Markov Functional Interest Rate Models with Stochastic Volatility

Markov Functional interest rate

models with stochastic volatility

New College University of Oxford

A thesis submitted in partial fulﬁllment of the MSc in Mathematical Finance

December 9, 2009

To Rahel

Acknowledgements

I would like to thank my supervisor Dr Jochen Theis for advising me throughout the project and proof–reading of this dissertation. Furthermore I want to extend my gratitude to d–ﬁne GmbH for giving me the opportunity to attend the MSc in Mathematical Finance programme. But above all I am indebted to my family, especially to my wife Rahel, for their great support and patience.

Abstract

With respect to modelling of the (forward) interest rate term structure under consideration of the market observed skew, stochastic volatility Libor Market Models (LMMs) have become predominant in recent years. A powerful rep- resentative of this class of models is Piterbarg’s forward rate term structure of skew LMM (FL–TSS LMM). However, by construction market models are high– dimensional which is an impediment to their eﬃcient implementation.

The class of Markov functional models (MFMs) attempts to overcome this in- convenience by combining the strong points of market and short rate models, namely the exact replication of prices of calibration instruments and tractability. This is achieved by modelling the numeraire and terminal discount bond (and hence the entire term structure) as functions of a low–dimensional Markov process whose probability density is known.

This study deals with the incorporation of stochastic volatility into a MFM framework. For this sake an approximation of Piterbarg’s FL–TSS LMM is devised and used as pre–model which serves as driver of the numeraire discount bond process. As a result the term structure is expressed as functional of this pre–model. The pre–model itself is modelled as function of a two–dimensional Markov process which is chosen to be a time–changed brownian motion. This approach ensures that the correlation structure of Piterbarg’s FL–TSS is imposed onto the MFM, especially the stochastic volatility component is inherited.

As part of this thesis an algorithm for the calibration of Piterbarg’s FL–TSS LMM to the swaption market and the calibration of a two–dimensional Libor MFM to the (digital) caplet market was implemented. Results of the obtained skew and volatility term structure (Piterbarg parameters) and numeraire discount bond functional forms are presented.

Contents

1 Introduction 1

2 A review of Libor Market and Markov Functional Models 3 2.1 TheLiborMarketModel...... 3 2.1.1 Non–log–normal forward Libor dynamics ...... 5 2.1.2 Incorporation of stochastic volatility ...... 6 2.2 MarkovFunctionalModels...... 7 2.2.1 Deﬁnition and examples of Markov Functional Models ...... 8 2.2.2 ALiborMarkovFunctionalModel ...... 10 2.2.3 Multi–dimensional Markov Functional Models ...... 12 2.3 A Libor Market Model as pre–model for a Markov Functional Model . . . . 14

3 Piterbarg’s term structure of skew forward Libor model 17 3.1 TheforwardLibordynamics...... 17 3.2 Swap rate dynamics under the FL-TSS model ...... 19 3.2.1 Derivation of the forward swap volatility level ...... 20 3.2.2 Derivation of the forward swap skew ...... 21 3.3 The effective skew and volatility formulation ...... 23 3.3.1 Theeffectiveforwardswapskew ...... 23 3.3.2 The effective forward swap volatility ...... 26 3.4 CalibrationoftheFL-TSSmodel ...... 30 3.4.1 Forward rate volatility calibration ...... 31 3.4.2 Forwardrateskewcalibration ...... 35 3.4.3 Calibrationresults ...... 39

4 A Markov functional model with stochastic volatility 41 4.1 Piterbarg’s FL–TSS Libor Market Model as pre–model ...... 42 4.2 The pre–model with two Brownian drivers ...... 43 4.3 A simpliﬁcation of the pre–model process ...... 44

i 4.4 Construction of a two–dimensional Libor Markov functional model . . . . . 47 4.4.1 A two–dimensional Libor Markov functional model in the terminal measure...... 48 4.4.2 Calibrationresults ...... 50

5 Conclusion 53

A Mathematical details 55 A.1 ThedrifttermintheLiborMarketModel ...... 55 A.2 The derivative of the forward swap rate w.r.t the forward Liborrates. . . . 58

A.3 Derivation of the coeﬃcient cmn ...... 59 A.4 Proofofcorollary3.3.2...... 60 A.5 A recursion scheme for a system of time dependent Riccati equations . . . . 63

A.5.1 An analytic solution for Di t, Ti+1 ...... 64

A.5.1.1 The case gi =0 ...... 65 6 A.5.1.2 The case gi =0 ...... 66

A.5.2 An analytic solution for Ai t, Ti+1 ...... 66

A.5.2.1 The case gi =0 ...... 66 6 A.5.2.2 The case gi =0 ...... 68 A.5.3 Summaryofthesolution...... 70 A.6 Derivationofrelation(3.28) ...... 70 A.7 2d–Markov functional integration ...... 72

B The Heston Model 75 B.1 Speciﬁcationofthemodeldynamics ...... 75 B.2 Thecharacteristicfunction ...... 77 B.3 ThesolutionoftheHestonODE ...... 78 B.3.1 Boundaryconditions ...... 78 B.3.2 AsystemofRiccatiODEs...... 79 B.4 Option pricing by transformation techniques ...... 80 B.5 CalibrationoftheHestonModel ...... 86

C Tables and ﬁgures 87

Bibliography 93

ii List of Figures

3.1 Volatility level λ10(t) and skew β10(t) of forward rate F10(t) for times T0 = 0y t < T = 10y...... 40 ≤ 10 ¯ 4.1 Proxy forward rate F10(T10) (4.11) as function of zt = 0,z2) at reset time T = 10y. This corresponds to the zero correlation case, Γ(s) 0...... 47 10 ≡ ¯ ¯ 4.2 The numeraire discount bond as functional of F10(T10, zT 10): D(T10, T11; F10(T10, zT 10)). 51

¯ ¯ C.1 The numeraire discount bond as functional of F1(T1, zT 1): D(T1, T11; F1(T1, zT 1)). 89 ¯ ¯ C.2 The numeraire discount bond as functional of F3(T3, zT 3): D(T3, T11; F3(T3, zT 3)). 89 ¯ ¯ C.3 The numeraire discount bond as functional of F5(T5, zT 5): D(T5, T11; F5(T5, zT 5)). 90 ¯ ¯ C.4 The numeraire discount bond as functional of F10(T7, zT 7): D(T7, T11; F7(T7, zT 7)). 90

iii

Chapter 1

Introduction

This study is dedicated to the incorporation of stochastic volatility into a Markov functional framework. The class of Markov functional models (MFM) was introduced by Hunt, Kennedy and Pelsser in [15]. A major motivation which lead to their development was the desire to have models that can exactly replicate prices of liquid calibration instruments in a similar fashion to market models while maintaining the efficiency of short rate models in calculating derivative prices [13], [18]. Latter are formulated in terms of the short rate or instantaneous forward rate which cannot be traded in the market. As a consequence the prices of derivatives in these models are quite involved functions of the underlying process which is being modelled. This fact makes it difficult to capture the most characteristic features of a derivative product with models of this kind. However, their strong point is that the short rate process is easy to follow and hence implemention is straightforward [13]. Unlike short rate models the class of market models is formulated in terms of market rates which are directly related to tradable assets. Thus they exhibit better calibration properties than short rate models. However, as these models capture the joint distribution of market rates, they are high–dimensional by construction and tedious to implement. The first formulation of a market model was provided by Brace, Gatarek and Musiela in the context of forward Libors (LMM) [4]. A forward swap market model was developed by Jamshidian in 1997 [16]. In these approaches the underlying rates are modelled as log– normal martingales under their own probability measure. However, the presence of a volatility skew in the caplet and swaption markets indicate that a pure log–normal forward dynamics is not appropriate. In this respect modified forward rate dynamics were introduced, e.g in the context of constant elasticity of variance (CEV) and displaced diffusion models in which mixtures of pure normal and log–normal dynamics are considered [5], [23]. Aiming at a proper modelling of the skew term structure, stochastic volatility extensions of Libor and Swap Market models were introduced. This was

1 achieved by modelling the forward rate variance as CEV process. Approaches of this kind are the ones by Andersen and Andreasen [2] and Piterbarg [19], [20]. Especially Piterbarg’s stochastic volatility extension accounts for a Libor speciﬁc term structure of forward rate skews and volatilities. In the study at hand this forward Libor term structure of skew model (FL–TSS) is used in the construction of a Markov functional model with stochastic volatility. The MFM framework is based on formulating the numeraire and the terminal discount bond as functionals of a low dimensional Markov process whose dynamics can be followed easily. The functional forms in turn are obtained by calibration to prices of liquid derivatives at particular dates which are relevant to the product being priced. As the discount bonds at earlier times are obtained by applying the martingale property of numeraire rebased assets the resulting model is arbitrage free by construction. Thus, MFMs combine the strong points of market and short rate models. The incorporation of stochastic volatility into a MFM is based on the concept of a pre–model which depends on a low–dimensional Markov process [14],[17]. By regarding the forward Libors and hence the discount bond processes as functions of the pre–model process, the calibration can be formulated in terms of the latter. Thus the correlation structure of the pre–model is incorporated into the MFM. In this study a pre–model is selected which is an approximation of a calibrated Piterbarg FL–TSS model and depends on a two–dimensional brownian motion. This proxy is then employed in constructing a Libor MFM which inherits the stochastic volatility structure of Piterbarg’s FL–TSS. Tractability is maintained since calibration involves the integration of the known probability distribution of the two–dimensional Markov process. The thesis is structured as follows: In chapter 2 the concepts of LMMs and MFMs are reviewed. Working under the terminal (forward) measure drift terms for a stochastic volatility LMM are derived. Furthermore multi–dimensional extensions of MFMs are discussed. Chapter 3 is dedicated to the detailed study of Piterbargs FL–TSS LMM. This model is calibrated to the swaption market, and the resulting term structure of skews and volatilites is presented. The incorporation of stochastic volatility into a MFM is the topic of chapter 4. Here a two–dimensional Libor MFM is constructed which uses an approximation of Piterbarg’s FL–TSS as pre–model. The model is calibrated to the (digital) caplet market, and resulting numeraire discount bond funtionals are presented. The thesis concludes with chapter 5. Besides result tables and ﬁgures, the appendices contain mathematical details and a thorough presentation of the Heston Model.

2 Chapter 2

A review of Libor Market and Markov Functional Models

In this chapter we review the class of Libor Market and Markov functional models which have become prevalent in the last ten to ﬁfteen years. In section 2.1 we discuss the LMM under consideration of non–log–normal forward rate dynamics and stochastic volatility. In particular, working under the terminal (forward) measure forward rate drift terms are derived. The Markov functional framework is introduced in section 2.2. Therein multi– dimensional extensions are discussed as well. In section 2.3 the idea of constructing MFMs in terms of a pre–model is introduced.

2.1 The Libor Market Model

As already mentioned in the introduction the Libor Market Model focusses on modelling the dynamics of forward Libor rates Fi t; Ti, Ti+1 which reset at times Ti, i = 1,...,N. In a deterministic volatility setting employing K independent Brownian drivers these are modelled as log–normal variables with respect to their martingale measure Qi+1 which is induced by taking the discount bond D t, Ti+1 as numeraire, K i+1 T i+1 dFi t; Ti, Ti+1 = λi(t)Fi t; Ti, Ti+1 σi,k(t) dWk (t)= λi(t)Fi t; Ti, Ti+1 σi(t) dWt , k=1 X i+1 for times T0 t < Ti TN where dW is a K–dimensional vector of orthogonal Brow- ≤ ≤ t i+1 nian motions under Q and λi(t) are positive continuous, real valued functions. In particular the relation dWk,dWl = δkl dt holds. The K–dimensional vector σi(t) con- tains the load factors of the orthogonal brownian motions onto forward rate Fi which K 2 satisfy k=1 σi,k(t) = 1. Thus they deﬁne a correlation matrix through the relation T 1 ρij(t) =Pσi(t)σj(t) = ρij i=1,...,K . Indeed, in this formulation the covariance of for- { }j=1,...,K

1We assume that K = N, i.e., that each forward rate has its own driving brownian motion.

3 ward Libor yields is given by K dFi(t) dFj(t) i+1 i+1 , = λi(t)λj(t) σi,k(t)σj,l(t) dWk (t),dWl (t) Fi(t) Fj(t) k,l=1 X δkl dt K | {z } = λi(t)λj(t) σi,k(t)σj,k(t) dt = λi(t)λj(t)ρij(t) dt, Xk=1 =ρij (t) from where it becomes apparent that the| brownian{z correlatio} ns ρij(t) as well as the volatilities λi(t) contribute to the forward rate correlation. Since the forward Libors follow a log–normal process in their own martingale measure caplet prices are given by the Black76 formula, comp. [3]. Therefore the volatilities λi(t) can be obtained by calibration to quoted caplet prices. For this a widespread approach is to assume a parametric shape for the volatilities as function of time to expiry which correctly captures their dynamics observed in the market. The parameters are then determined in the course of the calibration process, comp. [21]. With respect to the correlation function the most reasonable approach is to model it in parametric form as well. The reason for this is that it is not easy at all to extract information on the instantaneous correlation ρij(t) out of quoted derivative prices, e.g., swaption volatilities, because the latter depend on the history of λi(t)λj(t)ρij(t) on the time interval which starts at T0 and ends on swaption expiry. The parametric form proposed by Rebonato in [21] and [22] is given by

ǫ ǫ ρij(t)= ρ + (1 ρ )exp δ (Ti t) (Tj t) , (2.1) ∞ − ∞ − − − − with constants 1 ρ 1, δ,ǫ> 0, and will be adopted in what follows. − ≤ ∞ ≤ Of course, to use the LMM in practice the dynamics of all forward Libor rates have to be formulated in a single measure. In this respect a convenient choice is the terminal measure N+1 Q which is induced by taking the terminal discount bond D t, TN+1 as numeraire. As a consequence only the forward Libor FN is a martingale, and according to Girsanov’s theorem all other forward Libors will be modiﬁed by additional drift terms, comp. [13]. Indeed, when N+1 changing from the terminal measure Q to the martingale measure of forward rate Fi N+1 N+1 i+1 the process dWk (t)+ µi (t) dt also is a brownian motion under Q . Hence in the terminal measure the forward rate processes become

T N+1 N+1 dFi t; Ti, Ti+1 = λi(t)Fi t; Ti, Ti+1 σi(t) µi (t) dt + dWt , N+1 T N+1 = Fi t; Ti, Ti+1 λi(t) µi (t) dt + σi(t) dWt , (2.2) N N+1 αl λl(t) ρil(t) Fl N+1 µi (t)= e , µN (t) = 0, − 1+ αlFl(t) l=Xi+1 ,e T0 t < Ti TN , i =e 1,...,N ≤ ≤

4 N+1 where the drifts µi (t) are given by (A.10) which is derived in appendix A.1.

2.1.1 Non–log–normale forward Libor dynamics

The forward Libor process (2.2) presented above models the forward Libors Fi as a log– normal processes with respect to their martingale measure. In the terminal measure a log–normal behaviour is maintained for forward Libor FN . However, the caplet/floorlet market displays a volatility surface (in terms of terms of implied Black volatilities), i.e., the volatility varies as option expiries and moneyness changes. Specifically, the observed volatilities are monotone decreasing functions of the forward Libor level, a behaviour which is denoted as volatility skew. Its presence indicates that the forward Libors do not follow a log–normal process, for in that case the implied Black volatilities should be constant. One proposal for an alternate forward rate dynamics provided by Rubinstein [23] is a displaced diffusion which combines a log–normal and a normal process. Another approach is to model the forward Libors as constant elasticity of variance (CEV) processes which was proposed by Cox and Ross [5]. In both cases the forward Libor change can be written in terms of a volatility function ϕ Fi which imposes a rate level dependence onto the forward rate volatilities, i.e the forward Libors are modelled as

T i+1 dFi t; Ti, Ti+1 = ϕ Fi λi(t)σi(t) dWt , i = 1,...,N. For example a displaced diﬀusion model is established with a function of the form

ϕ Fi t; Ti, Ti = β Fi t; Ti, Ti + 1 β Fi T0; Ti, Ti , +1 +1 − +1 where the displacement parameter β is a real valued constant. Obviously the case β = 1 corresponds to a log–normal dynamics. For β = 0 a normal process is recovered. A generalization of this with time dependent paramter β(t) will be considered in the next chapter where Piterbarg’s forward Libor term structure of skew model (FL–TSS) will be presented. A CEV–model is obtained by deﬁning

β ϕ Fi t; Ti, Ti+1 = Fi t; Ti, Ti+1 , (2.3) with 0 β 1. As with the displaced diﬀusion model the case β = 1 corresponds to ≤ ≤ log–normal dynamics whereas β = 0 results in a normal model. In this model the yield or dF ϕ(F ) β 1 percentage volatility, i.e., the volatility of i is given by i λ (t)= λ (t)F t; T , T − Fi Fi i i i i i+1 which for 0 <β< 1 is a monotone decreasing function of forward Libor Fiin accordance with the observed market behaviour. A model with this kind of local volatility function was ﬁrst introduced by Dupire in modelling equities, comp. [7].

5 Based on these ideas a generalised forward Libor process can be formulated in the terminal measure QN+1,

T N+1 N+1 dFi t; Ti, Ti+1 = ϕ Fi λi(t)σi(t) µi (t) dt + dWt , N+1 T N+1 = ϕFi λi(t) µi (t) dt + σi(t) dWt , (2.4) N N+1 αl λl(t) ρil(t) ϕ Fl N+1 µi (t)= e , µN (t) = 0, − 1+ αlFl(t) l=i+1 X e T0 t < Ti TN , i = 1,...,Ne ≤ ≤ which was proposed by Andersen and Andreasen, comp. [1]. The drift terms are given by equation (A.9) which is derived in appendix A.1.

