906) (0.004926) (0.006627) c¯ - 0.00247 - (0.001018) (-) κ - - 1.664 (0.3125) η - - 0.1134 (0.06212) ǫ - - 0.09344 (0.009094) AP E 0.02871 0.02867 0.02691 (0.0005743) (0.0005681) (0.00057461) This table shows the average parameter values for the self-similar additive models. Parameters are estimated by minimizing the SSE for each Wednesday from May 7th 2003 to November 10th 2004 and then averaged. The APE denotes the average absolute percentage caplet pricing error. Standard errors are denoted in parenthesis

141 Table 3: Time-Changed Levy Models: Single day calibration Θ VGSV VGSVC VGSVD C 4.464 2.812 2.673 (0.9362) (0.1534) (0.165) G 4.285 3.888 2.793 (0.1708) (0.1114) (0.09465) M 7.467 7.078 14.05 (0.1544) (0.09654) (0.9767) κ 1.287 1.197 1.537 (0.1088) (0.09786) (0.09272) η 0.0141 0.007875 0.0002765 (0.001885) (0.00154) (7.928e-005) ǫ 0.4633 0.478 0.5014 (0.04138) (0.02326) (0.02886) λ1 0.6833 0.6891 0.7202 (0.009835) (0.009266) (0.01259) c¯ - 0.01549 - (0.002142) (-) κ˜ - - 2.432 (0.07716) η˜ - - 0.001761 (0.0001926) ǫ˜ - - 0.3886 (0.01588) AP E 0.02551 0.0253 0.02457 (0.0005838) (0.00058789) (0.00070581) This table shows the average parameter values for the time-changed L´evy models. Parameters are estimated by minimizing the SSE for each Wednesday from May 7th 2003 to November 10th 2004 and then averaged. The APE denotes the average absolute percentage caplet pricing error. Standard errors are denoted in parenthesis

142 Table 4: Average Percentage Errors of Classic Models Moneyness/Maturity 0 1 2 3 4 5 6 7 8 910121520 Panel A: Average Percentage Errors of DDSV 0.6 - - 0.180 0.172 0.170 0.167 0.160 0.155 0.141 0.129 0.111 0.078 0.044 0.7 - 0.133 0.137 0.125 0.122 0.122 0.121 0.119 0.107 0.096 0.082 0.048 0.015 0.8 - 0.098 0.083 0.066 0.063 0.067 0.068 0.071 0.065 0.050 0.045 0.010 -0.016 0.9 0.008 0.059 0.031 0.014 0.013 0.017 0.019 0.024 0.021 0.009 0.010 -0.022 -0.043 1 -0.010 0.024 -0.004 -0.018 -0.017 -0.015 -0.014 -0.006 -0.007 -0.015 -0.010 -0.039 -0.054 1.1 0.002 0.005 -0.020 -0.029 -0.024 -0.022 -0.023 -0.012 -0.013 -0.017 -0.009 -0.034 -0.042

