On the Calibration of the

SABR–Libor Market Model Correlations

Master’s Thesis

Dr. Elidon Dhamo Christ Church College University of Oxford

Submitted in Partial Fulfillment for the MSc in Mathematical Finance

September 2011

To Migena ...

Abstract

This work is concerned with the SABR-LMM model. This is a term structure model of interest forward rates with stochastic volatility that is a natural extension of both, the LIBOR market model (Brace-Gatarek-Musiela [1997]) and the SABR stochastic volatility model of Hagan et al. [2002].

While the seminal approximation formula (developed by Hagan et al. [2002]) to implied Black volatility using the SABR model parameters allows for a successful calibration of each forward rate dynamics to the volatility smile of the respective caplets/floorlets, an adequate calibration of the rich correlation structure of SABR-LMM (correlations among the forward rates, the volatilities and the cross correlations) is a challenging topic and of great interest in practice. Although widely used for calibration, it is well known that swaptions’ volatilities carry only little information about correlations among the forward rates. As practically successful for the classical LMM, desirable would be to take the market swap rate correlations into account for the model calibration.

In this study we develop a new approach of calibrating the model correlations, aiming at incorporating the market information about the forward rate corre- lations implied from more correlation-sensitive products such as CMS spread derivatives, in which also swap rate correlations are involved. To this end we derive a displaced-diffusion model for the swap rate spreads with a SABR stochastic volatility. This we achieve by applying the Markovian projection technique which approximates the dynamics of the basket of forward rates, in terms of the terminal distribution, by a univariate displaced-diffusion. The CMS spread derivatives can then be priced using the SABR formulas for the implied volatility, taking the whole market smile of CMS spread options into consideration. For the ATM values in the payoff measure of the projected SDE we use a standard smile-consistent replication of the necessary convexity adjustment with swaptions.

Numerical simulations conclude the work, giving a comparison between this method and the classical one of calibrating the model correlations to swaption volatilities. Furthermore, we study the performance of different parameteriza- tions of the correlation (sub-)matrices.

v

Acknowledgements

First of all, I would like to thank Dr. Christoph Reisinger for agreeing to supervise this thesis, his support and his encouragements.

I would like to express my gratitude towards my former employer, d-fine GmbH, for offering me the opportunity to take part in the Mathematical Finance course at the University of Oxford, and for providing financial support.

Last but not least I am particularly indebted to my family for their understand- ing, their great moral support and their patience over the numerous weekends I did not spend with them.

vii

Contents

Introduction 1

Chapter1. ForwardLiborandSwapMarketModels 4 1.1 A Review of the Classical Libor and Swap Market Models ...... 4 1.1.1 LiborDynamicsUndertheForwardMeasure ...... 7 1.1.2 ValuationinLMM ...... 8 1.1.3 CovarianceandCorrelationsinLMM ...... 9 1.1.4 SwapRateModelsandMeasures ...... 10 1.1.5 Incompatibility Between the LMM and the SMM ...... 11 1.2 The Convexity Adjustment and CMS Derivatives ...... 13 1.2.1 Constant Maturity Swaps and Related Derivatives ...... 14 1.2.2 ValuationofCMSDerivatives ...... 16 1.3 ParameterizationandCalibration ...... 22 1.3.1 Parametric Forms of the Instantaneous Volatilities ...... 23 1.3.2 CalibrationtotheCap/FloorMarket ...... 24 1.3.3 The Structure of Instantaneous Correlations ...... 25 1.3.4 Calibration of LMM Correlations to Swaptions Volatilities . . 30 1.3.5 Calibration to Correlations Implied From CMS Spread Options 30

Chapter2. TheSABRModelofForwardRates 32 2.1 GeneralModelDynamics...... 32 2.1.1 TheTime-HomogeneousModel ...... 34 2.1.2 Joint Dynamics of the SABR Forward Rates and Their Volatilities 35 2.2 ValuationintheSABRModel ...... 36

Chapter3. PricingCMSDerivativesinSABR 37 3.1 TheMarkovianProjectionMethod ...... 37 3.2 A Displaced SABR Diffusion Model for CMS Derivatives ...... 38 3.2.1 Projection of CMS-Spreads to Displaced SABR Diffusion . .. 38 ix Contents x 3.2.2 Pricing of CMS-Spread Options in a SABR Displaced Diffusion Model ...... 45

Chapter4. TheSABR-LMMModelandItsCalibration 48 4.1 SABR–Consistent Extension of the LMM and Its Calibration ..... 48 4.2 CalibratingtheVolatilityProcess ...... 50 4.3 TheSABRCorrelationStructure ...... 51 4.4 Calibration of the SABR–LMM Correlations to Swaption Implied Volatil- ities ...... 52 4.5 Calibrating to Correlations Implied From CMS Spread Options ... 55 4.6 NumericalSimulations ...... 56

Chapter 5. Conclusion and Outlook 59

AppendixA.ClassicalModelsandSABR-LMM 60 A.1 ValuationinLMM ...... 60 A.2 SwapRateDynamicsandtheChoiceofNumeraire ...... 63 A.3 Valuation in the Log-Normal Swap Market Model ...... 64 A.4 DriftApproximationinLMMandSimulations ...... 65 A.5 SABRImpliedVolatility ...... 66

Appendix B. Calibration Details 69 B.1 BootstrappingtheMarketData ...... 69 B.2 Parameterization of SABR–LMM and Its Calibration ...... 71 B.2.1 ParameterizationoftheSABR–LMMModel ...... 71 B.2.2 CalibrationProcedure ...... 72

Bibliography 77 Introduction

While the Brace-Gatarek-Musiela (BGM) or Libor1 Market Model (LMM), based on the assumption that forward term rates follow lognormal processes under their cor- responding forward measure, has established itself as a benchmark model for pricing interest rate derivatives, it is less successful in recovering other essential characteris- tics of interest rate markets, particularly volatility skews and smiles. The presence of these volatility skew and smiles in the market, however, indicates that a pure lognormal forward rate dynamics is not appropriate. In the last decade several extensions of the BGM model have been proposed, in which various versions of the volatility structures of forward Libor rates have been designed to match the observed volatility smile effects in the market. The set of extensions covers local volatility, jump-diffusion and stochastic volatility models with and without time-dependent parameters. The calibration procedures in most of these models are complicated and computationally expensive, and are performed on a best- fit basis. One of the most successful and popular extensions of the LMM, the SABR2 model, models the forward rate process under its forward measure using a correlated log- normal stochastic volatility process. Its success is indebted to two main properties of the model: the crucial property of taking into account the quality of prediction of the future dynamics of the volatility smile, meeting the observations from the market reality, and the seminal asymptotic expansion formula, developed by Hagan et al. [2002], to approximate the implied Black volatility using the SABR model parame- ters. Hence, prices of options, such as caps and floors, can be calculated using the well known Black pricing framework but taking the volatility smile surface via the SABR parameters into account. The SABR model and the LMM, although modeling the same assets, ”do not directly talk to each other”3. The SABR does not link the snapshots of the caplet smiles into well-defined joint dynamics. To overcome this Rebonato [2007] introduced a natural extension of the LMM, the SABR-LMM, that recovers the SABR caplet prices almost exactly for all strikes and maturities. The dynamics of the volatility in this model is chosen so as to be consistent across expiries and to make the evolution of the implied volatilities as time-homogeneous as possible. While the approximation to implied Black volatility using the SABR model parame- ters allows for a successful calibration of each forward rate dynamics to the volatility

1Libor = London Inter-Bank Offered Rates 2Launched by Hagan et al. [2002] 3Rebonato [2007]

1 Introduction 2 smile of the respective caplets and floorlets, an adequate calibration of the rich cor- relation structure of SABR-LMM (comprising correlations among the forward rates, the volatilities and the cross correlations) is a challenging topic and of great inter- est in practice. Although widely used for calibration, it is well known that swap- tion volatilities carry only little information about correlations among the forward rates4. Facing the richness of the correlation structure of SABR-LMM, the need for a calibration approach to more correlation-sensitive products is obvious and of wide practical interest. As already successfully applied for the classical LMM5, it is desirable to additionally take the swap rate correlations into account for the model calibration, which consistently are to be implied from the market prices of appropri- ate products. A broadly known and traded class of interest rate derivatives meeting these requirements, particularly incorporating information about the swap rate cor- relations, is the one of Constant Maturity Swap (CMS) spread derivatives. While valuation (in all the mentioned forward rate models) of these products is typically done straight-forwardly by Monte-Carlo simulation, the calibration of the models to these products, by finding accurate and fast analytical approximations to reproduce the prices of these instruments, has been always subject to research6, even within the simpler LMM framework7.

Scope of the Present Work and Contribution

In this work we develop a novel approach for the calibration of the rich correlation structure in the SABR-LMM model of forward rates. Given the reasons above, our scope is to extract the information about the forward rate correlations from the market prices of correlation-sensitive derivatives, such as CMS spread options, and fit the model correlation parameters to those. To this end the derivation of analytical pricing formulas for these products in the SABR framework is necessary. The Markovian projection (MP) technique8 is an effective technique to volatility calibration that seeks to optimally approximate a complex underlying process with a simpler one, keeping essential properties of the initial process, and is, in principle, applicable to any diffusion model. Starting with the SABR-LMM we apply the MP technique to the CMS spreads and derive a displaced–diffusion SABR model for the spread between the swap rates with different maturities. To achieve this we adapt the recent work of Kienitz-Wittkey [2010], carried out in a SABR swap rate framework, to our SABR-LMM model. Consequently, we can price the CMS spread options in the resulting SABR swap spread model by making use of the seminal SABR formula of Hagan et al. [2002]. In this way we can calibrate the SABR swap spread model parameters to the market implied (normal) volatilities of the corresponding CMS spread options. For the ATM

4We refer at this point to the works of Alexander [2003], Brigo-Mercurio [2007], Rebonato [2002], Schoenmakers [2002, 2005], Schoenmakers-Coffey [2003]. 5B¨orger-van Heys [2010]. 6Antonov-Arneguy [2009], Castagna-Mercurio-Tarenghi [2007], Lutz [2010], Kienitz-Wittkey [2010], etc. 7Belomestny-Kolodko-Schoenmakers [2010], B¨orger-van Heys [2010], etc. 8MP has been introduced in this context by Piterbarg [2003, 2005a,b] and formalized in Piterbarg [2007]. Introduction 3 values in the expiry forward measure of the projected SDE (i.e. expiry time of the corresponding CMS spread option) we use a standard smile-consistent replication9 of the necessary convexity adjustment with swaptions. By this means we shall be able to implicitly retrieve the important information about swap rate correlations contained in the market smile of CMS spread options and embed it into the SABR-LMM model correlations. This work is concluded with the numerical implementation of this new calibration procedure and some numerical simulations. We shall discuss different parameteri- zations of the sub-matrices of the model correlation matrix, in particular the Doust parameterizations, and study the performance of this calibration approach, in terms of pricing errors for swaptions, in comparison to the approach of calibrating the model correlations to swaptions’ implied volatilities, given in Rebonato [2007]. The performed simulations shall also illustrate the effectiveness and robustness of this approach, and provide information about expected and possible drawbacks.

Outline

The thesis is organized as follows. In the first chapter we review the classical forward LIBOR and swap rate market models. Particular focus is set on the introduction of convexity and the different approaches to carry out the convexity correction. We incorporate these methods to the pricing of CMS derivatives, in particular, the CMS caps and spread options. We also describe the calibration of LMM and introduce the different parameterizations of the correlation matrix. At the end of the chapter a recent approach to imply correlations from CMS spread options is presented. The second chapter is devoted to introduction and general properties of the SABR model. The application of the Markovian Projection (MP) method to the basket of forward rates is carried out in the third chapter. Here we derive a displaced-diffusion model for the swap rate spread with a SABR stochastic volatility. For the pricing of CMS spread caplets in the payoff forward measure we use a smile-consistent replication of the convexity adjustment for the ATM spreads via swaptions, and apply the Hagan et al. [2002] formulae. The SABR-LMM model is introduced in the fourth chapter which is mainly concerned with the parameterization and the calibration of the model. Two different approaches to correlation calibration are presented. Numerical simulations give a comparison between the these approaches, taking different parameterizations of the sub-matrices of the correlation matrix into account. We conclude the work by presenting a short summary of our analysis, and give an outlook of possible future directions of the discussed topics.

9According to Hagan [2003]. Chapter 1

Forward Libor and Swap Market Models

1.1 A Review of the Classical Libor and Swap Market Models

Over the past two decades the Brace-Gatarek-Musiela [1997] model (BGM) has es- tablished itself as a benchmark model for pricing and risk managing interest rate derivatives. It is based on the assumption that the forward rates follow lognormal processes with deterministic (time-dependent) volatilities under their correspond- ing measures, and it is widely known as the lognormal forward Libor Market Model (LMM). The popularity of this model is indebted to its compatibility with the seminal Black model which establishes a direct relationship between caplets’ prices and local (implied) volatilities of forward rates and constitutes the standard market convention for quoting benchmark instruments. While the LMM model has established a standard for incorporating all available at- the-money information, it is less successful in recovering other essential characteristics of interest rate markets, particularly volatility skews and smiles1. Various extensions of the LMM model, designed to incorporate skew and smile effects, have been pro- posed. The set of extensions covers local volatility, jump-diffusion and stochastic volatility models with and without time-dependent parameters. In the next chap- ters we shall discuss one of the most successful extensions of the LMM, the SABR model, which enjoys increasing popularity among the practitioners and academics alike, again due to its compatibility with the Black’s framework. Nevertheless, one of the biggest challenges in using these models is the calibration of the forward rate correlations which are not covered by the caplets markets, but rather incorporated in other benchmark instruments, such as European swaptions or CMS spread derivatives. While the valuation in all these forward rate models is

1The volatility tends to rise if the option is out of the money. This results in the so called volatility smile describing the fact that implied Black volatility is strike-dependent.

4 Chapter 1. Forward Libor and Swap Market Models 5 typically done straight-forwardly by Monte-Carlo simulation, calibration by finding accurate and fast analytical approximations to prices of these benchmark instruments has always been subject to research. In this chapter we briefly describe the construction of forward Libor models as given in Brace-Gatarek-Musiela [1997] and Jamshidian [1997], whereby we follow the approach proposed by Musiela-Rutkowski [2005]. Assuming, for the time being, that there are no smile effects present in the interest rate markets, the formal model setup we present here is based on assumptions made in Musiela-Rutkowski [2005]. In the following sections we shall present some of the mostly used parameterizations and calibration methods for the forward rate volatilities and their correlations, which will prepare the ground for introducing the SABR-LMM model in the next chapters and its calibration approaches. We shall also briefly mention the lognormal swap rate model (SMM), putting emphasis on the incompatibility between the two models. Furthermore, a separate section is dedicated to the approximation and pricing of CMS derivatives via the convexity correction technique which paves the way to calibrating the forward rate correlations to the prices of CMS spread derivatives.

Let T ∗ > 0 represent a fixed time horizon. Given a filtered probability space T ∗ Ω, t t [0,T ∗], P which satisfies the basic assumptions made in Musiela-Rutkowski {F } ∈ ∗ T ∗ [2005], let Wt t [0,T ] denote a d-dimensional standard Brownian motion (Wiener { } ∈ T ∗ process) and assume that the filtration t t [0,T ∗] is the usual P augmentation of ∗ {F } ∈ − the filtration generated by WT (cf. Hunt-Kennedy [2004]). { t } In the given probability space an interest rate system formally consists of a system of zero-coupon bonds = B(t,T ) 0

The system of zero-coupon bond prices = B(t,T ) 0 B(t,S) for any t T

T ∗ The equivalent martingale measure P can be interpreted as the time T ∗-forward measure and implies that the bond price dynamics is arbitrage-free. It follows from the Martingale Representation Theorem (cf. Karatzas-Shreve [1991]) that for every T [0,T ∗) the forward interest rate process F (t,T,T ∗) has the repre- sentation ∈ T ∗ (1.1.2) dF (t,T,T ∗) = F (t,T,T ∗)γ(t,T,T ∗) dW , 0 t T, t ≤ ≤ 2This definition is derived from the self-financing portfolio of zero bonds: at time t we sell B(t,T ) B(t,T ) ∗ B(t,T ) ∗ and buy ∗ B(t,T ) at total of zero, B(t,T ) ∗ B(t,T ) = 0. This leads to the definition B(t,T ) − B(t,T ) ∗ ∗ ∗ B(t,T ) of F (t,T,T ), satisfying 1+(T T )F (t,T,T )= ∗ . The latter is the interest amount received − B(t,T ) at time T ∗. Chapter 1. Forward Libor and Swap Market Models 6 ∗ PT ∗ T T ∗ T ∗ Rd under , where Wt = Wt,1 ,...,Wt,d is a -valued (element-wise independent) ∗ PT -Brownian motion and γ(t,T,T )isa Rd-valued, -adapted3 volatility process ∗ t T ∗ T 2 {F } satisfying the condition P γ(u,T,T ∗) du < =1. 0 d ∞ Given these assumptions and the representation of the forwa rd rate process (1.1.2), we can in principle construct an interest rate model with an exogenously specified volatility structure process γ(t,T,T ∗).

The volatility structure γ(t,T,T ∗) might be, in general, a stochastic process. In the special case where γ(t,T,T ∗) is a deterministic, bounded, piecewise continuous func- tion, the forward Libor rate F (t,T,T ∗) is a lognormal martingale under its equivalent martingale measure. The construction of a model of forward rates as presented by Brace-Gatarek-Musiela [1997] starts by postulating that the dynamics of the forward T ∗ rates F (t,T,T ∗) under the equivalent martingale measure P are governed by the stochastic differential equation (1.1.2), where the deterministic volatility function is exogenously given. The model for the forward rates (1.1.2) is referred to in the liter- ature as BGM (Brace-Gatarek-Musiela) Model or the lognormal LiborMarket Model (LMM). In practice, however, we do not model a continuum of forward rates with a fixed compounding period τ but only a finite number of simple forward rates, which in the following will be termed forward Libor rates.

Definition 1.1 (d-factor Libor Market Model). Let T0,...,TN be the set of ex- piries and B(t,T ),..., B(t,T ) the corresponding set{ of zero coupon} bond prices. { 0 N } Let d be the fixed number of independent driving Brownian motions in the model. For each i 0,...,N 1 the d-factor Libor Market Model (LMM) assumes the ∈ { − } following GBM dynamics for forward rate Fi(t) := Fi(t,Ti,Ti+1), under its payoff martingale measure Pi+1 := PTi+1 :

(1.1.3) dF (t) = F (t)γ (t) dWi+1, 0 t T , i i i t ≤ ≤ i i+1 where Wt is a standard d-dimensional Brownian motion under the forward mea- Pi+1 i+1 i+1 sure , with d(Wt,k ,Wt,l ) = δk,l dt, k,l 1,...,d (δk,l is the usual Kronecker ∈4 { } T Delta). γi(t) is a deterministic vector process given by γi(t)=(σi,1(t),...,σi,d(t)) , with σ (t) := γ (t) . i i d Using the Itˆo’s lemma, the GBM equation (1.1.3) can be solved by

t 1 t (1.1.4) F (t) = F (0) exp γ (s) dWi+1 ds γ (s) 2 ds , 0 t T . i i i s − 2 i d ≤ ≤ i 0 0 The dynamics in (1.1.3) does not yet distinguish between the correlations and the volatility of the forward rates. To make this clearer we re-formulate the equation (1.1.3) in the form (cf. Rebonato [1999a] for more details)

(1.1.5) dF (t) = F (t)σ (t) b (t) dWi+1, i i i i t ∗ 3 WT The filtration t is consistently generated by t . 4 {F } { } The interpretation for γi(t) is that it contains the responsiveness of the i’th forward rate for d different independent random shocks. Chapter 1. Forward Libor and Swap Market Models 7 where b (t) Sd Rd (Sd the unit hypersphere in Rd) is given by i ∈ ⊂ d σi,k(t) 2 (1.1.6) bi,k(t) = , bi,k = 1. γi(t) d k=1 In this way (1.1.5) formally separates the volatility σi of the forward rate Fi from the correlation structure ρ between the forward rates, which can be equivalently defined via its pseudo square root b = b0,..., bN 1 , containing the vectors bi as columns: { − } γi(t) γj(t) (1.1.7) ρ(t) = b(t)⊥ b(t) = , 0 t min T ,T . γ (t) γ (t) ≤ ≤ { i j} i d j d i,j Rebonato [1999a] presented a significant and efficient way to reduce to a very large extent the difficulties in the simultaneous calibration of the volatilities and the corre- lation matrix thanks to straightforward geometrical relationships and matrix theory. The notation so far with the introduction of loading vectors in (1.1.6) shall simplify the understanding and the usage of these results.

Remark 1.2 Once the fixing time Ti is reached the forward rate becomes constant, which means that Fi(t) remains constant for all t Ti. Although obvious, let it be mentioned that the instantaneous volatility function≥ then satisfies σ (t) = γ (t) 0, t T , i 1,...,N. i i d ≡ ∀ ≥ i ∈

1.1.1 Libor Dynamics Under the Forward Measure

Of course, in order to use the LMM in practice the dynamics of all forward Libor rates have to be formulated in a single measure. In this respect convenient choices are either the terminal measure PN which is induced by taking the terminal discount bond B(t,TN ) as numeraire, or the spot measure which is defined by the numeraire given in (1.1.10). As a consequence only one of the forward Libor rates is a martingale and, according to Girsanov’s theorem, all other forward rate processes will have be modified by additional drift terms (cf. Hunt-Kennedy [2004]). In practice, the standard approaches to construct the system of forward Libor rates rely either on the forward induction as in Brace-Gatarek-Musiela [1997] or on the so-called backward induction as in Musiela-Rutkowski [2005]. k+1 Concretely, under the measure P the forward rate process Fj reads (1.1.8) dF (t) = (t,T ,T ) dt + F (t)γ (t) dWk+1, j j k j j t where the drift term (t,Tj,Tk) is determined by requiring lack of arbitrage. For 0 t T the drifts are given by (cf. Brace-Gatarek-Musiela [1997]) ≤ ≤ j (τ = T T ): i i+1 − i k τiFi(t) σi(t)σj (t)ρi,j (t) for j < k − 1+τiFi(t)  i=j+1 (1.1.9) (t,T ,T ) = F (t) 0 for j = k j k j   j  τiFi(t) σi(t)σj (t)ρi,j (t) for j >k. 1+τiFi(t) i=k+1    Chapter 1. Forward Libor and Swap Market Models 8 The spot measure (cf. Jamshidian [1997]) is induced by the rolling bond numeraire5

ξ(t) 1 − (1.1.10) Gt = B(t,Tξ(t)) (1 + τjFj(t)), j=0 where the left-continuous function ξ : [0,T ] 1,...,N gives the next reset date N → { } at time t:

k 1 − (1.1.11) ξ(t) = inf k N T + τ t = inf k N T t . ∈ | 0 i ≥ ∈ | k ≥ i=0 The forward Libor process for F , j =0,...,N 1, is then given by j − j τiFi(t) σi(t)σj(t)ρi,j(t) (1.1.12) dFj(t) = Fj(t) γj(t) dWt∗ + Fj(t) dt, 1+ τiFi(t) i=ξ(t) P with Wt∗ denoting a standard Brownian motion under the spot measure ∗.

We see that in the spot Libor measure Fj in (1.1.12) contains j ξ(t)+1 drift terms, whereas in the terminal measure it contains N j 1 drift terms,− cf. (1.1.9). For numerical reasons it is important to keep the calculation− − costs of the Libor drifts as small as possible. Therefore, for products involving only short maturity Libors the dynamics in the spot Libor measure (1.1.12), involving repeatedly the rolling of the bond with the shortest time to maturity available, is preferable, whereas for longer dated products the representation in the terminal measure may be recommended. In both cases the numeraire process remains alive throughout the time span of the N tenor structure Tn n=0. This is particularly necessary for the evaluation of deriva- tive securities that{ } involve random payoffs at any date in the tenor structure. Our simulations in Sec. 4.6 are carried out by calculating the drifts with respect to the spot measure.