2.1.2 Incorporation of stochastic volatility

As the market observed volatility skews cannot be solely captured by the introduction of a (local) volatility function stochastic volatility extensions of the LMM were devised. In this respect one approach is to extend the forward Libor process (2.4) with a stochastic variable which accounts for the volatility level and as such modulates the local volatility function

ϕ(Fi). A convenient choice is the square root of a variance process Σt which follows a one–dimensional CEV–process, comp. [1]. Assuming that the brownian driver of the variance process is correlated with each for- N+1 N+1 ward rate driver, i.e., dV (t),dWk (t) = Γk(t) dt (k = 1,...,N), the process (2.4) can be generalised to

N+1 T N+1 dFi t; Ti, Ti+1 = ϕ Fi λi(t) Σt µi (t) dt + σi(t) dWt N+1 T T N+1 T N+1 = ϕFi λi(t)pΣt µi (t) dt + σi(t) Ω(t) dZt + σi(t) Γ(t) dVt , e (2.5a) p e N+1 b dΣt =Θ Σ0 Σt dt + η Σt dV , (2.5b) − t p N+1 N+1 N+1 N+1 with dV (t),dWk (t) =Γk(t) dt,

and T 0 t < Ti TN , i = 1, . . . , N, k = 1,...,N, ≤ ≤ N+1 where the K–dimensional vector of brownian drivers dWt was decomposed into orthog- N+1 N+1 onal components dZt and dVt according to (A.3). The drift terms are given by

6 equation (A.8) which is derived in appendix A.1:

N N+1 αl λl(t) ϕ Fl µi (t)= Σt − 1+ αlFl(t) l=i+1 p X T T e σl(t) Ω(t) Ω(t)σi(t) + σl(t)Γ(t) σi(t)Γ(t) , (2.5c) × µN+1(t) = 0,h i N b b 2 with matrix Ω(t)= 1 Γk(t) δkj k=1,...,K and vector Γ(t)= Γk(t) k=1,...,K . e − j=1,...,K { } np o b Of course, the introduction of an additional Brownian driver increases the dimensionality of the model, and the additional correlation coeﬃcients Γk(t) enlarge the parameter space. However, for the stochastic volatility model we will work with in the following chapters, namely Piterbarg’s FL–TSS LMM, the rate and variance processes are independent. Thus

Γk(t) = 0 and the drift reduces to

N N+1 αl λl(t) ρil(t) ϕ Fl N+1 µi (t)= Σt , µN (t) = 0. (2.6) − 1+ αlFl(t) l=i+1 p X e e Nevertheless the process (2.5) will be referenced in section 2.3 where the idea of a pre–model in a Markov functional context is discussed.

2.2 Markov Functional Models

The class of Markov functional interest rate models was originally introduced by Hunt, Kennedy and Pelsser in [15]. A major motivation which lead to their development was the desire to have models that can fit observed prices of liquid instruments in a similar fashion to the market models while maintaining the efficiency of short rate models in calculating derivative prices, comp. [13], [18]. This is achieved by specifying a low dimensional process which is Markovian in some martingale measure and formulating pure discount bond prices as functions of this process. Since efficient algorithms to compute conditional distribution functions are known for this set up, the valuation of derivatives in a Markov functional framework is much more efficient when compared to pricing using market models. Although market models are Markovian as well, they are naturally of high dimension. Moreover an essential feature of these types of models is the freedom to choose the functional form of the discount bond prices in such a way that market prices of calbration instruments are replicated. This distinguishes Markov functional from short rate models in which the functional form of discount bond prices with respect to the Markovian short rate is fixed. Therefore Markov functional models combine the strong points of market as well as short rate models, namely the fitting to observed prices of liquid instruments and tractability.

7 2.2.1 Deﬁnition and examples of Markov Functional Models

In this section we want to give a formal deﬁnition of Markov functional models and also present some examples. We begin by citing the deﬁnition given by Hunt and Kennedy:

Deﬁnition 2.2.1 (Hunt and Kennedy [13]). An interest rate model is said to be Markov– functional if there exists some numeraire pair (N, N) and some process x sucht that:

1. the process x is a (time–inhomogeneous) Markov process under the measure N;

2. the pure discount bond prices are of the form

DtS = DtS (xt), T0 t ∂S S, ≤ ≤ ≤

for some boundary curve ∂S : [0,∂ ] [0,∂ ] and some constant ∂ ; ∗ → ∗ ∗ 3. the numeraire N, itself a price process, is of the form

Nt = Nt(xt) T0 t ∂∗. ≤ ≤

Obviously the boundary curve ∂S is introduced so that the model does not need to be deﬁned over the entire time domain 0 t S. The most common choice for the boundary ≤ ≤ curve is S, if S T ∂S = ≤ , T, if S > T for some constant T. Thus the main ingredients of a Markov functional model are the driving Markov process xt which describes the state of the economy and the functional forms of

1. the discount bond D∂ S x∂ D ∂S, S; x∂ on the boundary curve ∂S; S S ≡ S 2. the numeraire Nt(xt) for times T0 t ∂ . ≤ ≤ ∗

The reason for this is that the functional forms of bonds at earlier times t<∂S are determined by the functional form of the discount bond on the boundary curve by the martingale property of numeraire rebased assets,

N D(∂S, T ; x∂S ) D t, T ; xt = Nt(xt) E t , T0 t ∂S T, (2.7) N (x ) F ≤ ≤ ≤ ∂S ∂S comp. [10]. One particular choice of measure is the (terminal) forward measure N = QN+1 which is induced by taking the discount bond D t, TN+1 as numeraire. Taking TN+1 as boundary we thus have

∂S = min S, TN+1 , Nt(xt)= D t, TN+1; xt , T0 t TN+1, ≤ ≤ 8 and the price of a discount bond maturing at S T becomes ≤ N+1

QN+1 D(S, S; xS ) D t, S; xt = D t, TN+1; xt E xt D S, T ; x N+1 S QN+1 1 = D t, TN+1; xt E xt , T0 t S TN+1, (2.8) D S, T ; x ≤ ≤ ≤ N+1 S where the expectation is conditioned on xt because of the Markov property of the underlying process2. If one considers interest rate derivatives like caplets/ﬂoorlets or swaptions expiring at time Tm with strike K their payoﬀ function Vm(Tm, K) depends on the discount bonds

D(Tm, Tj; xt), with j = m + 1 for caplets/ﬂoorlets and j > m in the case of swaptions, and by (2.8) is a function of the numeraire discount bond D(Tm, TN+1; xTm ) at time Tm. Thus

Vm(Tm, K) Vm Tm,K,D(Tm, TN+1; xT ) and by application of the fundamental theorem ≡ m of asset pricing the derivative value at time t is given by

QN+1 Vm Tm,K,D(Tm, TN+1; xTm ) Vm t, K; xt = D t, TN+1; xt E xt . (2.9) D T , T ; x m N+1 Tm Hence, if the Markov process xt and thus the conditional probability distribution p(xTm xt) | is specified, this relation provides a means to extract the functional form of the numeraire discount bond D(Tm, TN+1; xTm ) at time Tm from market observed derivative prices since the payoff function Vm Tm,K,D(Tm, TN+1; xTm ) is known. However, in order to proceed along these lines one has to assume that the discount bonds are monotone functions of the underlying Markov process. It has to be emphasized that due to the functional dependence the specified underlying

Markov process xt determines the probability distribution of discount bonds. As the prices of multi–temporal interest derivatives depend on the joint probability distribution of forward rates (and thus on the joint distribution of discount bonds) at those times relevant to the product at hand3, the driving Markov process encodes all information on the correlation structure. So in designing a Markov functional model for a speciﬁc product class the process xt has to be chosen in such a way as to capture the characteristic product features while retaining low dimensionality. Referring to the process dimension, the underlying Markov process should not have more than two brownian drivers. An example of a simple one dimensional underlying process is considered in the next section where a Libor Markov functional model is explored.

2 For a Markov process xt the relation E f(xt) Ft = E f(xt) xt holds. 3E.g., for a bermudan swaption the single exercise dates are the relevant times. Therefore the probability distribution of the process xt only needs to be known on these dates.

9 2.2.2 A Libor Markov Functional Model

We know consider the set of forward Libor rates Fi t; Ti, Ti+1 which reset a times Ti, i = 1,...,N, and specify a Markov functional model. For this we choose the ﬁnal time N+1 ∂S = TN+1 and work in the measure N = Q which is induced by taking the discount bond D t, TN+1 as numeraire.

The underlying Markov process xt is chosen to be a time changed brownian motion.

Assuming that σ(t) is a deterministic, positive real valued function on [T0, TN+1] we deﬁne t xt := σ(s) dWs, ZT0 N+1 where dWs is a brownian motion under Q . Clearly, due to the Markovian character of the brownian motion xt is a Markov process whose conditional probability distribution is normal, 1 1 (x x )2 p x x s t , s t. s t = 1 exp s − 2 (2.10) 2 −2 σ(u) du ≥ (2π)2 s σ(u)2 du t t R Having speciﬁed the underlyingR process the functional form of the numeraire discount bond D t, TN+1; xt remains to be determined. This will be done at discrete times Ti (i = 1,...,N) according to a recursion scheme in which the functional from will be determined by calibration to digital caplet prices.

To start with we observe that at time TN we observe that the forward rate FN t; TN , TN+1; xTN N+1 is a log–normal martingale under Q . Thus its dynamics is governed by

dFN t; TN , TN+1 = σ(t)FN t; TN , TN+1 dWt, T0 t TN , ≤ ≤ N+1 where dWt is a brownian motion under Q , which integrates to

TN TN 1 2 FN TN ; TN , TN+1; xT = FN T0; TN , TN+1 exp σ(s) ds + σ(s) dWs N −2 ZT0 ZT0 1 TN = F T ; T , T exp σ(s)2 ds + x . N 0 N N+1 −2 TN ZT0 With this relation at hand the functional form of forward rate FN at time TN is known. Because forward rates and discount bond prices are related by

1 D(Tm, Tm+1; xTm ) Fm Tm; Tm, Tm+1; xTm = − , 1 m N, αm(Tm, Tm+1)D(Tm, Tm+1; xTm ) ≤ ≤ the functional form of D TN , TN+1; xTN unfolds itself as monotone decreasing function of xTN , 1 D TN , TN+1; xTN = . 1 TN 2 1+ αN (TN , TN+1)FN T0; TN , TN+1 exp σ(s) ds + xT − 2 T0 N R (2.11)

10 This result serves as the basis for the recursive calculation of functional forms at earlier times Tm < TN which will be extracted from market observed digital caplet prices. The payoﬀ of a digital caplet expiring at time Tm with strike K is given by

V T , K; x = D T , T ; x 1 , m m Tm m m+1 Tm Fm(Tm;Tm,Tm+1;xTm )>K and at time T0 < Tm < TN the numeraire rebased derivative value is therefore

Vm T0, K; xT N+1 Vm Tm,K,D(Tm, TN+1; xT ) 0 = EQ m x D T , T ; x D T , T ; x T0 0 N+1 T0 m N+1 Tm

N+1 D Tm, Tm+1; xTm = EQ 1 x D T , T ; x Fm(Tm;xTm )>K T0 m N+1 Tm

QN+1 QN+1 1 = E E xTm 1F (T ;x )>K xT , D T , T ; x m m Tm 0 m+1 N+1 Tm+1

(2.12) where in the last line the martingale property of numeraire rebased discount bonds (2.8) was used. Thus if the functional form of the numeraire bond D Tm+1, TN+1; xTm+1 is known at time Tm+1, the expected value on the right hand side can be calculated (2.12) because the conditional probability distribution of xt is known. Indeed, since the forward rate Fm is a monotone function of xTm through its dependence on the discount bond D Tm, Tm+1; xTm , there exists a unique value xT∗m for which the forward rate matches the digital caplet strike 4 value K, Fm t; Tm, Tm+1; xT∗m = K . Thus (2.12) is equivalent to

Vm T0, K; xT0 ∞ ∞ 1 = p xTm+1 xTm dxTm+1 p xTm xT0 dxT0 , D T0, TN+1; xT x D Tm+1, TN+1; xT 0 Z T∗m Z m+1 −∞ (2.13) which provides a relation between xT∗m and market derived derivative values. Indeed, because D T0, TN+1; xTi and Vm T0, K; xTi can be observed in the market at time T0, the left hand side of (2.13) is known. With the known functional form D Tm+1, TN+1; xTm+1 the intergrals on the right hand side can be calculated numerically for varying values of xT∗m . That value of xT∗m for which the left and right hand side of (2.13) are equal is the 5 desired target value which satisfies the relation Fm Tm; Tm, Tm+1; xT∗m = K. Conducting this matching procedure for a series of options with differen t strike values Kj (j = 1,...M) and values Vm T0, Kj; xT0 results in a set xT∗m,j Fm Tm; Tm, Tm+1; xT∗m,j = Kj 1 D Tm, Tm+1; xT∗m,j − = x∗ D Tm, TN+1; x∗ = 1+ αN (TN , TN+1)Kj , Tm,j Tm,j D T , T ; x m N+1 ∗Tm,j (2.14) 4 As mentioned above we assume that the discount bonds are monotone functions of xt. Due to their relation this behaviour transfers to the forward Libor rate. 5 In practice the solution for x∗Tm is found by a numerical root finding method, e.g., the Brent algorithm.

11 where in the second line the relation

1 D(Tm, Tm+1; xT∗m,j ) Kj = Fm Tm; Tm, Tm+1; xT∗m,j = − αm(Tm, Tm+1)D(Tm, Tm+1; x∗Tm,j ) D(T ,T ;x ) 1 m m+1 T∗m,j

D Tm,TN+1;x∗ − D Tm,TN+1;x∗ = Tm,j Tm,j , 1 m N, D(T ,T ;x ) m m+1 T∗m,j ≤ ≤ αm(Tm, Tm+1) D T ,T ;x m N+1 T∗m,j

D(T ,T ;x ) was used. Because m m+1 T∗m,j corresponds to the inner integral (the bracket term) D T ,T ;x m N+1 T∗m,j of equation (2.13) its value has already been determined in the course of finding x . T∗m,j Therefore the set identity (2.14) defines the numeraire discount bond at time Tm as function of xT∗m,j . Obviously, because only a finite number of options with different strikes can be observed in the market the sets in (2.14) are discrete. Therefore continuous functional forms have to be obtained by interpolation between the single set elements.

Following the above reasoning the recursion scheme starts at time TN 1. Performing − the calibration according to (2.13) the expected value of the inverse of the already known functional D TN , TN+1; xTN given by (2.11) is calculated. As a result the set (2.14) and thus the functional forms D TN 1, TN+1; xT at time TN 1 are obtained. These in turn − N − serve as input for the calibration at time TN 2 where they enter the inner integral of (2.13). − Pursueing the recursion along these lines until time T1 the functional forms of the numeraire discount bond D Ti, TN+1; xT are established for times TN , TN 1,...,T1. i − 2.2.3 Multi–dimensional Markov Functional Models

In the previous section we presented the Libor Markov functional model in which a one– dimensional Markov process was used as underlying for the discount bond term structure. However it is also possible to consider higher dimensional underlying processes. As mentioned in the previous section the driving process should be selected in such a way that the essential features of the derivative product for which the model is designed are captured. As an example for which a two dimensional driving process is required the class of spread options can be considered. The payout structure of this kind of derivatives can depend on the level of two diﬀerent rate types which follow individual dynamics. Therefore a two– dimensional process is required in order to model the separate rate components. Another example is the incorporation of stochastic volatility which we focus on in this thesis. In this context a two dimensional Markov process would encompass a rate and a volatility component. As discussed above the speciﬁcation of the Markov process is only one part in the specif- cation of a Markov functional model. The second is the determination of the functional

12 forms for the numeraire and the discount bond on the boundary curve. These are obtained by a calibration procedure for which it is essential that the functionals are monotone functions of the driving Markov process. But when higher dimensional Markov processes zt are considered the monotonicity of functionals can no longer be maintained because e.g., in a two dimensional extension of the matching equation (2.13) more than one tuple zt = (zt,1,zt,2) would be obtained as target value. For an n–dimensional Markov process zt this problem can be overcome by the introduction of a function

n π : R R R, (t, zt) π(t, zt) =: xt, × −→ 7−→ which serves as projector to the one dimensional real axis, comp. [13]. In general, the process xt = π(t, zt) thereby deﬁned will not be Markovian. However this fact poses no impediment since the function π merely serves as a means to facilitate the calibration to derivative prices observed in the market. Following this approach the numeraire discount bond at time Tm becomes a functional of the multi–dimensional Markov process, D Tm, TN+1; zTm , and in the calibration procedure the expected values are calulated with respect to the conditional distribution of zt. Therefore the only modiﬁcation which needs to be applied to the calibration equation (2.12) is the change from one– to multi–dimensional integrals:

Vm T0, K; xT0 = π(T0, zT0 ) D T , T ; x = π(T , z ) 0 N+1 T0 0 T0

N+1 Vm Tm,K,D(Tm, TN+1; xT ) = EQ m x D T , T ; x T0 m N+1 Tm

QN+1 QN+1 1 = E E xTm 1F (T ;x )>K xT D T , T ; x m m Tm 0 m+1 N+1 Tm+1

QN+1 QN+1 1 = E E xTm 1x >x xT D T , T ; x Tm T∗m 0 m+1 N+1 Tm+1

QN+1 QN+1 1 = E E zTm 1π(Tm,z )>x zT D T , T ; z Tm T∗m 0 m+1 N+1 Tm+1

∞ ∞ 1 = p zTm+1 zTm d zTm+1 p zTm zT0 dzT0 , x =π(T ,z ) D Tm+1, TN+1; zT Z T∗m m T∗m Z m+1 −∞ (2.15) x F 1 T T , T K . n with T∗m = m− ( m; m m+1; ) The variable zT∗m describes a curve in the –dimensional state space for which xT∗m = π(Tm, zT∗m ) at time Tm.

Thinking about possible choices for the function π(t, zt) the concept of a pre–model was devised [14]. It is based on the idea that π can be deﬁned as approximation to a model which has already been calibrated to market prices, e.g., a Libor Market Model where the drift terms have been frozen to their initial values. Since the so deﬁned pre–model is an

13 approximation only it is not arbitrage free. However, since the no–arbitrage requirement is inherent in equation (2.15) a calibration to market quotes via the pre–model will result in an arbitrage free model.