143 1.2 0.000 -0.001 -0.021 -0.022 -0.013 -0.010 -0.012 -0.001 -0.001 -0.001 0.011 -0.010 -0.017 1.3 - - -0.012 -0.007 0.006 0.009 0.009 0.020 0.021 0.023 0.037 0.020 0.012 Panel B Average Percentage Errors of DDSVJ 0.6 - - 0.128 0.136 0.139 0.139 0.136 0.134 0.124 0.116 0.104 0.077 0.046 0.7 - 0.062 0.106 0.107 0.105 0.105 0.103 0.103 0.094 0.086 0.078 0.050 0.020 0.8 - 0.043 0.071 0.064 0.058 0.059 0.056 0.059 0.055 0.042 0.043 0.015 -0.010 0.9 0.001 0.019 0.034 0.024 0.017 0.015 0.012 0.015 0.013 0.003 0.009 -0.017 -0.035 1 -0.005 -0.002 0.008 -0.001 -0.007 -0.013 -0.019 -0.013 -0.015 -0.021 -0.011 -0.034 -0.045 1.1 0.001 -0.011 -0.003 -0.008 -0.013 -0.020 -0.028 -0.020 -0.022 -0.024 -0.011 -0.030 -0.034 1.2 - -0.011 -0.003 -0.003 -0.003 -0.009 -0.018 -0.011 -0.011 -0.010 0.007 -0.008 -0.009 1.3 - - 0.003 0.009 0.013 0.008 0.001 0.009 0.009 0.012 0.031 0.020 0.018 This table shows the average caplet pricing error across moneyness and strike when performing daily recalibration each Wednesday from May 7th 2003 to November 10th 2004 Table 5: Average Percentage Errors of Self-similar Additive Models Moneyness/Maturity 0 1 2 3 4 5 6 7 8 9 1012 15 20 Panel A: Average Percentage Errors of VGSSD 0.6 - - 0.037 0.047 0.055 0.058 0.059 0.064 0.062 0.063 0.065 0.052 0.043 0.7 - -0.021 0.036 0.045 0.048 0.049 0.049 0.052 0.049 0.048 0.049 0.031 0.019 0.8 - -0.020 0.034 0.038 0.037 0.037 0.032 0.035 0.034 0.025 0.030 0.005 -0.008 0.9 -0.020 -0.016 0.032 0.035 0.031 0.025 0.017 0.017 0.015 0.006 0.012 -0.017 -0.029 1 -0.004 -0.015 0.029 0.033 0.028 0.017 0.004 0.005 0.002 -0.004 0.002 -0.027 -0.035 1.1 0.002 -0.020 0.020 0.026 0.022 0.009 -0.006 -0.004 -0.007 -0.009 0.000 -0.023 -0.024 1.2 - -0.028 0.009 0.017 0.017 0.004 -0.011 -0.009 -0.010 -0.009 0.006 -0.010 -0.003 1.3 - - 0.000 0.012 0.014 0.002 -0.011 -0.008 -0.009 -0.005 0.013 0.004 0.015 144 Panel B Average Percentage Errors of VGSSDSV 0.6 - - 0.036 0.047 0.055 0.058 0.059 0.064 0.062 0.062 0.064 0.050 0.040 0.7 - -0.021 0.038 0.047 0.050 0.051 0.051 0.054 0.051 0.049 0.049 0.030 0.016 0.8 - -0.015 0.039 0.042 0.041 0.041 0.037 0.040 0.039 0.030 0.034 0.007 -0.009 0.9 -0.027 -0.009 0.034 0.033 0.029 0.025 0.018 0.020 0.019 0.011 0.018 -0.012 -0.027 1 -0.009 -0.009 0.025 0.023 0.018 0.008 -0.002 0.002 0.001 -0.003 0.005 -0.023 -0.033 1.1 0.003 -0.013 0.014 0.014 0.010 -0.001 -0.013 -0.008 -0.009 -0.010 0.001 -0.022 -0.024 1.2 - -0.017 0.006 0.009 0.007 -0.003 -0.016 -0.010 -0.010 -0.007 0.009 -0.008 -0.005 1.3 - - 0.002 0.008 0.009 0.000 -0.011 -0.005 -0.004 0.001 0.020 0.008 0.014 This table shows the average caplet pricing error across moneyness and strike when performing daily recalibration each Wednesday from May 7th 2003 to November 10th 2004 Table 6: Average Percentage Errors of Time-Changed L´evy Models Moneyness/Maturity 0 1 2 3 4 5 6 7 8 9 10 12 15 20 Panel A: Average Percentage Errors of VGSV 0.6 - - 0.040 0.047 0.053 0.057 0.058 0.064 0.063 0.065 0.069 0.060 0.054 0.7 - -0.006 0.038 0.040 0.041 0.042 0.043 0.048 0.047 0.047 0.051 0.037 0.029 0.8 - 0.000 0.031 0.027 0.023 0.023 0.020 0.026 0.027 0.020 0.029 0.008 0.000 0.9 -0.015 0.006 0.024 0.016 0.010 0.005 -0.001 0.003 0.004 -0.002 0.008 -0.017 -0.024 1 0.005 0.006 0.018 0.011 0.004 -0.005 -0.014 -0.010 -0.010 -0.013 -0.003 -0.028 -0.033 1.1 0.000 0.000 0.011 0.008 0.003 -0.007 -0.018 -0.013 -0.012 -0.013 -0.001 -0.022 -0.023