1.1.2 Valuation in LMM

The key advantage of LMM in regard to model calibration as well as to pricing of the financial interest rate products is its compatibility of the forward rates’ modeling to the Black framework, due to the assumed lognormality of the forward rate dynamics and, of course, to the assumed deterministic (time-dependent) volatility. With regard to the scope of this work, we present in Appendix A.1 the valuation formulae of the basic benchmark instruments which will be used to calibrate the LMM and other models we will consider later in the next chapters. The spectrum of interest rate products which can be priced with LMM is huge. In practice, a lot of products are being priced with LMM by taking into consideration

5 Gt represents the wealth at the time t of a portfolio that starts at time 0 with one unit of cash invested in a zero-coupon bond of maturity T0, and whose wealth is then reinvested at each time Tj in zero-coupon bonds maturing at the next date Tj+1, cf. Schoenmakers [2005]. The process Gt is a continuous and completely determined by the Libors at the tenor dates, such that the spot ∗ measure P is then defined such that the relative bond prices B(t,Tj )/Gt, j = 1,...,N are local martingales. Chapter 1. Forward Libor and Swap Market Models 9 approximations. One family of popular approximations we shall discuss in detail in Chap. 1.2. Nevertheless, the main reason to develop a market model of forward rates is however to price exotic interest rate options, whose complex payoff can be expressed in terms of market observable Libor rates. In the most cases this is done by performing joint Monte-Carlo simulations of the forward rates in a calibrated LMM. The joint distributional evolution of the forward rates with respect to e.g. a payoff measure, though, results in solving a system of stochastic differential equations, as in (1.1.8)- (1.1.9), which involves state dependent drift terms. In Appendix A.4 we briefly present a standard method how to approximate the corresponding drifts in the joint evolution of forward rates.

1.1.3 Covariance and Correlations in LMM

In general, if one is interested in terminal correlations of forward rates at a future time instant (when pricing financial instruments with payoffs at future times), as implied by the LMM model, then the computation has to be based on a Monte Carlo simulation technique. Following Brigo-Mercurio [2007], let us assume we are interested in computing the terminal correlation between forward rates Fi and Fj at y time Tk,kk. Then we need to compute the terminal covariance Ey Ey Ey t (Fi(Tk) t [Fi(Tk)])(Fj(Tk) t [Fj(Tk)]) y − − Corr (Fi(Tk); Fj(Tk))(t)= . 2 2 Ey F (T ) Ey[F (T )] Ey F (T ) Ey[F (T )] t i k − t i k t j k − t j k We notice that, while the instantaneous correlations do not depend on the partic- ular probability measure or numeraire under which we are working, the terminal correlations do. y Recalling the dynamics of Fi and Fj under P , the expected values appearing in the above expression can be obtained by simulating the above dynamics of Fi and Fj up to time Tk. Fortunately, there exist approximated formulas that allow us to derive terminal correlations algebraically from the LMM parameters ρi,j(.) and σi(.). By partial freezing of the drift components in the log-normal dynamics of the forward rates with respect to Py, we can easily obtain (cf. Brigo-Mercurio [2007]):

Tk exp t σi(s)σj(s)ρi,j(s) ds 1 Corry(F (T ); F (T ))(t) − . i k j k ≈ Tk Tk exp σ (s)2 ds 1 exp σ (s)2 ds 1 t i − t j − This approach makes the terminal correlations independent of the chosen probabil- ity measure. Notice that a first order expansion of the exponentials appearing in the above formula yields a second formula for the terminal correlations (Rebonato [2004]):

Tk y t σi(s)σj(s)ρi,j(s) ds (1.1.13) CorrRR(Fi(Tk); Fj(Tk))(t)= . Tk 2 Tk 2 t σi(s) ds t σj(s) ds Chapter 1. Forward Libor and Swap Market Models 10 An immediate application of Schwartz’s inequality shows that terminal correlations, when computed via Rebonato’s formula, are always smaller, in absolute value, than instantaneous correlations. In agreement with this general observation, recall that through a careful repartition of integrated volatilities (caplets) in instantaneous volatil- y ities σi(t) and σj(t) we can make the terminal correlation CorrRR arbitrarily close to zero, even when the instantaneous correlation ρi,j is one.

1.1.4 Swap Rate Models and Measures

m,n n A probability measure P , induced by the annuity Bm,n(t) = τi 1B(t,Ti) ∗ i=m+1 − and equivalent to the measure PT , is said to be the forward swap probability mea- sure associated with the dates Tm and Tn, or simply the forward swap measure, if for every i = 0,...,N the relative bond price B(t,Ti) , for all t [0,T T ], fol- Bm,n(t) i m+1 m,n ∈ ∧ lows a local martingale process under P . Thus, the forward swap rate Sm,n(t) = B(t,Tm) B(t,Tn) m,n − , t [0,Tm], is a P -martingale (cf. Appendix A.2). Bm,n(t) ∈ Definition 1.3 If the (vector-valued) volatility process t γ (t) is a deterministic → m,n function we speak of a (lognormal) Swap Market Model (SMM) for Sm,n, assuming that forward swap rates follow a lognormal diffusion process of type

(1.1.14) dS (t) = S (t)γ (t) Wm,n, 0 t T , m,n m,n m,n t ≤ ≤ m where Wm,n denotes the corresponding d˜-dimensional Brownian motion under Pm,n.

As the correlations between the forward swap rates will not be focused on in this section, (1.1.14) can be alternatively expressed in an one-dimensional form6 as

(1.1.15) dS (t) = S (t)σ (t)dW m,n, 0 t T , m,n m,n m,n t ≤ ≤ m

m,n γm,n(t) m,n where σm,n(t)= γm,n(t) d˜, and Wt = γ (t) Wt being an one-dimensional m,n d˜ Brownian motion under Pm,n.

As an important consequence, European options on swap contracts over [Tm,Tn], called swaptions, can be priced exactly with the Black-Scholes formula ( see Appendix A.3). Moreover, as we will show, there exist very accurate swaption approximation formulas for swaptions in the LMM. While in the Libor model of forward rates there is only one degree of freedom for choosing the numeraire, see (1.1.9)–(1.1.12), for swap market models in general there are N degrees of freedom for a N +1 time grid. For instance, for a complete system of standard swaps it is possible to choose σ0,N ,...,σN 1,N simultaneously deterministic − (cf. discussions in Schoenmakers [2005]). In Appendix A.2 we shall briefly present some of swap rate models mostly used in practice: in particular, the co-terminal and the co-initial swap rate models. We refer to Galluccio et al. [2006] for their extensive studies on these and further swap rate models and their adequateness in practice.

6 We will consider the swap rate process Sm,n(t), if not otherwise explicitly specified, always in its ”natural” measure Pm,n. Chapter 1. Forward Libor and Swap Market Models 11 1.1.5 Incompatibility Between the LMM and the SMM

The cap and swaption markets are underpinned by the same state variables, either forward rates or, equivalently, swap rates which can be transformed to each other by simple bootstrapping methods. As a corollary, the instantaneous volatilities of for- ward rates and swaptions cannot be assigned independently. Once the instantaneous volatilities of, and correlations among, forward rates are given, then the correlations among and volatilities of swap rates are completely specified (cf. Rebonato [1999b]). It is market practice to price both sets of instruments (caps and swaptions) using the Black [1976] formula which is inconsistent as lognormal forward and lognormal swap rate models are incompatible; if simple forward rates are lognormal, swap rates can only be approximately so, and vice versa (cf. Brace [1997]). The Black model ceases to be arbitrage–free when it is assumed that, at the same time, forward rates and swap rates are all lognormal. Further discussions about the effects of this incompatibility in practice can be found in Rebonato [1999b], Brigo-Liinev [2005], Brigo-Mercurio [2007]. The next two paragraphs show how the two models interact with each other, and how far they are compatible.

Swap rate dynamics under the forward measure. Following Brigo-Mercurio [2007], the dynamics of the forward swap rate Sm,n in the SMM model (cf. (1.1.14)), under the Libor forward measure numeraire B(t,Tm) is given (after lengthy calcula- tions) by

(1.1.16) dS (t) = m (t)S (t)dt + S (t)γ (t) Wm, 0 t T . m,n m,n m,n m,n m,n t ≤ ≤ m The drift is defined by (1.1.17) n 1 − m,n νi,j (t)τiτjB(t,Tm,Ti+1)B(t,Tm,Tj+1)ρi,j(t)σi(t)σj(t)Fi(t)Fj(t) i,j=m m (t)= , m,n 1 B(t,T ,T ) − m n where B(t,T ,T )= B(t,Tk) denotes the price of the zero-coupon bond at time t for m k B(t,Tm) maturity Tk, as seen from expiry Tm, hence the forward price of the zero bond from Tm to Tk as seen at time t. m,n The weights νi,j are defined as

i n m,n B(t,Tm,Tn) τk 1B(t,Tm,Tk)+ τk 1B(t,Tm,Tk) ν (t) = k=m+1 − k=i+1 − i,j n 2 τk 1B(t,Tm,Tk) k=m+1 − n τk 1B(t,Tm,Tk). − k=j+1

Forward rate dynamics under the annuity measure. Symmetrically, it is pos- sible to work out the dynamics of the forward Libor rates under the SMM numeraire Bm,n. Applying the change-of-numeraire technique we have the following dynamics Chapter 1. Forward Libor and Swap Market Models 12

m,n for the forward rate Fi under P (cf. (1.1.15)):

(1.1.18) dF (t) = σ (t)F (t)(m,n(t)dt + F (t)σ (t)W m,n) , 0 t T . i i i i i i t ≤ ≤ i The drift is given by

n max i,k 1 { }− m,n B(t,Tk) τjFj(t)σj(t)ρi,j(t) (1.1.19) i (t)= (2χ k i 1 τk 1 .  { ≤ } − − Bm,n(t) 1+ τjFj(t)  k=m+1 j=min i,k { }   The details of these derivations can be found in Brigo-Mercurio [2007].

Approximating the Swap Rate Volatility in LMM

The dynamics of swap rates in a Libor market model, as seen in (1.1.16), is rather complicated due to the stochastic factors involved in the drifts. Therefore, closed form pricing of swaptions in the LMM is in general not possible, nonetheless it is possible to give surprisingly accurate swaption approximation formulas in LMM. To this end we write the swap rate again as a combination of forward rates and discount zero bonds (cf. (A.1.2))

n n 1 τi 1B(t,Ti)Fi 1(t) − m,n (1.1.20) S (t) = i=m+1 − − = w (t)F (t), t [0,T ], m,n B (t) i i ∈ m m,n i=m with the stochastic weights wm,n(t)= τiB(t,Ti+1) . The popular freezing of these weights, i Bm,n(t) which certainly simplifies the swap drifts in LMM, will also help us in approximating the swap rate variance in LMM. Following Schoenmakers [2005], the swap rate variance σ2 (t)= γ (t) 2 may be m,n m,n d˜ expressed in terms of the forward Libor volatilities by

n 1 n 1 1 − − (1.1.21) σ (t)2 = vm,n(t)vm,n(t)F (t)F (t)γ (t) γ (t) , m,n S2 (t) i j i j i j m,n i=m j=m m,n m,n with some weights vi (t) whose distance to the swap weights wi (t) is given via

0 0 m,n m,n Bi,n(t) Sm,n(t) Si,n(t) m,n (1.1.22) vi (t) wi (t) = τi − =:y ˆi (t), − Bm,n(t) 1+ τiFi(t)

0 where m i

0 B(t,Ti) B(t,Tn) Si,n(t) = n 1 − m,n . Bm,n(t) k=min− l m+l i wk (t) { | ≥ } m,n The termsy ˆi (t) have magnitudes comparable with differences of swap rates, hence, 0 0 they are usually rather small. They are zero when Si,n(t) = Sm,n(t) for myield curve is flat (see also Rebonato-Jaeckel [2003]). Integrating (1.1.21) over time to expiry we obtain (1.1.23) Tm n 1 Tm m,n m,n 1 − v (s)v (s)Fi(s)Fj(s) σ (s)2ds = i j γ (s) γ (s) ds. T t m,n S2 (s) i j m t i,j=m t m,n − We now note that the (stochastic) fractions in the r.h.s. of (1.1.23) add up to ap- proximately one and thus may be regarded as weights, tend to vary relatively slow in practice and therefore may be approximated by their values at t. Under this addi- tional assumption instantaneous swap volatilities may be considered as deterministic (though model inconsistent). This technique of ”freezing” all weights and forward rates in (1.1.23) to their initial value leads to Rebonato’s formula (Rebonato [2002]): (1.1.24) Tm n 1 m,n m,n Tm 1 − v (t)v (t)Fi(t)Fj(t) σ (s)2ds = i j γ (s) γ (s) ds, T t m,n S2 (t) i j m t i,j=m m,n t − with vm,n(t)= τiB(t,Ti+1) +ˆym,n(t). This formula can be used to calibrate of the model i Bm,n(t) i parameters to implied swaption volatilities according to (A.3.3).

1.2 The Convexity Adjustment and CMS Deriva- tives

In finance convexity is a broadly understood and non-specific term for nonlinear be- havior of the price of an instrument as a function of evolving markets. Such convex behaviors manifest themselves as convexity corrections/adjustments to various pop- ular interest rate derivatives. From the perspective of financial modeling they arise as the results of valuation done under the wrong martingale measure. Practitioners use various ad hoc rules to calculate convexity corrections for different products, often based on Taylor approximations (cf. Hunt-Pelsser [1998], Benhamou [2000], etc). However, Pelsser [2003] is the first to put convexity correction on a firm mathematical basis by showing that it can be interpreted as the side-effect of a change of numeraire. It can be understood as the expected value of an interest rate under a different probability measure than its own martingale measure. The well known Change of Numeraire Theorem, due to Geman et al. [1995], shows how in an arbitrage-free economy an expectation under a probability measure PN , generated by the numeraire N, can be represented as an expectation under a probabil- ity measure PM , generated by the numeraire M, times the Radon-Nikodym derivative dPN /dPM . For an expectation at time 0 of a random variable H at time T we have

N(T )M(0) (1.2.1) EN H(T ) = EM H(T ) . N(0)M(T ) Following Pelsser [2003], suppose we are given a forward interest rate F (t,T,T ∗) with maturity T

S follows. However, under the measure P the process F (t,T,T ∗) is in general not a martingale such that the expectation (1.2.2) can be expressed as F (0,T,T ∗) times a correction term. This correction term is known in the market as the convexity correction or convexity adjustment. Applying the change of numeraire technique ∗ (1.2.1) we can express (1.2.2) in terms of ET as follows

PS S T ∗ d T ∗ B(T,S)B(0,T ∗) E F (T,T,T ∗) = E F (T,T,T ∗) ∗ = E F (T,T,T ∗) 0 0 PT 0 d B(0,S)B(T,T ∗) ET ∗ = 0 F (T,T,T ∗)R(T ) , where R denotes the Radon-Nikodym derivative which is also a martingale under T ∗ the measure P . If we know the joint probability distribution of F (T,T,T ∗) and R(T ) the expectation can be calculated explicitly and we obtain an expression for the convexity correction. Only for very special cases exact expressions for the convexity correction can be obtained. In these special cases the Radon-Nikodym derivative of the change of mea- sure is equal to (a simple function of) the interest rate that determines the payoff. A prominent example where an exact expression for the convexity correction is possible is a Libor in Arrears contract, in which the payment is in arrears, i.e. at fixing time. ∗ Thus, we have for the Radon-Nikodym derivative dPT /dPT we have

T dP B(T,T )B(0,T ∗) 1+ τF (T,T,T ∗) ∗ ∗ (1.2.3) T = = , τ = T T, dP B(0,T )B(T,T ∗) 1+ τF (0,T,T ∗) − and hence,

T 2 R0 σT (s) ds ET 1+ τF (0,T,T ∗)e (1.2.4) 0 F (T,T,T ∗) = F (0,T,T ∗) . 1+ τF (0,T,T ∗) 1.2.1 Constant Maturity Swaps and Related Derivatives

The acronym CMS stands for constant maturity swap, and it refers to a swap rate with a pre-defined length which fixes in the future. CMS rates provide a convenient alternative to Libor as a floating index, as they allow market participants to express their views on the future levels of long term rates (for example, the 10 year swap rate). There are a variety of CMS based instruments, the simplest of them being CMS swaps and CMS caps / floors. A particularly known type of exotic European interest rate contract is a (fixed for floating) CMS swap. This is a swap where at every payment date a payment calcu- lated from a swap rate is exchanged for a fixed rate. The floating leg pays periodically a swap rate of fixed length (say, the 10 year swap rate) which fixes at the beginning of the accrual period. Chapter 1. Forward Libor and Swap Market Models 15 A CMS cap or floor is a basket of calls or puts on a swap rate of fixed tenor (say, 10 years) structured in analogy to a Libor cap or floor, cf. Sec. A.1. For example, a 5 year cap on 10 year CMS struck at K is a basket of CMS caplets over 5 years, each of which pays max(10 year CMS rate K;0), where the CMS rate fixes at the start of each accrual period. − Needless to say, a plethora of more sophisticated contracts are traded in the mar- kets, which may differ from the standard ones by differences in fixing and payment frequencies, whether the floating leg fixes in arrears or in advance, whether the term and payment frequency of the swap rate may be different from the specifications of the CMS swap itself and further market particularities. Moreover, the contracts can consist of even more complicated formulas involving algebraic expressions of CMS rates of different lengths.

CMS Swaps

As mentioned above the floating payments of a CMS swap are not based on the Libor forward rates but on some swap rate. Formally, at the settlement dates Ti+1, i 0,...,N m 1 7, the fixed payment K is exchanged for the variable payment S∈ { (T ) for− a preassigned− } length m 1. Let us consider one CMS swaplet only, i,i+m i ≥ paying at Ti+1 and based on a notional of 1. The discounted payment on the fixed leg as of t is obviously given by B(t,Ti+1)τiK, while for the floating leg

Ei+1 (1.2.5) CMS(t,Ti,m) = B(t,Ti+1)τi t Si,i+m(Ti) Ei+1 holds. t denotes the expectation at time t with respect to the forward measure Pi+1. For a fixed natural number n N m and k 0,...,N m 1 , we denote with ≤ − ∈ { − − } CMS(t,Tk,m,n), a n–period (forward starting) CMS swap rate which is defined by

n 1 − CMS(t,T ,m) (1.2.6) K = CMS(t,T ,m,n) = i=k i . k B (t) k,n

CMS Caps/Floors

The CMS caplets and CMS floorlets are built up analogously to their classical pen- dants, i.e. the interest rate caplets and floorlets. We can write

+ (1.2.7) CMSCPL(t,T ,m,κ) = B(t,T )τ Ek+1 S (T ) κ , k k+1 k t k,k+m k − + (1.2.8) CMSFLL(t,T ,m,κ) = B(t,T )τ Ek+1 κ S (T ) , k k+1 k t − k,k+m k where κ is obviously the optionlet strike. Not surprisingly, this implies a put-call parity relation for the CMS rate:

(1.2.9) CMSCPL(t,T ,m,κ) CMSFLL(t,T ,m,κ) = CMS(t,T ,m) κ. k − k k − 7 We are assuming that the total time horizon in our economy is up to TN . Chapter 1. Forward Libor and Swap Market Models 16 Analogously to the classical ones (Sec. A.1), for CMS caps (for CMS floors analo- gously) we have

n 1 − (1.2.10) CMSCAP(t,Tk,m,n,κ) = CMSCPL(t,Ti,m,κ). i=k CMS Spread Options

A holder of a CMS spread option(let) has the right to exchange for one period of time the difference between two CMS rates minus a spread κ. Hence, the payoff at expiry time Tk equals (1.2.11) + CMSSPO(T ,T ,n ,n ,κ) := τ a ωS (T )+ a ωS (T ) ωκ , k k 1 2 k 1 k,k+n1 k 2 k,k+n2 k − n1 = n2, k + ni < N, a1,a2 R, ω 1, 1 . A CMS spread option(let) can be seen as a special case of a CMS∈ basket∈ option(let) {− } . A generic CMS basket option(let), written on M CMS rates that reset at the option’s expiry date Tk, has the payoff

M + (1.2.12) CMSSPOB(T ,T , m M ,κ) := τ ωa S (T ) ωκ , k k { i}i=1 k i k,k+mi k − i=1 where ai R denote weights and mi, i = 1,...,M, (k + mi < N) are preassigned lengths of∈ reference swaps. A natural step further, far beyond the scope of this work, though, is to consider caps/floors of CMS spreads or even CMS basket options with periodic expiries/fixings whose payoff at every fixing/expiry time Tk reads as in (1.2.12).

1.2.2 Valuation of CMS Derivatives

We now come to the point where we can examine the pricing of CMS products we introduced previously. The most common characteristic of these products with respect to pricing is that their payoffs are functions of one or more CMS rates f(Si,i+m1 ,Si,i+m2 ,...), which are usually fixed at Ti and paid at Ti+1. As we know i,i+m the swap rate Si,i+m(t) is a martingale with respect to the measure P , induced i+1 by the annuity Bi,i+m(t). The forward measure P , associated with the payment i+1 date Ti+1, is not its natural measure, i.e. Si,i+m(t) is not a martingale w.r.t. P . Turning back to the CMS swaplet (1.2.5), by using the change of numeraire technique (1.2.1), we can write for i =0,...,N m 1: − − B (t) B(T ,T ) (1.2.13) Ei+1 S (T ) = i,i+m Ei,i+m S (T ) i i+1 . t i,i+m i B(t,T ) t i,i+m i B (T ) i+1 i,i+m i They are basically two ways how to deal with the expectation: either find a lognormal approximation for the CMS rate in the forward measure, by approximating the Libor drifts, or a convexity correction approach shall be applied by expressing the Radon- Nikodym derivative as a (simple) function of the interest rate that determines the payoff, as, for instance, in the case of the Libor in Arrears in the LMM (cf. (1.2.3)). Chapter 1. Forward Libor and Swap Market Models 17 In this section we want to discuss some of the methods used in the practice to ap- proximate (1.2.13). The first method exploits the idea of making the Radon-Nikodym derivative a function of the payout rate.