2.3 A Libor Market Model as pre–model for a Markov Func- tional Model

In this section an approximation of a forward Libor Market Model (LMM) will be considered which will then be used as pre–model for a Markov functional model. Working in the N+1 terminal measure Q the forward rate Fi t; Ti, Ti+1 is modelled according to (2.4), Fi t; Ti, Ti+1 = Fi T0; Ti, Ti+1 t t N+1 1 2 2 exp µ (s) λi(s) σi(s) ds + λi(s)σi(s) dW s , × i − 2 | | ZT0 ZT0 (2.16a) e K αl(Tl, Tl+1)λl(t)ρli(t)Fl t; Tl, Tl+1 µN+1(t)= , (2.16b) i − 1+ α (T , T )F t; T , T l=i+1 l l l+1 l l l+1 X e T0 t Ti, i = 1,...,N, 1 K N, ≤ ≤ ≤ ≤

N+1 where dW s is a K–dimensional Brownian motion under Q and σi(s) the load vector which encodes the eﬀect of the individual orthogonal Brownian drivers on forward rate Fi.

From this dynamics it is obvious that the change of forward rate Fi depends on the state of all forward rates Fl (l>i) at time t which is why the individual processes Fi(t) are not Markovian6. Therefore one usually resorts to Monte–Carlo methods in numerical evaluations which gets quite expensive as the number of factors increases. However, computations can be alleviated by referring to an approximation proposed by Rebonato due to which the forward rates Fl t; Tl, Tl+1 in the drift term (2.16b) are replaced by their time T0 values

Fl T0; Tl, Tl+1 , a process which is also denoted as partial freezing [22]. A further simpliﬁ- cation can be achieved by replacing the Brownian motion terms with normally distributed variables which exhibit the same mean and variance. Employing these ideas the forward rate vector can be approximated as

N+1 F(t) F(t)= F(T )exp µ0 (t)+ M(t) zt , ≈ 0 · 6However, the process for the entire forward rate vector F(t) is Markovian since its change at time t only depends on the state of the forward rate vector at time t.

14 with vectors t N+1 N+1 1 2 2 µ0 (t)= µ0i (s) λi(s) σi(s) ds , T0 − 2 i=1,...,N Z h i K eαl(Tl, Tl+1)λl(t)ρli( t)Fl T 0; Tl, Tl+1 where µ N+1(t)= , 0i − 1+ α (T , T )F T ; T , T l=i+1 l l l+1 l 0 l l+1 X e zt = t,i , where t,i 0,t T0 , Z i=1,...,K Z ∼N − and the matrix 1 t 2 1 2 2 M(t)= λi(s) σik(s) ds . t T0 T i=1,...,N − Z 0 k=1,...,K

Indeed, the terms Mik(t) t,k satisfy Z K K K t E Mik(t) t,k = Mik(t)E t,k =0= E λi(s)σik(s) dWk(s) , Z Z k=1 k=1 k=1 ZT0 X X t X = E λi(s)σi(s) dW s ZT0 t because λi(s)σik(s) dWk(s) are Ito integrals, ZT0 K K K and var Mik(t) t,k = E Mik(t)Mil(t) t,k t,l = Mik(t)Mil(t)E t,k t,l Z Z Z Z Z k=1 k,l=1 k,l=1 X X X K = Mik(t)Mil(t)cov t,k t,l Z Z k,l=1 X =δkl(t T0) − K t 2 | 2 {z } = λi(s) σik(s) ds T0 Xk=1 Z K t 2 2 = E λi(s) σik(s) ds T0 Xk=1 Z K t 2 = E λi(s)σik(s) dWk(s) by Ito’s isometry T0 Xk=1 Z K t 2 = λi(s) σik(s)σil(s) E dWk(s)dWl(s) k,l=1 ZT0 X =cov[ dWk(s),dWl(s)] K t | {z } = var λi(s)σik(s) dWk(s) k=1 ZT0 Xt = var λi(s)σi(s) dW s , i = 1,...,N. (2.17) ZT0 Above proxy processes can be related to the original forward rates Fi(t) by introducing monotone functions gi which act as perturbations on F¯i(t, zt). Thus Fi(t) = gi F¯i(t, zt) , 15 and one can deﬁne projector functions πi := gi F¯i by ◦ K πi : R R R, (t, zt) πi(t, zt) := gi F i(t, zt)), × −→ 7−→ K N+1 with F i(t, zt)= Fi(T0)exp µi (t)+ Mik(t)zt,k , for i = 1,...,N, Xk=1 which were introduced in the previous section. Hence the numeraire discount bond becomes a functional of the Markov process zTm through its dependence on Fm(Tm)= πm(Tm, zTm )= gm F m(Tm, zTm ) . Based on this approach the incorporation of stochastic volatility into a Markov functional framework will be devised in chapter 4.

16 Chapter 3

Piterbarg’s term structure of skew forward Libor model

This chapter is dedicated to a survey of Piterbarg’s term structure of skew forward Libor model (FL–TSS) which was introduced in [19]. As with other forward Libor Market Models the main motivation is to capture the dynamics of the joint distribution of forward Libors throughout time. To facilitate this the forward Libor dynamics has to be ﬂexible enough to capture information on the marginal distributions which is encoded in caplet and/or swaption prices. Since for these products the market implied volatilities exhibit a skew, i.e., the Black implied volatilities appear to be functions of the option strikes, Piterbarg considers a weighted sum of log–normal and normal dynamics for the forward Libors. Mathematically this is expressed by the introduction of a skew parameter. Reference to the swaption market necessitates this parameter to be time–dependent in order to reproduce swaption skews across expiries and underlying swap maturities. It is this time–dependent skew parameter which renders the model more ﬂexible over earlier (constant) skew models, e.g., Andersen and Andreasen [2]. Furthermore, to account for the market observed variability of volatility levels, similar to the formulation of [2] these are modelled as stochastic CEV processes whose Brownian components are assumed to be uncorrelated with the stochastic drivers of the forward rate processes. Hence the FL-TSS model belongs to the class of stochastic volatility models.

3.1 The forward Libor dynamics

Following the qualitative description of Piterbarg’s model we know set out to specify the dynamcis of forward Libors Fi t; Ti, Ti+1 resetting at times Ti, i = 1,...,N. Working in N+1 the measure Q induced by choosing the terminal discount bond D t, TN+1 as numeraire the forward rates are modelled as

17 K N+1 N+1 dFi(t)= βi(t)Fi(t)+ 1 βi T0 Fi T0 λi(t) Σt µ (t) dt + σi,l(t)dW (t) , − i l " l=1 # p X e (3.1a) N+1 dΣt =Θ Σ0 Σt dt + η Σt dV , (3.1b) − t p for times T0 t < Ti TN i = 1,...,N . All forward rates are driven by K independent ≤ ≤ N+1 Brownian motions dWl (t) which are assumed to be uncorrelated with the stochastic N+1 N+1 N+1 volatility driver dVt and therefore dVt ,dWl (t) = 0. Their influence on the forward Libors Fi is mediated by the load factors σi,l(t) which contain information on K forward Libor correlation since the relation l=1 σi,l(t)σj,l(t) = ρij(t) holds. The drift N+1 terms µi (t) i = 1,...,N result from workingP in the terminal measure and are given by expression (2.6). It vanishes for the last forward rate FN t; TN , TN+1 since it is a log–normal e N+1 N+1 martingale under Q . Hence µN (t) = 0. The model parameters are hence the time–dependent forward skews βi(t) and volatility e levels λi(t) as well as the time–independent volatility of variance η and mean reversion speed Θ. Piterbarg chooses the latter to be constant which reduces the degrees of freedom to the set of tuples βi(t), λi(t) i=1,...,N associated with forward Libors Fi t; Ti, Ti+1 for times T0 t Ti TN and i = 1,...,N. These parameters characterise the distribution ≤ ≤ ≤ of each forward Libor as they are not affected by a change of measure. Reference to (3.1a) clearly reveals the role of the skews βi(t) as parameters mixing a purely log–normal with a normal forward Libor dynamics. From there it also is apparent that the parameters λi(t) determine the level of the stochastic volatility √Σt which is governed by (3.1b). Together with correlations ρij(t) among forward rates Fi and Fj these parameters determine the joint distribution for times T t T . 0 ≤ ≤ N The term structure of forward Libor skews and volatility levels have to be obtained by calibration to market prices of caplets/floorlets or swaptions. As outlined above the FL- TSS model is calibrated to the swaption market. Since forward swap rates are the natural swaption underlyings, it is therefore necessary to formulate a consistent forward swap rate dynamics and relate the resulting forward swap skews and volatility levels to their forward Libor counterparts. This is done by requiring that the process governing the forward swap rates should have the same structure as the forward Libor dynamics. The detailing of this idea will be presented in the following section.

18 3.2 Swap rate dynamics under the FL-TSS model

To derive a consistent forward swap rate dynamics we ﬁrst observe that the par rate Smn(t) of a forward swap starting at time Tm >t T0 and maturing at Tn > Tm can be expressed ≥ as a weighted sum of its constituent forward rates Fm,...Fn 1. : −

D(t, Tm) D(t, Tn) Smn(t)= n 1 − l=−m αl(Tl, Tl+1)D(t, Tl+1) n 1 P− αl(Tl, Tl+1)D(t, Tl+1) = Fl t; Tl, Tl+1 n 1 α T , T D t, T l=m l=−m l( l l+1) ( l+1) X P =:wl(t) n 1 − | {z } = wl(t)Fl t; Tl, Tl+1 , (3.2) l=m X which follows from the forward rate deﬁnition

D(t, Tl) D(t, Tl+1) Fl t; Tl, Tl+1 = − , αl(Tl, Tl+1)D(t, Tl+1) where αl(Tl, Tl+1) denotes the year fraction of the period [Tl, Tl+1] and D(t, Tj) stands for the discount factor corresponding to time Tj.

Thus the forward swap rate is a function of its constituent forward rates, Smn(Fm,...,Fn 1), − and a stochastic diﬀerential equation is arrived at by application of Ito’s lemma:

n 1 n 1 2 − ∂Smn(t) 1 − ∂ Smn(t) dSmn(t)= dFl(t)+ dFl(t)dFk(t) ∂Fl(t) 2 ∂Fl(t)∂Fk(t) lX=m l,kX=0 n 1 − ∂Smn(t) = βl(t)Fl(t)+ 1 βl T0 Fl T0 λl(t) Σt ∂Fl(t) − l=m X =:ϕ(Fl(t)) p T m n m n | {z σ (t) µ}( , )dt + dW( , )(t) × l · l n 1 2 h i 1 − ∂ Smn(t) + ϕ(Fl(t))ϕ(Fk(t))ρlk(t)dt 2 ∂Fl(t)∂Fk(t) l,kX=m n 1 − ∂S (t) T m n = mn ϕ(F (t))λ (t) Σ σ (t)dW( , )(t) + drift terms. (3.3) ∂F (t) l l t l l m l X= p Above expression is formulated in the swap measure Q(m,n) which is induced by us- n 1 1 ing the present value of a basis point Pmn(t)= l=−m αl(Tl, Tl+1)D(t, Tl+1) as numeraire . (m,n) The K–dimensional drift vectors µl accountP for the change from the terminal measure m n QN+1 to Q(m,n), under which dW( , )(t)isa K–dimensional Brownian motion. Additional

1 Strictly speaking the quantity Pmn(t) refers to a notional of 1 and therefore represents the present value of 10,000 basis points.

19 T K drift terms arise from the non–zero correlations ρlk(t) = σl (t)σk(t) = j=1 σl,j(t)σk,j(t) between forward rates Fl and Fk. But since covariances and therefore volatilitiesP and correlations remain invariant under a change of measure, one can consider above swap dynamics under a new measure Q with associated K–dimensional Brownian motion dW(t) in which the drift terms vanish. Under Q (3.3) transforms into e f n 1 −e ∂S (t) T dS (t)= mn ϕ(F (t))λ (t) Σ σ (t)dW(t). (3.4) mn ∂F (t) l l t l l m l X= p f As was already mentioned in the introduction we are looking for a forward swap rate dynamics which has the same structure as the forward rate process. This is equivalent to requiring a dynamics of the form

dSmn(t)= βmn(t)Smn(t)+ 1 βmn T0 Smn T0 λmn(t) Σt σmn,l(t)dWl(t), − l=1 =:ϕ(S (t)) p X e mne e e f | {z } (3.5) with forward swap rate skews βmn(t), volatility levels λmn(t), and Brownian loadings σmn,l(t) (l = 1,...,K). In order to ensure consistency between the two formulations we set (3.4) e e e equal to (3.5) and in addition match the slopes of both expression with respect to all forward rates Fl. This reasoning results in direct relations between swap and forward rate volatility levels and skews, respectively, which will be presented in the following subsections.

3.2.1 Derivation of the forward swap volatility level

Matching of both expressions for the forward swap rates results in the requirement

n 1 − ∂S (t) T T mn ϕ(F (t))λ (t) Σ σ (t)dW(t)= ϕ(S (t))λ (t) Σ σ (t)dW(t) ∂F (t) l l t l mn mn t mn l m l X= p p f e f from which the forward swap volatility level can be derived: e

n 1 T − ∂Smn(t) ϕ(Fl(t)) T λmn(t)σmn(t)= λl(t) σl (t) (3.6) ∂Fl(t) ϕ(Smn(t)) lX=m e n 1 e − ∂Smn(t) ϕ(Fl(t)) T = λmn(t)= σl (t)σmn(t)λl(t), (3.7) ⇒ ∂Fl(t) ϕ(Smn(t)) lX=m T e e since σmn(t)σmn(t) = 1. Whereas the load factors of the orthogonal forward Libor drivers can be extracted from the correlation matrix 2 this does not apply to the Brownian motions e e 2 We assume that the correlations between forward Libors Fi and Fj are given in the parametric form ǫ ǫ (2.1) propsed by Rebonato [21]: ρij (t) = ρ + (1 − ρ ) exp −δ (Ti − t) − (Tj − t) ∞ ∞

20 driving the swap rate. In above case of K > 1 Brownian swap rate drivers we therefore have to retrieve the swap rate volatility by referring to (3.6) instead of the volatility level

λmn(t) itself. It is obvious that forward skew and volatility parameters as well as their swap counterparts are intertwined in above expressions, which is due to the presence of the skew e functions ϕ(Fl(t)) and ϕ(Smn(t)). This dependency is resolved when (3.6) is considered at the money, i.e., at the time T0 forward Libor and swap rate points. In this case ϕ(Fl(t)) ∂Smn(t) and ϕ(Smn(t)) become skew independent, and by further freezing (which are given ∂Fl(t) by (A.11)) at their initial values (3.6) simpliﬁes to

n 1 T − ∂Smn(t) Fl(T0) T λmn(t)σmn(t)= λl(t) σl (t), (3.8) ∂Fl(t) Smn(T0) l=m t=T0 X e which is the expression one which calibration will be based.

3.2.2 Derivation of the forward swap skew

The forward swap and Libor skew can be interpreted as slopes of the respective skew functions ϕ(Smn(t)) and ϕ(Fl(t)) which are linear in the underlying rate variable. Hence we have

∂ βmn(t)Smn(t)+ 1 βmn T0 Smn T0 ∂ϕ(Smn(t)) − βmn(t)= = , ∂Smn(t) ∂ϕ(Smn(t)) e e e ∂ βl(t)Fl(t)+ 1 βl T0 Fl T0 ∂ϕ(Fl(t)) − βl(t)= = for l = m,...,n 1, ∂Fl(t) ∂F l(t) − and a relation between both quantities can be established by matching the slope of dSmn(t) in formulations (3.4) and (3.5) with respect to the forward rates Fk(t). Referring to (3.4) and assuming that the derivatives ∂Smn(t) (l = m,...,n 1) do not vary signiﬁcantly as ∂Fl(t) − Fl(t) varies with time, we thus obtain

n 1 2 ∂ dSmn(t) − ∂Smn(t) ∂ϕ(Fl(t)) ∂ Smn(t) T =  + ϕ(Fl(t)  λl(t) Σ σ (t)dW(t) ∂F (t) ∂F (t) ∂F (t) ∂F (t)∂F (t) t l k l=m  l k k l  X   p 0 ≈ f n 1   − ∂Smn(t) | T {z } βk(t)δlkλl(t) Σt σl (t)dW(t) ≈ ∂Fl(t) l=m t=T0 X p ∂Smn(t) T f = βk(t)λk(t) Σt σk (t)dW(t), (3.9) ∂Fk(t) t=T0 p f where our assumptions took eﬀect in the second line by freezing ∂Smn(t) at their initial values ∂Fl(t) and neglecting second derivatives of the swap rate with respect to the forward Libors.

21 Focussing on the formulation in terms of swap volatility levels and skews (3.5), a similar analysis yields

∂ dSmn(t) ∂ϕ(S (t)) ∂S (t) T = mn mn λ (t) Σ σ (t)dW(t) ∂F t ∂S t ∂F t mn t mn k( ) mn( ) k( ) p ∂Smn(t) e T f βmn(t) λmn(t) Σt σemn(t)dW(t). (3.10) ≈ ∂Fk(t) t=T0 p e e f Matching of the forward swap slopes with respect to Liborse Fm,...,Fn 1 in equations − (3.10) and (3.9) results in a system of n m equations between forward swap and Libor − skews:

βmn(t)λmn(t)σmn(t)= βk(t)λk(t)σk(t) (k = m,...,n 1), − to which no uniquee solutione exists.e Therefore one has to revert to a least squares optimisation to ﬁnd an approximate solution. For this sake we consider the functional

n 1 − 2 J βmn(t) = βmn(t)λmn(t)σmn(t) βk(t)λk(t)σk(t) , − k=m X e e e and obtain the optimal solution for the forwarde swap skew by requiring the derivative of J with respect to βmn(t) to vanish:

n 1 dJe T − 0= = 2 λmn(t)σmn(t) βmn(t)λmn(t)σmn(t) βk(t)λk(t)σk(t) dβmn(t) − kX=m n 1 T e e− λ (t)σe (t)e λ (t)eσ (t) e 1 mn mn k k = βmn(t)= T βk(t). (3.11) ⇒ n m λ (t)σ (t) λ (t)σ (t) − k=m mn mn mn mn X e e e As the central result of this sectione wee hence havee establishe ed the relation between forward swap skews and volatilities to their forward Libor counterparts for each swap rate

Smn(t) over time interval T0 t Tm. It is this time dependence of the forward Libor ≤ ≤ parameters which deﬁnes the central idea of a skew term structure in Piterbarg’s FL–TSS model. As mentioned earlier, in calibrating the FL–TSS model to the swaption market Piterbarg uses skew and volatility parameters of a Heston model as market input. By construction Hestons’s model parameters are time–independent which necessitates the introduction of a time-averaging method for the swap rate parameters within the FL–TSS. This requirement results in the formulation of eﬀective skew and volatility parameters which are central in paving the way to calibration.