145 1.2 - -0.008 0.004 0.005 0.005 -0.004 -0.016 -0.010 -0.008 -0.005 0.012 -0.004 -0.001 1.3 - - -0.001 0.006 0.009 0.001 -0.008 -0.002 0.000 0.006 0.026 0.015 0.021 Panel B Average Percentage Errors of VGSVD 0.6 - - 0.027 0.033 0.038 0.041 0.041 0.044 0.041 0.042 0.044 0.040 0.035 0.7 - -0.017 0.028 0.030 0.029 0.029 0.029 0.032 0.030 0.030 0.036 0.029 0.020 0.8 0.000 0.001 0.033 0.027 0.022 0.020 0.015 0.019 0.020 0.014 0.024 0.009 -0.001 0.9 -0.023 0.015 0.035 0.027 0.020 0.013 0.005 0.006 0.007 0.001 0.011 -0.010 -0.021 1 -0.001 0.010 0.026 0.022 0.016 0.005 -0.006 -0.003 -0.004 -0.008 0.001 -0.020 -0.029 1.1 0.001 -0.002 0.012 0.014 0.011 0.001 -0.012 -0.009 -0.010 -0.011 0.000 -0.019 -0.020 1.2 - -0.012 0.001 0.008 0.010 0.001 -0.012 -0.008 -0.009 -0.007 0.007 -0.007 -0.003 1.3 - - -0.006 0.005 0.012 0.005 -0.005 -0.001 -0.001 0.002 0.018 0.008 0.014 This table shows the average caplet pricing error across moneyness and strike when performing daily recalibration each Wednesday from May 7th 2003 to November 10th 2004 Table 7: Out-of-Sample Errors Model Today 1 week 1 month 3 months 6 months DDSV 0.0466 0.0501 0.0565 0.0648 0.072 DDSVJ 0.0391 0.0436 0.0502 0.0583 0.0664 VGSSD 0.0287 0.0331 0.0423 0.0476 0.057 VGSSDC 0.0287 0.0332 0.0424 0.0478 0.0572 VGSSDSV 0.0269 0.0317 0.0415 0.0466 0.0561 VGSV 0.0255 0.0315 0.0406 0.0444 0.0552 VGSVC 0.0253 0.0313 0.0399 0.0437 0.0547 VGSVD 0.0246 0.0296 0.0385 0.0436 0.0522 This table shows the average caplet pricing error. The SSE is minimized for every day in the sample and then the error is recorded 1 week to 6 months ahead and averaged across each day in the sample

Table 8: Classic Models: Sample Wide Estimation Θ DDSV DDSVJ α 0.5395 0.3415 (0.001916) (6.652e-005) κ 0.9348 0.8601 (0.03145) (0.004776) η 0.0001027 4.257e-012 (0.001979) (0.0003069) ǫ 0.005296 0.008384 (0.1526) (0.1964) λ(1) 0.9942 0.8589 (0.01337) (0.0002484) λCP - 15.97 (1.795) µCP - -0.003644 (0.008604) σCP - 0.0004898 (0.0004087) AP E 0.06064 0.05717 This table shows the estimated parameters from the sample-wide analysis. Parameters are estimated by Generalized Method Moments for the entire sample. Standard errors are denoted in parenthesis

146 Table 9: Self-similar Additive Models: Sample Wide Estimation Θ VGSSD VGSSDC VGSSDSV C 1.333 1.333 1.257 (0.005949) (0.003396) (0.04389) G 2.847 2.847 2.336 (0.01039) (0.004659) (0.09208) M 6.277 6.277 7.821 (0.01169) (0.004226) (0.1773) γ 0.07492 0.07492 0.05086 (0.0002244) (0.0001615) (0.005208) λ1 0.491 0.491 0.5114 (0.003487) (0.002321) (0.0006514) c¯ - 8.044e-009 - (31.18) (-) κ - - 1.183 (0.004361) η - - 4.918e-011 (7.244e-005) ǫ - - 0.1453 (0.01066) AP E 0.04912 0.04912 0.04751 This table shows the estimated parameters from the sample-wide analysis. Parameters are estimated by Generalized Method Moments for the entire sample. Standard errors are denoted in parenthesis