Approximating the Radon-Nikodym Derivative and the Convexity Cor- rection

For evaluating (1.2.13) we here recall the convexity approach in Pelsser [2003], based on the assumption of a lognormal SMM, cf. Hunt-Kennedy [2004]:

B(Ti,Ti+1) (1.2.14) a + bi+1Si,i+m(Ti), Bi,i+m(Ti) ≈ where a and bi+1 are constants which are determined as follows. As the Radon- Nikodym derivative is a martingale w.r.t. to the annuity measure, by taking the expectation we obtain B(t,T ) B(T ,T ) i+1 = Ei,i+m i i+1 = a + b S (t). B (t) t B (T ) i+1 i,i+m i,i+m i,i+m i Hence, 1 B(t,T ) (1.2.15) b = i+1 a . i+1 S (t) B (t) − i,i+m i,i+m On the other hand we have by summing up

i+m 1 i+m 1 − τkB(t,Tk+1) − 1 = = τk (a + bk+1Si,i+m(t)) . Bi,i+m(t) k=i k=i

Replacing bi+1 by (1.2.15),

Bi,i+m(t) B (t) i+m−1 i+1 τ 1 − Pk=i k (1.2.16) a = i+m 1 , bi+1 = − τ Bi(t) Bi+m(t) k=i k − hold. Finally, we can rewrite (1.2.13) as

b Vari,i+m[S (T )] (1.2.17) Ei+1 S (T ) = S (t) 1+ i+1 i,i+m i . t i,i+m i i,i+m S (t)(a + b S (t)) i,i+m i+1 i,i+m The linear approximation in (1.2.14) does seem very crude at first, but can be justified by the following argument (cf. Pelsser [2003]). Convexity corrections only become sizable for large maturities. However, for large maturities the term structure almost moves in parallel. Hence, a change in the level of the long end of the curve is well described by the swap rate. Furthermore, for parallel moves in the curve, the ratio B(Ti,Ti+1)/Bi,i+m(Ti) is closely approximated by a linear function of the swap rate, which is exactly what the approach does. This leads to a good approximation of the convexity correction for long maturities. With these formulas we can easily price linear CMS products like in (1.2.5) – (1.2.6). Chapter 1. Forward Libor and Swap Market Models 18 In his seminal work Hagan [2003] discusses general approximations of functional form to the Radon-Nikodym derivative given through (1.2.13). He writes for the CMS caplets:

+ B(T ,T )/B (T ) CMSCPL(t,T ,m,κ)= B(t,T )τ Ei,i+m S (T ) κ i i+1 i,i+m i i i+1 i t i,i+m i − B(t,T )/B (t) i+1 i,i+m + = B(t,T )τ Ei,i+m S (T ) κ i+1 i t i,i+m i − + B(T ,T )/B (T ) + B(t,T )τ Ei,i+m S (T ) κ i i+1 i,i+m i 1 . i+1 i t i,i+m i − B(t,T )/B (t) − i+1 i,i+m The first term is exactly the price of a European swaption (cf. (A.3.2)) with notional B(t,Ti+1)/Bi,i+m(t), regardless of how the swap rate is modeled. The last term is the convexity correction. Following the argumentation in Hagan [2003], since Si,i+m is a martingale in the annuity measure and B(Ti,Ti+1)/Bi,i+m(Ti) 1 is zero on average, this B(t,Ti+1)/Bi,i+m(t) − term goes to zero linearly with the variance of the swap rate, and is much smaller than the first term. Giving the ratio a general form

(1.2.18) B(Ti,Ti+1)/Bi,i+m(Ti) = G(Si,i+m(Ti)), for some function G, we then have

+ CMSCPL(t,T ,m,κ)= B(t,T )τ Ei,i+m S (T ) κ i i+1 i t i,i+m i − + G(S (T )) + B(t,T )τ Ei,i+m S (T ) κ i,i+m i 1 . i+1 i t i,i+m i − G(S (t)) − i,i+m Using the general property for smooth functions f with f(κ) = 0 (integration by parts):

+ ∞ + f(S) for S>κ (1.2.19) f ′(κ) S κ + S x f ′′(x) dx = − − 0 for S <κ, κ and choosing

G(x) (1.2.20) f(x) = x κ 1 , − G(S (t)) − i,i+m we obtain by simple transformations

B(t,Ti+1) CMSCPL(t,T ,m,κ) = τ 1+ f ′(κ) PSWO(t,T ,T ,κ) i i B (t) i i+m i,i+m ∞ (1.2.21) + PSWO(t,Ti,Ti+m,x)f ′′(x) dx . κ This formula replicates the value of the CMS caplet in terms of European swaptions at different strikes It takes into account the presence of a market smile, incorporating consistently the information coming from the quoted swaption Black-volatilities. We refer to Mercurio-Pallavicini [2006] for discussions about the approximation of the Chapter 1. Forward Libor and Swap Market Models 19 integral above on a practically plausible, in general not negligible, strike interval [0,K], with K ”large enough”. We will come back to this approximation of the convexity adjustment when considering the SABR model. The formula for CMS floorlets (1.2.8) is a slight adaption of (1.2.21), replacing PSWO with RSWO (cf. (A.3.4)):

B(t,Ti+1) CMSFLL(t,T ,m,κ) = τ 1+ f ′(κ) RSWO(t,T ,T ,κ) i i B (t) i i+m i,i+m κ (1.2.22) RSWO(t,T ,T ,x)f ′′(x) dx . − i i+m −∞ The value of the CMS swaplet is easily derived from the CMS put-call parity (1.2.9). The method of replicating the CMS caplets/floorlets by means of swaptions is opaque and computationally intensive. Hagan [2003] gives simpler approximate formulas for the convexity correction, as an alternative to the replication method. Expanding at first order the function G around Si,i+m(t) makes f quadratic

G′(Si,i+m(t)) (1.2.23) f(x) (x Si,i+m(t))(x κ), ≈ G(Si,i+m(t)) − − and f ′′(x) constant. Together with the equality

∞ ∞ + PSWO(t,T ,T ,x) dx = B (t)Ei,i+m S (T ) x dx i i+m i,i+m t i,i+m i − κ κ 1 2 = B (t)Ei,i+m S (T ) κ)+ , 2 i,i+m t i,i+m i − 2 we have, by considering (S (T ) κ)(S (T ) κ)+ = S (T ) κ)+ , i,i+m i − i,i+m i − i,i+m i − B(t,Ti+1) CMSCPL(t,Ti,m,κ) = τi PSWO(t,Ti,Ti+m,κ) Bi,i+m(t) i,i+m + +τ G′(S (t))B (t)E (S (T ) S (t))(S (T ) κ) , i i,i+m i,i+m t i,i+m i − i,i+m i,i+m i − B(t,Ti+1) CMSFLL(t,Ti,m,κ) = τi RSWO(t,Ti,Ti+m,κ) Bi,i+m(t) i,i+m + τ G′(S (t))B (t)E (S (t) S (T ))(κ S (T )) . − i i,i+m i,i+m t i,i+m − i,i+m i − i,i+m i For a CMS swaplet,

(1.2.24) CMS(t,Ti,m) = τiB(t,Ti+1)Si,i+m(t) i,i+m 2 +τ G′(S (t))B (t)E (S (T ) S (t)) i i,i+m i,i+m t i,i+m i − i,i+m holds. The SMM (cf. Hunt-Kennedy [2004]) gives

i,i+m Ti 2 (1.2.25) E (S (T ) S (t))2 = S (t)2 eRt σi,i+m(s)ds 1 . t i,i+m i − i,i+m i,i+m − Chapter 1. Forward Libor and Swap Market Models 20

Given that G(Si,i+m(t)) approximates the ratio B(t,Ti+1)/Bi,i+m(t) (cf. (1.2.18)) linear as in (1.2.14)), the equation (1.2.24) perfectly matches with (1.2.17). Hagan [2003] suggests to use for CMS swaps the volatility of at-the-money swaptions, since the expected value includes high and low strike swaptions equally. For out-of- the-money CMS caplets and floorlets, the strike-specific volatility should be used, while for in-the-money options, the largest contributions come from swap rates near the mean value. Accordingly, call-put-parity should be used to evaluate in-the-money caplets and floorlets as a CMS swap payment plus an out-of-the-money CMS floorlet or caplet. The function G has been considered a general smooth and slowly varying function, regardless of the model used to obtain it. Hagan [2003] develops simpler approximate formulas for the convexity correction, by specifying G. We shall present here the market standard method for computing convexity corrections which uses bond math approximations and goes as follows. Let the yield curve be flat, fixed at a level y. Then, given an equidistant time grid with step size ∆T and discrete discounting, we can write i+m i+m B(t,Ti+1) Bi,i+m(t) = τk 1B(t,Tk) = ∆T k i 1 . − (1+∆Ty) − − k=i+1 k=i+1 The standard formula for the geometric sum gives then

B(t,Ti+1) 1 Bi,i+m(t) = (1+∆TSi,i+m(t)) m 1 , Si,i+m(t) − (1+∆TSi,i+m(t)) − where the par swap rate y = Si,i+m(t) was taken as discount rate, since it represents the average rate over the life of the reference swap. Thus,

std Si,i+m(t) (1.2.26) G (Si,i+m(t)) = 1 . (1+∆TSi,i+m(t)) m−1 − (1+∆T Si,i+m(t)) A more accurate lognormal approximation of the swap rates and their correlations in the forward measure was introduced by Belomestny-Kolodko-Schoenmakers [2010], based on the method of freezing the weights (as in Section 1.1.5) but assuming a more sophisticated approximation. The approximation (1.2.17) is model independent and quite accurate especially for flat yield curves and highly correlated rates. Such constraints are not necessarily unrealistic, since adjustments are mostly relevant for long maturities (and tenors), where (forward) rates tend to be constant and to move in parallel fashion. Assuming lognormal-type dynamics for the swap rates as in SMM we obtain the classical Black-like adjustment with at-the-money implied volatilities given through (1.2.24)–(1.2.26).

Pricing CMS Spread Options

Once more than one CMS rate is part of a payoff, as in case of CMS spread options, the correlation between the CMS rates in the forward measure starts playing an important role in pricing these products. Chapter 1. Forward Libor and Swap Market Models 21 Let us first focus on CMS spread option(let)s with zero strike whose payoff at expiry time Ti equals (cf. (1.2.11))

+ (1.2.27) CMSSPO(T ,T ,n ,n , 0) := τ S (T ) S (T ) . i i 1 2 i i,i+n1 i − i,i+n2 i The arbitrage-free value of the payoff (1.2.27) at time t is given by

+ (1.2.28) CMSSPO(t,T ,n ,n , 0) := τ B(t,T )Ei S (T ) S (T ) , i 1 2 i i t i,i+n1 i − i,i+n2 i which can be calculated as soon as we know the joint distribution of the pair of swap Pi rates Si,i+n1 and Si,i+n2 under the forward measure . Apart from the fact that the expectation is taken in the non-natural forward mea- sure, this payoff is the one of an exchange option. Therefore, the simplest valuation procedure is based on assuming that the logarithms of the swap rates are jointly nor- mally distributed as in the Black-Scholes model of two underlying assets. A formal justification of this approach is given by resorting to the SMM (cf. Def. 1.3) and suitable approximations. Thus, let assume that both swap rates evolve according to

i (1.2.29) dSi,i+n1 = i,i+n1 (t)Si,i+n1 dt + σi,i+n1 Si,i+n1 dWt ˜ i (1.2.30) dSi,i+n2 = i,i+n2 (t)Si,i+n2 dt + σi,i+n2 Si,i+n2 dWt ,

i ˜ i Pi i ˜ i where Wt and Wt are Brownian motions under , correlated via d(Wt , Wt )=ˆρ(t)dt, withρ ˆ(t) assumed to be given (estimated historically or approximated, for instance, as in Belomestny-Kolodko-Schoenmakers [2010]. The drifts i,i+n1 and i,i+n2 of the corresponding swap rates with respect to Pi are motivated by (1.1.17) and assumed to be frozen or deterministic. The formula for pricing exchange options, developed by Margrabe [1978] using the change of numeraire technique, can now be applied to obtain: (1.2.31) + Ei S (T ) S (T ) = BS Ei[S (T )], Ei[S (T )], σ¯ T t, 1 . t i,i+n1 i − i,i+n2 i t i,i+n1 i t i,i+n2 i i − With the swap rate dynamics given in (1.2.29)–(1.2.30) we obtain explicitly:

Ti Ei Rt µi,i+nk (s)ds t[Si,i+nk (Ti)] = Si,i+nk (t)e , k =1, 2, and

1 Ti (1.2.32)σ ¯2 = σ2 (s)+ σ2 (s) 2ˆρ(s) σ (s)σ (s) ds. T t i,i+n1 i,i+n2 − i,i+n1 i,i+n2 i − t There are several ways how to approximate drifts deterministically under Pi, for instance:

they can be inferred from the convexity adjustments, i.e. from the approxima- • Ei tions to t Si,i+nk (Ti) , k = 1, 2 (discussed in the previous section). We note that the convexity adjustment technique does not give the correlation between the swap rates w.r.t. Pi; Chapter 1. Forward Libor and Swap Market Models 22 the classical method of ”freezing the coefficients” can be applied to (1.1.17), • Pi Ei to make the -dynamics of the swap rates lognormal, i.e. t Si,i+nk (Ti) = i µ (t)(Ti t) i S (t)e i,i+nk − , with (t) given in (1.1.17); i,i+nk i,i+nk complex lognormal approximations, as in Belomestny-Kolodko-Schoenmakers • [2010] for instance, can be applied to the swap rates under Pi.

i Assuming the dynamics (1.2.29)–(1.2.30) with drifts i,i+nk (t), k =1, 2, and the cor- relationρ ˆ(t) between the two swap rates frozen at evaluation time t, Brigo-Mercurio [2007] gives a formula for the more general case of a time Ti payoff (1.2.11) with a strike κ = 0: + + i ∞ 1 1 v2 (1.2.33) E aωSi,i+n (Ti)+ bωSi,i+n (Ti) ωκ = e− 2 f(v) dv, t 1 2 − √2π −∞ where 1 f(v) = aωS (t)exp i (t) ρˆ(t)2σ2 (t) τ +ρ ˆ(t)σ (t)√τv i,i+n1 i,i+n1 − 2 i,i+n1 i,i+n1

aSi,i+n1 (t) 1 2 2 ln + i,i+n1 (t)+( ρˆ(t) )σi,i+n (t) τ +ρ ˆ(t)σi,i+n1 (t)√τv Φ ω h(v) 2 − 1  2  × σi,i+n1 (t)√τ 1 ρˆ(t) −  aSi,i+n1 (t) 1 2  ln + i,i+n1 (t) σi,i+n (t) τ +ρ ˆ(t)σi,i+n1 (t)√τv ωh(v)Φ ω h(v) − 2 1 ,  2  − σi,i+n1 (t)√τ 1 ρˆ(t) −   1 2 (µi,i+n (t) σ (t))τ+σi,i+n (t)√τv h(v) = κ bS (t)e 2 − 2 i,i+n2 2 , τ = T t. − i,i+n2 i − A straight-forward calculation shows that for κ = 0 the equation (1.2.31) is recovered. By no log-normality of the swap rates, an analytical solution for the case κ = 0 is only feasible if the spread is modeled as a normal distributed random variable: (1.2.34) S (t) S (t) = S¯(t) with dS¯(t) =σd ¯ W¯ (t). i,i+n1 − i,i+n2 This framework is too simple to consistently price CMS spread options since implicitly a perfect correlation is assumed. And it is also not taking into account the smile and the skew effects. The market quotes spread options by their implied normal volatilities, similar to swaptions which are quoted by their implied Black volatility.

1.3 Parameterization and Calibration

The general form of the forward Libor model (cf. Def. 1.1) is merely a framework which becomes a model once the forward volatility structure γi(t),i 0,...,N 1 , is specified, which determines both the level of the forward rates and∈ {the correlation− } between the forward rates via γ (t) γ (t) ρ (t) = i j , 0 t min T ,T . i,j γ (t) γ (t) ≤ ≤ { i j} i j Chapter 1. Forward Libor and Swap Market Models 23 The selected covariance structure should match the observable dynamics of the Libor rates, such as the number and the shape of the underlying principal components, cf. Rebonato [2002]. Once the forward volatility structure is specified, the chosen model is calibrated to the current forward rate curve and to liquid market instruments.

Since the current (at t0) forward rates Fi(t0) are initial conditions, and hence inputs for the forward LMM, the calibration to the current forward rate curve is automatic. Calibration to cap and European swaption prices is achieved by choosing the for- ward volatility structure such that the model prices of these derivatives match their market prices as closely as possible. As shown in (A.1.3)–(A.1.7), the lognormal as- sumption in the forward Libor model allows for the pricing of caplets by the ”market convention” Black-Scholes formula, and, as we will see, it enables the derivation of good closed-form approximations of European swaption prices, which then leads to efficient calibration of the model correlations to swaption market prices. Nevertheless, with regard to the valuation of correlation-sensitive products such as CMS spread options, the calibration of instantaneous Libor correlation has always been a challenging point of the LMM which has not been satisfactorily fulfilled by the classical way of approximating the swap rate implied volatilities. At the end of this section we shall present two approaches to calibrate the LMM correlations: by approximating the swap rate volatilities implied from the swaption quotes, • Sec. 1.3.4; by approximating the swap rate correlations implied from the prices of CMS • spread options, Sec. 1.3.5.

1.3.1 Parametric Forms of the Instantaneous Volatilities

Driven by empirical observations many authors and practitioners put special emphasis to the desideratum that the term structure of instantaneous volatilities should evolve in a time–homogeneous manner, assuming ”by default” that it is desirable for a instantaneous volatility function to be able to reproduce (at least approximately) the current term structure of volatilities in the future (cf. Rebonato [2002]). As a result the instantaneous volatility function should be modeled not as a function of calendar time, but rather as a function of left time to maturity σi(t)= g(Ti t). It is important to point out that the result does not depend on the details of the− functional form of the instantaneous volatility function; the future smile surface will exactly ”look like” today’s smile surface. Apart from the time-homogeneity, Rebonato [1998, 1999a] states that the volatility function should have a flexible functional form to be able to reproduce either a humped or a monotonically decreasing instantaneous volatility, and allow for an easy analytical integration of its square (facilitating the evaluation of the necessary variance and covariance elements). Rebonato suggests in his works the following parametric form:

(1.3.1) g(T t) = [a + b(T t)]exp c(T t) + d, c,d> 0, a + d> 0, i − i − {− i − } which fulfills these criteria to an acceptable degree (see Rebonato [2002] for examples and further explanations). Chapter 1. Forward Libor and Swap Market Models 24 (1.3.1) can be extended to a richer parametric form; the extended linear-exponential volatility model (cf. Rebonato [2002], Brigo-Capitani-Mercurio [2003]):

(1.3.2) g (T t)= k g(T t), k(T ) > 0. ext i − i i − i N Rebonato [2002] models the vector k R as ki =1+ ǫ(Ti), being ideally close to one and flexible enough to allow for a∈ better fit of volatility function to the market implied volatilities of different maturities. Assuming no smile and skews in the caplet markets, any choice of the parameters a,b,c,d will only approximately satisfy the ATM caplet condition, { }

Ti (1.3.3) (σBlack)2 (T t) = g(u)2 du, i i − t 8 across all forward rates . The parameters ki then allow for the Libor rate specific adjustment to exactly fit the market implied volatility:

Black 2 2 (σi ) (Ti t) (1.3.4) ki = − . Ti 2 t g(u) du The caplet condition (1.3.3) is then fulfilled by construction everywhere along the curve. A good and extensive overview of the volatility parameterizations used in practice can be found in Brigo-Mercurio [2007].

1.3.2 Calibration to the Cap/Floor Market

The market convention to quote caps and floors is to use the (implied) Black- volatility which plugged into the Black formula gives the market price of the cap/floor. In a smile-less world, we know from the Black-Scholes theory that for the instantaneous volatility function σi(t) of a forward rate Fi(t) in the lognormal LMM the implied Black Black volatility σi is given by (cf. (A.1.4)),

Ti (1.3.5) (σBlack)2 (T t) = σ (u)2 du. i i − i t Ideally, in case of given market prices of ATM caplets, their implied Black volatility constitutes the right value to fit with the model volatility parameters. The market prices are unfortunately a bit more involved. The market quotes flat volatilities for caps of different maturities, T and strikes, K. Thus, an implied volatil- Black ity surface σcap (T,K) is quoted at any point in time. So what are the implied volatilities of the caplets that make up the caps with different strikes and maturi- ties that are consistent with the quoted cap volatility surface? Alexander [2003] gives a brief and good overview about the particularities in stripping the information out of cap market prices. For instance, each fixed strike caplet in a cap with the

8Depending on the calibration target one can choose the model volatility parameters to meet condition (1.3.3) even not (only) for ATM Black volatilities. Chapter 1. Forward Libor and Swap Market Models 25

ATM strike K has a different moneyness. Each Ti maturing caplet is assumed to be ATM if Fi(Ti)= K, but since each caplet has a different underlying forward rate, it will have a different ATM strike. So the different caplets in an ATM cap are only approximately ATM. One of the popular iterative methods, used to back out these caplet volatilities from the cap market implied volatility surface, is the vega-weighted interpolation technique, for which we refer to Alexander [2003]. Several stripping algorithms9 to extracting caplet volatilities out of quoted cap volatil- ities are presented in detail in the technical work by Hagan-Konikov [2004]. Finally, in a smile-less world, the calibration to caplets’ and floorlets’ implied volatil- ities (once extracted from the market quotes of caps and floors) for the LMM model is straight-forward via (1.3.5), and the correlations between forward Libor rates have no impact on the cap/floor prices. The application of the calibration requirement (1.3.5) to the volatility parameterizations given above is straight-forward as well. Therefore, in the sequel we will focus on the parameterization given in (1.3.2)–(1.3.4). The first step in the calibration procedure is to find the solution parameters a,b,c,d for the { } minimizing problem (1.3.6) N 1 2 − Ti 2 Black min σi (Ti t) [a + b(Ti t)]exp c(Ti t) + d du . a,b,c,d − − − {− − } i=0 t Additionally, the free Libor rate specific parameters ki in (1.3.4) can be used to exactly fit the respective (ATM) Black caplet volatilities.

1.3.3 The Structure of Instantaneous Correlations

As discussed in Sec. 1.1.3, both, the instantaneous volatility formulation as well as the chosen instantaneous correlation, can contribute to terminal correlations. The qualities and properties an instantaneous correlation matrix ρ associated with a LMM should have are (cf. Brigo-Mercurio [2007]):

Symmetry and ones on the diagonal: • ρ = ρ , ρ = 1, for all i, j 0,...,N 1 10; i,j j,i i,i ∈ { − }

ρi,j 0 for all i,j, and the map i ρi,j has to be decreasing for i j, • thus,≥ moving away from the diagonal along→ a column or row the entries become≥ monotonically decreasing as joint movements of far away rates are less correlated than movements of the rates with close maturity; When moving along the yield curve, the larger the tenor, the more correlated • the adjacent forward rates are. Hence, the sub-diagonals, i ρi+p,i, will be increasing for a fixed p. →

9The idea behind the boot-stripping algorithms is that if we know, for instance, the 1 and 2 year flat volatilities we know the 1 year and 2 year cap prices. Their price difference is by no arbitrage arguments the second caplet in 2 year cap contract. It is thus required to solve a volatility that implies this caplet price. The same procedure is continued iteratively further. 10For ease of notation we will be numbering the elements of the correlation matrix by beginning N−1 with zero, ρ = ρi,j , coinciding with the numbering of the forward rates and their expiries. { }i,j=0 Chapter 1. Forward Libor and Swap Market Models 26 A variety of parameterization functions have been introduced over the past years that allow for expressing a given correlation matrix of forward rates in a functional form. There are several advantages to this: of course, it is computationally convenient to work with an analytical formula. But also noise, such as bid-ask spreads, and illiquidity are removed by focusing on general properties of correlation. Furthermore, the rank and the positive semi-definiteness of the correlation matrix can be controlled through the functional form. The parameterizations we shall present here are full-rank parameterizations. We will also discuss how to reduce their rank depending on the number of underlying Brownian motions of the model. One property that is implicitly present in all pa- rameterizations is the desirable time-homogeneity of the correlations.

Full-rank correlation parameterization

In general, the full instantaneous correlation matrix is characterized by N(N 1)/2 − entries, given the symmetry and the ones on the diagonal. This number of entries may be too high for practical purposes, thus, a parsimonious parametric form with reduced number of parameters has to be found. In the literature a vast number of correlation parameterizations is presented; to be mentioned here are the works of Schoenmakers-Coffey [2003], Wu-Zhang [2003], Morini-Webber [2006], and lately the papers of B¨orger-van Heys [2010] and Lutz [2010]. We will focus in the sequel on some of the parameterizations which will be used in our model calibration later on. Three-parameters full-rank exponential parameterization. For 0 t min T ,T Rebonato [2004] proposed a parameterization of the form ≤ ≤ { i j}

(1.3.7) ρi,j(ρ ,α,β; t) = ρ + (1 ρ )exp Ti Tj (β α max i, j ) , ∞ ∞ − ∞ −| − | − { } which fulfils the desirable properties given above. Thus, it may produce for a given tenor structure realistic market correlations for properly chosen ρ ( 1, 1),β> 0 ∞ and (small) 0 α β/(N 1) (cf. Rebonato [1999a]). A slight∈ modification− of (1.3.7), also given≤ in≤ Rebonato− [2004], reads:

(1.3.8)ρ ˜i,j(ρ ,α,β; t) = ρ + (1 ρ )exp β Ti Tj exp α max i, j , ∞ ∞ − ∞ − | − | − { } with ρ ( 1, 1),β> 0 and α R. ∞ ∈ − ∈ A special case of (1.3.7) is the Rebonato’s two-parameters full-rank exponential pa- rameterization:

(1.3.9) ρi,j(ρ , β; t) = ρ + (1 ρ )exp β Ti Tj , β> 0, ρ ( 1, 1). ∞ ∞ − ∞ − | − | ∞ ∈ − However, it should be noted that for a particular choice of parameters it is not directly guaranteed that (1.3.7) or (1.3.8) defines valid correlation structure indeed (it might violate the positive semi-definiteness of the correlation matrix). The special case with ρ = 1 (cf. Rebonato [2002]), ∞ (1.3.10) ρ (β; t) = exp β T T , t [0,T T ], i,j − | i − j| ∈ i ∧ j Chapter 1. Forward Libor and Swap Market Models 27 assures a symmetric correlation matrix with positive eigenvalues. This parameteri- zation is analytically very attractive and fulfills the basic modeling requirements. Apart from their parameter poorness which might turn out to be a handicap when fitting to market quotes, these parameterizations do not distinguish on the distance between two different forward rates such that different pairs of forward rates with the same distance to each other are correlated to the same degree. Since an unconstrained optimization is preferable to a constrained one, Schoenmakers-Coffey [2003] param- eterizations might be preferred from this point of view. Nonetheless, the Rebonato’s parameterizations are widely used in practice because of their analytical tractability and the easy calibration. Doust’s multiplicative correlations (cf. Doust [2007], Rebonato-McKay-White [2009]). To overcome the above-mentioned problem that the decorrelation (brought into the model by the constant exponential decay factor β) only depends on the distance between two rates, the challenge will be to introduce a dependence of the decorrelation factor on the expiries of the forward rates, β = βi,j, in such a way that the resulting correlation matrix does not loose any of the desired properties of being a valid correlation matrix. Similar in spirit to the construction in Schoenmakers-Coffey [2003], Doust proposes the following parametric structure (cf. Doust [2007]): 11 12 For ai [ 1, 1], i =1,...,N 1 the elements of the correlation matrix are defined recursively:∈ − − - First define the trivial diagonal elements, ρ = 1, i = 0,...,N 1; i,i − - Then define the elements of the first row by respecting the symmetry as

j ρ = a = ρ , j = 1,...,N 1; 0,j k j,0 − k=1 - By inspection, assuming that i>j, fill the lower triangle part by

i ρ0,i ρi,j = = ak. ρj,0 k=j+1 The upper triangle part is then defined by the symmetry relationship:

1 a1 a1a2 a1a2a3 ... a1 aN 1 − a1 1 a2 a2a3 ... a2 aN 1  −  a1a2 a2 1 a3 ... a3 aN 1 − ρ(a1,...,aN 1; t)=  . . . . . . −  ......     a1 aN 2 ... aN 2 1 aN 1   − − −   a1 aN 1 ... aN 1 1   − −    Given the N 1 quantities ai, Doust [2007] proves that the resulting matrix is always a real− symmetric positive definite matrix which admits a simple Cholesky

11 In the most cases ai will be positive, 0 j

i

(1.3.12) ρi,j(β1,...,βN 1; t) = exp βk∆T , 0 t min Ti,Tj . − − ≤ ≤ { } k=j+1 The dependence of βk on k allows us to specify the degree of decorrelation be- tween rates with same distance but different expiries. A decreasing property of βk : βk > βk+1, is empirically evident. Obviously, for a constant βk β, the simple pa- rameterization in (1.3.10) will be recovered. The flexibility can be≡ even increased by introducing functional forms to describe the dependence of βk > 0 on k. Polynomial forms of degree M N 1, , like ≤ − PM M (1.3.13) β = (k) = g /kl, k PM l l=0 with positive parameters gl, easily guarantee the desired properties of the matrix. The correlation parameterization then takes for i > j the shape

i M (1.3.14) ρ (g ,...,g ; t) = exp ∆T g /kl , t [0,T T ]. i,j 0 M − l ∈ i ∧ j k=j+1 l=0 Finally, Rebonato-McKay-White [2009] go one step further imposing an additional long-term decorrelation among the forward rates, preventing that the decorrelation goes asymptotically to zero with increasing distance between the rates, but rather to some finite economically plausible level ρ > 0: ∞

(1.3.15) ρi,j(ρ ,g0,...,gM ; t) = ρ + (1 ρ )ρi,j(g0,...,gM ; t), t [0,Ti Tj]. ∞ ∞ − ∞ ∈ ∧ Most of the introduced parameterizations are discussed with great detail in Rebonato [2004], while Schoenmakers [2002] is a good reference for the particularities of their numerical implementation and performance.