22 3.3 The eﬀective skew and volatility formulation

This section is dedicated to the formulation of swap rate eﬀective skews and volatilities within Piterbarg’s FL–TSS model. In summary, these facilitate the transition from a swap rate dynamcis with time–dependent parameters (3.5),

T dSmn(t)= βmn(t)Smn(t)+ 1 βmn T0 Smn T0 λmn(t) Σt σ (t)dW(t), − mn p to a formulation basede on time–independente ones, e e f

T dSmn(t)= β Smn(t)+ 1 β Smn T0 λmn Σt σ (t)dW(t), mn − mn mn p where for each swap rate Smn(t) the effective skews βmn and volatilitiese λmnfare in principle given as weigthed time averages of the time–dependent quantities (3.8) and (3.11). The detailing of this central concept will be provided in the following, where at first attention will be paid to the effective skew in subsection 3.3.1 after which effective volatility is covered in 3.3.2.

3.3.1 The eﬀective forward swap skew

Piterbarg arrives at the effective swap rate skew by considering two diffusion processes where one has a time–dependent local volatility function and the other a time–independent one. The latter is defined as a weighted average of its time–dependent counterpart. Focussing on the weight function w(t) on time interval [0, T ], Piterbarg derives an explicit expression such that the average of differences between european swaption prices across an infinite range of strikes, calculated with respect to the respective processes, tends to zero as valuation time approaches T0. In detail, the following theorem holds:

Theorem 3.3.1 (Piterbarg [19]). For T > 0, let f C1 [0, T ] R, R+ be a local volatility ∈ × function satisfying the usual growth requirements. Let σ(t), t [0, T ] be a function of time ∈ only. Fix x R. For any ǫ> 0 deﬁne a re–scaled local volatility function 0 ∈ 2 fǫ(t,x)= f tǫ ,x + (x x )ǫ , 0 − 0 and assume without loss of generality

f(t,x ) 1, t [0, T ] 0 ≡ ∈ which implies

fǫ(t,x ) 1, t [0, T ]. 0 ≡ ∈

23 Let w(t), t [0, T ] be a weight function satisfying ∈ T w(t) dt = 1, Z0 and deﬁne an averaged local volatility function

T 2 2 f ǫ(x) = fǫ(t,x) w(t) dt. (3.12) Z0 Further deﬁne two families of diﬀusions indexed by ǫ,

dXǫ(t)= fǫ t, Xǫ(t) λ(t) Σ(t)dW (t), (3.13a)

dYǫ(t)= f ǫYǫ(t) λ(t) pΣ(t)dW (t), (3.13b)

Xǫ(0) = x0, p (3.13c)

Yǫ(0) = x0, (3.13d) for t [0, T ] with ∈

dΣt =Θ Σ0 Σt dt + η Σt dV (t), dV (t),dW (t) = 0. − p If the weights w(t) are given by the expression

2 2 v(t) λ(t) 2 2 w(t)= , with v(t) = E Σ(t) X0(t) x0 , (3.14) T 2 2 − 0 v(t) λ(t) dt h i then R

∞ + + 2 E Yǫ(T ) K E Xǫ(T ) K dK = O ǫ (3.15) − − − Z −∞ for ǫ 0.3 →

By applying above theorem to the swap rate Smn(t) with local volatility function ϕ(Smn(t)) f t, Smn(t) = and replacing time zero with T0, a formulation with time–independent Smn(T0) eﬀective skew parameters is obtain according to

T dSmn(t)= βmn(t)Smn(t)+ 1 βmn T0 Smn T0 λmn(t) Σt σ (t)dW(t), − mn T = S (T )f t, S (t) λ (t) Σ σ (t)dW(t), p mne 0 1 mn mne t mn e e f T Smn(T0)f Smn(t) λmn(t) pΣt σ (t)dW(t), ≈ 1 e mne f T = β Smn(t)+ 1 β Spmn T0 λmn(t) Σt σ (t)dW(t), mn −e mn e f mn 3 p While above theorem provides a relation between the twoe diﬀusions, ite has to bef observed that (3.15) is not formulated in terms of absolute values. Hence the fact that Yǫ(t) complies with relation (3.15) in the limit ǫ → 0 does not generally ensure convergence to Xǫ(t) in probability.

24 Tm with βmn = 0 βmn(t)w(t) dt. By referring to above theorem one has to bear in mind that the derived relationR between the time–dependent and time averaged re–scaled local volatility e function holds in the limit ǫ 0. This corresponds to regarding the local volatility function → at time zero, although we are interested in a time–independent proxy for f(t,x) which would be obtained in the limit ǫ 1. In this sense the so derived eﬀective skew parameters have → to be regarded as approximate results since their derivation is based on the consideration of f1(t,x), the re-scaled local volatility function with ǫ = 1. Their derivation is presented as corollary:

Corollary 3.3.2 (Piterbarg [19]). The eﬀective skew βmn for the equation

T dSmn(t)= β Smn(t)+ 1 β Smn T0 λmn(t) Σt σ (t)dW(t), mn − mn mn p over a time horizon [T0, Tm] is given by e e f

Tm βmn = βmn(t)wmn(t) dt, ZT0 where the weights w(t) are given by e

2 2 vmn(t) λmn(t) wmn(t)= , (3.16) Tm 2 2 vmn(t) λmn(t) dt 0 e t t eΘ(s T0) e Θ(s T0) 2 R 2 2 2 Θ(t T0) 2 − − − vmn(t) = Σ T0 λmne (s) ds + Σ T0 η e− − λmn(s) − ds. 2Θ ZT0 ZT0 e e (3.17) Proof. The proof is provided in section A.4 of Appendix A.

In calibrating the model to the market the swap rates Smn(t) will be considered at discrete points in time Tk, T Tk < TK = t Tm, between which the volatility λmn(t) 0 ≤ ≤ will be assumed to be constant. For such a piecewise constant swap rate volatility function e

25 2 vmn(t) simpliﬁes to:

t t eΘ(s T0) e Θ(s T0) 2 2 2 2 Θ(t T0) 2 − − − v (t) = Σ T λ (s) ds + Σ T η e− − λ (s) − ds mn 0 mn 0 mn 2Θ ZT0 ZT0 K 1 2 − e 2 e = Σ T0 λmn(Tk) Tk+1 Tk − Xk=0 K 1 e − Tk+1 eΘ(s T0) e Θ(s T0) 2 Θ(t T0) 2 − − − + Σ T η e− − λ (Tk) − ds 0 mn 2Θ k=0 ZTk X K 1 e 2 − 2 = Σ T0 λmn(Tk) Tk+1 Tk − Xk=0 K 1 η2 e − Θ(t T0) 2 Θ(Tk+1 T0) Θ(Tk+1 T0) + Σ T e− − λ (Tk) e − + e− − 0 2Θ2 mn k=0 X Θ(Tek T0) Θ(Tk T0) e − e− − . (3.18) − − Above corollary veers towards a formulation in terms of time–independent, eﬀective parameters in providing a link between time–dependent and eﬀective swap rate skews. As a result the swap dynamics takes the form

T dSmn(t)= β Smn(t)+ 1 β Smn T0 λmn(t) Σt σ (t)dW(t), (3.19) mn − mn mn p which is time–independent as far as the skews are concernede bute whichf still refers to the time–dependent swap volatility levels λmn(t). This gap will be closed in the following by deriving an eﬀective volatility which completes our work towards a swap rate dynamics with e time–independent parameters.

3.3.2 The eﬀective forward swap volatility

We begin by observing that (3.19) represents a displaced diﬀusion process with displacement

1 βmn parameter γmn = − Smn T0 , comp. [23]. Indeed (3.19) can be recasted, βmn

T dSmn(t)= d Smn(t)+ γmn = Smn(t)+ γmn βmnλmn(t) Σt σmn(t)dW(t), (3.20) p and because the Brownian driver of the variance process,e dV (t), ise independentf of dW(t) it is obvious that for a given volatility level √Σt the shifted swap rate follows a log–normal f process. Therefore the terminal variance of the swap rate Smn Tm conditioned on a par- Tm ticular variance path Σt is given by T0

Tm 2 2 2 σ(Tm) = βmn λmn(s) Σs ds. (3.21) ZT0 e

26 Also the mentioned independence of Brownian drivers enables us to value european options with strike K by integrating the well known Black76-formula against the distribution of the stochastic terminal swap rate variance. Focussing on expected values only and working in the measure Q we have

+ + 2 e E Smn Tm K = E E Smn Tm K σ(Tm) , − − h h ii where the inner expectation directly yields the displaced Black76 formula with volatility

σ(Tm) [3]. For at the money options it becomes especially simple and, omitting the numeraire once again, the option value becomes

+ + 2 E Smn Tm Smn T0 = E E Smn Tm Smn T0 σ(Tm) − − h h 2 ii = E g σ(Tm) , (3.22)

with the displaced ’at the money’ Black76 function

1 Tm 2 2 1 2 2 g σ(Tm) = Smn T0 + γmn 2 Φ β λmn(s) Σs ds 1 2 mn − ZT0 S T mn 0 1 2 e = 2 Φ σ(Tm) 1 . βmn 2 − h p i To arrive at an eﬀective volatility λmn Piterbarg considers the swap variance (3.21)

Tm 2 2 2 σ(Tm) = βmn λmn Σs ds, (3.23) ZT0 and again formulates values of european at the money options as integrals against the 2 probability distribution of σ(Tm) , following the same analysis which lead to (3.22). Since the option values calculated either way have to match, this requirement results in a deﬁning equation for the eﬀective swap rate volatility:

2 2 E g σ(Tm) = E g σ(Tm)

Tm Tm 2 2 2 2 E g β λmn(s) Σs ds = E g β λ Σs ds (3.24) ⇐⇒ mn mn mn ZT0 ZT0 Both sides of equation (3.24)e involve expected values of the cumulative normal distribution which makes it diﬃcult to solve for the eﬀective volatility. The situation can be alleviated by considering an approximation to (3.24) obtained through the formulation of an equivalent matching condition. It is based on using the analytic function

cmnx h(x)= amn + bmne−

27 as approximation for g(x) around the expected value of the stochastic variable umn(Tm) := 2 σ(Tm) . From (3.21) the expected value of the latter reads βmn Tm Tm 2 2 umn := E[umn(Tm)] = E λmn(s) Σs ds = λmn(s) E Σs ds ZT0 ZT0 Tm e 2 e = ΣT0 λmn(s) ds. ZT0 e With this at hand the coeﬃcients amn, bmn, and cmn are determined as solutions of the system of matching equations,

cmnumn h(umn)= amn + bmne− = g(umn),

dh cmnumn dg = ( bmncmn)e− = , dx x=umn − dx x=umn 2 2 d h 2 cmnumn d g 2 = (bmncmn)e− = 2 , dx x=umn dx x=umn 2 d g dg 1 1 = cmn = 2 = + 2 2 , (3.25) ⇒ − dx x=umn dx x=umn 8 2 T T β λ m 0 mn mn − where the derivation of coeﬃcient cmn is provided in Appendix A (comp. expression (A.13)).

For values x around the expected variance level umn the relation

h(x) g(x) ≃ holds. The approximate matching condition for the expected values (3.24) then becomes:

2 2 E h σ(Tm) = E h σ(Tm)

Tm 2 2 amn + bmn E exp cmn β λmn(s) Σs ds ⇐⇒ − mn ZT0 T e 2 2 m = amn + bmn E exp cmn β λ Σs ds − mn mn ZT0 Tm Tm 2 2 2 2 = E exp cmn β λmn(s) Σs ds = E exp cmn β λ Σs ds . ⇒ − mn − mn mn ZT0 ZT0 e (3.26) The expected values on either side of equation (3.26) can be represented in terms of the function

ϕ µ Σ0 = E exp µ f(s) Σs ds Σ0 , (3.27) − ZT0 2 2 2 2 if µ and f(s) are deﬁned as µ = cmn β , f(s)= λmn(s) and µ = cmn β λ , f(s) 1= mn mn mn ≡ const., respectively. e

28 Since the process Σt is affine, i.e., the drift and variance terms in (3.1b) are affine t T0 ≥ functions of Σt, ϕ µ Σ 0 can be expressed as ϕ µ Σ0 = exp Aµ,f T0, Tm + Bµ,f T0, Tm Σ0 , (3.28) where the functions Aµ,f t, Tm and Bµ,f t, Tm obey the Riccati system of ordinary differential equations

dAµ,f t, Tm = Θ Σ Bµ,f t, Tm , dt − 0 2 dBµ,f η 2 t, Tm = µ f(t)+Θ Bµ,f t, Tm Bµ,f t, Tm , dt − 2

Aµ,f Tm, Tm = Bµ,f Tm, Tm = 0, (3.29) for times T t T . Obviously, the coeﬃcients A and B depend on the function f(t) over 0 ≤ ≤ m the considered time interval which is the reason why the system does not possess an analytic solution. Therefore one has to refer to numerical methods, e.g., a Runge-Kutta scheme, or deﬁne a discretization of the considered time span [T0, Tm] in sub intervals [Ti, Ti+1] on each of which the function f(t) is assumed to be constant. Since in the case of a constant function f(t) above system can be solved analytically, time T0 solutions Aµ,f T0, Tm and

Bµ,f T0, Tm can be obtained by a recursion scheme in which the analyticalsolutions of adjacent sub intervals are concatenated. With (3.28) the matching condition deﬁning the eﬀective forward swap volatility becomes

exp A 2 2 T0, Tm + B 2 2 T0, Tm Σ0 cmn βmn,λmn(t) cmn βmn,λmn(t) e = exp A 2 e2 T0, Tm + B 2 2 T0, Tm Σ0 cmn βmnλmn,1 cmn βmnλmn,1

= A 2 2 T0, Tm + B 2 2 T0, Tm ⇒ cmn βmn,λmn(t) cmn βmn,λmn(t)

= A 2 2 T0, Tm + B 2 2 T0, Tm , (3.30) e cmn βmnλemn,1 cmn βmnλmn,1 because Σ0 = 1. The left hand side involves the time–dependent forward swap volatility

λmn(t), whereas the coefficients on the right hand side depend on λmn, the effective forward swap volatility which is constant over the entire time interval [T0, Tm]. Therefore the right e hand side coefficients A 2 2 T0, Tm and B 2 2 T0, Tm are given by analytic c βmnλmn,1 c βmnλmn,1 expressions (comp. (A.40) and (A.41) in Appendix A). In the calibration process the time–dependent forward swap volatility λmn(t) entering the left hand side of (3.30) will be determined iteratively as described above in order to e match a market derived effective volatility level. As the former in turn depend on the forward swap rate volatilities through (3.8), equation (3.30) provides a means to calibrate these

29 to given market parameters. The detailing of the calibration procedure will be presented next.

3.4 Calibration of the FL-TSS model

In calibrating the FL–TSS model Piterbarg assumes that the market information is encoded in a set of time–independent skew and volatility parameters of a Heston model which already is calibrated to a set of swaptions across expiries and underlying swap maturities. Thus calibration of the FL-TSS model is done in an indirect fashion by considering the Heston parameters as primary market input rather than market implied swaption prices. H H So let us assume that the Heston parameters (λmn,βmn) m=1,...,N are known. Each { }n=m+1,...,2N tuple encodes the market prices of swaptions on the underlying swap rate Smn expiring at time Tm for a range of strikes (comp. Appendix B), and is thus considered to be constant on time interval [T0, Tm]. Calibration is then based on minimisation of the squared differences between effective and market derived (Heston) swap rate volatilities and skews. Because the effective volatilities are derived at the money (comp. (3.22)) they depend on the swap rate skews only weakly. Therefore calibration of swap rate volatilities and skews can be carried out separately. Specifically, the quantity

M = M1 + M2, (3.31a) N m+N 2 H with M = λmn λ (3.31b) 1 − mn m=1 n=m+1 X X N m+N 2 and M = β βH , (3.31c) 2 mn − mn m=1 n=m+1 X X will be minimised by an iteration scheme in which the skews are kept constant during the volatility calibration step and vice versa. Hence the process rests on a sequential minimisation of M1 and M2 that will be repeated until convergence of M is established. The aim of above calibration scheme is to extract forward rate volatilities and skews out H H of swaption market data. As we are assuming knowledge of Heston parameters (λmn,βmn) for m = 1,...,N and n = m + 1,...,m + N as market data input, TN represents the latest time for which information on forward rate volatilities and skews can be obtained. Indeed, as the swap rates consist of underlying forward rates, i.e., Smn Smn(Fm, Fm+1,...,Fn 1), ≡ − the provided market data only allow us to obtain volatilities and skews of forward rates

F1,...,F2N 1 for times T0 t TN . Further market data for swaption expiries TN+1,...,T2N − ≤ ≤ would be needed to extract forward rate volatilities and skews for times T

30 3.4.1 Forward rate volatility calibration

Focussing on the calibration of forward rate volatilities, the process can be broken down into two steps:

1. The calibration of (time–dependent) forward rate volatilities to (time–independent) eﬀective swap rate volatilities.

2. Variation of forward rate volatilities (and thus via 1. eﬀective swap rate volatilities)

to minimise M1 for ﬁxed forward swap skews.