147 Table 10: Time-Changed Levy Models: Sample Wide Estimation Θ VGSV VGSVC VGSVD C 2.383 2.069 2.23 (0.01169) (0.02239) (0.1605) G 4.1 3.67 3.245 (0.009628) (0.02409) (0.1821) M 7.515 7.083 15.01 (0.007561) (0.02217) (0.3459) κ 0.8159 0.8251 0.9738 (0.009497) (0.0007558) (0.003811) η 0.007727 0.0005804 6.786e-008 (0.0002186) (0.001099) (0.0001121) ǫ 0.4651 0.4688 0.4997 (0.00139) (0.01958) (0.04134) λ1 0.7093 0.7102 0.7759 (0.0005323) (0.0001673) (0.001157) c¯ - 0.02905 - (0.002724) (-) κ˜ - - 2.327 (0.003144) η˜ - - 0.001577 (3.863e-005) ǫ˜ - - 0.4655 (0.001141) AP E 0.0474 0.04713 0.04522 This table shows the estimated parameters from the sample-wide analysis. Parameters are estimated by Generalized Method Moments for the entire sample. Standard errors are denoted in parenthesis

148 Table 11: Diebold-Mariano Test Results Θ DDSV DDSVJ VGSSD VGSSDC VGSSDSV VGSV VGSVC VGSVD DDSV 0------DDSVJ -18.20 ------VGSSD -3.21 -1.74 0 - - - - - VGSSDC -3.21 -1.74 -0.00621 0 - - - - 149 VGSSDSV -3.9 -2.44 -12 -12 0 - - - VGSV -4.18 -2.77 -11.2 -11.2 -5.48 0 - - VGSVC -4.23 -2.83 -12 -12 -6.38 -15.6 0 - VGSVD -4.63 -3.24 -15.9 -15.9 -10.2 -9.66 -8.75 0

This table shows the Diebold-Mariano test statistics. The test statistics are critical at a 5% levels for the null hypethis of equal performance when smaller

than -1.645 and larger than 1.645. The entrys in the table indicate modeli vs. modelj in the (i,j)’th entry. A negative value indicates that modeli is superior

to modelj 180

160

140

120

100

Price 80

60

40

20

0 0.4 0.6 0.8 20 1 15 1.2 10 5 1.4 0 Moneynes(X/F) Maturity

Figure 1: Average Caplet Prices This figure shows the average Euribor caplet price in basis points for our weekly sample covering May 7th 2003 to May 4th 2005. Prices are interpolated in terms of strike to get a common moneyness throughout the sample

150 0.4

0.35

0.3

0.25

0.2 Implied vol

0.15 0.5

0.1 1

0.05 0 2 4 6 8 10 1.5 12 14 16 18 20 Moneynes(X/F)

Maturity

Figure 2: Average Caplet Implied Volatilities This figure shows the average Euribor caplet implied vol for our weekly sample covering May 7th 2003 to November 10th 2004. Volatilities are found by inverting the Black-Scholes formula for the prices in Figure 1.

151 0.4

0.35

0.3

0.25

Implied Vol. 0.2

0.15

0.1

0.05 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 Moneynes(X/F)

Figure 3: Average Caplet Implied Volatilities This figure shows the average Euribor caplet implied vol for our weekly sample covering May 7th 2003 to November 10th 2004. Volatilities are found by inverting the Black-Scholes formula for the prices in Figure 1. The two curves slightly shorter than the rest are the 1 and 2 year maturity. The remaining curves are declining with respect to maturity.

152 ITM X/F=0.6

0.4

0.2 May03

Implied Vol. 0 Dec03 0 2 4 6 8 Jul04 10 12 14 16 18 20 Time ATM X/F=1

0.4

0.2 May03

Implied Vol. 0 Dec03 0 2 4 6 8 Jul04 10 12 14 16 18 20 Time OTM X/F=1.3

0.4

0.2 May03

Implied Vol. 0 Dec03 0 2 4 6 8 Jul04 10 12 14 16 18 20 Time Maturity

Figure 4: ITM/ATM/OTM Caplet Implied Volatilies This figure shows the in-the-money, at-the-money and out-of-money term structure of Euribor caplet implied vol for our weekly sample covering May 7th 2003 to November 10th 2004. Volatilities are found by inverting the Black-Scholes formula for the prices.

153 0.065

0.06

0.055

0.05

0.045

0.04

0.035 6m LIBOR rate 0.03

0.025

0.02

0.015 May03 Aug03 Dec03 Apr04 20 15 Jul04 10 Nov04 5 0 Time Maturity

Figure 5: Term Structure of the 6 month Libor Rate This figure shows the shows the 6 month Libor rate for all maturities from 1 to 20 years for our weekly sample covering May 7th 2003 to November 10th 2004.