Rank-Reduced Correlations

From the standard matrix calculus it is well known that any positive semi-definite N N symmetric matrix ρ R × can be diagonalized by means of a real and orthogonal N N ∈ matrix P R × : ∈ (1.3.16) ρ = PDP⊥, with PP⊥ = P⊥P = I,

N N where D R × is the diagonal matrix containing the positive eigenvalues of the original matrix∈ ρ, whereas the columns of P are the eigenvectors of ρ. Setting B := P√D we obtain

(1.3.17) ρ = BB⊥ and B⊥B = D. Chapter 1. Forward Libor and Swap Market Models 29 Rebonato [1999a] mimics the decomposition (1.3.17) by means of a suitable matrix N d B R × of rank d

B = b⊥(t) = b0(t),... bN 1(t) ⊥ at any time t. − Here we shall present the two mostly used rank reduction techniques.

The Hypersphere Decomposition. Rebonato-Jaeckel [1999] suggests the fol- lowing form of the i-th column vector of the matrix b in (1.1.7):

k 1 cos(θi,k(t)) j=1− sin(θi,j(t)) if k =1,...,d 1 (1.3.18) bi,k(t) = k 1 − − sin(θ (t)) if k = d, j=1 i,j where the angles θi,j(t) constitute the parameters to be altered within the fitting optimization algorithm. We denote the resulting low-rank correlation matrix by ρθ.

The Spectral Decomposition. Following Rebonato-Jaeckel [1999] an alternative and effective way of rank-reduction is the so-called spectral decomposition. Given a target number of driving factors d and assuming (1.3.16) we can arrange the eigen- values in D in descending order and rearrange in P the corresponding eigenvectors such that their numbering in columns corresponds to the order of eigenvalues in the new D˜ : ρ˜(t) = P˜ (t)D˜ (t)P˜ ⊥(t). Then the smallest N d eigenvalues will be set to zero and the corresponding eigen- vectors will be taken− off the matrix P˜ (t), resulting in the approximative correlation matrix (d) (d) (d) (d) ρˆ (t) = P˜ (t)D˜ (t)(P˜ )⊥(t). In general the resulting matrix ρˆ(d)(t), although positive semidefinite, does not nec- essary feature ones on its diagonal. The solution is to interpret ρˆ(d)(t) as a covariance matrix and to derive the correlation matrix associated with it by normalizing it:

(d) (d) ρˆi,j (t) ρi,j (t) = . (d) (d) ρˆi,i (t)ˆρj,j (t) Chapter 1. Forward Libor and Swap Market Models 30 By following this procedure we obtain an acceptable correlation matrix ρ(d)(t) which is a d-rank approximation and intuitively similar to the target one. This methodology can be found in the literature as the principal component analysis (PCA).

Approach to Optimizing on a Low Rank Parametric Form Once the target full-rank correlation matrix ρmod is given as input, we can minimize over the angle parameters θi,j(t) (cf. (1.3.18)) the norm of the difference between the target matrix ρmod and the low-rank matrix ρθ: N mod θ 2 min ρi,j (t) ρi,j(t) . θi,j (t) | − | i,j=1 Rebonato-Jaeckel [1999] proved empirically that the differences between the reduc- tion over the angles and the spectral decomposition is typically very small. In general, when we calibrate the LMM to swaptions using the instantaneous corre- lations ρ as fitting parameters, as we will see below, we are free to select a-priori a parametric form for the correlation matrix. Once the model matrix of instantaneous correlations ρmod is defined we can use one of the introduced rank reduction algorithms to reduce the degrees of freedom for the random shocks we will use to simulate the forward term structure for future times and price interest rate derivatives.

1.3.4 Calibration of LMM Correlations to Swaptions Volatil- ities

The market is quoting the swaptions in terms of their Black implied volatilities. For instance, a Tm (Tn Tm) ATM payer swaption with expiry at Tm on the swap rate × − Black Sm,n(t) is conventionally quoted as σm,n .

Calibrating the LMM consists of finding the instantaneous volatility σi(t) i and correlation parameters ρ(t) in the LMM dynamics that reflect the swaptions{ } prices observed in the market. Combining the equations (A.3.3) and (1.1.23) we obtain

n 1 Tm m,n m,n 2 − v (s)v (s)Fi(s)Fj(s) (1.3.19) σBlack (T t) = i j γ (s) γ (s) ds. m,n m − S2 (s) i j i,j=m t m,n A sufficiently good and market proven approximation of (1.3.19) is the following:

n 1 m,n m,n Tm 2 − v (t)v (t)Fi(t)Fj(t) (1.3.20) σBlack (T t) = i j γ (s) γ (s) ds, m,n m − S2 (t) i j i,j=m m,n t which we shall consider as the standard approach.

1.3.5 Calibration to Correlations Implied From CMS Spread Options

Among practitioners and academics alike, there is consensus on the fact that even the low-parametric LMM correlation parameterizations can hardly be calibrated reliably Chapter 1. Forward Libor and Swap Market Models 31 to market data due to the fact that swaptions carry only little information about cor- relations (Alexander [2003], Brigo-Mercurio [2007], Rebonato [2002], Schoenmakers [2002, 2005], Schoenmakers-Coffey [2003]). The market of structured interest rate products, in particular pay-off structures in- cluding derivatives of constant maturity swaps, has undergone an enormous growth during the past few years. As already discussed in Sec. 1.2.2, such structures equipped with call rights such as CMS spread options, require a realistic modeling of not only the development of swap rates, but also their correlation. Such correlation infor- mation is meanwhile available, because a separate market has developed for spread options used for hedging purposes almost like plain vanilla instruments. Assuming jointly lognormal swap rates we can work out an implied correlation of the involved swap rates in a CMS spread option. This correlation can then be taken as target to be reproduced by a pricing model and, therefore, has to be included in a model calibration procedure of LMM. Motivated by the general formula for the valuation of the CMS spread optionlet given i in (1.2.33), assuming log-normality of the swap rates under P , with Ti the expiry time of the optionlet, B¨orger-van Heys [2010] propose a calibration of the parameterized instantaneous forward rate correlations to prices of CMS spread options. Regarding (1.2.33), the value of a CMS spread optionlet can be written as (1.3.21) + Ei ωS (T ) ωS (T ) κ = F S (t),i (t),σ (t),ρ(t),T ,κ , t i,i+n1 i − i,i+n2 i − i,i+nk i,i+nk i,i+nk i i where k =1, 2. Applying the typical freezing for the drifts i,i+nk (t) of the swap rates Pi under , and deriving the swap rate volatilities σi,i+nk (t) from the given market quotes of the swaption ATM-implied volatilities (as in (A.3.3)), (1.3.21) can now be considered as a target function for the swap rate correlation ρ(t), and also for the forward rate correlations through

i+n1 1 i+n2 1 i,i+n i,i+n 1 − − v 1 (t)v 2 (t)F (t)F (t) ρ(t) = k l k l (Ti t)σi,i+n1 (t)σi,i+n2 (t) Si,i+n1 (t)Si,i+n2 (t) − k=i l=i Ti (1.3.22) σ (s)σ (s)ρ (s) ds. × l k l,k t Here we used a standard approximation, cf. Belomestny-Kolodko-Schoenmakers [2010], for the swap rate correlation. This approach to the calibration of the Li- bor rate correlations was treated as a improvement of the typical market practice of calibration to swaption volatilities, showing satisfactory results for certain parame- terization of the instantaneous correlations, such as in Schoenmakers-Coffey [2003]. We refer for the details to B¨orger-van Heys [2010]. The latter underpins our motivation for calibrating the forward rate correlation of the more involved SABR–LMM model we shall introduce in Chap. 3.2. Chapter 2

The SABR Model of Forward Rates

2.1 General Model Dynamics

As already mentioned in Chap. 1.1, one problem encountered when modeling deriva- tives like caplets in the LMM and therefore using the Black formula is, that the market prices for caplets over different strikes cannot be obtained with a constant volatility parameter as the model demands1. The presence of these volatility skews and smiles in the market is however evidence that the underlying is driven by some process other than a lognormal one. With this in mind Dupire [1994] proposed the local volatility model, which has the advantage that the model perfectly replicates the current mar- ket situation. But the approach behaves poorly in forecasting future dynamics and option pricing is not possible in closed form. Thus, it became of practical interest to develop stochastic versions of the volatility structures of forward Libor models capable of matching the observed volatility in the markets of caps and swaptions, by 2 considering a more general volatility process of the form γ(t,T,S; F (t,T,T ∗)) . It was the seminal work of Hagan et al. [2002] who launched the so called SABR model, where the forward rate process is modeled under its forward measure using a correlated lognormal stochastic volatility process. Hagan et al. [2002] explain clearly why ”just fitting the today’s market prices” is not good enough. Taking into account the quality of prediction of the future dynamics of the volatility smile, meeting the observations from the market reality, is as crucial as the best achievable fitting to the today’s market prices. Hagan and his colleagues were not the first to equip BGM-type models with stochastic volatility, see for instance the Cox-Ingersoll-Ross (CIR)-type models of Andersen- Andreasen [2002], Andersen-Brotherton-Ratcliffe [2005], Piterbarg [2003, 2005a,b] or models of jump-diffusion. All these models lack the ability to fit accurately the

1It should be mentioned that the generic BGM framework does not necessarily require the forward volatility functions of forward rates to be deterministic functions; they may be adapted processes or some deterministic or random functions of the underlying forward Libor rates. 2Nevertheless, the class of models of practical interest are mainly characterized by a separa- ble volatility structure, cf. Andersen-Andreasen [2002], Andersen-Brotherton-Ratcliffe [2005], Andersen-Piterbarg [2007], Piterbarg [2003, 2005a,b], Wu-Zhang [2006], etc. 32 Chapter 2. The SABR Model of Forward Rates 33 appropriate market smile surface in a simple fast and robust manner. It is the ability to do so which constitutes one major advantage of the SABR model, as there exists an approximation formula to implied Black volatility using the SABR parameters, introduced by Hagan et al. [2002]. Hence, option prices, such as of caps and floors, can be calculated using the well known Black pricing framework but taking into account the volatility surface using a strike dependent volatility function. Nowadays the SABR model has become a reference stochastic volatility framework for modeling smiles in the financial industry, because of the described properties and its easy application. The SABR model attempts to capture the dynamics of a single forward rate. De- pending on the context, this forward rate could be a Libor forward rate, a forward swap rate, the forward yield on a bond, etc. However, we shall focus in the following on the SABR model of forward Libor rates.

Definition 2.1 (multifactor SABR Model). Assume that the number of model fac- tors, that is the number of independent driving Brownian motions, is d + d˜ under the forward measure Pi+1, for the index i 0,...,N 1 . ∈ { − } Building on the preliminary framework presented in Chap. 1 for the classical LMM, the (d d˜)–factor SABR model (SABR) assumes the following dynamics for the × i+1 forward rate Fi under its payoff measure P :

(2.1.1) dF (t) = F βi(t)(t)σ (t) b (t) dWi+1, 0 t T , i i i i t ≤ ≤ i where 0 βi(t) 1 and σi(t) is a stochastic variable following a diffusion process of type3: ≤ ≤

i+1 (2.1.2) dσ (t) = σ (t)ν (t) b˜ (t) dW˜ , 0 t T , i i i i t ≤ ≤ i i+1 ˜ i+1 ˜ where Wt is a d-dimensional and Wt a d-dimensional independent standard Brownian motion under i+1, and ν (t) the exogenously given deterministic volatility P i of volatility function4. 5 Rd ˜ Rd˜ ˜ The loading vectors bi(t) and bi(t) satisfy bi(t) d = bi(t) d˜ = 1 and for 0 t min T ,T : ∈ ∈ ≤ ≤ { i j} T (2.1.3) bi(t) bj(t) = ρi,j(t), min d,d˜ { } (2.1.4) bik(t)˜bjk(t) = φi,j(t), φi,i(t) = ξi(t), k=1 (2.1.5) b˜ (t)T b˜ (t) = θ (t), j =0,...,N 1, i j i,j − 3Note that there is no mean reversion for the volatility process. Since we are looking at one forward rate at a time, this is not necessarily a problem as long as the correct terminal distribution of the forward rate is obtained. 4 i+1 While , induced by B(t,Ti+1), is a natural martingale measure for the forward rate Fi(t), due to its definitionP in (1.1.1), the fact that the volatility process in (2.1.2) is defined to be driftless under i+1 is rather ”artificial” and can be seen as a model assumption. 5ForP ease of notation we will be numbering the elements of all correlation matrices by beginning with zero such that they coincide with the numbering of the forward rates. Chapter 2. The SABR Model of Forward Rates 34 where the matrices ρ(t), φ(t), θ(t) RN RN are exogenously defined, with the par- ∈ × ticularity that φ(t) has got ξi(t)=(ξ(t))i, the correlation between the forward rate and its own volatility process, on the diagonals (instead of ones)6.

Imposing the initial conditions for the forward rate process Fi(0) and its initial volatil- SABR ity σi(0) = σi , the model becomes fully specified.

Investigating the qualitative behavior of the (Black) volatility implied from the SABR model, Rebonato-McKay-White [2009] spotted the following properties for the SABR parameters:

SABR a change (upwards) of the SABR initial volatility σi causes an almost shift • (upwards) of the implied volatility smile across strikes, and a modest steepening of the smile (low strikes increase more than high strikes);

when the exponent βi goes from 1 to 0, it causes a progressive steepening of • the smile and introduces modest curvature to it, while, on the other hand, an increasing β lowers the level of the smile; a similar effect as for decreasing β can be spotted when the correlation parame- • i ter ξi moves from 0 to 0.5; the smile becomes negatively slopped, accompanied with small decrease in− curvature. There seems to be a pronounced redundancy in the resulting effects in choosing the parameters βi and ξi; the prevalent mar- ket practice is to fix the exponent βi (usually at 0.5) and optimize the fitting over the other parameters;

finally, νi caters for the curvature of the smile (increasing νi increases the cur- • vature), with certain secondary effect on the steepness of the smile.

However, only the interaction of all these parameters together makes the model suc- cessfully capable to capture the different market volatility smile surfaces. For deeper discussions about the solvability of the SABR system for different CEV exponents β and the property of the dynamics whether zero or negative rates are attainable we refer to the extensive analysis done in Rebonato-McKay-White [2009]. Rebonato-McKay-White [2009] provides a good empirical overview of the strengths of the SABR model with respect to recovering the dynamics of the smile evolution when the underlying changes, emphasizing, in particular, the aspects of hedging of the interest rate risk.

2.1.1 The Time-Homogeneous Model

Definition 2.2 The multifactor time–homogeneous SABR model is a special case of the model described in Def. 2.1. The alteration consists of the constant parameters over time:

β (t) β [0, 1], i ≡ i ∈ ν (t) ν R, i ≡ i ∈ 6Here we assume that the matrices ρ(t) and θ(t) are valid correlation matrices (cf. Sec. 1.3.3). The slightly modified matrix φ˜(t)=(φi,j (t))i= j and (φ˜(t))i,i = 1 is assumed to be a valid correlation matrix as well. Chapter 2. The SABR Model of Forward Rates 35 and time–homogeneous correlation matrices for 0 t min T ,T ≤ ≤ { i j} (2.1.6) b (t)T b (t) = ρ (T T ), i j i,j j − i min d,d˜ { } (2.1.7) b (t)˜b (t) = φ (T T ), φ (0) = ξ , ik jk i,j j − i i,i i k=1 (2.1.8) b˜ (t)T b˜ (t) = θ (T T ). i j i,j j − i

Imposing the initial conditions for the forward rate process Fi(0) and its volatility SABR process σi(0) = σi , the model becomes fully specified.

2.1.2 Joint Dynamics of the SABR Forward Rates and Their Volatilities

Applying the same change of measure technique as in Sec. 1.1.1, in order to derive the arbitrage–free dynamics of the system of the SABR forward rates and their volatilities in a single measure, say Pk+1, able to be implemented in practice, we obtain for j < k (j > k analogously):

B(t,T ) B(t,T ) k τ d j+1 = j+1 i dF (t)+ (dt2), B(t,T ) B(t,T ) 1+ τ F (t) i O k+1 k+1 i=j+1 i i Consequently the following general drift formula for the forward rates in the SABR model at time 0 t T can be derived (cf. Sec. 1.1.1): ≤ ≤ j (2.1.9) k βi τiFi (t) σi(t)ρi,j , jk, 1+τiFi(t) i=k+1   Pk+1 where γj = σj(t)bj(t). Analogously, the arbitrage free dynamics with respect to of the volatility parameters in the SABR model is given by (νj(t)= νj b˜j(t)): (2.1.10) k βi τiFi (t) σi(t)φi,j , jk. 1+τiFi(t) i=k+1   Similarly, under the spot measure P∗ the dynamics is given by the stochastic system:

j βi βj s β τiFi (t) σi(t)σj(t)ρi,j (2.1.11) dFj(t) = Fj (t)γj(t) dWt + Fj (t) dt, 1+ τiFi(t) i=m(t) j βi ˜ s τiFi (t) σi(t)νj(t)φi,j (2.1.12) dσj(t) = σj(t)νj(t) dWt + σj(t) dt. 1+ τiFi(t) i=m(t) Chapter 2. The SABR Model of Forward Rates 36 2.2 Valuation in the SABR Model

The SABR model has meanwhile established itself as one of the most popular models for pricing and risk managing interest rate derivatives. One of the main virtues of this model is its ability to describe the smile effects in the volatility market quotations of the benchmark instruments which is the major limitation of the classical LMM. Although there are many models which try to catch the volatility smiles, the SABR’s popularity is indebted to its approximative compatibility with the Black formula (cf. Hagan et al. [2002]), the standard market practice of pricing benchmark instruments. It allows us to easily and accurately price benchmark instruments by making use of the SABR implied volatility in the Black–formula. In Appendix A.5 we give the formulas for the SABR implied Black and normal volatilities. It should be mentioned that the Hagan et al. [2002] asymptotic expansion to the SABR model to approximate the Black volatility, is an expansion in small volatility and small time, and was originally tested against short-dated Eurodollar options in a low-volatility high-rate environment. In the industry, the model is known to break down for high volatilities, high volatility-of-volatility, low rates, and long times to expiry. As mentioned above, the benchmark instruments can be accurately priced by mak- ing use of the SABR implied volatility in the Black formula. For instance, as- sume the SABR model of forward rates given in Def. 2.1, 2.2. The pricing of the caplets/floorlets is then approximated by7

BS Black √ (2.2.1) CPL(0,T,S,κ) = B(0,S)τ(T,S) F (0,T,S),κ, σ˜T (κ) T, 1 , with Black Black SABR σ˜T (κ) =σ ˜ (T,K,F (0,T,S),σT ,νT , βT ,ξT ) given in Appendix A.5. While in LMM for a certain forward rate we were able to directly recover the price of a caplet only for certain strike, mostly ATM, to which the model volatility was calibrated, in SABR we are able to recover the prices of the entire caplet smile using the SABR implied volatility. An other example of straight-forward pricing in SABR, by means of the SABR implied volatility, are the swaptions. Assume that the evolution of the swap rate is governed by the SABR dynamics given in (4.4.1). The payoff (A.3.1) can now be priced in SABR with respect to the measure Pm,n by BS Black PSWO(0,Tm,Tn,κ) = Bm,n(0) Sm,n(0),κ, σ˜m,n (κ) Tm, 1 , with Black Black SABR σ˜m,n (κ) =σ ˜ (Tm,κ,Sm,n(0),σm,n ,νm,n, βm,n,ξm,n) given in Appendix A.5. Here again, the whole swaption smile can be accurately recovered by the model parameters. We will see in the next chapters that the fitting of the SABR model parameters to the respective smile constitutes the basis of any calibration of the SABR models to be considered. 7The notation corresponds to the notation in Sec. A.1. Chapter 3

Pricing CMS Derivatives in SABR

3.1 The Markovian Projection Method

The Markovian projection method, first introduced by Piterbarg [2003, 2005a,b] and formalized in Piterbarg [2007], is an approach to volatility calibration and repre- sents a way of deriving efficient, analytical approximations to European-style option prices on various underlyings. This generic framework is applicable to a wide range of diffusion models and its power has been demonstrated on a number of examples, including spread and basket options, relevant to practical applications (cf. Antonov- Arneguy [2009], Antonov-Misirpashaev [2006], etc.). As we shall see, this method is also capable to incorporate stochastic volatility models with a correlation structure between all stochastic variables/processes. We will apply this technique to approx- imate a basket of SABR variables by an univariate model, aiming at pricing CMS derivatives analytically, in particular the CMS spread contracts.

The Mimicking Theorem

The term Markovian projection (MP) refers to a technique that is based on a theorem by Gyoengy [1986] which explains how a complicated, usually non-Markovian process can be replaced by a Markovian process, the mimicking process, with the same one- dimensional marginal distributions as the original process.

Theorem 3.1 (Gyoengy [1986]) Given a filtered probability space Ω, , P , let X(t) be an Ito˜ process governed by {Ft} (3.1.1) dX(t) = α(t)dt + β(t) dW(t), where W(t) is a d-dimensional -Brownian motion and α(t) Rn, β(t) Rn,d are Ft ∈ ∈ bounded measurable t-adapted processes. Let (3.1.1) admit a unique solution. F n n n Then there exist bounded measurable functions a : R+ R R and b : R+ R Rd,d, defined for every (t, y) R Rn through × → × → ∈ + × 1/2 (3.1.2) a(t, y) = E[α(t) X(t)= y], b(t, y) = E β(t)T β(t) X(t)= y , | | 37 Chapter 3. Pricing CMS Derivatives in SABR 38 such that the following SDE:

(3.1.3) dY(t)= a(t, Y(t))dt + b(t, Y(t)) dW(t), Y(0) = X(0), admits a weak solution Y(t) that has the same one-dimensional marginals as X(t).

Since X(.) and Y(.) have the same one-dimensional marginal distributions, the prices of European-style options on X(.) and Y(.) for all strikes K and expiries T are the same. Thus, for the purpose of European option valuation or the purpose of calibration to European options, one can replace a potentially complicated process X(.) with a much simpler Markov process Y(.). The correspondence between the processes is called Markovian projection of X(.) onto Y(.), where Y(.) follows a local volatility process. The function b(t, x) is often called Dupire’s local volatility. It should be noted that the Markovian projection is exact for European options but, of course, does not preserve the dependence structure of the underlying at different times. Thus, the prices of securities dependent on sampling at multiple times, such as barriers, American options and so on, are different between the original model and the projected model (cf. Piterbarg [2007]). A direct application of this result is however not possible, since the ”simpler” equiv- alent Markovian process is usually due to the calculation of the conditional expected values in (3.1.2) still too complicated to enable analytical tractability. The Marko- vian process then needs to be approximated by e.g. a displaced-diffusion which is a linear function of state, possibly with time-dependent parameters.