Starting with 1., we recall that the forward swap volatility is given by (3.8),

n 1 T − ∂Smn(t) Fl(T0) T λmn(t)σmn(t)= λl(t) σl (t), (3.32) ∂Fl(t) Smn(T0) l=m t=T0 X e =: dmn,l e T | {z } where the vector σl (t) is obtained by a singular value decomposition of the known correlation matrix ρmn(t). Also, the time T0 values of the swap rate derivatives and hence dmn,l are known. Hence the forward swap rate volatility level λmn(t) is a function of forward rate volatility levels λm(t), λm+1(t),...,λn 1(t). − e The ﬁtting of time–dependent forward rate volatilities to eﬀective swap rate volatilities is then achieved by imposing the matching condition

exp A 2 2 T0, Tm + B 2 2 T0, Tm Σ0 cmn βmn,λmn(t) cmn βmn,λmn(t) e = exp A 2 e2 T0, Tm + B 2 2 T0, Tm Σ0 cmn βmnλmn,1 cmn βmnλmn,1

= A 2 2 T0, Tm + B 2 2 T0, Tm ⇒ cmn βmn,λmn(t) cmn βmn,λmn(t)

= A 2 2 T0, Tm + B 2 2 T0, Tm , Σ0 = 1, e cmn βmnλemn,1 cmn βmnλmn,1 (3.33) for times T0 t Tm and m = 1,...,N, n = m + 1,...,m + N, where TN corresponds ≤ ≤ to the latest swaption expiry. Here the left hand side involves the time–dependent forward swap volatility λmn(t), whereas the coefficients on the right hand side depend on λmn, the effective forward swap volatility which is constant over the entire time interval [T0, Tm]. e The problem (3.33) is discretised with piecewise constant forward rate volatility levels and 4 skews . Specifically, time interval [T0, Tm] is split into sub intervals Ii := [Ti, Ti+1] for

4As mentioned above, the forward rate skews are kept constant during the volatility calibration.

31 i = 0,...,m 1. On each Ii the forward rate volatilities are then assumed to take on − (i) (i) constant values λm(Ti), λm+1(Ti),...,λn 1(Ti), and enter coefficients A , B . − The calibration of forward rate volatilities to effective forward swap rate volatilities is performed according to the following iteration, in the course of which λm(Ti) will be determined for m = 1,..., 2N 1 and times T0 Ti < Tm < TN . Hence for a fixed m, − ≤ the forward rate volatility λm(t) is obtained on intervals I0,I1,...,Im 1 where the values −

λm(T0), λm(T1),...,λm(Tm 1) are attained. At the beginning of the iteration, all forward − rate volatilities λm (m = 1,..., 2N) are set to a constant value, e.g., the average of all H Heston volatility parameters λmn.

On interval I0 the volatilities are determined as follows:

T T 1. λ1(T0) is obtained by fitting λ12(T0)σ12(T0) = d12,1 λ1(T0) σ1 (T0) to λ12. This is done by a one–dimensional root finding method, e.g., Brent’s algorithm. e e T T 2. λ2(T0) is obtained by a fitting to λ13, since λ13(T0)σ13(T0) = d13,1 λ1(T0) σ1 (T0)+ T d13,2 λ2(T0) σ2 (T0), and λ1(T0) was obtained in the first iteration step. Hence the e e swap rate volatility only depends on λ2(T0) and the optimisation problem only is one–dimensional.

3. Assuming that λ1(T0), λ2(T0),...,λn 2(T0) have already been determined, λn 1(T0) is − − T n 2 T derived from a calibration of λ1n(T0)σ1n(T0)= − d1n,l λl(T0) σl (T0) + d1n,n 1 l=1 − T λn 1(T0) σn 1(T0) to λ1n. Because the sum termP is known, the swap rate volatility × − − e e only depends on λn 1(T0) and again the optimisation problem is one–dimensional. − Therefore a one–dimensional root ﬁnding algorithm can be applied.

As a result of performing this iterative calibration at time T0 for underlying swap maturities T2,...,T2N 1, forward rate volatilities λ1(T0),...,λ2N 1(T0) are obtained which are − − calibrated to the eﬀective volatilities λ12,..., λ1,2N . These results will be used in further iteration steps. 5 On interval I1 = [T1, T2] the goal is to obtain volatilities λ2(T1),...,λ2N 1(T1). This is − done as follows:

1. The interval [T0, T2] is split into sub–intervals I0 and I1. From the iteration procedure

on I0 the time T0 volatilities are already known. Hence, λ1(T1) is obtained by ap-

plication of matching condition (3.33) at time T1 with respect to the forward swap T T rate λ23(T1). Due to the relation λ23(T1)σ23(T1)= d23,2 λ2(T1) σ2 (T1), the forward rate

5 As forward rate F1 resets at time T1, λ1 only lives for times T0 ≤ t < T1. e e e

32 volatility level λ2(T1) can be obtained by calibration to the eﬀective volatility λ23. The matching condition results in equations

A(0) T = A(1) T , 2 2 1 2 2 1 c23 β23,λ23(T0) c23 β23,λ23(T1) B(0) T = B(1) T , 2 e 2 1 2 e 2 1 c23 β23,λ23(T0) c23 β23,λ23(T1) with terminal condition e e

A(1) T = B(1) T = 0, 2 2 2 2 2 2 2 2 c23 β23(T1) ,λ23(T1) c23 β23(T1) ,λ23(T1) and initial condition e e

(0) A T = A 2 2 T , 2 2 0 c β ,λ 0 c23 β23,λ23(T0) 23 23 23 (0) B T = B 2 2 T , 2 e 2 0 c β ,λ 0 c23 β23,λ23(T0) 23 23 23 where the coeﬃcients A(i), B(i) fore i = 0, 1 are given by expressions (A.40) and (A.41)

derived in Appendix A.6. Since the volatility λ2(T0) and hence λ23(T0) are known from (0) (0) the previous iteration step, coefficients A (T1), B (T1) are given as well. Therefore e the matching condition together with the initial and terminal conditions enable us (1) (1) to determine coefficients A (T1), B (T1) which depend on λ23(T1) and thus λ2(T1) by a one–dimensional optimisation algorithm. Through this matching of coefficients, e λ2(T1) is obtained by a fitting to λ23.

2. With λ2(T1) at hand, λ3(T1) is obtained by a ﬁtting to eﬀective volatility λ24. Again

this is done by a matching of coeﬃcients at time T1, however here forward swap T rate λ24(T1) is serving as reference function. Due to the relation λ24(T1)σ24(T1) = T T d24,2 λ2(T1) σ2 (T1)+ d24,3 λ3(T1) σ3 (T1) and the fact that λ2(T1) has been obtained e e e previously, the forward swap rate solely depends on λ3(T1). Hence, the calibration to

λ24 via a matching of coeﬃcients yields the desired forward rate volatility λ3(T1).

3. Assuming that λ2(T1), λ3(T1),...,λn 2(T1) have already been determined, λn 1(T1) − − T is derived by a matching of coeﬃcients at time T1 with respect to λ2,n(T1)σ2,n(T1)= n 2 T T l=1− d2n,l λl(T1) σl (T1)+d2n,n 1 λn 1(T1) σn 1(T1) and calibration to λ2n. As the sum − − − e e Pterm is known from previous iteration steps, the forward swap rate solely depends on

λn 1(T1). Hence, the calibration to λ2n via a matching of coeﬃcients yields the desired −

forward rate volatility λn 1(T1). − As a result of performing this iterative calibration to the eﬀective forward swap volatilities

λ23,..., λ2,2N at time T1 for underlying swap maturities T3,...,T2N 1, forward rate volatili- − ties λ2(T1),...,λ2N 1(T1) are obtained. These are referenced in further calibration steps at − later times Ti > T1.

33 Assuming now that the forward rate volatilities have already been determined at times

T1,...,Tm 2 by a sequential application of matching condition (3.33) at times T1,...,Tm 3, − − 6 on interval Im 1 =[Tm 1, Tm] the goal is to obtain volatilities λm(Tm 1),..., λ2N 1(Tm 1). − − − − − This is done as follows:

1. The interval [T0, Tm] is split into sub–intervals I0,I0,...,Im 1. From the iteration pro- −

cedure on interval Im 2 the time Tm 2 volatilities are already known. Hence, λm(Tm 1) − − −

is obtained by application of matching condition (3.33) at time Tm 1 with respect to − T the forward swap rate λm,m+1(Tm 1). Due to the relation λm,m+1(Tm 1)σm,m+1(Tm 1)= − − − T d(m,m+1),m λm(Tm 1) σm(Tm 1), the forward rate volatility level λm(Tm 1) can be ob- − e − e e− tained by calibration to the eﬀective volatility λm,m+1. The matching condition is a special case of the general formulation which is given under number 3..

2. Having determined λm(Tm 1) in the previous step, λm+1(Tm 1) is obtained by a ﬁtting − −

to eﬀective forward swap rate volatility λm,m+2. Here matching condition (3.33) is T applied at time Tm 1 with respect to λm,m+2(Tm 1)σm,m+2(Tm 1)= d(m,m+2),m λm(Tm 1) − − − − T T σm(Tm 1)+d(m,m+2),m+1 λm+1(Tm 1) σm+1(Tm 1), which only depends on λm+1(Tm 1) × − − e − e − since the ﬁrst term is known from 1.. Hence a one–dimensional optimisation with

λm,m+2 as target variable yields the desired forward rate volatility λm+1(Tm 1). The − matching condition is the special case n = m + 2 of the general formulation presented next.

3. Assuming that λm(Tm 1), λm+1(Tm 1),...,λn 2(Tm 1) have already been determined, − − − −

λn 1(Tm 1) follows by application of matching condition (3.33) at time Tm 1 with re- − − − T n 2 T spect to forward swap rate λm,n(Tm 1)σm,n(Tm 1)= − d(m,n),l λl(Tm 1) σl (Tm 1) − − l=m − − T + d(m,n),n 1 λn 1(Tm 1) σn 1(Tm 1). Because the sumP term is already known, λn 1(Tm 1) − − − − e − e − − can be obtained by calibration to the eﬀective swap rate volatility λm,n. The matching condition results in equations

(m 2) (m 1) A − T = A − T , (3.34a) 2 2 m 1 2 2 m 1 cmn βmn,λm,n(Tm 2) − cmn βmn,λm,n(Tm 1) − − − (m 2) (m 1) B − T = B − T , (3.34b) 2 e 2 m 1 2 e 2 m 1 cmn βmn,λm,n(Tm 2) − cmn βmn,λm,n(Tm 1) − − − with terminal conditione e

(m 1) (m 1) A − T = B − T = 0, 2 2 2 m 2 2 2 m cmn βmn(Tm 1) ,λm,n(Tm 1) cmn βmn(Tm 1) ,λm,n(Tm 1) − − − − 6 As forward rates Fm−1 reset ate time Tm−1, the λm−1 only live for timese T0 ≤ t < Tm−1.

34 and initial condition

(0) A T = A 2 2 T , 2 2 0 c β ,λ 0 cmn βmn,λm,n(T0) mn m,n m,n (0) B T = B 2 2 T , 2 e 2 0 c β ,λ 0 cmn βmn,λm,n(T0) mn m,n m,n where the coefficients A(i), B(ie) for i = 0,...,m 1 are given by expressions (A.40) − and (A.41) derived in Appendix A.6. Since all forward rates are known at time (m 2) (m 2) Tm 2 the forward swap rate λm,n(Tm 2) and hence the coefficients A − , B − − − (m 1) (m 1) are known as well. Furthermore, the coefficients A − , B − on the right hand e side of the matching conditions (3.34a), (3.34b) only depend on λn 1(Tm 1) because − −

all λj (Tm 1) with j

this matching of coeﬃcients, λn 1(Tm 1) is obtained by a ﬁtting to λmn. Performing − − this iterative calibration for n = m + 3,..., 2N results in calibrated forward rate

volatilities λm+2(Tm 1),...,λm+2(T2N 1). − −

Above iterative calibration scheme enables us to extract forward rate volatilities λm(Tm 1), −

...,λ2N 1(Tm 1) out of effective forward swap volatilities λm,m+1,..., λm,2N for m = 1,...,N. − − H By identifiying the effective volatilities with market derived Heston volatilities λmn for n = m + 1,..., 2N and n = 1,...,N, expression M1 (comp. (3.31b)) is automatically minimised by performing the one–dimensional optimisations at times T1,...,TN 1 as detailed − above. During the entire forward volatility calibration the forward (and hence the swap rate) skews are kept constant as their calibration can be conducted separately. With respect to this, the detailing is provided in the following section.

3.4.2 Forward rate skew calibration

As with the forward rate volatilities, the calibration process for forward rate skews can be sub–divided into two steps:

1. The calibration of (time–dependent) forward rate skews to (time–independent) eﬀec- tive swap rate skews.

2. Variation of forward rate skews (and thus via 1. eﬀective swap rate skews) to minimise

M2 for ﬁxed forward swap volatilities.

35 Starting with 1. and assuming that forward rate skews are piecewise constant, from (3.11) we known that the swap skews are related to their forward rate counterparts by

n 1 T n 1 − 1 λmn(Tj)σmn(Tj) λk(Tj)σk(Tj) − βmn(Tj)= T βk(Tj)= gmn,k(Tj) βk(Tj), n m λ (T )σ (T ) λ (T )σ (T ) k=m − mn j mn j mn j mn j k=m X e e X e =:gmn,k(Tj ) e e e e | {z } for times T0 Tj < Tm, where the gmn,k are known from the preceding volatility calibration. ≤ From corollary 3.3.2 the eﬀective forward swap skew is given by

Tm βmn = βmn(t)wmn(t) dt, ZT0 with weights w(t) of the form e

2 2 vmn(t) λmn(t) wmn(t)= , Tm 2 2 0 vmn(t) λmn(t) dt t e 2 R 2 2 vmn(t) = Σ T0 λmne (s) ds ZT0 t eΘ(s T0) e Θ(s T0) e 2 Θ(t T0) 2 − − − + Σ T η e− − λ (s) − ds, 0 mn 2Θ ZT0 for m = 1,...,N and n = m+1,...,m+N. Since thee forward rate skews are assumed to be piecewise constant, the following discretised expression for the eﬀective skew is considered in the calibration process:

m 1 − β = βmn(Tl)wmn(Tl) Tl Tl , mn +1 − l=0 X e 2 2 vmn(TK 1) λmn(TK 1) − − wmn(TK 1)= , − K 1 2 2 − vmn(Tl) λmn(Tl) (Tl Tl) k=0 +1 − K 1 e 2 P 2 − 2 vmn(TK 1) = Σ T0 λmn(eTk) Tk+1 Tk − − Xk=0 K 1 η2 e − Θ(t T0) 2 Θ(Tk+1 T0) Θ(Tk+1 T0) + Σ T e− − λ (Tk) e − + e− − 0 2Θ2 mn k=0 X Θ(Tek T0) Θ(Tk T0) e − e− − , (3.35) − − 2 for T0 TK 1 < Tm, where the expression for vmn(TK 1) follows from (3.18). As the ≤ − − forward swap rate volatilities are known from the volatility calibration, the weights wmn(t) and vmn(t) can be evaluated straight away. The calibration of forward rate to eﬀective forward swap rate skews is performed according to the following iteration, in the course of which βm(Ti) will be determined for m = 1,..., 2N 1 and times T0 Ti < Tm < TN . Hence for a ﬁxed m, the forward rate skew − ≤

36 βm(t) is obtained on intervals I0,I1,...,Im 1 where the values βm(T0),βm(T1),...,βm(Tm 1) − − are attained. At the beginning of the iteration, all forward rate skews βm (m = 1,..., 2N) are set to a constant value, e.g., the average of all market derived Heston skew parameters H βmn.

On interval I0 the skews are determined as follows:

1. β1(T0) is obtained by ﬁtting β = β (T0)w12(T0) T1 T0 = g , (T0) β1(T0)w12(T0) 12 12 − 12 1 H T1 T0 to β12 . This is done by a one–dimensional root ﬁnding method, e.g., Brent’s × − e algorithm.

2. With β (T ) at hand, β (T ) is obtained by ﬁtting β = β (T )w (T ) T T = 1 0 2 0 13 13 0 13 0 1 − 0 H g13,1(T0) β1(T0) + g13,2(T0) β2(T0) w12(T0) T1 T0 to β13 . Since the weights and − e β 1(T0) is already known from 1., β13(T0) only depends on β2(T0), and the calibration H to the market input β13 is performend by a one-dimensional root ﬁnding algorithm.

3. Assuming that β1(T0),β2(T0),...,βn 2(T0) have already been determined, βn 1(T0) − − n 2 follows by ﬁtting β1n = β1n(T0)w1n(T0) T1 T0 = l=1− g13,l(T0) βl(T0)+g13,n 1(T0) − − H βn 1(T0 w1n(T0) T1 T0 to β1n. Because the sumP term is already known, β1n(T0) × − −e only depends on βn 1(T0) and the forward rate skew is obtained by a one–dimensional − H root ﬁnding with β1n as target variable.

As a result of performing this iterative calibration at time T0 for underlying swap maturities

T2,...,T2N 1, forward rate skews β1(T0),...,β2N 1(T0) are obtained which are calibrated to − − H H the Heston skews β12 ,...,β1,2N . These results will be used in further iteration steps. 7 On interval I1 =[T1, T2] the goal is to obtain forward rate skews β2(T1),...,β2N 1(T1). − This is done as follows:

1. β (T ) is obtained by ﬁtting β = β (T )w (T ) T T +β (T )w (T ) T T = 2 1 23 23 0 23 0 1 − 0 23 1 23 1 2 − 1 H g23,2(T0) β2(T0) w23(T0) T1 T0 + g23,2(T1) β2(T1) w23(T1) T2 T1 to β23 . Because − e e − β 2(T0) is known form the iteration at time T0, this expression solely depends on β2(T1) H and calibration to β23 is performed with a one–dimensional root ﬁnding method.

2. With β (T ) at hand, β (T ) is obtained by ﬁtting β = β (T )w (T ) T T 2 1 3 1 24 24 0 24 0 1 − 0 +β24(T1)w24(T1) T2 T1 = g24,2(T0) β2(T0) + g24,3(T0) β3(T0) w24(T0) T1 T0 + − e − H g24,2(T1) β2(T1)+ g24,3(T1) β3(T1) w24(T1) T2 T1 to β24 . In this expression β3(T1) e − H is the only unknown which is why a one–dimensional optimisation with β24 as target variable yields the desired forward rate skew.