154 Average Implied Variance 0.7 2003−05−07 to 2003−09−17 2003−09−24 to 2004−02−04 2004−02−11 to 2004−06−23 0.6 2004−06−30 to 2004−11−10

0.5

0.4 Variance 0.3

0.2

0.1

0 0 2 4 6 8 10 12 14 16 18 20 Time

Figure 6: This figure shows the average implied variance as a function of time backed out from the caplet dataset. Variances are backed out for each week in the sample using Variance Gamma model for each time to maturity. The variances are then averaged across the four different periods in the legend.

155 Average Implied Skewness 0 2003−05−07 to 2003−09−17 2003−09−24 to 2004−02−04 2004−02−11 to 2004−06−23 2004−06−30 to 2004−11−10 −0.5

−1

−1.5 Skewness

−2

−2.5

−3 0 2 4 6 8 10 12 14 16 18 20 Time

Figure 7: This figure shows the average implied skewness as a function of time backed out from the caplet dataset. The skewness’s are backed out for each week in the sample using a Variance Gamma model for each time to maturity. They are then then averaged across the four different periods in the legend.

156 Average Implied Kurtosis 14 2003−05−07 to 2003−09−17 2003−09−24 to 2004−02−04 2004−02−11 to 2004−06−23 12 2004−06−30 to 2004−11−10

10

8

6 Excess Kurtosis

4

2

0 0 2 4 6 8 10 12 14 16 18 20 Time

Figure 8: This figure shows the average implied excess kurtosis as a function of time backed out from the caplet dataset. The kurtosis’s are backed out for each week in the sample using a Variance Gamma model for each time to maturity. They are then then averaged across the four different periods in the legend.

157 Classic Models

0.05 ← DDSV 0.04 ← DDSVJ 0.03

0.02 Percentage Pricing Error 0.01 May03 Dec03 Jul04 Time Selfsimilar Additive Models

0.05

0.04 ← VGSSDC ← VGSSD 0.03 ← VGSSDSV 0.02 158 Percentage Pricing Error 0.01 May03 Dec03 Jul04 Time Time−Changed Levy Models 0.06

0.05

0.04

0.03 ← VGSV ← VGSVC 0.02 ← VGSVD Percentage Pricing Error 0.01 May03 Dec03 Jul04 Time

Figure 9: Daily Recalibration: Percentage Errors This figure shows the average percentage error for each day in the sample covering May 7th 2003 to November 10th 2004 SCHOOL OF ECONOMICS AND MANAGEMENT UNIVERSITY OF AARHUS - UNIVERSITETSPARKEN - BUILDING 1322 DK-8000 AARHUS C – TEL. +45 8942 1111 - www.econ.au.dk

PhD Theses:

1999-4 Philipp J.H. Schröder, Aspects of Transition in Central and Eastern Europe.

1999-5 Robert Rene Dogonowski, Aspects of Classical and Contemporary European Fiscal Policy Issues.

1999-6 Peter Raahauge, Dynamic Programming in Computational Economics.

1999-7 Torben Dall Schmidt, Social Insurance, Incentives and Economic Integration.

1999 Jørgen Vig Pedersen, An Asset-Based Explanation of Strategic Advantage.

1999 Bjarke Jensen, Five Essays on Contingent Claim Valuation.

1999 Ken Lamdahl Bechmann, Five Essays on Convertible Bonds and Capital Structure Theory.

1999 Birgitte Holt Andersen, Structural Analysis of the Earth Observation Industry.

2000-1 Jakob Roland Munch, Economic Integration and Industrial Location in Unionized Countries.

2000-2 Christian Møller Dahl, Essays on Nonlinear Econometric Time Series Modelling.

2000-3 Mette C. Deding, Aspects of Income Distributions in a Labour Market Perspective.

2000-4 Michael Jansson, Testing the Null Hypothesis of Cointegration.

2000-5 Svend Jespersen, Aspects of Economic Growth and the Distribution of Wealth.

2001-1 Michael Svarer, Application of Search Models.

2001-2 Morten Berg Jensen, Financial Models for Stocks, Interest Rates, and Options: Theory and Estimation.

2001-3 Niels C. Beier, Propagation of Nominal Shocks in Open Economies.

2001-4 Mette Verner, Causes and Consequences of Interrruptions in the Labour Market.

2001-5 Tobias Nybo Rasmussen, Dynamic Computable General Equilibrium Models: Essays on Environmental Regulation and Economic Growth.