3.2 A Displaced SABR Diffusion Model for CMS Derivatives

3.2.1 Projection of CMS-Spreads to Displaced SABR Diffu- sion

A detailed discussion about the CMS spread options and their pricing in LMM is given in Sec. 1.2.2. The scope of this chapter is to develop a approximation formula for pricing of these derivatives in the SABR model of forward rates. In what follows we shall treat these derivatives as basket options, thus, options where the underlying is a basket of SABR forward rates. In the setting of a basket of forward price processes, an option on the basket can only be valued analytically by the formula of Margrabe [1978] and its derivations in the case of two assets. For higher dimensions the arbitrage-free price needs to be computed numerically, usually by Monte Carlo simulations, which can become in the case of stochastic volatility very time consuming and impracticable for calibration purposes. Therefore, the necessity for approximation formulas for calibration pur- poses can not be circumvented. Following Kienitz-Wittkey [2010], our aim is to make use of the Markovian Projec- tion (MP) to approximate, in terms of the terminal distribution, a basket of diffusions by a univariate diffusion. In the case of multivariate SABR diffusions for a basket of Chapter 3. Pricing CMS Derivatives in SABR 39 forward rates, as the CMS spreads can be formulated as, we show how these CMS spreads can be approximated by a displaced diffusion model of Rubinstein [1983] with a SABR stochastic volatility, aiming at valuing the spread option in closed form by taking into account the volatility cube and a full correlation structure of the SABR model of forward rates. To this end we shall apply the techniques developed by Piter- barg [2007] with the applications in Antonov-Arneguy [2009], Antonov-Misirpashaev [2006], and later adopted by Kienitz-Wittkey [2010] in the case of a more generic basket of SABR diffusions. We start with the time–homogeneous SABR model introduced in Def. 2.1 – 2.2 and make the simplification i+1 (3.2.1) dW i+1 = b (t) dWi+1, dW˜ i+1 = b˜ (t) dW˜ , i =0,...,N 1, t i t t i t − i+1 ˜ i+1 Pi+1 with Wt , Wt rate-specific one-dimensional Brownian motions under . The loading factors bi and b˜i will come into play when a given correlation model matrix needs to be factorized in the spirit of the Sec. 1.3.3 by a rank reduction algorithm, in order to match its rank (2N) to the number of Brownian drivers (d + d˜) to be used in simulations. Summarized the model dynamics reads for i =0,...,N 1: − βi i+1 (3.2.2) dFi(t) = Fi (t)σi(t) dWt , (3.2.3) dσ (t) = ν σ (t) dW˜ i+1, 0 t T , i i i t ≤ ≤ i (3.2.4) dW i,dW j = ρ (T T )dt = ρ dt, j =0,...,N 1 t t i,j j − i i,j − (3.2.5) dW i,dW˜ j = φ (T T )dt = φ dt, with φ (0) = ξ , t t i,j j − i i,j i,i i j (3.2.6) dW˜ i,dW˜ = θ (T T )dt = θ dt, 0 t min T ,T , t t i,j j − i i,j ≤ ≤ { i j} imposing the initial conditions for the forward rate process Fi(0) and its volatility SABR process σi(0) = σi . In the following we will consider the spread between two swap rates with different lengths: S (t) := S (t) S (t). k,n1,n2 k,k+n1 − k,k+n2 As in (1.1.20) the difference between the two swap rates can be written as

k+n1 1 k+n2 1 − − (3.2.7) S (t) = wk,k+n1 (t)F (t) wk,k+n2 (t)F (t), t [0,T ], k,n1,n2 i i − i i ∈ k i=k i=k with the stochastic weights wm,n(t)= τiB(t,Ti+1) . i Bm,n(t) Let us assume, without loss of generality, that n = max n ,n , and the weights 1 { 1 2} belonging to the shorter swap with length n2 are set to zero beyond k + n2, i.e. wk,k+n2 =0, for i k + n , such that the two weight vectors are of the same length i ≥ 2 n1. This eases the notation for the difference in (3.2.7) to

k+n1 1 − S (t) = wk,k+n1 (t) wk,k+n2 (t) F (t), t [0,T ], k,n1,n2 i − i i ∈ k i=k k+n1 1 − k,n1,n2 (3.2.8) = υi (t)Fi(t), i=k Chapter 3. Pricing CMS Derivatives in SABR 40 where υk,n1,n2 is defined by υk,n1,n2 (t) := wk,k+n1 (t) wk,k+n2 (t). i i i − i The more general swap spread, given in (1.2.11), can be written as (3.2.9) k+n1 1 − k,n1,n2 Sk,n1,n2 (t; a,b,ω) = aωSk,k+n1 (Tk)+ bωSk,k+n2 (Tk) = υi (t; a,b,ω)Fi(t), i=k with a,b R, ω 1, 1 and ∈ ∈ {− } k,n1,n2 k,k+n1 k,k+n2 (3.2.10) υi (t; a,b,ω) := aωwi (t)+ bωwi (t). For the sake of readability of what follows we will be abbreviating the notation for the swap spread in (3.2.8) or (3.2.9) to

M 1 − (3.2.11) S(t) = υ (t)F (t), 0 t T , i i ≤ ≤ 0 i=0 assuming that the swap rates are fixed at time T0 and M = max n1,n2 . For the more general case the formulas shall be given subsequently in Sec.{ 3.2.2. } Using the methodology of Markovian Projection (MP) we shall project the multidi- mensional diffusion process of the spread basket of the SABR forward rates onto an one-dimensional displaced SABR-type diffusion model. The stochastic weights υi(t) of this basket will be frozen at the initial time, thus, for the following approximations they are constants. Formally, we approximate the diffusion of the basket (3.2.11) of forward rates, which evolve according to (3.2.2)–(3.2.6), with a displaced diffusion model with stochastic volatility under an abstract spread measure.

Lemma 3.2 Given the multivariate SABR model of forward rates (3.2.2)–(3.2.6), applying the Markovian Projection technique leads to an approximation of the dy- namics of the swap spread S(t) by the system

(3.2.12) dS(t) = U(t)f(S(t))dW , t [0,T ], t ∈ 0 (3.2.13) dU(t) = U(t)ΥdZ , t [0,T ], t ∈ 0 (3.2.14) dW , dZ = Γdt, t t (3.2.15) f(S0) = p, f ′(S0) = q, with the start values S(0) = S0, U(0) = 1. f denotes a deterministic function, while p and q are given in (3.2.22) and (3.2.32), respectively. The parameters Υ and Γ are defined in (3.2.36)–(3.2.37).

The function f(S(t)) may be for instance a linear function f(S(t)) = p+q(S(t) S ), − 0 cf. Sec. 3.2.2.

Proof. Let the stochastic weights υi(t) of this basket be frozen at the initial time, υ (t) υ (0) =: υ , thus, for the following approximations they are constants. Define i ≡ i i σi(t) ui(t) := , s.t. ui(0) = 1, σi(0) Chapter 3. Pricing CMS Derivatives in SABR 41 and the function φ(.) by βi φ(Fi(t)) := σi(0)Fi (t).

Furthermore, define the parameters pi and qi by

βi (3.2.16) pi := φ(Fi(0)) = σi(0)Fi (0), βi 1 (3.2.17) qi := φ′(Fi(0)) = σi(0)βiFi − (0). Let us rewrite equation (3.2.2) as

i+1 (3.2.18) dFi(t) = ui(t)φ(Fi(t)) dWt . By applying the freezing the weights υ (t) υ to (3.2.11) we obtain i ≡ i M 1 M 1 − − i+1 (3.2.19) dS(t) = υi dFi(t) = υiui(t)φ(Fi(t)) dWt . i=0 i=0 We now write the dynamics of S in (3.2.19) as a single diffusion with stochastic variance. The dynamics can be seen as being given with respect to an abstract spread measure, where it becomes driftless.

Define the process Wt as

M 1 1 − dW := υ u (t)φ(F (t)) dW i+1, t σ(t) i i i t i=0 with σ(t) given through

M 1 M 1 − − 2 2 2 2 σ (t) = υi ui (t)φ(Fi(t)) + 2 υiυjui(t)uj(t)φ(Fi(t))φ(Fj(t))ρi,j. i=0 i,j=0, i

(3.2.20) dS = σ(t)dWt.

To apply the result of Gyoengy [1986] as presented in Thm. 3.1 (with α = 0) we need to compute the variance of (3.2.12)–(3.2.15) on which the spread dynamics will be projected. We compute U 2(t) as

M 1 M 1 1 − − (3.2.21) U 2(t) = υ2u2(t)p2 + 2 υ υ u (t)u (t)p p ρ , p2 i i i i j i j i j i,j i=0 i,j=0, i

M 1 M 1 − − (3.2.22) p = υ2p2 + 2 υ υ p p ρ . i i i j i j i,j i=0 i,j=0, i

1/2 b(t,y)= ES σ2(t) S(t)= y , 0 | and on the other hand b2(t,y)= ES U 2(t) S(t)= y f 2(y), 0 | where the expectation is taken with respect to the spread measure, with respect to which Wt is a Brownian motion. The function f has therefore to fulfill ES [σ2(t) S(t)= y] (3.2.23) f 2(y) = 0 | . ES [U 2(t) S(t)= y] 0 | To compute the conditional expectations of the nominator and the denominator we observe that σ2(t) and U 2(t) are linear combinations of the form:

M 1 M 1 − − 2 2 (3.2.24) σ (t) = υi fi,i(t)+ 2 υiυjfi,j(t)ρi,j, i=0 i,j=0, i

(3.2.26) fi,j(t) := φ(Fi(t))φ(Fj(t))ui(t)uj(t), p p u (t)u (t) (3.2.27) g (t) := i j i j . i,j p2 A first order Taylor expansion leads to

qi qj fi,j(t) pipj 1+ (Fi(t) Fi(0)) + (Fj(t) Fj(0)) + (ui(t) 1)+(uj(t) 1) . ≈ pi − pj − − − Analogously, p p g (t) i j 1+(u (t) 1)+(u (t) 1) . i,j ≈ p2 i − j − Thus, to compute the conditional expectations of (3.2.23) we need simple expressions for ES [F (t) F (0) S(t)= y] and ES [u (t) 1 S(t)= y] , j =0,...,M 1. 0 j − j | 0 j − | − To find a simple formula we apply the Gaussian approximation, introduced in this context by Piterbarg [2007], to compute the expectations. As discussed in Piterbarg’s works, the Gaussian approximation is a simple but reasonable approximation. Here it is given by: dF (t) dF˜ (t) = p dW i+1, i ≈ i i t du (t) du˜ (t) = ν dW˜ i+1, i ≈ i i t dS(t) dS˜(t) = pdW˜ , ≈ t M 1 1 − (3.2.28) dW˜ := υ p dW i+1, t p i i t i=0 Chapter 3. Pricing CMS Derivatives in SABR 43 with the correlation structure M 1 1 − (3.2.29) dW˜ ,dW i+1 = υ p ρ dt =: λ dt, t t p j j i,j i j=0 M 1 1 − (3.2.30) dW˜ ,dW˜ i+1 = υ p φ dt =: λ dt. t t p j j i,j i+M j=0 The expected values with respect to the spread measure can now be computed by Gaussian calculus, cf. Piterbarg [2007]. We obtain

F˜j(t), S˜(t) ES ˜ ˜ y S(0) 0 Fj(t) Fj(0) S(t)= y = (y S(0)) = pjλj − , − | S˜(t), S˜(t) − p and y S(0) ES u˜ (t) 1 S˜(t)= y = ν λ − . 0 j − | j j+M p Using these expressions we compute f 2(y) by ES [σ2(t) S(t)= y] p2 + A (y S(0)) (3.2.31) f 2(y) = 0 | n − , ES [U 2(t) S(t)= y] ≈ 1+ A (y S(0)) 0 | d − with M 1 2 − A = υ2p2(q λ + ν λ ) n p i i i i i i+M i=0 M 1 − + υiυjpipjρi,j(qiλi + qjλj + νiλi+M + νjλj+M ) , i,j=0, i

1 2 f(S(0)) = p, f ′(S(0)) = (A p A ) = q, 2p n − d with p given in (3.2.22) and q defined by

M 1 M 1 1 − − (3.2.32) q := υ2p2q λ + υ υ p p ρ (q λ + q λ ) . p2 i i i i i j i j i,j i i j j i=0 i,j=0, i

(3.2.34) dU(t) := ΥU(t)dZt, where for the Brownian Motion Zt under the spread measure we have M 1 1 − (3.2.35) dZ = υ p λ ν dW˜ i+1, t Υp i i i i t i=0 and M 1 1 − Υ2 = Var υ p λ ν dW˜ i+1 p i i i i t i=0 M 1 M 1 1 − − (3.2.36) = υ2p2λ2ν2 + 2 υ υ p p λ λ ν ν θ , p2 i i i i i j i j i j i j i,j i=0 i,j=0, i

Remark 3.3 For νi = 0, i = 0,...,M 1, we end up with the projection of CEV diffusions since all stochastic volatility and− cross correlation terms in the calculation cancel out. If additionally β = 1 the basket of SABR diffusion even simplifies to a basket of Geometric Brownian Motions. Chapter 3. Pricing CMS Derivatives in SABR 45 3.2.2 Pricing of CMS-Spread Options in a SABR Displaced Diffusion Model

We now apply the MP method to the case of the CMS spread (3.2.11), projecting the basket diffusions to a displaced diffusion of SABR type as in (3.2.12)-(3.2.15), with f defined as a linear function p (3.2.38) f(S) := (S(t)+ A)q, with A = S(0). q −

In the case of the spread Sk,n1,n2 (t) = Sk,n1,n2 (t;1, 1, 1), n1 > n2, (cf. (3.2.9)) we then have for t [0,T ]1: − ∈ k ¯ d (Sk,n1,n2 (t)+ Ak,n1,n2 ) = dSk,n1,n2 (t) = U(t)(Sk,n1,n2 (t)+ Ak,n1,n2 ) dWt, ¯ ¯ ¯ (3.2.39) dU(t) = U(t)Υk,n1,n2 dZt, U(t) := qk,n1,n2 U(t), dW , dZ = Γ dt, t t k,n1,n2 where

pk,n1,n2 (3.2.40) Ak,n1,n2 = Sk,n1,n2 (0), qk,n1,n2 −

k+n1 1 k+n1 1 − − (3.2.41) p = (υk,n1,n2 )2p2 + 2 υk,n1,n2 υk,n1,n2 p p ρ , k,n1,n2 i i i j i j i,j i=k i,j=k, i

k+n1 1 1 − (3.2.42) q = (υk,n1,n2 )2p2q λk,n1,n2 k,n1,n2 p2 i i i i k,n1,n2 i=k k+n1 1 − k,n1,n2 k,n1,n2 k,n1,n2 k,n1,n2 + υi υj pipjρi,j(qiλi + qjλj ) . i,j=k, i

k+n1 1 1 − (3.2.43) Γ = (υk,n1,n2 )2p2ν λk,n1,n2 ξ k,n1,n2 Υ p2 i i i i i k,n1,n2 k,n1,n2 i=k k+n1 1 − k,n1,n2 k,n1,n2 k,n1,n2 + υi υj pipjνiλi φi,j , i,j=k, j=i 1We will keep the notation to make clear the dependencies of the parameters on the model variables in the following formulas . 2The swap spread weights are frozen at the initial time, t = 0. Chapter 3. Pricing CMS Derivatives in SABR 46 and

k+n1 1 1 − (3.2.44) Υ2 = (υk,n1,n2 )2p2(λk,n1,n2 )2ν2 k,n1,n2 p2 i i i i k,n1,n2 i=k k+n1 1 − k,n1,n2 k,n1,n2 k,n1,n2 k,n1,n2 +2 υi υj pipjλi λj νiνjθi,j . i,j=0, i

Black σk,n1,n2 (κ + Ak,n1,n2 ) σ˜Black(T ,κ + A , Ek S (k) + A ,q , Υ , 1, Γ ) ≈ k k,n1,n2 0 k,n1,n2 k,n1,n2 k,n1,n2 k,n1,n2 k,n1,n2 =:σ ˜Black (κ + A ), k,n1,n2 k,n1,n2 as given in Appendix A.5, the solution of the projected SDE can be written as an asset in a Black [1976] framework and therefore a closed form solution can be derived. Nevertheless, the expectation of the payoff has to be computed under the Tk-forward measure. We can formulate the pricing equation as:

+ (3.2.45) CMSSPO(0,T ,n ,n ,κ) = τ B(0,T )Ek S (T ) κ k 1 2 k k 0 k,n1,n2 k − + = τ B(0,T )Ek S (T )+ A (κ + A ) k k 0 k,n1,n2 k k,n1,n2 − k,n1,n2 BSEk Black = τkB(0,Tk) 0 Sk,n1,n2 (Tk) + Ak,n1,n2 ,κ + Ak,n1,n2 , σ˜k,n1,n2 (κ + Ak,n1,n2 ) Tk, 1 . Therefore, the pricing depends on the chosen measure of the projected SDE as an expectation of the CMS spread at the maturity of the option needs to be computed. Hence, choosing the forward risk adjusted measure, Pk, which coincides with the options maturity, the CMS spread is not a martingale under this measure. The solution in this case shall be either a lognormal approximation of the CMS rates with respect to Pk, or an appropriate convexity correction. These approximations have been discussed in detail in Chapter 1.2.2.

SABR-Consistent Approximations of CMS-Spreads in the Forward Mea- sure

To compute the price of the CMS spread option using the SABR-DD model, we need to compute the expectation of the approximated spread at option expiry. Using the convexity correction (CC) we obtain

Ek S (T ) = Ek S (T ) Ek S (T ) 0 k,n1,n2 k 0 k,k+n1 k − 0 k,k+n2 k = S (0) + CC(S (T )) CC(S (T )). k,n 1,n2 k,k+ n1 k − k,k+n2 k Since we assume stochastic volatility when considering the SABR-DD projection of the CMS spread diffusion we have to incorporate this into computation of the convex- ity adjustment. One of the approximation which consider this is the Hagan [2003] Chapter 3. Pricing CMS Derivatives in SABR 47 convexity adjustment given in general terms in (1.2.21)–(1.2.22)) which together with the put-call-parity (1.2.9) give for the CMS rates:

(3.2.46) Ek S (T ) = S (0) + Ek (S (T ) S (0))+ 0 k,k+ni k k,k+ni 0 k,k+ni k − k,k+ni Ek + (Sk,k+ni (0) Sk,k+ni (Tk)) − 0 − B(0,Tk) = τ (1 + f ′(S (0)))PSWO(0,T ,T ,S (0)) k B (0) k,k+ni k k+ni k,k+ni k,k+m ∞ + PSWO(0,Tk,Tk+ni ,x)f ′′(x) dx S (0) k,k+ni (1 + f ′(S (0)))RSWO(0,T ,T ,S (0)) − k,k+ni k k+ni k,k+ni Sk,k+ni (0)

+ RSWO(0,Tk,Tk+ni ,x)f ′′(x) dx , −∞ with f given through (1.2.23), and (cf. Appendix. A.3)

BS Black PSWO(0,Tk,Tk+ni ,κ) = Bm,n(0) Sk,k+ni (0),κ,σk,k+ni (κ) Tk, 1 , BS Black RSWO(0,Tk,Tk+ni ,κ) = Bm,n(0) Sk,k+ni (0),κ,σk,k+n (κ)Tk, 1 . i − We shall use the standard model of replication of the CMS caplets/floorlets given in (1.2.23)–(1.2.26) for the numerical implementation. Chapter 4

The SABR-LMM Model and Its Calibration

4.1 SABR–Consistent Extension of the LMM and Its Calibration

The SABR model and the LMM, although modeling the same assets, ”do not directly ’talk to each other’ (Rebonato [2007]). The SABR does not link the snapshots of the caplet smiles into well-defined joint dynamics. To overcome this shortcoming1, Rebonato [2007] introduced an extension of the LMM that recovers the SABR caplet prices almost exactly for all strikes and maturities. The dynamics of the volatility is chosen so as to be consistent across expiries, and to make the evolution of the implied volatilities as time homogeneous as possible. This chapter is concerned with the description of this model and its calibration.

Consider again the (single) SABR dynamics of the forward rate FT (t) = F (t,T,T ∗) ∗ under its martingale measure PT given in (A.5.1)–(A.5.2). The SABR dynamics SABR for FT is fully described by the initial conditions, FT (0), σT (0) = σT and by the further forward rate specific parameters βT , νT and ξT . We now move to working ∗ under the same terminal measure PT under which the forward rate is driftless. The choice of a different measure will simply introduce the familiar drift correction terms (see Sec. 2.1.2). The SABR parameters above are assumed to be available from a previous SABR fitting to the market prices of caplets/floorlets for all expiries T, i.e. forward rates, and strikes K, and are assumed as given in what follows. Note that to calibrate the SABR parameters, by means of the implied SABR volatility function given in

1The issue of reconciling the SABR dynamics with the LMM setting was also addressed by Labordere [2006, 2007], Mercurio-Morini [2009]. Labordere presents a possible unification of the LMM and the SABR models using concepts borrowed from hyperbolic geometry, where he proves that approaches based on the ”freezing” of suitably chosen stochastic quantities, as extensively used here, cannot be correct away from the at-the-money strike. Nevertheless, they shall be acceptable in the range of parameters normally found in typical pricing applications.

48 Chapter 4. The SABR-LMM Model and Its Calibration 49 Appendix A.5, to the market (implied) Black volatilities of caplets/floorlets usually an optimization is carried out for each forward rate. For instance, the minimization problem may in general read

2 Black SABR Black (4.1.1) min σ˜ (T,K,FT (0),σ ,νT , βT ,ξT ) σ (K) , SABR T T σ ,νT ,βT ,ξT − T K Black σT (K) denoting the implied Black-volatility of a caplet with expiry T and strike K. As discussed above, due to redundancy in information stored in βT and ξT , usually βT is kept constant while altering the other parameters in (4.1.1). The SABR forward rate parameters implicitly determine the caplet prices for all strikes for the maturity for which they have been fitted. We want to determine the parameters of an LMM model such that the LMM caplet prices for all the same strikes and maturities are as close as possible to the SABR caplet prices. Sticking to the notation given in Def. 2.1 we introduce the Rebonato’s SABR- extension of the LMM, the (multi-factor) SABR–Libor Market Model (SABR–LMM).

Definition 4.1 The d d˜-factor time–homogeneous SABR Libor Market model (SABR– × LMM) assumes the following dynamics for forward rate Fi, i =0,...,N 1, and its volatility under payoff measure i+1 − P dF (t) = F βi (t)s (t) b (t) dWi+1, i i i i t (4.1.2) dsi(t) = gi(t)dki(t), i+1 dk (t) = k (t)h (t) b˜ (t) dW˜ , 0 t T , i i i i t ≤ ≤ i i+1 ˜ i+1 ˜ with Wt , Wt independent standard d, respectively d-dimensional Brownian mo- i+1 d tions under P , and the exponential βi(t) and the loading vectors bi(t) R , ˜ ∈ b˜ (t) Rd as defined in Def. 2.1 (2.1.6)–(2.1.8). The functions g and h will be i ∈ i i described below. Imposing the initial conditions for the forward rate process Fi(0) and its volatility process si(0), the model becomes fully specified.

The function gi(t) is given by gi(t)= g(Ti t), preserving the time homogeneity of the system. To retain time-homogeneity as much− as possible, Rebonato [2007] imposes that the volatility of volatility should also have the functional form

(4.1.3) h (t) = h(T t), i i − where g and h are assumed to be given deterministic functions. This leads to

t t i+1 1 (4.1.4) k (t) = k (0) exp h(T s)b˜ (s) dW˜ ds h2(T s) ds , i i i − i s − 2 i − 0 0 and therefore,

si(t) = gi(t)ki(t) t t i+1 1 = g (t)k (0) exp h(T s)b˜ (s) dW˜ ds h2(T s) ds i i i − i s − 2 i − 0 0 Chapter 4. The SABR-LMM Model and Its Calibration 50 holds. In the sequel we shall use again the notational simplification, introduced with (3.2.1),

i+1 dW i+1 = b (t) dWi+1, dW˜ i+1 = b˜ (t) dW˜ , i =0,...,N 1, t i t t i t − which makes the SABR–LMM dynamics equivalent to the Def. 4.1 if we assume the Pi+1 i+1 ˜ i+1 rate-specific one-dimensional -Brownian motions Wt and Wt to be correlated at 0 t min T ,T via (cf. (2.1.6)–(2.1.8)): ≤ ≤ { i j} (4.1.5) dW i,dW j = ρ (T T )dt = ρ dt, t t i,j j − i i,j i j (4.1.6) dW ,dW˜ = φi,j(Tj Ti)dt = φi,jdt, with φi,i(0) = ξi, t t − j (4.1.7) dW˜ i,dW˜ = θ (T T )dt = θ dt, i, j =0,...,N 1. t t i,j j − i i,j − 4.2 Calibrating the Volatility Process

SABR Given a pre-calibration of the SABR parameters σi , βi, νi and ξi to the market quotes of caplets volatilities, in this section the focus will be on the specification of the volatility parameterization we shall use in our further work. In the spirit of (1.3.2)–(1.3.4), let the instantaneous volatilities si(t) of the forward rates in our model (4.1.2) be defined by the deterministic function g(T t) of the residual time i − to maturity, and the forward rate specific functions ki as

(4.2.1) s (t) = k (t)g(T t), i i i − where the function g shall be parameterized as

g g cgu g g g g g (4.2.2) g(u) = [a + b u]e− + d , c ,d > 0, a + d > 0.