7 As forward rate F1 resets at time T1, β1 only lives for times T0 ≤ t < T1.

37 3. Assuming that β2(T1),β3(T1),...,βn 2(T1) have already been determined, βn 1(T1) − −

follows by ﬁtting β = β n(T0)w2n(T0) T1 T0 + β n(T1)w2n(T1) T2 T1 2n 2 − 2 − n 2 n 2 = l=1− g(2,n),l(T0) βl(T0)+g(2,n 1),l(T0) βn 1(T0) w2n(T0) T1 T0 + l=1− g(2,n),l(T1) e − − e − H βl(TP1)+ g(2,n 1),l(T1) βn 1(T1) w2n(T1) T2 T1 to β2n. Since all weightP factors and − − − skews at time T0 are known, and βn 1(T1) is the only unknown at time T1, a one– − H dimensional optimisation with β2n as target variable yields the desired forward rate skew.

H H As a result of performing this iterative calibration to the Heston skews β23 ,...,β2,2N at time

T1 for underlying swap maturities T3,...,T2N 1, forward rate skews β2(T1),...,β2N 1(T1) − − are obtained. These are referenced in further calibration steps at later times Ti > T1. Assuming now that the forward rate skews have already been determined at times

T1,...,Tm 2, on interval Im 1 = [Tm 1, Tm] the goal is to obtain skews βm(Tm 1),..., − − − − 8 β2N 1(Tm 1). This is done as follows: − −

1. The interval [T0, Tm] is split into sub–intervals I0,I0,...,Im 1. From the iteration pro- −

cedure on interval Im 2 the time Tm 2 skews are already known. Hence, βm(Tm 1) is − − − m 1 obtained by ﬁtting β = − βm,m (Tl)wm,m+1(Tl) Tl Tl m,m+1 l=0 +1 +1 − m 2 = l=0− g(m,m+1),m(Tl) βm(TlP) wm,m+1(Tl) Tl+1 Tl + g(m,m+1),m(T m 1) βm(Tm 1) e − − − H wm,m+1(Tm 1) Tm Tm 1 to βm,m+1. By assumption βm(Tl) is known for l = × P − − − 1,...,m 2. As βm,m+1 only depends on the unknown skew βm(Tm 1), latter is ob- − − H tained by a one–dimensional optimisation with βm,m+1. as target variable.

m 1 2. With βm(Tm 1) at hand, βm+1(Tm 1) is obtained by ﬁtting β = − βm,m+2(Tl) − − m,m+2 l=0 m 2 wm,m+2(Tl) Tl+1 Tl = − g(m,m+2),m(Tl) βm(Tl) + g(m,m+2),m+1(Tl) βm+1(Tl) − l=0 P e wm,m+1(Tl)Tl+1 Tl + g(m,m+2),m (Tm 1) βm(Tm 1)+g(m,m+2),m+1(Tm 1) βm+1(Tm 1) − P − − − − H wm,m+1(Tm 1) Tm Tm 1 to βm,m+2. Because the skews βm(Tl) are known for l = − − − 1,...,m 1, and β (T ) is given for previous times corresponding to l = 1,...,m − m+1 l − 2, β only depends on βm+1(Tm 1). Hence the latter is obtained by a one– m,m+2 − H dimensional optimisation with βm,m+2 as target variable.

3. Assuming that βm(Tm 1),βm+1(Tm 1),...,βn 2(Tm 1) have already been determined, − − − − m 1 βn 1(Tm 1) follows by ﬁtting βm,n = l=0− βm,n(Tl)wm,n(Tl) Tl+1 Tl − − − m 1 n 2 m 2 = l=0− k=−m g(m,n),k(Tl) βk(Tl)wm,nP(Tl) Tl+1 Tl + l=0− g(m,n),n 1(Tl) βn 1(Tl) e − − − H wm,n(Tl) Tl+1 Tl +g(m,n),n 1(Tm 1) βn 1(Tm 1)wm,n (Tm 1) Tm Tm 1 to βm,n. By P P − − − − − P− − − assumption the ﬁrst term is known. Also, as all forward rates have been determined

at earlier times Ti < Tm 1, the second sum is known as well. Hence βm,n only depends − 8 As forward rates Fm−1 reset at time Tm−1, the βm−1 only live for times T0 ≤ t < Tm−1.

38 H on βn 1(Tm 1), which is then obtained by a one–dimensional optimisation with βm,n − − as target variable.

Above iterative calibration scheme enables us to extract forward rate skews βm(Tm 1), − H H ...,β2N 1(Tm 1) out of market derived Heston skews βm,m+1,...,βm,2N for m = 1,...,N. − − In doing so expression M2 (comp. (3.31c)) is automatically minimised by performing the one–dimensional optimisations at times T1,...,TN 1 as detailed. − As mentioned in the introduction, the forward rate volatility and skew calibration are carried out sequentially. Hence the error measures M1 (comp. 3.31b) and M2 (comp.

3.31c) are minimised separately with respect to a given volatility (for M2) or skew level

(for M1). The resulting volatility and skew vectors are then used in the next iteration step. This process is repeated until the error measure M (comp. 3.31a) is minimised. Results obtained by this calibration procedure are presented in the next section.

3.4.3 Calibration results

H H In the following we present results of the calibration to a set Heston parameters (λm,n,βm,n) corresponding to swaptions with expiries Tm and underlying swap maturities Tn which was performed according to the iteration schemes outlined in the previous sections. The Heston parameters were obtained by calibration to Euro swaption prices of May 7th, 2008, on underlying swaps of a one year tenor with maturities between one and ten years. Swaption expiries ranged from one to ten years. Hence Heston parameters were available for m = 1,..., 10 and n = m + 1,...,m + 10. With these data at hand, the Piterbarg skew and volatility term structure were obtained for forward reset times T1,...,T10 with T0–forward rates taken from Table C.2. For the calibration the forward rate skews were initialised with a constant value of 0.5. Furthermore the volatility of variance and mean reversion speed of the variance process (3.1b) were set to η = 0.59, Θ = 0.15, respectively. Forward rate volatilities were initially set to 0.02.

Table (C.3) presents βi(t) and λi(t) for times T0 = 0y t < Ti = i y and i = 1,..., 10. ≤ The following ﬁgure displays the skew and volatility of forward rate F10 over time period [0y, 10 y]. Here it becomes apparent that in Piterbarg’s FL–TSS model the skew can become negative.

39 Forward rate volatility and skew

λ10(t)

β10(t) 0.4

0.2 ) t ( 1 β,

) 0.0 t ( 0 1 0 λ

-0.2

-0.4

0 2 4 6 8 10 t (years)

Figure 3.1: Volatility level λ10(t) and skew β10(t) of forward rate F (t) for times T =0y t < T = 10y. 10 0 ≤ 10

40 Chapter 4

A Markov functional model with stochastic volatility

This chapter is dedicated to the formulation of a Markov functional model with stochastic volatility. Following the ideas presented in section 2.2.3 of chapter 2 where multi– dimensional Markov functional models were discussed, a two–dimensional Libor Markov functional model is devised which is calibrated to the (digital) caplet market. With a model of this kind the forward rate dynamics can be described by reference to the distribution of a two–dimensional Markov process. As swap rates are functions of their constituent forward rates, the pricing of derivatives on underlying swaps, e.g., Bermudan swaptions, can be conducted more eﬃciently. Our formulation is based on an approximation to Piterbarg’s FL–TSS Libor Market Model which is used as pre–model. That is, the numeraire discount bonds are expressed as functionals of pre–model processes. Since the latter have a stochastic volatility component, this approach enables us to incorporate the concept of stochastic volatility into a Markov functional framework. The outline of this chapter is as follows: Based on the FL–TSS model, two–dimensional proxy processes for the forward rates are devised in sections 4.1 to 4.3. This is done in three steps, which can be summarised as

1. Approximate forward rates Fi(t) by displaced diﬀusion processes Fi(t) with partially frozen drift terms in section 4.1 b

2. Find a two–dimensional formulation of Fi(t), i.e., a formulation with two stochastic components in section 4.2 b

3. Replace the two–dimensional proxies to Fi(t) with processes F¯i(t, zt) exhibiting the same mean and variance, which are functions of a two–dimensional Brownian motion b zt with independent components in section 4.3

41 The forward rates themselves are then formulated as functionals of the processes F¯i(t).

Thus Fi(t) = gi F¯i(t, zt) with monotone functions gi, and the functional dependence of the forward rates on the two–dimensional Markov process is only through the pre–model process F¯i(t, zt . The latter allows for a non–zero correlation between the Brownian drivers of the forward rate and variance components. The construction and calibration of a two–dimensional Libor Markov functional model is demonstrated in section 4.4. Due to the dependence on independent Brownian motions, the calibration involves Gaussian integrals which are easier to handle in numerical imple- mentations.

4.1 Piterbarg’s FL–TSS Libor Market Model as pre–model

The incorporation of stochastic volatility into MFMs will be based on a calibrated Piterbarg FL–TSS model,

N+1 T N+1 dFi(t)= βi(t)Fi(t)+ 1 βi T0 Fi T0 λi(t) Σt µ (t) dt + σi(t) dW , − i t h i =:ϕ (F (t)) p i i e | {z } (4.1a) N N+1 αl λl(t) ρil(t) ϕl Fl N+1 µi (t)= Σt , µN (t) = 0, (4.1b) − 1+ αlFl(t) l=i+1 p X N+1 e dΣt =Θ Σ0 Σt dt + η Σt dV , e (4.1c) − t p for times T0 t < Ti TN i = 1,...,N . By calibrated we mean that the skew and ≤ ≤ volatility parameters βi(t), λi(t) are known on the considered time interval. Following the ideas presented in 2.3 an approximation to (4.1) will be introduced which is used as pre–model. As such it will serve as prerequisite for the derivation of numeraire discount bond functional forms from market observed derivative prices. First we observe that the FL–TSS model can be formulated as displaced diﬀusion model with time dependent displacement parameter γi(t), comp. [23],

N+1 T N+1 dFi(t)= Fi(t)+ γi(t) βi(t)λi(t) Σt µi (t) dt + σi(t) dWt , (4.2a) p N+1h i dΣt =Θ Σ0 Σt dt + η Σt dV , (4.2b) − t e 1 βi T0 p γ (t)= − F T , i = 1,...,N, T t < T T . (4.2c) i β t i 0 0 i N i() ≤ ≤ Introducing the displaced forward rate Fi(t) by deﬁning Fi(t) := Fi(t)+ γi(t), by reference

e e

42 to (4.2) its dynamics is given by

γi(t) dβi(t) dFi(t)= d Fi(t)+ γi(t) = dFi(t) dt − βi(t) dt e N+1 T N+1 γi(t) dβi(t) = Fi(t)βi(t)λi(t) Σt µi (t) dt + σi(t) dWt dt, − βi(t) dt p h i dγi(t) 1 β (T ) dβi(t) γi(t) dβi(t) e − i 0 because dt = 2 dt Fi T0e = dt . Now we approximate the dynamics of − βi(t) − βi(t) the process Fi(t) by freezing the local volatility functions ϕl(Fl(t)) at their time T0 values, and omitting the additional drift which is due to the time dependence of the displacement e parameter γi(t). Hence ϕl(Fl(T0)) = Fl(T0), and we can deﬁne the approximate displaced diﬀusion dynamics

N+1 T N+1 dFi(t) := Fi(t)βi(t)λi(t) Σt µ0i (t) dt + σi(t) dWt , (4.3a) K p h i N+1b b αl(Tl, Tl+1)λl(t)ρli(t)Fl T0; Tl, Tl+1 N+1 µ0 (t)= Σt e , µ0 (t) = 0, (4.3b) i − 1+ α (T , T )F T ; T , T N l=i+1 l l l+1 l 0 l l+1 p X e 1 βi T0 e F (t) F (t)+ γ (t), γ (t)= − F T , (4.3c) i i i i β t i 0 ≈ i() N+1 bdΣt =Θ Σ0 Σt dt + η Σt dV , (4.3d) − t p for i = 1,...,N and times T0 t < Ti TN , which serves as starting point for a two– ≤ ≤ dimensional pre–model.

4.2 The pre–model with two Brownian drivers

As already discussed above, a key motivation of Markov functional models is to formulate the numeraire discount bond process in terms of a low dimensional driving process. Here we want to deﬁne a pre-model with only two Brownian drivers which is based on the displaced diﬀusion approximation (4.3). For this we consider the one dimensional Brownian drivers of the rate and variance N+1 N+1 processes, dWt and dVt , and assume that their correlation is expressed by a function Γ(t). Hence 0 Γ(t) 1, and dW N+1, dV N+1 = Γ(t) dt. By introducing the independent ≤ ≤ t t N+1 N+1 Brownian driver dZt the process dWt can be decomposed into

dW N+1 = Γ(t) dV N+1 + 1 Γ(t)2 dZN+1, (4.4) t t − t N+1 N+1 pN+1 N+1 and the correlation dWt , dVt = Γ(t) dVt , dVt = Γ(t) dt is recovered since dZN+1, dV N+1 = 0. Also, dW N+1,dW N+1 = Γ(t)2 + 1 Γ(t)2 dt = dt. By reference t t t t − N+1 to (4.3d) dVt can be expressed in terms of the variance level,

N+1 Θ Σ0 Σt 1 N+1 dVt = − dt + dΣt . (4.5) − η√Σt η√Σt

43 Applying above expressions to the process (4.3a) with only one Brownian driver one obtains

N+1 N+1 dFi(t)= Fi(t)βi(t)λi(t) Σt µ0i (t) dt + dWt

p h N+1 Ni +1 2 N+1 = Fi(t)βi(t)λi(t) Σt µ0 (t) dt + Γ(t) dV + 1 Γ(t) dZ b b e i t − t p h N+1 Θ p i = Fbi(t)βi(t)λi(t) Σt µ0 (t) Γ(t) Σ0 Σt dt e i − η − hp Γ(t) N+1 2 N+1 b e + dΣ + Σt 1 Γ(t) dZ , (4.6) η t − t p p i N+1 where the dynamics of Fi(t) is formulated in terms of the variance process dΣt and an N+1 independent Brownian motion dZt . By Ito’s lemma (4.6) integrates to b t N+1 1 Fi(t)= Fi(T0)exp βi(s)λi(s) µ0 (s) (βi(s)λi(s)) Σs ds i − 2 ZT0 t h i b b Θ Γ(s)βi(s)λi(bs) Σ0 Σs ds − η − ZT0 t t 1 N+1 2 N+1 + Γ(s)βi(s)λi(s) dΣ + 1 Γ(s) βi(s)λi(s) Σs dZ η s − s ZT0 ZT0 p p (4.7) with

N+1 N µ (t) αl(Tl, Tl+1)λl(t)Fl T0; Tl, Tl+1 µ N+1(t) := 0i = , µ N+1(t) = 0. (4.8) 0i √Σ − 1+ α (T , T )F T ; T , T 0N t l=i+1 l l l+1 l 0 l l+1 e X b b Note that in this one dimensional setting the forward rates Fl(t) are perfectly correlated and thus ρli(t) 1 for all l,i = 1,...,N. ≡ N+1 The processes Fi(t) involve integrals of the stochastic variables Σt, √Σt and dZt . In order to obtain a pre–model which is of practical use these expressions have to be simpliﬁed. b This will be done in the following section where the stochastic entities will be replaced by appropriate approximations.

4.3 A simpliﬁcation of the pre–model process

The aim of this section is to replace the stochastic integrals in (4.7) with simpler stochastic variables which can be controlled more easily, thereby deﬁning a process F¯i(t) as proxy to

Fi(t). Speciﬁcally, we intend to ﬁnd approximations in terms of variables whose probability density is given by a standard normal distribution, comp. [12]. b For this we consider the integrals

t t N+1 N+1 Ii,2(t) := gi(s) dΣs and Ii,3(t) := hi(s) Σs dZs , (4.9) T0 T0 Z Z p

44 with deterministic functions gi(s), hi(s) and calculate their expected values and variances.