2001-6 Søren Vester Sørensen, Three Essays on the Propagation of Monetary Shocks in Open Economies.

2001-7 Rasmus Højbjerg Jacobsen, Essays on Endogenous Policies under Labor Union Influence and their Implications.

2001-8 Peter Ejler Storgaard, Price Rigidity in Closed and Open Economies: Causes and Effects.

2001 Charlotte Strunk-Hansen, Studies in Financial Econometrics.

2002-1 Mette Rose Skaksen, Multinational Enterprises: Interactions with the Labor Market.

2002-2 Nikolaj Malchow-Møller, Dynamic Behaviour and Agricultural Households in Nicaragua.

2002-3 Boriss Siliverstovs, Multicointegration, Nonlinearity, and Forecasting.

2002-4 Søren Tang Sørensen, Aspects of Sequential Auctions and Industrial Agglomeration.

2002-5 Peter Myhre Lildholdt, Essays on Seasonality, Long Memory, and Volatility.

2002-6 Sean Hove, Three Essays on Mobility and Income Distribution Dynamics.

2002 Hanne Kargaard Thomsen, The Learning organization from a management point of view - Theoretical perspectives and empirical findings in four Danish service organizations.

2002 Johannes Liebach Lüneborg, Technology Acquisition, Structure, and Performance in The Nordic Banking Industry.

2003-1 Carter Bloch, Aspects of Economic Policy in Emerging Markets.

2003-2 Morten Ørregaard Nielsen, Multivariate Fractional Integration and Cointegration.

2003 Michael Knie-Andersen, Customer Relationship Management in the Financial Sector.

2004-1 Lars Stentoft, Least Squares Monte-Carlo and GARCH Methods for American Options.

2004-2 Brian Krogh Graversen, Employment Effects of Active Labour Market Programmes: Do the Programmes Help Welfare Benefit Recipients to Find Jobs?

2004-3 Dmitri Koulikov, Long Memory Models for Volatility and High Frequency Financial Data Econometrics.

2004-4 René Kirkegaard, Essays on Auction Theory.

2004-5 Christian Kjær, Essays on Bargaining and the Formation of Coalitions.

2005-1 Julia Chiriaeva, Credibility of Fixed Exchange Rate Arrangements.

2005-2 Morten Spange, Fiscal Stabilization Policies and Labour Market Rigidities.

2005-3 Bjarne Brendstrup, Essays on the Empirical Analysis of Auctions.

2005-4 Lars Skipper, Essays on Estimation of Causal Relationships in the Danish Labour Market.

2005-5 Ott Toomet, Marginalisation and Discouragement: Regional Aspects and the Impact of Benefits.

2005-6 Marianne Simonsen, Essays on Motherhood and Female Labour Supply.

2005 Hesham Morten Gabr, Strategic Groups: The Ghosts of Yesterday when it comes to Understanding Firm Performance within Industries?

2005 Malene Shin-Jensen, Essays on Term Structure Models, Interest Rate Derivatives and Credit Risk.

2006-1 Peter Sandholt Jensen, Essays on Growth Empirics and Economic Development.

2006-2 Allan Sørensen, Economic Integration, Ageing and Labour Market Outcomes

2006-3 Philipp Festerling, Essays on Competition Policy

2006-4 Carina Sponholtz, Essays on Empirical Corporate Finance

2006-5 Claus Thrane-Jensen, Capital Forms and the Entrepreneur – A contingency approach on new venture creation

2006-6 Thomas Busch, Econometric Modeling of Volatility and Price Behavior in Asset and Derivative Markets

2007-1 Jesper Bagger, Essays on Earnings Dynamics and Job Mobility

2007-2 Niels Stender, Essays on Marketing Engineering

2007-3 Mads Peter Pilkjær Harmsen, Three Essays in Behavioral and Experimental Economics

2007-4 Juanna Schrøter Joensen, Determinants and Consequences of Human Capital Investments

2007-5 Peter Tind Larsen, Essays on Capital Structure and Credit Risk

2008-1 Toke Lilhauge Hjortshøj, Essays on Empirical Corporate Finance – Managerial Incentives, Information Disclosure, and Bond Covenants

2008-2 Jie Zhu, Essays on Econometric Analysis of Price and Volatility Behavior in Asset Markets

2008-3 David Glavind Skovmand, Libor Market Models - Theory and Applications