As far as caplets are concerned, the calibration problem consists of choosing the parameters of the function g and the functions ki adequately. In a deterministic volatility setting the quantities ki would be fully determined by the requirement that each caplet should be perfectly priced. This is no longer the case in the stochastic volatility setting. Following Rebonato [2007] we therefore heuristically impose that the parameters ag,bg,cg and dg should be chosen in such a way as to match as closely SABR as possible the expectation at time 0 of si(t), namely σi . This can be done by minimizing the squared discrepancies (cf. (1.3.6)) by performing a minimization:

N 1 2 − 1 Ti 2 SABR g g cgu g (4.2.3) min σi [a + b u]e− + d du , ag,bg,cg,dg − T i=0 i 0 where the sum runs over the N caplet expiries. Once the parameters of the function g have been found, the forward rate specific initial values ki(0) are then chosen so as SABR to provide exact recovery of the quantities σi , analogously to (1.3.4):

Ti SABR ki(0) 2 (4.2.4) σi = g(u) du. √T 0 i Chapter 4. The SABR-LMM Model and Its Calibration 51

SABR If the chosen function g allows for a good fit to the initial SABR values σi for different maturities Ti, these correction factors will all be close to 1.

The variable ki(t) follows itself an other stochastic process given in (4.1.4). The time– homogeneous deterministic function h shall also be parameterized as (recommended in Rebonato [2007]):

h h chu h h h h h (4.2.5) h(u) = [a + b u]e− + d , c ,d > 0, a + d > 0.

The parameters ah,bh,ch,dh are chosen to minimize the sum of the squared discrep- ancies:

N 1 2 − 1 Ti 2 h h chu h (4.2.6) min νi [a + b u]e− + d du . ah,bh,ch,dh − T i=0 i 0 Afterwards, rate–specific small correction factors, εi, are applied to ensure the exact recovery of the volatility of volatility, νi (analogous to (4.2.4)):

Ti εi(0) 2 (4.2.7) νi = h(u) du. √T 0 i To accurately take into account the stochasticity of the volatility, Rebonato-White [2009] suggests a better approximation to the function h which is given by

1/2 k (0) Ti u u (4.2.8) ν = i 2 g(u)2 h(T τ)2 dτ du , i σSABRT i − T i i 0 0 i reflecting the relationship between the market–given quantity νi and the parameters of our SABR–LMM model. Details and the derivation can be found in Rebonato- White [2009] and Rebonato-McKay-White [2009].

4.3 The SABR Correlation Structure

The prices of caplets only depend on the correlation among the forward rates and the volatilities. For products with more complex payoffs (e.g., swaptions), however, we must fully specify the parameters in (4.1.5)–(4.1.7), i.e., we must also define the correlations among the forward rates ρ(t), among the volatilities θ(t) and the non- diagonal elements of the correlations matrix between the forward rates and volatilities φ(t), as only the correlation between the forward rate and its own volatility, ξ , is { i} available from the SABR caplet-related fitting. In the spirit of Rebonato-White [2009], let us therefore define the full correlation matrix (occasionally referred to as the super-correlation matrix) as

ρ φ (4.3.1) Σ(t) := (t). φ⊥ θ There are many ways to parameterize the super-correlation matrix. Based on the discussions in Sec. 1.3.3 we will give two examples. Chapter 4. The SABR-LMM Model and Its Calibration 52 Example 4.2 Rebonato-White [2009]: Full-rank exponential parameterizations at 0 t min T ,T (cf. (1.3.9)): ≤ ≤ { i j}

ρi,j(ρ ,λ; t) = ρ + (1 ρ )exp λ Ti Tj , λ,ρ > 0, ∞ ∞ − ∞ − | − | ∞ θi,j(θ ,; t) = θ + (1 θ )exp Ti Tj , ,θ > 0, ∞ ∞ − ∞ − | − | ∞ φ (ϑ ,ϑ ; t) = sign(ξ ) ξ ξ exp ϑ (T T )+ ϑ (T T )+ , ϑ ,ϑ > 0. i,j 1 2 i | i j| − 1 i − j − 2 i − j 1 2 Example 4.3 Full-rank mixed parameterizations2 (cf. (1.3.9), (1.3.14)):

i 1 M − (4.3.2) ρ (g ,...,g ; t) = exp ∆T g /kl , g > 0, i,j 0 M − l l k=j l=0 (4.3.3) θi,j(θ ,; t) = θ + (1 θ )exp Ti Tj , ,θ > 0, ∞ ∞ − ∞ − | − | ∞ and for ϑk > 0, k =1, 2:

(4.3.4) φ (ϑ ,ϑ ; t) = sign(ξ ) ξ ξ exp ϑ (T T )+ ϑ (T T )+ . i,j 1 2 i | i j| − 1 i − j − 2 i − j In principal, the existence of three different correlation sub-matrices, independent from each other, leaves enough space to test a lot of possible combinations of param- eterizations. In the chapter about numerical results we shall analyze and illustrate the accuracy of different parameterizations.

4.4 Calibration of the SABR–LMM Correlations to Swaption Implied Volatilities

We stress that a good choice of the correlation matrix is important to study the congruence between the caplet and the swaption markets. However, the results in this section do not depend on the particular parametrization chosen, as they are all expressed simply in terms of the elements of the super–correlation matrix Σ (cf. (4.3.1)). In what follows we assume that the SABR dynamics of the swap rate S 3, m,n N, m,n ≤ is given by (cf. (A.5.1)–(A.5.2), replacing FT by the swap rate Sm,n):

βm,n m,n dSm,n(t) = σm,n(t)Sm,n (t)dWt , ˜ m,n (4.4.1) dσm,n(t) = νm,nσm,n(t)dWt , m,n ˜ m,n dWt ,dWt = ξm,ndt, m,n ˜ m,n Pm,n where Wt and Wt are the Brownian motions with respect to the measure , induced by the annuity numeraire Bm,n(t). The SABR dynamics is therefore fully

2Without loss of generality here we assume constant spacing between the forward rate maturities, i.e. equidistant time grid. 3As already mentioned the SABR model can be used to model the dynamics of an arbitrary asset if appropriate to do so. Here, the swap rates are being modeled with SABR. Chapter 4. The SABR-LMM Model and Its Calibration 53

SABR described by the initial conditions, Sm,n(0),σm,n(0) = σm,n and by the swap rate dependent parameters βm,n, νm,n and ξm,n. It is obvious that the formula, discussed in Appendix A.5, to approximate the implied Black volatilities by the set of SABR model parameters, holds for swap rates as well, assuming the system is embedded into its the natural probability space, measured with Pm,n. In the sequel we shall assume that the parameters for the SABR dynamics of the swap rates have been calibrated, in the simplest case, to a set of given market quotes for implied European swaption volatilities. The calibration can be carried out in complete analogy to the forward rates’ case given in (4.1.1). Following the approach proposed by Rebonato [2007], for a given set of SABR forward rate parameters our goal in this section is to approximate analytically the SABR parameters of the swap rates implied by the SABR–LMM dynamics of the forward rates. Therefore we need to express the SABR parameters of the swap rate dynamics in (4.4.1) in terms of the SABR–LMM parameters of our forward rate SABR family. Concretely, we need to formulate the parameters βm,n, σm,n , νm,n and ξm,n in terms of g, h, Fi(0), ki(0), βi, εi, for i = m,...,n 1, and of course the elements of the super-correlation matrix Σ. − Writing the swap rate as a sum of forward rates as in (1.1.20) and applying the trick m,n ∂wi (0) of freezing the weights, =0, we obtain for t [0,Tm]: ∂Fi(t) ∈ n 1 − ∂S (t) σ (t)Sβm,n (t)dW m,n = dS (t) = m,n dF (t) m,n m,n t m,n ∂F (t) i i=m i n 1 − wm,n(0)F βi (t)s (t) b (t) dWi+1. ≈ i i i i t i=m

2 βm,n 2 Considering the expression (σm,n(t)) Sm,n(t) we can write

n 1 − 2 βm,n 2 m,n m,n βi βj (σm,n(t)) Sm,n (t) = wj (0)wi (0)Fi (t)Fj (t)si(t)sj(t)ρi,j(t), i,j=m leading to

n 1 − 1 m,n m,n βi βj (4.4.2) σm,n(t) = w (0)w (0)F (t)F (t)si(t)sj(t)ρi,j(t). βm,n j i i j Sm,n (t)i,j=m βi βm,n Assume additionally that the variation over time of the ratio Fi (t)/Sm,n (t) can be considered small compared with the variation in the functions si(t) because swap rates are strongly correlated with the underlying forward rates. As usual practice in this context we are tempted to ”freeze” these ratios and end up having the simplified formula

n 1 − (4.4.3) σ (t) = W m,nW m,ns (t)s (t)ρ (t), m,n j i i j i,j i,j=m Chapter 4. The SABR-LMM Model and Its Calibration 54 where F βi (0) W m,n := wm,n(0) i , i = m,...,n 1. i i βm,n Sm,n (0) − See Labordere [2006] for a discussion about the limitations of this approach. Having (4.2.1) in mind and approximating

Tm 2 σSABR T (σ (t))2 dt, m,n m ≈ m,n 0 we obtain for t [0,T ], ∈ m

n 1 1 − Tm σSABR = W m,nW m,n s (t)s (t)ρ (t) dt m,n T j i i j i,j m i,j=m 0 n 1 1 − Tm (4.4.4) = W m,nW m,nk (0)k (0) g(T t)g(T t)ρ (t) dt. T j i i j i − j − i,j m i,j=m 0 Analogously to equation (4.2.8) for the forward rate volatility of volatility, Rebonato- White [2009] propose a more accurate approximation for the volatility of volatility νm,n, given by

n 1 1 − m,n m,n νm,n = 2 W W ki(0)kj(0) σSABRT j i m,n m i,j=m 1 Tm t t 2 (4.4.5) g(Ti t)g(Tj t)ρi,j(t)θi,j(t) h(Ti τ)h(Tj τ) dτ dt . × − − − − T T 0 0 i j In order to approximate the correlation between a swap rate and its volatility ξm,n, Rebonato-White [2009] derive, after some simplification and approximations, the following simplistic expression

n 1 − (4.4.6) ξm,n = Ωi,jφi,j, i,j=m with the matrix Ω defined as m,n m,n 2Wj Wi ki(0)kj(0) Ωi,j = SABR 2 νm,nσm,n Tm Tm t t (4.4.7) g(Ti t)g(Tj t)ρi,j(t)φi,j(t) h(Ti τ)h(Tj τ) dτ dt. × − − − − T T 0 0 i j n 1 From Equation (4.4.5) we then have Ω 0, and − Ω =1, i.e. the quantities i,j ≥ i,j=m i,j Ωi,j have the desired properties of weights. The last bit is the approximation of the swap rate exponent βm,n. Rebonato-White [2009] make the heuristic ansatz Chapter 4. The SABR-LMM Model and Its Calibration 55

n 1 − βm,n = ωkβk, k=m as the (approximate) sum of CEV-variables with exponent βk is in general not a CEV-variable with the same exponent. We know, however, that in the lognormal case the approximation is good (see Rebonato [2004]), and that it is exact in the normal case. Concluding this chapter, with the equations (4.4.4)–(4.4.7) we have the ingredients for calibrating a parameterization of the super–correlation matrix Σ to the SABR- parameters for (a chosen set of) swap rate processes. Hence, the super–correlation matrix can in principle be calibrated to rich market data of swaptions over different strikes, maturities and tenors. Nevertheless, as we shall see, a sensible choice of market swaptions has to be considered in order to prevent from parameter instability. An additional advantage of these approximations is that the calibration of different parts of the super–correlation matrix can be carried out successively. With (4.4.4), the correlations between the forward rates can be calibrated, which then entered into equation (4.4.5) allow for the calibration of correlations between the forward rate volatilities. Finally, (4.4.6)-(4.4.7) summarize the gained information so far to make possible the calibration of correlations between the forward rates and the foreign volatilities (as the diagonals of the sub–matrix φ are already determined by the SABR parameters of the previous fit to the caplets’ market, cf. (4.1.1)).

4.5 Calibrating to Correlations Implied From CMS Spread Options

As for the classical LMM there is consensus among practitioners on the fact that swaptions carry only little information about correlations of the involved forward rates (cf. Sec. 1.3.5). The presented approach of calibrating the super–correlation matrix to swaption prices in Sec. 4.4, although easy implementable, suffers from the lack of sensitivity of the calibrated parameters against different market states of the yield curve, when, for instance, whole swaption smile cubes are taken into consideration. In other words one can hardly fine-tune the model correlations via this approach. The scope is to gain more precise information about the SABR-LMM correlations from correlations between the swap rates which are completely missing in the previous approach. Motivated by the corresponding analogies for the LMM (cf. Sec. 1.3.5), we propose here a new approach of calibrating the SABR correlations; a calibration to the SABR implied volatilities of CMS spread options. The gain is to include the precious market information about the swap rate correlation in the calibration of the SABR–LMM model correlations. CMS spread structures are particularly suitable for this scope; apart from being available and liquid interest rate instruments in today’s markets, they are one of the easiest financial products to contain information about the swap rate correlations. This approach of calibrating the SABR-LMM super–correlation matrix to the swap Chapter 4. The SABR-LMM Model and Its Calibration 56 rate correlations implied from the CMS spread options will consist of the following steps. Consider the SABR model (3.2.39)-(3.2.44) for the swap spread. First we need to imply the SABR model parameters for the CMS swap spread from the market data for CMS spread options4. To this end the equation (3.2.45) will be used to calibrate the SABR CMS spread parameters through

Black (4.5.1) σ˜k,n1,n2 (κ + Ak,n1,n2 ) = Black Ek =σ ˜ (Tk,κ + Ak,n1,n2 , 0 Sk,n1,n2 (Tk) + Ak,n1,n2 ,qk,n1,n2 , Υk,n1,n2 , 1, Γk,n1,n2 ), Ek where the expectation 0 Sk,n1,n2 (Tk) is approximated via the convexity replication approach (3.2.46), discussed in detail in Sec. 1.2.2. In order to adapt the SABR–LMM parameters to the SABR framework used in Chap. 3.2, we express the parameters pi in (3.2.16), qi in (3.2.17) and νi in terms of SABR- LMM parameters as in Sec. 4.2 (via (4.2.4)-(4.2.8)). Let us emphasize at this point that, through the Hagan’s implied SABR volatility formula, this approach takes the broad information of the whole smile of CMS spreads options into account. Finally, we use then the determined SABR parameters for the CMS spreads to imply, via the formulas given in Sec. 3.2.2, the correlation sub-matrices of (4.3.1), i.e., the correlations among the forward rates ρ(t), among the volatilities θ(t) and the correlations between the forward rates and volatilities φ(t) (the non-diagonal elements only as the diagonals ξ are available from the SABR caplet-related fitting). { i}

4.6 Numerical Simulations

In this section we aim at illustrating the discussed calibration methods and perform simulations with real market data for a certain date5. We took as an example stan- dard EUR market data (yield curve, caps smiles, swaption smiles and CMS spread option smiles as of September 12, 20116. The chosen time grid step ∆T for the imple- mentation setup is 6 months (6M) and the time horizon is 20 years (20Y) ahead, i.e. 40 Tk k=0, with T39 = today+20Y as the last fixing time. In Appendix B.1 the general {setup} is introduced, the market data and briefly the algorithms used to pre-process the market data, i.e. adapt these to our setup, are presented. In Appendix B.2.1 the concrete parameterization of the model is described. We use Doust parameterizations for the forward rate correlations ρ(t) and for the correlations among the volatilities θ(t). In the following we will denote by ”(m x n) Doust param- eterizations” parameterizations with M = m 1 in (1.3.14) for ρ(t), and M = n 1 in (1.3.14) for θ(t). The correlation matrix between− the forward rates and volatilities− φ(t) is parameterized with the two-parameter exponential parameterization given in (4.3.4) throughout the examples and simulations.

4The market usually quotes these products in terms of implied normal volatilities. In this case the Hagan’s formula for the implied SABR normal volatility (cf. Sec. A.5) has to be used instead of (4.5.1) for the calibration of the SABR model parameters for the CMS swap spread. 5All numerical work is implemented in C++, where we made use of the free and open-source QuantLib framework, to be found at quantlib.org. 6The date was chosen randomly. Market data was mainly taken from Thomson Reuters c , with a special thank to Dr. J¨org Kienitz for the market data for CMS spread options. Chapter 4. The SABR-LMM Model and Its Calibration 57 The Appendix B.2.2 is concerned with the calibration of the SABR parameter sets for the Libor rates according to (A.5.1)–(A.5.2), for the SABR model of co-terminal swap rates given in (4.4.1), as well as the SABR parameters for the CMS spread optionlets as described in the Sec. 3.2.2, which are needed for SABR-LMM model calibration. Illustrations of the fitting of the SABR parameter sets complete the prepared data for the model calibration. The first step in the calibration of the SABR-LMM model is the calibration of the time-dependent volatility variables as discussed in Sec. 4.2 (cf. Appendix B.2.2). Then, the calibration of the super-correlation matrix is carried out according to the two methods. Here we denote, for simplistic reasons, with ”new” method the one given in Sec. 4.5, while the ”old” method denotes the approach discussed in Sec. 4.4. It should be mentioned that, while in Sec. 4.4 (”old” method) the calibration of the sub-matrices is carried out sequentially (the nature of the approximations allows this), in the ”new” approach we calibrate the super-matrix as a whole. This lack of flexibility might be, in certain circumstances, disadvantageous. On the other hand it points to self-consistent dependencies in the super-correlation structure. In Appendix B.2.2 the correlation sub-matrices, calibrated according to the ”new” approach, and their difference to the ones calibrated according to the ”old” approach, are depicted. As a benchmark for testing the performance of the calibrated model we consider the (interpolated) market prices of co-terminal swaptions of our setup. Although the ”old” approach explicitly fits the model correlations to the market data of swaptions, in our example the ”new” calibration approach performs slightly better in recovering the swaption prices, by taking the finer information about the correlations from the CMS spreads into account.

Differences betw. simulated prices of co−terminal swaptions ("new" approach, 4x3 factors) Errors become smaller for finer (Doust) parameterizations

−4 x 10

0.02 10

0 5 −0.02

−0.04

Price diffs 0 −0.06

−0.08 −5 0.02 0.02 20 20 0 15 0 15 10 10 −0.02 −0.02 5 5 −0.04 0 −0.04 0 strike spread fixing times strike spread fixing times

Figure 4.1: Differences betw. simulated and market prices of co-terminal swaptions using the ”new” approach for calibrating the correlations: for (4x3) Doust parame- terizations (left), and its slight amelioration for a finer (6x4) Doust parameterization (right). For the right picture we took the difference between the prices simulated with the finer parameterization versus those shown in the left picture. For the simulations we used a 4x2 factor model.

Concerning the simulations, the model is evolved under the spot Libor measure asso- ciated with the discretely rebalanced bank account numeraire in (1.1.10). The drifts for the system of forward rates and volatilities are approximated via the predictor– corrector algorithm, analogous to the algorithm introduced in Appendix A.4 for the Chapter 4. The SABR-LMM Model and Its Calibration 58

Comparison betw. the two approaches (4x3 factors) Comparison betw. the two approaches (3x1 factors)

−4 x 10

−4 x 10 10

10 5

5

0

0

20 −5

−5 15 0.02 20 0.02 10 0 15 0 10 5 −0.02 −0.02 5 −0.04 0 fixing times −0.04 0 strike spread strike spread fixing times

Figure 4.2: These two pictures show the performance of the ”new” method vs. the ”old” one (the parts above zero show the better performance of the ”new” method) for different factorizations. In the left picture the simulations were performed with a 4x2 factor model, while in the right picture a 3x1 factor model was taken. For the correlations we used a (4x3)-Doust parameterization for both cases. LMM. For the simulations we used different number of stochastic factors for the forward rates and their volatilities7, and made use of the spectral decomposition dis- cussed in Sec. 1.3.3 for the correlation matrices. In all the runs the number of Monte Carlo simulations was 20.000. In the Fig. 4.1 the difference between simulated and market prices of co-terminal swaptions using the ”new” approach is shown. The performance of a (4x3) Doust pa- rameterization (left) and a finer (6x4) Doust parameterization (right) are compared. The finer parameterization performs slightly better (right-hand picture). This ame- lioration, though, is very small, and can be interpreted as an evidence for the effec- tiveness and robustness of the method8. The pictures in Fig. 4.2 shall illustrate the performance of the two methods of cal- ibrating the correlations for different factorizations, i.e. different number of model stochastic drivers. They show the difference between the simulated swaption prices with the ”new” correlation calibration method against the ones simulated with cor- relations calibrated according to the ”old” method. We shall identify three effects out of these pictures: the first is that for long-term expiries there is (almost) no difference between the two methods, i.e. they are equally good performing. This fact can easily be explained with the ”freezing” of swap weights which has been applied to the derivation of both approaches. The second effect is that, although in a very small scale, the ”new” approach apparently provides a slightly smaller error between the simulated and market prices of swaptions (the positive parts of the surface). Finally, with increasing the number of stochastic drivers, i.e. retrieving more information from the correlation matrices via the used decomposition, the model errors become smaller; an evidence more of the importance of the information stored in the correlation structure.

7It is a rather standard market usance that in the simulations the number of factors for volatilities (in the stochastic volatility models) is smaller than the one for the Libor rates, attributing ”less importance” to the volatility sub-model. 8For a better proof one would need to have data enough to be able to sketch the dependence of the degree of parameterization fineness on the simulation error. Chapter 5

Conclusion and Outlook

In this work we have developed and discussed a novel approach of calibrating the rich correlation structure in the SABR-LMM model of forward rates, implying market information on correlations from correlation-sensitive derivatives such as CMS spread options, which additionally incorporate information on the swap rate correlations. To this end the derivation of analytical pricing formulas for these products is necessary. Applying the effective Markovian projection (MP) technique we have derived a displa- ced–diffusion SABR model for the spread between the swap rates with different ma- turities. Consequently, we can apply the seminal formula of Hagan et al. to calibrate the SABR model parameters for the CMS spread to the market implied (normal) volatilities of the CMS spread options. For the ATM values in the payoff measure of the projected SDE we have used a standard smile-consistent replication of the necessary convexity adjustment with swaptions. In this way we are able to retrieve the additional information about the swap rate correlations from the whole market smile of CMS spread options and embed it into the SABR-LMM model correlations. Numerical simulations with different parameterizations of the correlation sub-matrices have evidenced effectiveness and robustness of this calibration approach. Simulations with real market data have shown so far a slightly better performance, (even) in terms of pricing errors of swaptions, of this approach in comparison to the rather standard one of calibrating the model correlations to swaptions’ implied volatilities. As ex- pected, the used ”freezing” of the swap rate coefficients turns out to be restrictive for the usage of this method for longer expiries. This is a drawback in both approaches. Nevertheless, the simulations were based only on a snapshot of market data, such that further numerical analysis is certainly necessary to study the properties of this approach for different market movements and different constellations of volatility skews/smiles of the market segments involved before it can be used in practice. The market data involved in the proposed calibration is very broad and these empirical studies would have gone far beyond the scope of this project. Also not presented in this thesis is a performance analysis on the valuation of more complex correlation- sensitive products. Future research should also encompass the parameterizations and the interplay of the different sub-matrices of the model correlation structure, taking also an weighting of the sub-matrices into account for the calibration. A comparison to other stochastic volatility models which use the MP technique for their calibration, such as e.g. SV-LMM and FLTSS-LMM, may also be objective for studies and future developments in this area, particularly regarding the MP technique. 59 Appendix A

Classical Models and SABR-LMM

A.1 Valuation in LMM

With regard to the scope of this work, this section is concerned with the analyt- ical pricing of basic benchmark instruments which are used to calibrate the LMM (and partly also the SABR-LMM). The notation chosen for the different financial instruments is aligned with the notation used in Brigo-Mercurio [2007].