Starting with Ii,2 we have

t t t N+1 N+1 E gi(s) dΣs = E Θ gi(s) Σ0 Σs ds + E η Σs dVs = 0, T0 T0 − T0 Z Z Z p =0 =0

| {z t } | N+1 {z } because E Σs = E ΣT E Σ0 = 1, and Mt := √Σs dV is a martingale E Σ(s) < 0 ≡ 0 s which is why E Mt MT ΣT = 0. Furthermore, ∞ − 0 | 0 R t t 2 t 2 N+1 N+1 N+1 var gi(s) dΣ = E gi(s) dΣ E gi(s) dΣ s s − s ZT0 ZT0 ZT0 =0 t 2 2 = E η gi(s) Σs ds by Ito’s| isometry,{z } ZT0 t t t 2 2 2 2 2 2 = η gi(s) E Σs ds = Σ0 η gi(s) ds = η gi(s) ds. ZT0 ZT0 ZT0 =1

Turning to Ii,3 we obtain |{z}

t N+1 E hi(s) Σs dZs = 0, T0 Z p 2 because the integral is a martingale E hi(s) Σ(s) < . Secondly, ∞ t t 2 t 2 N+1 N+1 N+1 var hi(s) Σs dZs = E hi(s) Σs dZs E hi(s) Σs dZs T0 T0 − T0 Z p Z p Z p =0 t 2 = E hi(s) Σs ds by Ito’s isometry,| {z } ZT0 t t t 2 2 2 = hi(s) E Σs ds = Σ0 hi(s) ds = hi(s) ds. ZT0 ZT0 ZT0 =1

Hence the stochastic integrals Ii,2 and Ii,3 can be|{z} approximated as

t t 2 2 2 Ii, (t) var Ii, (t) = η gi(s) ds and Ii, (t) var Ii, (t) = hi(s) ds , 2 ≃ 2 U U 3 ≃ 3 Z Z ZT0 ZT0 with independent standard normal variables , (0, 1). In particular , = 0, and U Z∼N U Z the independence of Ii,2 and Ii,3, i.e., Ii,2,Ii,3 = 0, is preserved. The desired approximation to Fi(t) is arrived at by applying these results to expression (4.7) where the functions gi(s) and hi(s) are deﬁned by b

Γ(s) 2 gi(s) := βi(s)λi(s) and hi(s) := 1 Γ(s) βi(s)λi(s). (4.10) η − p 45 Furthermore the stochastic drift terms will be replaced by their expected values, i.e.,

t N+1 1 Θ Θ βi(s)λi(s) µ0 (s) (βi(s)λi(s)) + Γ(s) Σs Γ(s) Σ0 ds i − 2 η − η ZT0 t nh i o N+1 1 Θ Θ E βi(sb)λi(s) µ0 (s) (βi(s)λi(s)) + Γ(s) Σs Γ(s) Σ0 ds −→ i − 2 η − η ZT0 t nh i o N+1 1 Θ Θ = βi(s)λi(s) µb0 (s) (βi(s)λi(s)) + Γ(s) E Σs Γ(s) Σ0 ds i − 2 η − η ZT0 nh i =Σ 0 o t b N+1 1 |Θ{z } Θ = Σ0 βi(s)λi(s) µ0i (s) (βi(s)λi(s)) + Γ(s) Γ(s) ds T0 − 2 η − η =1 Z nh i o t b |{z} N+1 1 Θ Θ = βi(s)λi(s) µ0i (s) (βi(s)λi(s)) + Γ(s) Γ(s) ds. T − 2 η − η Z 0 nh i o b Utilising (4.10), the ﬁnal result for the proxy F¯i(t) Fi(t) can be summarised as ≃ t ¯ ¯ N+1 1 b Θ Θ Fi(t)= Fi(T0)exp βi(s)λi(s) µ0 (s) (βi(s)λi(s)) + Γ(s) Γ(s) ds i − 2 η − η ZT0 t nh t i o 2 2 2 2 + Γ(s) (βi(bs)λi(s)) ds + 1 Γ(s) (βi(s)λi(s)) ds , U − Z ZT0 ZT0 =:W =:W 1,σ1(t) 2,σ2(t) | {z } | {z (4.11)} with Brownian motions W1,σ1(t) and W2,σ2(t) of variance

t t 2 2 2 2 2 2 σ1(t) := Γ(s) (βi(s)λi(s)) ds and σ2(t) := 1 Γ(s) (βi(s)λi(s)) ds, (4.12) − ZT0 ZT0 for i = 1,...,N and times T0 t Ti. At time T0 the forward rates Fi(T0) are known and ≤ ≤ ¯ Fi(T0) we have Fi(T0)= Fi(T0)+ γi(T0) = for i = 1,...,N. βi(T0) According to the discussion of multi–dimensional Markov functional models in 2.2.3, ¯ T Fi(t) deﬁnes a projection function for the two–dimensional Markov process zt := W1,σ1(t),W2,σ2(t) N+1 ¯ in the terminal measure Q . Hence Fm(t, zt) is of Markov functional form and deﬁnes a one dimensional stochastic process. Process (4.11) is general enough to account for a correlation function Γ(s) between the Brownian drivers of the rate and volatility processes.

For the zero correlation case Γ(s) 0, and the proxy process F¯m becomes one–dimensional ≡ (comp. Figure 4.1).

One can now deﬁne functional relationships between forward rates Fi(t) and their approximations F¯i(t, zt) by introducing monotone functions gi for which Fi(t)= gi F¯i(t, zt) . ¯ The gi can be interpreted as perturbation functions which act on the proxy processes Fi so that they fall in line with the arbitrage free Fi for i = 1,...,N, comp. [14]. Re- calling that forward discount bonds are functions of their constituent forward rates, i.e.,

46 Forward rate functional form 6 ¯ F10(T1 0 z, 2)

4 ) z,

13 2 T ( 0 1 0 ¯ F

0 0 1 2 3 4 5 6 z2

¯ Figure 4.1: Proxy forward rate F10(T10) (4.11) as function

of zt = 0,z2) at reset time T10 = 10y. This corresponds to the zero correlation case, Γ(s) 0. ≡

D(Tm, Tn) D(Tm, Tn; Fm(Tm),...,Fn 1(Tm) ), we are now in a position to consider the ≡ { − } forward discount bond functional

¯ D(Tm, Tn; gm Fm(Tm, zTm ) , Fm+1(Tm),...,Fn 1(Tm) ), { − } which is of Markov functional form through its dependence on the Markov process zt at time

Tm. This will be utilised in constructing a Libor Markov functional model in the following section.

4.4 Construction of a two–dimensional Libor Markov functional model

In section 2.2.2 we have already discussed the construction of a multi–dimensional Libor Markov functional model and its calibration to digital caplets. We know want to apply these ideas and demonstrate how a two–dimensional model can be formulated in terms of the one dimensional pre–model processes (4.11). The latter are based on Piterbarg’s FL–TSS model and depend on a Markov process which is given by a two–dimensional Brownian motion whose variance is entirely determined by the pre–model skew and volatility parameters. As

47 such, the pre–model processes incorporate the correlation structure of a Piterbarg Libor Market Model with stochastic volatility. The forward rates themselves are formulated as monotone functionals of the pre–model process

Fi(t)= gi F¯i(t, zt) , (4.13) T with monotone functions gi and zt = W1,σ1(t),W2,σ2(t) . Thus they depend on the two– ¯ dimensional Markov process only via the pre–model forward rates Fi(t, zt).

4.4.1 A two–dimensional Libor Markov functional model in the terminal measure

The construction of a two–dimensional Libor Markov functional model will be based on the calibration to digital caplets. A digital caplet expiring at time Tm with strike K has the payout proﬁle

Vm(Tm; K)= D(Tm, Tm+1) 1Fm(Tm;Tm,Tm+1)>K . (4.14)

Assuming that the forward rates Fm(t) follow a log–normal process in their individual martingale measures Qm+1 (m = 1,...,N), the digital caplet values at time T are given by the Black76 formula,

Vm(T0)= D(T0, Tm+1) Φ(dm− ), (4.15)

Fm(T0;Tm,Tm+1) log K 1 dm− = σ(Tm) Tm T0, σ(Tm)√Tm T0 − 2 − − p with constant volatility σ(Tm) and cumulative normal distribution function Φ(x). Based on this assumption market quotes for digital caplets are provided in terms of Black76 implied volatilities. As the Piterbarg FL–TSS and the pre–model process (4.11) are formulated in the terminal measure which is induced by the numeraire discount bond process D(t, TN+1) we construct the Libor Markov functional model under QN+1. By the fundamental theorem of asset pricing the numeraire rebased digital caplet process is a martingale. Referencing

(4.14), its value at time T0 is thus given by

Vm(T0; K) Vm(Tm; K) D(Tm, Tm+1) = E T0 = E 1Fm(Tm)>K T0 D(T , TN ) D(T , TN ) F D(T , TN ) F 0 +1 m +1 m +1 1 E E . = 1Fm(Tm)>K Tm T0 D(T , TN ) F F m+1 +1

As discussed in the introduction, according to (4.13) the forward rates are monotone functionals of the pre–model processes F¯m(t, zt) for m = 1,...,N. This transfers to the discount bond functionals with ¯ D Tm, Tm+1; gm Fm(Tm, zTm ) , 48 and hence ¯ D Tm, TN+1; gm Fm(Tm, zTm ) , Fm+1(Tm),...,FN (Tm) ¯ depending on Fm(Tm, zTm ) as driving variable, which is of Markov functional form due to the Markov property of zt. For clarity we will suppress the functional relationship ¯ ¯ gm Fm(Tm, zTm ) in the discount bond processes and directly refer to Fm(Tm, zTm ) instead. Thus the digital caplet value can be expressed in terms of the process zt whose probability density is known:

Vm(T0; K) 1 = E 1Fm(Tm)>K E Tm T0 D(T , TN ) D(T , TN ) F F 0 +1 m+1 +1

= E 1 ¯ z gm(Fm(Tm, Tm ))>K 1 E z z × D(T , T ; F¯ (T , z ), F (T ) ) Tm T0 m+1 N+1 m+1 m+1 Tm+1 i m+1 i { }

= E 1 ¯ z Fm(Tm, Tm )>x∗m 1 E z z × D(T , T ; F¯ (T , z ), F (T ) ) Tm T0 m+1 N+1 m+1 m+1 Tm+1 i m+1 i { }

= dzTm p zTm zT0 Zx∗m 1 p zTm+1 zTm dzTm+1 , × 2 D(T , T ; F¯ (T , z ), F (T ) ) ZR m+1 N+1 m+1 m+1 Tm+1 i m+1 i { } (4.16) where xm∗ describes a boundary curve in two–dimensional state space,

¯ 1 2 ¯ F − (Tm,x∗ )= zT R Fm(Tm, zT )= x∗ , m m m ∈ m m and satisiﬁes gm(xm∗ )= K since gm is monotone. Details on the evaluations of the inner and outer expected values are given in Appendix A, where the two–dimensional integals above are given by expressions (A.45) and (A.46). In the course of the calibration this functional equation is solved for a series of strikes m m m K ,...,K m , where N stands for the number of caplet strikes available at expiry time 1 NS S Tm. As a result we obtain the set

m m Sm = xm,j∗ gm(xm,j∗ )= Kj for j = 1,...,NS , which deﬁnes the desired forward rate functional form Fm(Tm,xm) at reset time Tm. Indeed, ¯ m since Fm = gm Fm all xm,j Sm satisfy Fm(Tm,xm,j )= K . ◦ ∈ j The functional forms of the numeraire discount bond are determined recursively starting at time TN . At each forward reset date Tm TN equation (4.16) is solved numerically ≤

49 m by application of Brents’s algorithm for a set of digital caplets with strikes Kj for j = m 1,...,NS . Because

1 ¯ D(Tm, TN+1; Fm(Tm), Fi(Tm) i=m+1,...,N ) { } ¯ ¯ D(Tm, Tm+1; Fm(Tm)) = 1+ αm gm Fm(Tm) ¯ , (4.17) D(Tm, TN+1; Fm(Tm), Fi(Tm) i=m+1,...,N ) { } the time Tm numeraire discount bond is related to its functional form at time Tm+1 which was determined in the previous calibration step. By this ﬁtting to market quotes the sets

Sm are recovered for m = N,..., 1. Obviously, due to relation (4.17) this applies to the numeraire discount bond as well.

4.4.2 Calibration results

Here we present results of the construction of a two–dimensional Libor Markov functional model for ten forward rates F1,...,F10 on time grid T0 = 0y < 1y Tj 10 y for ≤ ≤ j = 1,..., 10. The considered underlying forward rate tenor was one year, and the model was calibrated to the Euro caplet market on May 7th, 2008. Caplet volatilities were available for strikes ranging from ATM 0.025 to ATM + 0.025 with a step size of 0.0025 and a one–year − spacing between caplet expiries. Hence the ﬁnal time of the model was TN+1 = T11 = 11y, and the numeraire discount bond given by D(t, T11). Digital caplet prices needed for the calibration of the model were calculated according to (4.15) with implied volatilities taken from the caplet market1.

In the implementation, at each time step Ti a grid of 10 nodes is set up in the x– and T y dimensions of the two–dimensional Markov process zt = (xt, yt) . Hereby the respective 2 maximum nodes are placed at values Mσ1(Ti) and Mσ2(Ti), where the variances σ1(Ti) 2 and σ2(Ti) are given by (4.12) and refer to the x, y–dimension, respectively. In our implementation we set the node spacing to three standard deviations, i.e., M = 3. Furthermore a constant correlation function Γ(s) 0.1 was assumed. Working backwards from time T ≡ 10 the inner expected values were calculated at each pair of nodes of the respective time slice ¯ Ti. The numeraire discount bond functional forms D(Ti, T11; F (Ti, zTi )) were then obtained by a cubic spline interpolation. Hence the calculation of the outer expected values in (4.16) involved the integration of third order polynomials which can be performed eﬃciently (SALI method, comp. [9]). Plots of the numeraire discount bond functional forms at various forward reset times can be found in Appendix C (Figures C.1 – C.4). Figure 4.2 displays the functional form of the numeraire discount bond at forward reset time T obtained on a grid of 10 10 nodes. 10 × 1 ∂VCaplet(K) This is justiﬁed since under the assumption of log–normality VDig. Caplet(K) = − ∂K .

50 1.0

0.8 )) 1 T (

10.6 0 ¯ F ; 1 0 T, 0 1 1 T (

D 0.4

0.2

0.0 0.030 0.035 0.040 0.045 0.050 0.055 0.060 ¯ F10(T1 0)

Figure 4.2: The numeraire discount bond as functional of ¯ ¯ F10(T10, zT 10): D(T10, T11; F10(T10, zT 10)).

51 52 Chapter 5

Conclusion

Within this study the incorporation of stochastic volatility into Markov functional models was investigated. It was explained that this can be achieved by introducing the idea of a pre–model which serves as driver of the numeraire discount bond process. For this sake Piterbarg’s forward Libor term structure of skew model (FL–TSS LMM) was discussed and calibrated to the swaption market. An approximation of this LMM was then used as pre–model in constructing a two–dimensional Libor Markov functional model. Following the introduction, the concept of market models was presented in chapter 2. The Libor Market Model was reviewed and generalisations to non log–normal dynamics were detailed. Special attention was paid to the derivation of drift terms under consideration of a stochastic volatility component. Subsequently the Markov Functional framework was introduced. After a formal deﬁnition it was shown how a model of this kind is calibrated to the digital caplet market. Furthermore the case of multi–dimensional Markov processes as drivers of the term structure was discussed. Because Piterbarg’s FL–TSS LMM was used as pre–model for a Libor Markov functional model, chapter three was dedicated to its detailed exposure. As this stochastic volatility Libor Market model is calibrated in the parameter domain, the ﬁtting to a grid of Hes- ton parameters which were derived form the Euro swaption market was demonstrated. A calibration algorithm was implemented and the resulting one–year forward rate skew and volatility term structure presented on an equally spaced time grid over a period of ten years. Chapter 4 focussed on the construction of a two–dimensional Libor Markov functional model. This was achieved by introducing approximations to the forward rate processes induced by Piterbarg’s FL–TSS dynamics. These were used as drivers of the numeraire discount bond process under the terminal measure. As the proxy processes were functions of a two–dimensional time–changed brownian motion, calibration involved the integration with respect to a Gaussian probability density.

53 As result of our work an implementation of a Libor Markov functional model is available which possesses the correlation structure of Piterbarg’s FL–TSS Libor Market Model. A such it can be used as an efficient tool for the pricing of exotic interest rate derivatives. Future research should encompass a comparison between Piterbarg’s FL–TSS LMM and the described pre–model based MFMs for a range of exotic products. Here the specific MFMs have to be calibrated to instruments which are relevant to the exotic product under consideration. In doing so also non–constant forms of the correlation function Γ(s) should be investigated. Although the interplay between pre–model choice and derivative pricing still offers much room for future reseach, our study illustrates that pre–model based MFMs are a promising alternative to Libor Market Models.

54 Appendix A

Mathematical details

A.1 The drift term in the Libor Market Model

In the following we derive an expression for the drift in equation (2.5) by referring to the fact that forward rate agreements (FRAs) are tradable and thus martingales by the fundamental theorem of asset pricing. In the following we work in the terminal measure QN+1 which corresponds to using the terminal bond D(t, TN+1) as numeraire. At time t the value of a

FRA for period [Ti, Ti+1] is given by

QN+1 D Ti, Ti+1 Fi Ti; Ti, Ti+1 K FRA(t)= D t, TN+1 E α Ti, Ti − , (A.1) +1 D T , T i N+1 where K denotes the agreed rate and α Ti, Ti+1 the accrual factor for period [Ti, Ti+1]. Since the numeraire rebased value of the FRA is a martingale it follows from the above N+1 F T ;T ,T D T ,T that EQ i i i i+1 i i+1 must be a martingale as well. Therefore the expression D T ,T i N+1 D(t,Ti+1) d Fi(t) must be driftless. As discount bonds are tradable and therefore martin- D(t,TN+1) gales under QN+1 this also applies to the expression d D(t,Ti+1) . Accordingly the relation D(t,TN+1 D(t, T ) D(t, T ) D(t, T ) d i+1 F (t) = i+1 dF (t) + d i+1 F (t) D(t, T ) i D(t, T ) i D(t, T ) i N+1 N+1 N+1 driftless driftless D(t, T ) | {z } | {z + d} i+1 , dF (t) , D(t, T ) i N+1 provides a means to determine an expression for the drift term because from the above it becomes apparent that the drift which results from the correlation term must cancel the forward Libor drift,

D(t, Ti+1) D(t, Ti+1) drift of dFi(t)= d , dFi(t) . (A.2) D(t, TN ) − D(t, TN ) +1 +1 Before we derive the drift term following (A.2) we reformulate the forward rate process in terms of independent Brownian drivers. Because the Brownian motion driving the variance

55 process is correlated with each Brownian forward Libor driver this will be achieved by N+1 introducing the K–dimensional vector of independent Brownian motions dZt which in N+1 N+1 addition is independent of the variance process. Therefore we have dZk (t), dVt = 0, N+1 N+1 N+1 N+1 dZl , dZj = δlj dt and dVt ,dWk (t) =Γk(t) dt for k = 1,...,K. Using these N+1 relations the Brownian motions dWk can be decomposed,

N+1 N+1 2 N+1 dW (t)=Γk(t) dV (t)+ 1 Γk(t) dZ (t). (A.3) k − k p Obviously hereby the relations

N+1 N+1 N+1 N+1 2 N+1 N+1 dV ,dW (t) =Γk(t) dV , dV + 1 Γk(t) dV , dZ (t) t k t t − t k =dt p =0 =Γk(t) dt,| {z } | {z } N+1 N+1 N+1 N+1 dWl (t),dWj (t) =Γl(t)Γj(t) dVt , dVt =dt 2 2 N+1 N+1 +| 1 {zΓl(t) 1} Γj(t) dZ (t), dZ (t) − − l j q p =δlj dt

2 2 = Γl(t)Γj(t)+ 1 Γl(t) 1 Γj(t)| δlj dt, {z } − − N+1 N+1 q and dWl (t),dWl (t) = dt, l,j = 1,...,K,p (A.4) are recovered. N+1 N+1 Following the decomposition (A.3) of dWt into uncorrelated Brownian motions dZt N+1 and dVt the forward Libor process (2.5) becomes