Forward rates and FRAs

An alternative definition of the forward rate is given in the literature (cf. Brigo- Mercurio [2007]) also through a Forward Rate Agreement (FRA). A FRA is a contract giving its holder an interest rate payment for the period between two time instants, T and S, with T

A generalization of the FRA is the interest rate swap. An interest rate (forward starting payer) swap (IRS) on a loan over a period [Tm,Tn], with notional amount 1, is a two-legs contract, starting from a future time instant Tm, to pay a fixed rate K and to receive spot Libor (floating) at the payment dates Tm+1,...,Tn. At every time Ti Tm+1,...,Tn the fixed leg pays out τi 1K, whereas the floating leg pays the ∈ { } − amount τi 1Fi 1(Ti), corresponding to the interest rate Fi, resetting at the previous 2− − time Ti 1 . The IRS paying the fixed leg and receiving the floating leg is termed payer − IRS (IRSp), whereas the opposite case is termed receiver IRS (IRSr). We assume again a notional of one in the sequel. The value of a receiver IRS at time t is given by (cf. Brigo-Mercurio [2007])

n n

IRSr(t,Tm,Tn) = FRA(t,Ti 1,Ti,K) = τi 1B(t,Ti)(K Fi 1(t)) − − − − i=m+1 i=m+1 n

= B(t,Tn) B(t,Tm)+ τi 1B(t,Ti)K. − − i=m+1

Making the contract ”fair” at time t, (i.e. setting IRSr(t,Tm,Tn) = 0) leads to the definition of the swap rate K as (A.1.2) B(t,Tm) B(t,Tn) B(t,Tm) B(t,Tn) K := Sm,n(t) = n − = − , t [0,Tm]. τi 1B(t,Ti) Bm,n(t) ∈ i=m+1 − n The amount Bm,n(t) := τi 1B(t,Ti) is called the swap annuity and represents i=m+1 − a natural numeraire for the swap rate process3. Similarly to caps and floors introduced in the next section, IRS contracts may be settled in arrears or, as assumed above, in advance.

Caps and Floors

A caplet/floorlet is a contract that can be viewed as a payer/receiver FRA where exchange payment is executed only if it has a positive value. The payoff profile of a caplet with strike level κ at reset time T and settlement time S reads

+ + B(T,S) FRA(T,T,S,κ) = B(T,S)τ(T,S) F (T,T,S) κ , − leading to the pricing formula at t T

T (A.1.4) (σBlack)2 (T t) = γ(u,T,S) 2 du5. T − t BS denotes the known Black-Scholes formula6, firstly introduced in the seminal work of Black-Scholes [1973], and given by the equation

(A.1.5) BS F,K,σ,ω = F ωΦ(ωd+(F,K,σ)) KωΦ(ωd (F,K,σ)), − − ln(F/K) σ2/2 (A.1.6) d (F,K,σ)) = ± , ω = +/ 1 (call/put), ± σ − where Φ denotes the standard Gaussian cumulative distribution function. Analogously to (A.1.3), the pricing equation for the corresponding floorlet with the same strike κ reads

(A.1.7) FLL(t,T,S,κ) = B(t,S)τ(T,S)BS F (t,T,S),κ,σBlack(κ)√T t, 1 . T − − We mention here that at time t a caplet7 is at–the–money /out–of–the–money/in– the–money when its strike κ = F (t,T,S) / κ>F (t,T,S) / κ

n

CAP(t,Tm,Tn,κ) = CPL(t,Ti 1,Ti,κ) − i=m+1 n BS Black (A.1.8) = B(t,Ti)τi 1 Fi 1(t),κ,σi 1 (κ) Ti 1 t, 1 , − − − − − i=m+1 4If not otherwise mentioned or notationally specified the implied Black volatility of an option will be an ATM one. 5We see already at this point the severe limitation of the LMM with respect to smile modeling. For caplets with different strikes the model volatility would have to match the equations

T 2 σBlack(κ) (T t) = γ(u,T,S) 2 du, κ. T − ∀ t The impossible task of accurately fitting the whole caplets’ smile surface with a desirable single volatility function (cf. Sec 1.3) has led many authors to introduce stochaticity in the model volatility. 6Widely known also as Black-76. 7For a floorlet the logic goes analogously. Appendix A. Classical Models and SABR-LMM 63

Black with σi 1 (κ) denoting the implied Black-volatility of the caplet CPL(t,Ti 1,Ti,κ), − respectively,− as seen at time t. The value for the corresponding interest rate floor (FLO) is analogous.

A CAP(t,Tm,Tn,κ) is said to be at–the–money at time t if and only if Sm,n(t)= κ. It is instead said to be in–the–money when Sm,n(t) > κ and out–of–the–money if Sm,n(t) < κ. For a floor the moneyness is defined analogously. We see already at this point that while the moneyness for caplets is natural, the one for cap/floors is rather a market convention. Being ATM for a cap/floor does not necessarily imply the same property for the caplets it is built upon. We shall come to this point in Chap. 1.3.2.

A.2 Swap Rate Dynamics and the Choice of Nu- meraire

Given the swap rate dynamics introduced in Def. 1.3, the commonly used swap rate models8 in the literature are the following (cf. Musiela-Rutkowski [2005]):

the co-terminal swap rate model (cf. Jamshidian [1997], Galluccio et al. • [2006]), seen as the more natural model analogous to the forward Libor market model. Within co-terminal SMM for every m =0,...,N 1 the swap rate − B(t,Tm) B(t,TN ) B(t,Tm) B(t,TN ) Sm,N (t) = N − = − , τi 1B(t,Ti) Bm,N (t) i=m+1 − m,N is considered and the Bm,N (t)-induced probability measure P is said to be the co-terminal swap measure for the date Tm if for every k =0,...,N the relative bond price B(t,T )/B (t), t [0,T T ] is a local martingale under Pm,N . k m,N ∈ k ∧ m the co-initial swap rate model (cf. Galluccio-Hunter [2003]) where, analogously • to the co-terminal one, for every m =1,...,N the swap rate S0,m(t), t T0 is considered (fixed starting point). The corresponding probability measure≤ P0,m is induced by the numeraire B0,m(t). The demand for the co-initial SMM is motivated by the desire to value and hedge exotic interest derivatives such as CMS spread options or forward starting swaptions.

the co-sliding swap rate model (cf. Galluccio et al. [2006]), where we no longer • assume that the swap agreements have different lengths but the same start or maturity date. In the co-sliding SMM, on the contrary, the constant length K 1 plays the role of the orientation variable. For every m =0,...,N K the ≥ − considered swap rates are then given by Sm,m+K (t),t Tm. The corresponding m,m+K ≤ probability measure P is induced by Bm,m+K (t). The LMM is the only admissible model of a co-sliding type as it can be seen as a special case of it. We refer to Galluccio et al. [2006] for their extensive studies on the three types of swap rate models and their adequateness in practice.

8Discussions about the differences between these models and further modeling issues can be found in Rebonato [1999b], Brigo-Liinev [2005] and the references therein. Appendix A. Classical Models and SABR-LMM 64 A.3 Valuation in the Log-Normal Swap Market Model

The second class of basic derivatives on interest rates is the class of the swap options, or more commonly swaptions. A European (payer) swaption over period [Tm,Tn] gives the right to enter at the future time Tm into an interest rate (payer) swap (IRS) with strike rate κ, starting from Tm. The underlying length Tn Tm is called the tenor of the swaption. Following Brigo-Mercurio [2007], conside−r the value of the underlying payer IRS at swaption expiry time Tm, i.e.

n

τi 1B(Tm,Ti) Fi 1(Ti 1) κ , − − − − i=m+1 the payoff of the (cash-settled) swaption can be written as

n + + (A.3.1) τi 1B(Tm,Ti) Fi 1(Ti 1) κ = (Sm,n(Tm) κ) Bm,n(Tm), − − − − − i=m+1 as the option will be exercised only if the swap value with strike κ is positive. A fundamental difference between the swaptions and the caps can be obviously derived from (A.3.1), this payoff cannot be decomposed in more elementary products, as for the cap case. As the additive decomposition is not feasible for the (.)+ operator we will need to consider the joint action of the forward rates involved in the contract payoff. Due to the convexity of the operator (.)+ the value of a (payer) swaption is always smaller than the value of the corresponding cap contract. The market convention is to price swaptions with the Black-Scholes formula. Taking the natural numeraire given by Bm,n(t), one can price the payoff (A.3.1) with respect to the measure Pm,n by

+ PSWO(t,T ,T ,κ) = B (t)Em,n S (T ) κ m n m,n t m,n m − (A.3.2) = B (t)BS S (t),κ,σBlack(κ) T t, 1 , t T , m,n m,n m,n m − ≤ m BS Em,n where the formula for is given in (A.1.5)–(A.1.6), t denotes the expectation Pm,n Black with respect to the probability measure , and σm,n (κ) the implied (Black–) volatility of the swaption quoted in the market. Black Within the SMM introduced in Chapter 1.1.4, the ATM Black volatility, σm,n (Sm,n(t)) = Black σm,n , satisfies

Tm 2 (A.3.3) σBlack (T t) = σ (u)2 du. m,n m − m,n t Analogously, the price for the receiver swaption is given by

+ RSWO(t,T ,T ,κ) = B (t)Em,n κ S (T ) m n m,n t − m,n m (A.3.4) = B (t)BS S (t),κ,σBlack(κ) T t, 1 , t T . m,n m,n m,n m − − ≤ m Appendix A. Classical Models and SABR-LMM 65 A.4 Drift Approximation in LMM and Simulations

As discussed in Chapter 1.1.1, the joint dynamics of forward rates under the ter- minal measure (let say Pk+1 as in (1.1.8)-(1.1.9)) is complex and the corresponding stochastic differential equations are high dimensional and contain state dependent drift terms. Hence, the joint distributional evolution of forwards in the terminal pricing measure is impossible to be computed explicitly. This leaves two natural alternatives:

The naive straightforward alternative is to discretize the corresponding stochas- • tic differential equation using some discretization scheme (Euler or Milstein, for example). After that, using a very small time step it is possible to evolve the forward rates on a fine time grid for the price sensitive events.

The second alternative is to try to approximate the state dependent stochas- • tic integrals by an efficient and accurate method which allows for solving the stochastic differential equation approximately analytically. Then the evolving the interest rates over long integration/simulation steps is possible.

The second alternative is more appealing and efficient as for each realization of the forwards only as many steps are needed as there are price sensitive events. There are several ways to improve the Euler scheme for the numerical integration of stochastic differential equations. Instead of using any of the well-known explicit, implicit, or standard predictor-corrector methods, here we are going for the hybrid technique presented in Hunter-J¨ackel-Joshi [2001]. Hereby we integrate the terms k+1 γk(t) dWt in (1.1.4) directly as if the drift coefficient is constant over any one time step. So far this is essentially consistent with the standard Euler method. However, in addition, we account for the indirect stochasticity of the drift term by using a Predictor-Corrector method. The algorithm for constructing one draw from the terminal distribution of the forward rates over one time step reads as follows.

Evolve the logarithms of the forward rates F (t) as if the drifts • j k τiFi(t) σi(t)σj (t)ρi,j (t) , jk 1+τiFi(t) i=k+1   were constant over [t,t +∆ t), and equal to their initial values at t, according to the log-Euler scheme:

d 1 log(F (t +∆t)) = log(F (t)) + √∆tσ (t) b z + k(t) σ (t)2 ∆t, j j j j,i i j − 2 j i=1

with the loading vector bj given in (1.1.6)–(1.1.7). z = z i denotes d-independent standard (0, 1)-random variables (w.r.t. Pk+1). { } N Appendix A. Classical Models and SABR-LMM 66 Compute the drifts at the terminal time with the so evolved forward rates, • k j (t +∆t). Average the initially calculated drift coefficients with the newly computed ones: • 1 ˜k(t) := k(t)+ k(t +∆t) . j 2 j j Re-evolve Fj(t +∆t) using the same normal variates as initially but using the • k new predictor-corrector drift terms ˜j (t).

This is a very natural way to incorporate the drift state-dependence. While further and potentially better approximations are possible (cf. Joshi [2003], Joshi-Stacey [2008]), this approach has the advantage of being very simple to understand and implement, and has been extensively studied with respect to its accuracy (cf. Hunter- J¨ackel-Joshi [2001]).

A.5 SABR Implied Volatility

The (single) SABR model for the forward rate FT (t)= F (t,T,T ∗) is an extension of the CEV model,

∗ βT T (A.5.1) dFT (t) = σT (t)FT (t) dWt , in which 0 β 19 and the volatility parameter σ is assumed to be stochastic, ≤ T ≤ T following

˜ T ∗ T ∗ ˜ T ∗ (A.5.2) dσT (t) = νT σT (t)dWt , dWt ,dWt = ξT dt. Except for the normal case of βT = 0 (cf. Hagan-Lesniewski [2001]), no explicit solution to this model is known. The general case can be solved approximately by means of an asymptotic expansion of (A.5.1)-(A.5.2) in the parameter ǫ = νT √T. In what follows the initial state is given by FT (0) and σT (0). In the sequel the SABR initial volatility parameter σT (0) will be denoted as

SABR σT := σT (0). Following the seminal work Hagan et al. [2002], we write t = Ts, and define X(s)= FT (Ts), Y (s)= σT (Ts)/νT such that the SABR dynamics can be recast in the form

∗ βT T dX(s) = ǫY (s)X(s) dWs , ˜ T ∗ dY (s) = ǫY (s)dWs ,

T ∗ T ∗ where we have also used the well known scaling law WTs = √TWs for Brownian motions. The initial conditions take the shape

SABR X(0) = FT (0), Y (0) = σT /νT .

9 We note that, according to Jourdain [2004], if βT = 1 and the rate-volatility correlation T ∗ parameter ξT > 0, then the process FT (t), while a local martingale, fails to be a martingale in P . Appendix A. Classical Models and SABR-LMM 67 Under typical market conditions, the parameter ǫ is small, and the asymptotic solu- tion is quite accurate and easy implementable in computer code. As a consequence, the asymptotic solution to the SABR model lends itself well to valuation and risk management of large portfolios of options in real time. In order to describe the asymptotic solution, we denote with

σNorm(K) σ˜Norm(T,K,F (0),σSABR,ν , β ,ξ ) =:σ ˜Norm T ≈ T T T T T T the implied normal volatility of an option struck at K and expiring T years from now (i.e. t = 0). The analysis of Hagan et al. [2002] of the model dynamics shows that the implied normal volatility is approximately given by:

1+ 1 log2(F (0)/K)+ 1 log4(F (0)/K)+ ... Norm SABR βT /2 24 T 1920 T σ˜ = σ (FT (0)K) 2 4 T T (1 βT ) 2 (1 βT ) 4 − − 1+ 24 log (FT (0)/K)+ 1920 log (FT (0)/K)+ ... SABR 2 SABR 2 ζ βT (2 βT )(σT ) ξT νT σT βT 2 3ξT 2 1+ − − + + − νT T + ... . δ(ζ) 24(F (0)K)1 βT 4(F (0)K)(1 βT )/2 24 T − T − The distance function entering the formula above is given by: (A.5.3) 2 νT (1 βT )/2 1 2ζξT + ζ ξT + ζ ζ = (F (0)K) − log(F (0)/K), δ(ζ) = log − − . σSABR T T 1 ξ T − T Hagan et al. [2002] describes how to approximatively obtain the implied lognormal volatility for Black’s model

σBlack(K) σ˜Black(T,K,F (0),σSABR,ν , β ,ξ ) =:σ ˜Black T ≈ T T T T T T by giving the following relationship to the normal volatility: σ˜Black(F (0) K) 1 (A.5.4)σ ˜Norm = T T − 1 (˜σBlack)2T + ... . T log(F (0)/K) − 24 T T Hence, the approximated implied Black volatility for the SABR model of an option struck on FT at K, expiring at T, reads then

SABR Black σT σ˜T = 2 4 (1 βT ) 2 (1 βT ) 4 (1 βT )/2 − − (FT (0)K) − 1+ 24 log (FT (0)/K)+ 1920 log (FT (0)/K)+ ... 2 SABR 2 SABR 2 ζ (1 βT ) (σT ) ξT νT σT βT 2 3ξT 2 1+ − + + − νT T + ... . δ(ζ) 24(F (0)K)1 βT 4(F (0)K)(1 βT )/2 24 T − T −

Remark A.1 Given the analogous SABR model for the swap rate Sm,n as in (4.4.1), the approximative implied SABR volatility of a swaption on the swap rate Sm,n, with tenor (T T ) and strike K reads n − m (A.5.5) σBlack(K) σ˜Black(T ,K,S (0),σSABR,ν , β ,ξ ). m,n ≈ m m,n m,n m,n m,n m,n It is obvious to mention that the approximative formula for the SABR implied volatil- ity can be extended to the SABR model of any underlying. In Sec. 3.2.2 we will see an application of this formula for the SABR model of a CMS swap spread. Appendix A. Classical Models and SABR-LMM 68 For practical use in interest rate options portfolio management, an important step is the calibration of the model parameters. For each benchmark option expiry and un- SABR derlying tenor we have to calibrate four model parameters: σT ,νT , βT ,ξT . In order to do it we need market implied volatilities for several different strikes. The experi- ence shows (cf. Rebonato-McKay-White [2009]) that there is a bit of redundancy between the parameters βT and ξT . As a result, one usually calibrates the model by fixing one of these parameters. Two common practices are:

SABR (a) Fix βT , say βT =0.5, and calibrate σT ,νT ,ξT .

SABR (b) Fix ξT =0, and calibrate σT ,νT , βT . Calibration results show a persistent term structure of the model parameters as func- tions of the expiration and underlying tenor. Typical is the shape of the parameter νT which start out high for short dated options and then declines monotonically as the option expiration increases. This indicates presumably that modeling short dated options should include a jump diffusion component (cf. Hagan-Lesniewski [2008]), an ongoing research topic. A good empirical analysis of the SABR parameters can be found in Rebonato-McKay-White [2009].

Special Cases.

Two cases are worthy of special treatment: the stochastic normal model (βT = 0) and the stochastic lognormal model (βT = 1). Both these models are simple enough that the expansion can be continued. For the stochastic normal model (βT = 0) the approximative implied volatilities of European calls and puts are: 2 3ξ σ˜Norm = σSABR 1+ − T ν2 T + ... , T T 24 T log(F (0)/K) ζ (σSABR)2 2 3ξ2 σ˜Black = σSABR T 1+ T + − T ν2 T + ... , T T F (0) K δ(ζ) 24F (0)K 24 T T − T 2 νT 1 2ζξT + ζ ξT + ζ ζ = SABR FT (0)K log(FT (0)/K), δ(ζ) = log − − . σT 1 ξT − For the stochastic log normal model (βT = 1) the approximative implied volatilities are: F (0) K ζ σ˜Norm = σSABR T − T T log(F (0)/K) δ(ζ) T (σSABR)2 ξ ν σSABR 2 3ξ2 1+ − T + T T T + − T ν2 T + ... , 24 4 24 T ζ ν σSABRξ 2 3ξ2 σ˜Black = σSABR 1+ T T T + − T ν2 T + ... , T T δ(ζ) 4 24 T ν 1 2ζξ + ζ2 ξ + ζ ζ = T log(F (0)/K), δ(ζ) = log − T − T . σSABR T 1 ξ T − T Appendix B

Calibration Details

B.1 Bootstrapping the Market Data

Here we describe the general setup, the original and the pre-processed market data.The time horizon chosen for our model setup is 20 years ahead with an equidistant 6 months (6M) time grid. The market data are taken from Thomson Reuters c as of September 12, 2011. The implementation has been carried out in C++ using QuantLib1. The raw market data and the interpolated data to be input to the mod- els are given as follows.

The yield curve: • – Market data: EUR/EURIB yield curve consisting of deposit rates, futures and swap rates; – Interpolated market quotes: set of 6M contiguous forward rates rolled for 20 years ahead. The discount term structure has been bootstrapped from the yield term structure and log-linearly interpolated2, see Fig. B.1.

Forward rates Forward co−terminal swap rates 0.05 0.05

0.045 0.045

0.04 0.04

0.035 0.035

0.03 0.03

0.025 0.025 forwards 0.02 0.02 co−term swap rates 0.015 0.015

0.01 0.01

0.005 0.005

0 0 0 5 10 15 20 0 5 10 15 20 fixing times fixing times Figure B.1: Model Libor and swap rates: 6M contiguous forward Libor rates for the time horizon of 20 years and the corresponding grid of co-terminal swap rates bootstrapped from market data as of Sept. 12, 2011.

1QuantLib Library Version 1.0.1., to be found in http://quantlib.org. 2Day count convention (DC) = Actual365Fixed, business days (BD) = ”Modified Following”. 69 Appendix B. Calibration Details 70 The caps/floors market: • – Market data: EUR/EURIB ATM 6M cap/floor (implied) volatility quotes and a matrix of 6M cap/floor smile spreads3 for strike spreads over ATM ranging within [-3%, +5%] and different expiries up to 30 years ahead; – Interpolated quotes: 6M grid of contiguous 6M ATM caplet series filling the 20 years time horizon. The resulting caplet volatility surface over strikes and expiries uses bi-cubic spline interpolation, see Fig. B.2.

Caplets smile Caplets prices

1.5 0.015

1 0.01

impl. vol 0.5 0.005 caplet prices

0 0 0 0 −0.02 −0.02 5 5 0 0 10 10 0.02 0.02 15 15 0.04 0.04 20 fixing times 20 fixing times strike spread strike spread Figure B.2: Model caplets: grid of implied volatilities and Black-prices of contiguous 6M caplet series rolled 20 years ahead for the given strikes, interpolated from the market data as of Sept. 12, 2011.

The swaption volatilities: • – Market data: EUR/EURIB ATM swaption volatilities for tenors from 1Y to 30Y, for different expiries up to 30 years ahead; EUR/EURIB swaption smile spreads for the strike spreads over ATM ranging in [-4%, +5%], and for tenors up to 30 years and expiries up to 20 years ahead; – Interpolated quotes: 6M spaced series of co-terminal swaps for the 20 years model time horizon, with 6M fixing and floating legs; ATM volatilities for the co-terminal swaps built by means of bilinear interpolation from the market data; series of 6M–expiring swaptions with the co-terminal swaps as underlyings, for each given absolute strike; resulting in a surface of absolute swaption volatilities with the axis (expiry strike), see Fig. B.3. × CMS Spread Optionlets4: • – Market data: implied ATM normal volatilities for the Single Look option- lets on the EUR/EURIB CMS spreads 10Y/2Y, 30Y/2Y and 30Y/10Y for different expiries from 2 weeks up to 20 years ahead; additionally nor- mal volatility smile spreads (in basis points (bp)) for the mentioned CMS spread optionlets with the absolute strike range [-0.5%, +1.5%] and ex- piries up to 20 years ahead;

3The cap/floor volatility smile is given in terms of vol spreads over the ATM volas. 4We thank Dr. J¨org Kienitz for providing the CMS spread options data used for the calibration examples in this chapter. Appendix B. Calibration Details 71

Swaptions smile Swaptions prices

1 0.5

0.8 0.4

0.3 0.6

impl vol 0.2 0.4

Swaption prices 0.1

0 0.2 0 −0.04 −0.04 0 5 −0.02 −0.02 5 10 0 10 0 15 0.02 15 0.02 20 fixing times 20 fixing times strike spread strike spread Figure B.3: Model swaptions: grid of implied volatilities and Black-prices of swap- tions on co-terminal swaps with 6M stepping expiries, interpolated from the market data as of Sept. 12, 2011.

– Interpolated quotes: normal volatility smile in bp of CMS spread optionlets on 10Y/2Y, 20Y/2Y and 20Y/10Y on a 6M grid of expiries, given in Table B.1, built by using 2D-interpolation.

B.2 Parameterization of SABR–LMM and Its Cal- ibration

B.2.1 Parameterization of the SABR–LMM Model

Starting from the time-homogeneous SABR–LMM model, given in Def. 4.1, we shall use the following parameterization of the model variables:

For the forward rate volatility we will use the parameterization (4.2.1) with the • time-homog. function g given in (4.2.2).

For the volatility of volatility function h we again will use a time-homog. pa- • rameterization given in (4.2.5).

For the super–correlation matrix (4.3.1) we will use full-rank mixed parameter- • izationsas follows:

– Doust’s correlation parametrization for the forward rate – forward rate submatrix, given in (1.3.14); – Doust’s parametrization for the volatility – volatility correlations, given (1.3.14), or alternatively, the two-parameter exponential parameterization given in (1.3.9); – The parametrization given in Example 4.3 (4.3.4) for the correlations be- tween the forward rates and the volatilities. Appendix B. Calibration Details 72

10Y/2Y ATM rate -0.5% -0.25% 0% 0.25% 0.5% 0.75% 1.0% 1.5% 6M 0.0117850 2 2 2 27 21 14 9 3 1Y 0.0123250 33 33 33 109 91 74 59 40 1Y6M 0.0114528 63 64 64 189 159 132 107 75 2Y 0.0109419 96 97 98 269 229 191 158 114 2Y6M 0.0096187 133 134 135 347 297 250 209 154 3Y 0.0082777 173 176 178 424 364 308 260 198 3Y6M 0.0073985 216 219 221 495 426 363 310 240 4Y 0.0065000 262 266 269 567 489 420 361 285 4Y6M 0.0061953 311 316 320 638 553 476 413 332 5Y 0.0059208 362 369 374 711 618 535 467 381 5Y6M 0.0059321 416 424 430 785 685 596 523 431 6Y 0.0059784 472 482 489 861 753 659 582 484 6Y6M 0.0060139 529 540 549 938 823 723 641 538 7Y 0.0060692 590 602 612 1018 896 790 704 595 7Y6M 0.0061427 650 664 675 1098 970 858 767 652 8Y 0.0063303 714 729 741 1181 1046 928 833 712 8Y6M 0.0063114 779 796 809 1264 1122 999 899 773 9Y 0.0064165 848 866 880 1349 1200 1072 968 836 9Y6M 0.0064623 916 936 951 1430 1276 1143 1035 898 10Y 0.0066037 989 1010 1026 1514 1354 1216 1105 963 10Y6M 0.0067320 1063 1085 1103 1596 1431 1289 1174 1028 20Y/2Y ATM rate -0.5% -0.25% 0% 0.25% 0.5% 0.75% 1.0% 1.5% 6M 0.0158146 4.59 4.34 4.35 3.96 33.69 26.93 20.30 9.61 20Y/10Y ATM rate -0.5% -0.25% 0% 0.25% 0.5% 0.75% 1.0% 1.5 6M 0.0040295 1.08 1.24 1.25 4.39 1.12 0.21 0.37 0.84

Table B.1: Interpolated 10Y/2Y, 20Y/2Y, 20Y/10Y CMS spread volatility smile (in bp) for the different strikes (first row), and the calculated ATM spread rates on the 6M grid of expiries (first left column) within the model time horizon, based on data as of Sept. 12, 2011.

B.2.2 Calibration Procedure

The calibration starts with bootstrapping and preprocessing the market data to fit a predefined (6M)-equidistantly spaced time grid, 0,T0,...,TN , (TN 1 = 20Y, the { } − last fixing date) of model forward rates and co-terminal swaps, briefly described in Appendix B.1. Then, the following calibration steps have to be carried out:

The Calibration of the SABR Forward and Swap Rate Parameters

As the next step we have to calibrate the (auxiliary) SABR models for the forward ((A.5.1)– (A.5.2)) and swap rates (4.4.1) separately, by using the formulas in Ap- pendix A.5 which can be split in the following steps:

Calibration of the time-homogeneous SABR model of forward rates F , i • i ∈ Appendix B. Calibration Details 73 0,...,N 1 , as defined in ((A.5.1)– (A.5.2)). The β parameters are set to { − } i the constant βi β =0.5, throughoutly. The calibration consists of optimizing ≡ SABR over the SABR parameters, σi ,νi and ξi, by using the Levenberg-Marquardt algorithm (cf. (4.1.1)): 2 Black SABR Black (B.2.1) min σ˜ (Ti,K,Fi(0),σ ,νi, 0.5,ξi) σ (K) , SABR i i σ ,νi,ξi − i K Black where the sum is taken over the available strikes K, σi (K) is the (inter- polated) market implied caplet volatility at strike K, and the SABR implied volatilityσ ˜Black is given in Chap. A.5. The goodness of the fit of the forward rate SABR parameters is illustrated in the Fig. B.4 (left picture).

SABR fitting to caplets prices SABR fitting to swaptions prices

0.04 0.08

0.03 0.06

0.02 0.04 rel. diffs 0.01 rel. diffs 0.02 0 20 0 0 −0.02 5 15 0 0.02 10 10 0 0.02 15 −0.02 5 0.04 20 0 fixing times strike spread −0.04 fixing times strike spread

Figure B.4: Relative errors to market Black-prices of model caplets (left) and co-term. swaptions (right) of the prices calculated using the calibrated SABR parameters.

Calibration of the SABR dynamics for the co-terminal swap rate Si,N , i • 0,...,N 1 , as given in (4.4.1). We again make use of the Levenberg-∈ Marquardt{ −algorithm} to calibrate the SABR parameter for the grid of co- terminal swaps, where again the exponent βi,N 1/2 is kept constant and SABR ≡ the other three parameters, σi,N , νi,N and ξi,N , are calibrated to match the bootstrapped swaption smile, 2 Black SABR Black min σ˜ (Ti,κ,Si,N (0),σ ,νi,N , 0.5,ξi,N ) σ (κ) , SABR i,N i,N σ ,νi,N ,ξi,N − i,N κ Black with κ representing the available swaption strikes, σi,N (κ) the market implied swaption volatility at strike κ, and the SABR implied volatilityσ ˜Black given in Chap. A.5. The fit of the SABR parameters for the swap rates is illustrated in the Fig. B.4 (right picture). Calibration of the SABR-DD dynamics (cf. (3.2.39)) of the CMS spreads • Sk,n1,n2 , k 0,..., max n1,n2 1 , to the given implied (normal) volatil- ity smile given∈ { in Tab. B.1,{ by}− using} the formulae in Sec. A.5, together with Hagan [2003] replication approach (3.2.46) for the expectation of the spread in the corrsp. forward measure. The determination of the SABR parame-

ters, Γk,n1,n2 , Υk,n1,n2 and qk,n1,n2 , (βk,n1,n2 = 1), is done by using the standard Levenberg-Marquardt optimization algorithm. Appendix B. Calibration Details 74 The following Fig. B.5 shows the fitting of the SABR parameters for the CMS spread 10Y/2Y, i.e. the difference between the normal volatilities calculated with the calibrated SABR parameters and the market volatilities given in Table B.1. It should be mentioned that the fitting implicitly includes the approxima- tion errors of the Hagan’s convexity adjustment method in (3.2.46).

Smile fitting of the CMS Spread 10y − 2y

0.08

0.06

0.04

0.02 10

0 8 15 6 10 4 −3 x 10 5 2 0 fixing times strike −5 0 Figure B.5: Fitting error of the normal volatilities after the calibration of the SABR parameters for the 10Y/2Y CMS spread.

The Calibration of the SABR–LMM Model

Finally, in what follows we describe briefly the calibration of the time-homogeneous SABR-LMM for the forward rates, as given in Def. 4.1.

To calibrate the forward rate volatility in the SABR–LMM model we follow the • approach discussed in Chap. 4.2, by using the parameterization given in Sec. B.2.1. To this end we should

– perform (4.2.3) to calibrate the time-homogeneous function g and adjust the initial rate-dependent values ki(0) according to (4.2.4); – calibrate the function h as in (4.2.6), by taking the restriction (4.2.8) into account.

Using the full-factor parameterizations introduced in Sec. B.2.1, the calibration • of the super–correlation matrix is carried out in the following two different ways:

– Given the calibrated SABR parameters for the swap rate dynamics (cf. (4.4.1)), we follow the approach given in Chap. 4.4 (”old” approach), where the correlation submatrices of (4.3.1) are successively calibrated according to the equations (4.4.4)–(4.4.7); – Following the (”new”) approach in Chap. 4.5, we use the determined SABR-DD parameters for the CMS spreads to derive via (3.2.41)–(3.2.44) the parameters of the super–correlation matrix.

In both cases we use square differences between the model parameters and the target values as objective function and apply the Levenberg-Marquardt opti- mization algorithm. Appendix B. Calibration Details 75 The next Fig. B.6 - B.8 show the correlation sub-matrices, calibrated according to the ”new” approach, and their difference to the ones calibrated according to the ”old” approach. As introduced in Sec. B.2.1 we used for these examples a Doust-parameterization of degree M = 3 (4 parameters) for the forward rate correlations, and a Doust-parameterization of degree M = 2 (3 parameters) for the volatility correlations. Finally, (4.3.4) is used for the cross correlations between the forward rates and the volatilities. The pictures show that, while for short-term expiries the correlations between the Libor rates is lower than in the ”old” approach, the other sub-matrices, i.e. vol. - vol. and Libor rate - vol. correlations, generally are higher. The full picture of the super-correlation matrix (cf. (4.3.1)) is given in the last Fig. B.9, calibrated according to the ”new”-approach.

Forward Libor rates correlations (new method) Forward Libor rates correlations (diff. betw. "new" and "old" approaches)

1 0

0.8 −0.005

0.6 −0.01

0.4 −0.015

0.2 −0.02

0 −0.025 20 20 15 20 15 20 10 15 10 15 10 10 5 5 5 5 0 0 0 0 fixing times fixing times fixing times fixing times

Figure B.6: Matrix of correlations betw. the Libor rates (left), calibrated to the CMS spread options (”new” approach), and its difference to the corresponding correlation matrix computed according to Chap. 4.4 (”old” approach).

Volatility − volatility correlations (new method) Volatility − volatility correlations (diff. betw. "new" and "old" approaches)

1 0.01

0.008 0.8

0.006 0.6 0.004 0.4 0.002

0.2 0 20 20 0 15 20 15 20 10 15 15 10 10 10 5 5 5 5 0 0 fixing times 0 0 fixing times fixing times fixing times

Figure B.7: Matrix of correlations betw. the Libor rate volatilities (left), calibrated to the CMS spread options (”new” approach), and its difference to the corresponding correlation matrix computed according to Chap. 4.4 (”old” approach). Appendix B. Calibration Details 76

Libor rate − volatility correlations (new method) Libor rate − volatility correlations (diff. betw. "new" and "old" approaches)

0.25

0 0.2

−0.1 0.15 −0.2 0.1 −0.3 0.05 −0.4 20 20 0 20 15 15 15 20 10 10 10 15 10 5 5 5 5 fixing times 0 0 fixing times 0 0 fixing times fixing times

Figure B.8: Matrix of cross correlations betw. the Libor rates and their volatilities (left), calibrated to the CMS spread options (”new” approach), and its difference to the corresponding correlation matrix computed according to Chap. 4.4 (”old” approach).

Super correlation matrix (new method)

1

0.5

0

−0.5 80 60 80 60 40 40 20 20 0 0

Figure B.9: The full super-correlation matrix (4.3.1), calibrated according to the ”new”-approach. Bibliography

C. Alexander: Common Correlation and Calibrating the Lognormal Forward Rate Model. Wilmott Magazine, March 2003, pp. 68-78. L. Andersen and J. Andreasen: Volatilite Volatilities. Risk, Vol. 15:12 (2002), pp. 163–168. L. Andersen and R. Brotherton-Ratcliffe: Extended Libor Market Models With Stochastic Volatility. Journal of Computational Finance, Vol. 9:1 (2005), pp. 1– 40. L. Andersen and L. Piterbarg: Moment Explosions in Stochastic Volatility Models. Finance and Stochastics, Vol. 11 (2007), pp. 29-50. A. Antonov and M. Arneguy: Analytical Formulas for Pricing CMS Products in the Libor Market Model with Stochastic Volatility. Working paper, 2009. A. Antonov and T. Misirpashaev: Markovian Projection Onto a Displaced Diffusion. Generic formulas with applications. International Journal of Theor. and Appl. Fi- nance, Vol. 12:4 (2009), pp. 507–522. D. Belomestny, A. Kolodko and J. Schoenmakers: Pricing CMS Spread Options in a Libor Market Model. International Journal of Theor. and Appl. Finance, Vol. 13:1 (2010), pp. 45–62. E. Benhamou: A Martingale Result for Convexity Adjustment in the Black Pricing Model. Working Paper, 2000. F. Black: The Pricing of Commodity Contracts. Journal of Financial Economics, Vol. 3 (1976), pp. 167-179. F. Black and M. Scholes: The Pricing of Options and Corporate Liabilities. Journal of Political Economy, Vol. 81:3 (1973), pp. 637-654. R. H. B¨orger; J. v. Heys: Calibration of the Libor Market Model Using Correlations Implied by CMS Spread Options. Applied Mathematical Finance, Vol 17:5 (2010), pp. 453–469. A. Brace: Arbitrage-Free Lognormal Interest Rate Models. Risk Seminars, 2nd draft, 1997. A. Brace, D. Gatarek and M. Musiela: The Market Model of Interest Rate Dynamics. Mathematical Finance, Vol. 7:2 (1997), pp. 127–155. 77 BIBLIOGRAPHY 78 D. Brigo, C. Capitani and F. Mercurio: Different Covariance Parametrizations of the Libor Market Model and Joint Caps/Swaptions Calibration. Working paper, 2003.

D. Brigo and J. Liinev: On the Distributional Distance Between the Lognormal LI- BOR and Swap Market Models. Quantitative Finance, Vol. 5:5 (2005), pp. 433-442.

D. Brigo and F. Mercurio: Interest Rate Models - Theory and Practice. With Smile, Inflation and Credit. 2’nd edition. Springer Finance, 2007.

A. Castagna, F. Mercurio and M. Tarenghi: Smile-Consistent CMS Adjustment in Closed Form: Introducing the Vanna-Volga Approach. Working paper, 2007.

R. Cont and P. Tankov: Financial Modelling for Jump Processes. Chapman and Hall, 2004.

J. Cox: Notes on Option Pricing I: Constant Clasticity of Variance Diffusions. Jour- nal of Portfolio Management, Vol. 22 (1996), pp. 15-17.

J. C. Cox, J. E. Ingersoll and S. A. Ross: A Theory of the Term Structure of Interest Rates. Econometrica, Vol. 53 (1985), pp. 385-407.

P. Doust: Modelling Discrete Probabilities. Working paper, 2007.

B. Dupire: Pricing with a smile. Risk, Vol. 7:1 (1994), pp. 18–20.

B. Dupire: A Unified Theory of Volatility. Working Paper, 1994.

S. Galluccio, J.-M. Ly, Z. Huang and O. Scaillet: Theory and Calibration of Swap Market Models. Mathematical Finance, Vol. 17:1 (2006), pp. 111–141.

S. Galluccio and C. Hunter: The Co-Initial Swap Market Model. Working paper, 2003.

D. Gatarek, P. Bachert and R. Maksymiuk: The LIBOR Market Model in Practice. Wiley Finance, 2006.

H. Geman, N. El Karoui, J.C. Rochet: Changes of Num´eraire, Changes of Probability Measure and Option Pricing. Journal of Applied Probability, Vol. 32:2 (1995), pp. 443–458.

P. Glasserman: Monte Carlo Methods in Financial Engineering. Springer, Berlin, 2003.

I. Gyoengy: Mimicking the One-Dimensional Marginal Distributions of Processes Having an Ito Differential. Probability Theory and Related Fields, Vol. 71 (1986), pp. 501–516.

P. Hagan, A. Lesniewski and D.E. Woodward: Probability Distribution in the SABR Model of Stochastic Volatility. Working paper, 2001.

P. S. Hagan: Convexity conundrums: Pricing CMS Swaps, Caps and Floors. Wilmott Magazine, March 2003, pp. 38–44. BIBLIOGRAPHY 79 P. S. Hagan and M. Konikov: Interest Rate Volatility Cube: Construction and Use. Working paper, 2004.

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

P. Hagan and A. Lesniewski: LIBOR Market Model With SABR Style Stochastic Vvolatility Models. Working paper, 2008.

D. Heath, R. Jarrow and A. Morton: Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation. Econometrica, Vol 60 (1992), pp. 77–105.

J. C. Hull: Options, Futures and Other Derivatives. 6’th edition. Prentice Hall, 2005.

J. C. Hull and A. White: Efficient Procedures for Valuing European and American Path Dependent Options. Journal of Derivatives, Vol. 1 (1993), 21–31.

J. C. Hull and A. White: Forward Rate Volatilities, Swap Rate Volatilities, and the Implementation of the Libor Market Model. Journal of Fixed Income, Vol. 10:2 (2000), pp. 46-62.

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

P. J. Hunt and A. Pelsser: Arbitrage-Free Pricing of Quanto-Swaptions. Journal of Financial Engineering, Vol. 7:1 (1998), pp. 25–33.

C. J. Hunter, P. J¨ackel and M. S. Joshi: Drift Approximations in a Forward-Rate- Based LIBOR Market Model. Published as Getting the Drift, Risk, July 2001, pp. 81-84.

F. Jamshidian: LIBOR and Swap Market Models and Measures. New Products and Strategic Trading. Finance and Stochastics, Vol. 1 (1997), pp. 293–330.

M. S. Joshi: Rapid Computation of Drifts in a Reduced Factor LIBOR Market Model. Wilmott Magazine, May 2003, pp. 84–85.

M. Joshi and A. Stacey: New and Robust Drift Approximations for Libor Market Model. Quantitative Finance, Vol. 8 (2008), pp. 427–434.

B. Jourdain: Loss of Martingality in Asset Price Models with Lognormal Stochastic Volatility. Working paper, 2004.

I. Karatzas and S.E. Shreve: Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, 2’nd Ed., Springer, 1991.

J. Kienitz and M. Wittkey: Option Valuation in Multivariate SABR Models - with an Application to the CMS Spread. Working Paper. 2010.

P. H. Labordere: A General Asymptotic Implied Volatility for Stochastic Volatility Models. Working paper, 2005. BIBLIOGRAPHY 80 P. H. Labordere: Unifying the BGM and SABR Models: A Short Ride in Hyperbolic Geometry. Working paper, 2006.

P. H. Labordere: Combining the SABR and LMM models. Risk, October 2007, pp. 102-107.

Y. Lu and S. Neftci: Convexity Adjustment and Forward Libor Model: Case of Con- stant Maturity Swaps. FINRISK - Working Paper Series, 2003.

M. Lutz: Extracting Correlations From the Market: New Correlation Parameteriza- tions and the Calibration of A Stochastic Volatility LMM to CMS Spread Options. Working paper, 2010.

W. Margrabe: The value of an Option to Exchange One Asset for Another. Journal of Finance, Vol. 33:1 (1978), pp. 177–186.

F. Mercurio and A. Pallavicini: Smiling at Convexity. Risk, August 2006, pp. 64-69.

F. Mercurio and M. Morini: Joining the SABR and Libor Models Together. Risk, March 2009, pp. 80–85.

R. C. Merton: Theory of Rational Option Pricing. The Bell Journal of Economics and Management Science, Vol. 4:1 (1973), pp. 141-183.

M. Morini and N. Webber: An EZI Method to Reduce the Rank of a Correlation Matrix in Financial Modelling. Applied Mathematical Finance, Vol. 13:4 (2006), pp. 309–331.

M. Musiela and M. Rutkowski: Martingale Methods in Financial Modelling, Appli- cation of Mathematics: Stochastic Modelling and Applied Probability. 2’nd edition, Springer Verlag, 2005.

A. Pelsser: Mathematical Foundation of Convexity Correction. Quantitative Finance, Vol. 3:1 (2003), pp. 59–65.

V. Piterbarg: A Stochastic Volatility Forward LIBOR Model with a Term Structure of Volatility Smiles. Working Paper, 2003.

V. Piterbarg: Time to Smile. Risk, Vol. 18:5 (2005), pp. 71–75.

V. Piterbarg: Stochastic Volatility Model With Timedependent Skew. Applied Math- ematical Finance, Vol. 12:2 (2005), pp. 147–185.

V. Piterbarg: Markovian Projection Method for Volatility Calibration. Risk, Vol. 20:4 (2007), pp. 84–89.

R. Rebonato: Interest Rate Option Models. 2’nd edition, John Wiley Publ., 1998.

R. Rebonato: On the Simultaneous Calibration of Multifactor Lognormal Interest- Rate Models to Black Volatilities and to the correlation matrix. Journal of Comp. Finance, Vol. 2:4 (1999), pp. 5–27. BIBLIOGRAPHY 81 R. Rebonato: On the Pricing Implications of the Joint Lognormal Assumption for the Swaption and Cap Markets. Journal of Comp. Finance, Vol. 2:3 (1999), pp. 57–76.

R. Rebonato: Modern Pricing of Interest Rate Derivatives: The Libor Market Model and Beyond. 1’st edition. Princeton University Press, 2002.

R. Rebonato: Volatility and Correlation: The Perfect Hedger and the Fox. 2’nd edi- tion, Wiley Finance, 2004.

R. Rebonato: A Time-Homogeneous, SABR-Consistent Extension of the LMM: Cal- ibration and Numerical Results. Risk, Vol. 20:11 (2007), pp. 92–97.

R. Rebonato and P. Jaeckel: The Most General Methodology to Create a Valid Corre- lation Matrix for Risk Management and Option Pricing Purposes. Journal of Risk, Vol. 2:2, (1999), pp. 17–28.

R. Rebonato and P. Jaeckel: Linking Caplet and Swaption Volatilities in a BGM/J Framework: Approximate Solutions. Journal of Comp. Finance, Vol. 6:4 (2003), pp. 41–60.

R. Rebonato and R. White: Linking Caplets and Swaption Prices in the LMM-SABR model. Journal of Comp. Finance, Vol. 13:2 (2009), pp. 19–46.

R. Rebonato, K. McKay and R. White: The SABR/LIBOR Market Model: Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives. Wiley Finance, 2009.

M. Rubinstein: Displaced Diffusion Option Pricing. Journal of Finance, Vol. 38:1 (1983), pp. 213–317.

J. Schoenmakers and B. Coffey: Systematic Generation of Parametric Correlation Structures for the LIBOR Market Model. International Journal of Theor. and Appl. Finance, Vol. 6:4 (2003), pp. 1–13.

J. Schoenmakers: Calibration of LIBOR Models to Caps and Swaptions: A Way Around Intrinsic Instabilities Via Parsimonious Structures and a Collateral Market Criterion. Weierstrass-Institut f. Angewandte und Stochastik. WIAS Preprint 740, Work presented at Risk Europe 2002, Paris 23 & 24 April 2002.

J. Schoenmakers: Robust Libor Modelling and Pricing of Derivative Products. CRC Financial Mathematics Series, Chapman & Hall / CRC, 2005.

L. Wu and Z.Y. Zhang: Optimal low-rank approximation of correlation Matrices. Linear Algebra and Its Application, Vol. 364 (2003), pp. 161–187.

L. Wu and F. Zhang: Libor Market Model with Stochastic Volatility. Journal of In- dustrial and Management Optimization, Vol 2:2 (2006), pp. 199–227. List of Figures

4.1 Differences betw. simulated and market prices of co-terminal swap- tions using the ”new” approach for calibrating the correlations: for (4x3) Doust parameterizations (left), and its slight amelioration for a finer (6x4) Doust parameterization (right). For the right picture we took the difference between the prices simulated with the finer param- eterization versus those shown in the left picture. For the simulations weuseda4x2factormodel...... 57 4.2 These two pictures show the performance of the ”new” method vs. the ”old” one (the parts above zero show the better performance of the ”new” method) for different factorizations. In the left picture the simulations were performed with a 4x2 factor model, while in the right picture a 3x1 factor model was taken. For the correlations we used a (4x3)-Doust parameterization for both cases...... 58

B.1 Model Libor and swap rates: 6M contiguous forward Libor rates for the time horizon of 20 years and the corresponding grid of co-terminal swap rates bootstrapped from market data as of Sept. 12, 2011. ... 69 B.2 Model caplets: grid of implied volatilities and Black-prices of con- tiguous 6M caplet series rolled 20 years ahead for the given strikes, interpolated from the market data as of Sept. 12, 2011...... 70 B.3 Model swaptions: grid of implied volatilities and Black-prices of swap- tions on co-terminal swaps with 6M stepping expiries, interpolated fromthemarketdataasofSept. 12,2011...... 71 B.4 Relative errors to market Black-prices of model caplets (left) and co- term. swaptions (right) of the prices calculated using the calibrated SABRparameters...... 73 B.5 Fitting error of the normal volatilities after the calibration of the SABR parametersforthe10Y/2YCMSspread...... 74 B.6 Matrix of correlations betw. the Libor rates (left), calibrated to the CMS spread options (”new” approach), and its difference to the corre- sponding correlation matrix computed according to Chap. 4.4 (”old” approach)...... 75 B.7 Matrix of correlations betw. the Libor rate volatilities (left), calibrated to the CMS spread options (”new” approach), and its difference to the corresponding correlation matrix computed according to Chap. 4.4 (”old”approach)...... 75

82 LIST OF FIGURES 83 B.8 Matrix of cross correlations betw. the Libor rates and their volatilities (left), calibrated to the CMS spread options (”new” approach), and its difference to the corresponding correlation matrix computed according toChap.4.4(”old”approach)...... 76 B.9 The full super-correlation matrix (4.3.1), calibrated according to the ”new”-approach...... 76