N+1 T N+1 dFi t; Ti, Ti+1 = ϕ Fi λi(t) Σt µi (t) dt + σi(t) dWt N+1 T T N+1 = ϕFi λi(t)pΣt µ (t) dt + σi(t) Ω(t) dZ ei t T N+1 p + σi(t) Γ(t) dV , (A.5a) e b t N+1 dΣt =Θ Σ0 Σt dt + η Σt dV , (A.5b) − t N+1 N+1 dV (t), dZ (t) = 0, T0 t < Ti pTN , i, k = 1,...,N, k ≤ ≤

2 where the matrix Ω(t) = 1 Γk(t) δkj k=1,...,K , and the vector Γ(t) = Γk(t) k=1,...,K − j=1,...,K { } p b

56 were deﬁned. Hence the covariance between forward rates changes dFi and dFj is given by

dFi t; Ti, Ti+1 , dFj t; Tj, Tj+1

= ϕ Fi ϕFj λi(t)λj(t)Σt K 2 2 σi,k(t)σj,l(t) 1 Γk(t) 1 Γk(t) δkl +Γk(t)Γl(t dt × − − k,l=1 X hp p i = ϕ Fi ϕ Fj λi(t)λj(t)Σt K K 2 σi,k(t)σj,k(t) 1 Γk(t) + σi,k(t)σj,l(t)Γk(t)Γl(t) dt × − k=1 k,l=1 X X i = ϕ Fi ϕ Fj λi(t)λj(t)Σt T T σi(t) Ω(t) Ω(t)σj(t) + σi(t)Γ(t) σj(t)Γ(t) dt. (A.6) × h i Focussing now on the right handb sideb of (A.2) we ﬁrst recall that the numeraire rebased discount bonds are related to the forward Libors by

N D(t, Ti+1) = 1+ αl(Tl, Tl+1)Fl(t) Fl(t, Tl, Tl+1), D(t, TN+1) l=Yi+1 which leads to N D(t, Ti+1) D(t, Ti+1) αl d = dFl(t) + drift terms, D(t, TN+1) D(t, TN+1) 1+ αlFl(t) l=Xi+1 1 where the time reference in the accrual factors αl has been omitted for simplicity . In D(t,Ti+1) N+1 N+1 calculating d , dFi(t) only terms proportional to < dV (t), dV (t) >= D(t,TN+1) N+1 N+1 dt and < dZi (t), dZj (t) >= δij dt contribute. Hence

N D(t, Ti+1) D(t, Ti+1) αl d , dFi(t) = dFl(t), dFi(t) D(t, TN+1) D(t, TN+1) 1+ αlFl(t) l=i+1 X N D(t, Ti+1) αl λl(t) ϕ Fl = Σt λi(t) ϕ Fi D(t, TN+1) 1+ αlFl(t) l=i+1 X T T σl(t) Ω(t) Ω(t)σi(t) + σl(t)Γ(t) σi(t)Γ(t) dt, × h i(A.7) b b D(t,Ti+1) where (A.6) was used in the second line. By reference to (A.5) the drift of dFi(t) D(t,TN+1) is given by

D(t, Ti+1) D(t, Ti+1) T N+1 drift of dFi(t)= ϕ Fi λi(t) Σt σi(t) µi (t) dt, D(t, TN+1) D(t, TN+1) p 1Since D(t,Ti+1) is a martingale the drift terms cancel those drifts which are contained in the individual D(t,TN+1) forward Libor changes dFl.

57 and employing (A.7) the drift now follows from (A.2):

N N+1 T N+1 αl λl(t) ϕ Fl µi (t) := σi(t) µi (t)= Σt − 1+ αlFl(t) l=i+1 p X T T e σl(t) Ω(t) Ω(t)σi(t) + σl(t)Γ(t) σi(t)Γ(t) × h (A.8)i b b

In a deterministic volatility setting we have Σt 1, Ω(t) = 1K and Γk = 0 for all ≡ k = 1,...,K. In this case the decomposition (A.3) collapses to dW N+1(t)= dZN+1(t) and b k k the drift vector (A.8) becomes

N N N+1 αl λl(t) ϕ Fl T αl λl(t) ρil(t) ϕ Fl µi (t)= σl(t) )σi(t) = . (A.9) − 1+ αlFl(t) − 1+ αlFl(t) l=i+1 l=i+1 X =ρli(t)=ρil(t) X e | {z } If one chooses ϕ to be the identity function, ϕ Fl = Fl (l = 1,...,N), a lognormal dynamics is recovered for forward Libor FN and the drift vector becomes

N N+1 αl λl(t) ρil(t) Fl µi (t)= . (A.10) − 1+ αlFl(t) l=Xi+1 e A.2 The derivative of the forward swap rate w.r.t the forward Libor rates

In this section we derive the derivative of the forward swap rate Smn(t) with respect to the forward Libor rates Fk(t). The swap rate is given by

D(t, Tm) D(t, Tn) D(t, Tm) D(t, Tn) S (t)= − = − mn P t n 1 mn( ) l=−m αl(Tl, Tl+1)D(t, Tl+1) where αl(Tl, Tl+1) denotes the year fraction of theP period [Tl, Tl+1] and D(t, Tk) stands for the discount factor corresponding to time Tk. Obviously the forward swap rate Smn(t) is a function of the discount bonds D(t, Tk) which in turn depend on the forward Libor rates

Fl t; Tl, Tl+1 because

k 1 − 1 D(t, Tk)= . 1+ αl(Tl, Tl+1)Fl t; Tl, Tl+1 Yl=0

58 Thus we arrive at the relations

∂Pmn(t) = αk 1(Tk 1, Tk)1m

∂D(t, Tk) αj (Tj , Tj+1)D(t, Tk) = 10 j k 1 ≤ ≤ − ∂Fj(t) −1+ αj (Tj , Tj+1)Fj t; Tj, Tj+1 D(t, Tj+1) = αj (Tj , Tj+1) D(t, Tk) 10 j k 1, − D(t, Tj) ≤ ≤ −

∂Smn(t) (δmk δnk)Pmn(t) D(t, Tm) D(t, Tn) αk 1(Tk 1, Tk)1m

αk 1(Tk 1, Tk)D(t, Tk)10 j k 11m

αk 1(Tk 1, Tk)D(t, Tk)10 j k 1 , (A.11) × − − ≤ ≤ − k=Xm+1 for j = 1,... . Speciﬁcally, ∂S (t) mn = 0 for j

A.3 Derivation of the coeﬃcient cmn

Here we provide the derivation of coeﬃcient cmn in (3.25) used in the matching condition (3.26) for the eﬀective volatility. We have

Smn T0 1 g x = 2 Φ √x 1 , βmn 2 − h i

59 z 1 1 2 where Φ(z) = 2π exp 2 x dx denotes the cumulative normal distribution. Taking −∞ − derivatives withR respect tox we obtain:

dg Smn T0 1 1 1 2 1 1 1 Smn T0 1 1 1 = 2 exp √x x 2 = exp x x 2 , dx π − π − βmn 2 −2 2 4 2 βmn 2 −8 2 h h i i h i d g 1 Smn T0 1 1 1 1 1 3 = = exp x x 2 x 2 , dx2 π − − ⇒ 2 βmn 2 −8 −8 − 2 h i1 3 1 2 1 2 d2g dg 8 x− 2 x− 1 1 = = − − = . ⇒ dx2 dx 1 −8 − 2 x x− 2 2 σ(Tm) With umn(Tm) := which has an expected value of βmn Tm Tm 2 2 umn := E[umn(Tm)] = E λmn(s) Σs ds = λmn(s) E Σs ds ZT0 ZT0 Tm e 2 e = ΣT0 λmn(s) ds, ZT0 e the coeﬃcient cmn becomes

d2g dg 1 1 cmn = 2 = + . (A.12) − dx x=umn dx x=umn 8 2 umn

The expected value umn also satisiﬁes

2 σ(Tm) 2 2 2 2 umn = ΣT Tm T0 β λ = Tm T0 β λ , ≃ β 0 − mn mn − mn mn mn 2 2 Tm since σ(Tm)= β λ Σs ds and ΣT = 1. Inserting this into (A.12) we arrive at mn mn T0 0 R 1 1 c = + , (A.13) mn 8 2 2 2 Tm T0 β λ − mn mn which will be used in the process of calibrating the forward rate volatilities.

A.4 Proof of corollary 3.3.2

Identify X0(t) from theorem 3.3.1 with the swap rate Smn(t) and deﬁne the local volatility function 1 f(t, Smn(t)) = βmn(t)Smn(t)+ 1 βmn(t) Smn T0 . Smn T0 − Consider further the re-scaled local volatilitye function of theoreme 3.3.1 with parameter ǫ set to 1, ǫ = 1. Then we have

f1(t,x)= f(t,x), and f(t,x0)= f t, Smn T0 = 1,

60 so that from theorem 3.3.1 the time–independent eﬀective volatility function becomes 1 1 Tm 2 2 f (Smn(t)) = βmn(t)Smn(t)+ 1 βmn(t) Smn T0 wmn(t) dt . 1 S T − mn 0 ZT0 e e Approximating f 1(Smn (t)) with a linear function g(Smn(t)) matching the eﬀective local volatility function and its slope at Smn(t)= Smn T0 , f (Smn(t)) g Smn(t) 1 ≈ dg(Smn(t)) = g Smn T0 + Smn(t) Smn T0 dSmn(t) − Smn(t)=Smn(T0) df 1(Smn(t)) = f 1 Smn T0 + Smn(t) Smn T0 , dSmn(t) − Smn(t)=Smn(T0) =1 and observing that | {z } Tm df (S (t)) 1 βmn(t)Smn(t)+ 1 βmn(t) Smn T0 βmn(t)wmn(t) dt 1 mn = T0 − dS t 1 mn( ) Smn T0 R Tm 2 2 e βmn(t)Smn(t)+ 1 e βmn(t) Smn T0 e wmn(t) dt T0 − h Tm Tm i df 1(Smn(t)) R 1 = = e βmn(t) wmne (t) dt, since wmn(t) dt = 1, ⇒ dSmn(t) Smn(T0) Smn(t)=Smn(T0) ZT0 ZT0 we arrive at e

Tm Smn(t) Smn(T0) f (Smn(t)) 1+ − βmn(t) wmn(t) dt 1 ≈ S (T ) mn 0 ZT0 1 e = βmnSmn(t)+ 1 βmn Smn T0 , Smn T0 − with Tm βmn = βmn(t)wmn(t) dt, ZT0 as asserted. e 2 Turning to the function v(t) we can write, using conditional independence of X0(t) and Σ(t),

v(t)2 = E Σ(t) X (t) x 2 by definition (3.14), 0 − 0 h i 2 = E Σ(t)E X (t) x Σ(t) by conditional independence, 0 − 0 h h t ii 2 = E Σ(t) E λ(s) Σ( s)dW (s) Σ(t) by (3.13a) since f0 t, X0(t) = 1, ZT0 t p 2 = E Σ(t) E λ(s) Σ(s) ds Σ(t) by Ito’s isometry, ZT0 t 2 = λ(s) E Σ(t)Σ(s) ds by linearity. (A.14) T Z 0 h i 2 Since X0(t) is identified with swap rate Smn(t) its volatility λ(t) corresponds to the swap rate volatility λmn(t). For clarity we suppress the subscripts in the following. e 61 Now, defining the process U(t)= eΘtΣ(t), its dynamics is given by

Θt Θt Θt Θt dUt =Θe Σ(t)+ e dΣt = Θ Σ(T0)e dt + η Σ(t)e dV (t) t t Θs Θs p = Ut = U0 + Θ Σ(T0) e ds + η e Σ(s) dV (s) . ⇒ T0 T0 Z Z p =:Mt

Θs 2 | {z } Since E ηe Σ(s) < , the term denoted by Mt is a martingale and therefore we ∞ have forhs t : i ≤ p t Θu E Us Ut = E Us Us + Θ Σ(T0) e du + Mt Ms − Zs t 2 Θu = E Us + Us Θ Σ(T0) e du s Z t 2 Θu = E Us + E Us Θ Σ(T0) e du. Zs From this we deduce that for s t : ≤ Θs Θt E Σ(s) Σ(t) = e− e− E Us Ut t Θs Θt 2Θs 2 Θs Θt Θs Θu = e− e− E e Σ(s) + e− e− E e Σ(s) Θ Σ(T0) e du Zs t Θ(t s) 2 2 Θt Θu = e− − E Σ(s) + Σ(T0) Θ e− e du, because E Σ(s) = Σ(T0), Zs Θ(t s) 2 2 Θ(t s) = e− − E Σ(s) + Σ(T ) 1 e− − (A.15) 0 − To arrive at an analytic expression for v(t)2 the ﬁrst term of (A.15) needs to be further analysed. For this we ﬁrst consider the quantity Σ(t)2 and derive its dynamics:

d Σ(t)2 = 2Σ(t) dΣ(t)+ dΣ(t)2 3 2 = 2Σ(t)Θ Σ(T ) Σ(t) dt + 2ηΣ(t) 2 dV (t)+ η Σ(t)dV (t) 0 − t t = Σ(t)2 Σ T 2 = (2Θ Σ T + η2 Σ(s) ds 2Θ Σ(s)2 ds ⇒ − 0 0 − ZT0 ZT0 t 3 + 2η Σ(s) 2 dV (s), ZT0 and since the last term is a martingale we obtain, upon taking expectations and bearing in mind that E Σ(t) = Σ T0 , t E Σ(t)2 = Σ T 2 + 2Θ Σ T 2 + η2Σ T (t T ) 2Θ E Σ(s)2 ds. (A.16) 0 0 0 − 0 − ZT0 Deﬁning

u(t)= E Σ(t)2 , 62 it follows from (A.16) that u(t) solves the inhomogeneous diﬀerential equation

du(t) = 2Θ Σ T 2 + η2 Σ T 2Θ u(t), dt 0 0 − where the solution can be expressed as

2 u(t)= g(t) + Σ T0 , (A.17a) dg(t) = η2 Σ T 2Θ g(t). (A.17b) dt 0 − Equation (A.17b) is solved by

t 2 2Θ t 2Θ u g(t)= η Σ T0 e− e du, ZT0 so that the expectation of the squared volatility becomes:

t 2 2 2 2Θ t 2Θ u u(t)= E Σ(t) = Σ T0 + η Σ T0 e− e du, ZT0 2Θ (t T ) 2 1 e 0 = Σ T + η2 Σ T − − − . (A.18) 0 0 2Θ Finally an expression for v(t) can be derived by inserting expression (A.18) into (A.15) and carrying out the integration in (A.14):

t v(t)2 = λ(s)2 E Σ(t)Σ(s) ds ZT0 t h i 2 2 Θ(t s) 2 Θ(t s) = λ(s) Σ T0 e− − + Σ T0 1 e− − T − Z 0 2Θ (s T0) 2 1 e− − Θ(t s) + η Σ T − e− − ds 0 2Θ t t eΘ(s T0) e Θ(s T0) 2 2 2 Θ(t T0) 2 − − − = Σ T λ(s) ds + η Σ T e− − λ(s) − ds, 0 0 2Θ ZT0 ZT0 which is, upon replacing λ(s) with λmn(s), the result (3.17) we wanted to derive.

A.5 A recursion schemee for a system of time dependent Ric- cati equations

In this section we consider the system of Riccati equations dA t, T = α(t) D t, T , dt N N dD 2 t, T = f(t)+ Θ D t, T + D t, T , dt N N N A T , T = D T , T = 0, (A.19) N N N eN 63 for times t [T , T ], where α :[T , T ] R is a real valued differentiable function, f(t) a ∈ 0 N 0 N → differentiable complex valued function, f :[T , T ] C, and Θ a complex valued constant, 0 N → Θ C. Due to the time dependence of f, the system does not possess an analytic solution ∈ e on [T0, TN ]. However, above problem can be approximated by dividing the considered time e span into sub intervals [T , T ], i = 0,...,N 1, on each of which the functions α and f i i+1 − are assumed to take on constant values α Ti and f Ti . As in this case each sub problem does allow for an analytic solution, the overall solution on time interval [T0, TN ] is obtained by concatenating the individuals solutions of adjacent sub intervals by imposing separate matching conditions on coefficients Ai and Bi, respectively. The so defined discretisation of system (A.19) can be expressed as dA i t, T = α(T ) D t, T , (A.20a) dt i+1 i i i+1 dDi 2 t, T = f(T )+ Θ D t, T + D t, T , (A.20b) dt i+1 i i i+1 i i+1 A T , T = A T , T , D T , T =D T , T , (A.20c) i i+1 i+1 i+1 i+1e i+1 i i+1 i+1 i+1 i+1 i+1

AN 1 TN , TN = DN 1 TN , TN = 0, (A.20d) − − for t [Ti, Ti+1] and i = 0,...,N 1 with αi := α(Ti) const., fi := f(Ti) const.. We ∈ − ≡ ≡ know derive analytic solutions for functions Ai t, Ti+1 and Di t, Ti+1 on each sub interval. In doing so we start with equation (A.20b).

A.5.1 An analytic solution for Di t, Ti+1

In solving (A.20b) on time interval [Ti, Ti+1] we ﬁrst deﬁne the function g(t) by

g(t)= Θ2 4 f(t), − and abbreviate gi = g(Ti) const., since f(Ti) const.. Furthermore, we introduce the ≡ e ≡ complex valued constants 1 d± = Θ √gi , i 2 ± and observe the relations e dD i t, T = D t, T 2 + Θ D t, T + f dt i+1 i i+1 i i+1 i 2 1 2 = Di t, Ti+1 + Θe Di t, Ti+1 + Θ gi 4 − 2 + + = Dit, Ti+1 d + d− Di t, Ti+1 + d d− − e i i e i i + = Di t, Ti+1 d Di t, Ti+1 d− , (A.21) − i − i + and d d− = √ gi. (A.22) i − i

Since the derivation of Di t, Ti+1 on time interval [Ti, Ti+1] for non–zero values of gi is diﬀerent to the gi = 0 case these two scenarios will be separated.

64 A.5.1.1 The case gi = 0 6

Now, if gi = 0 one can consider the total diﬀerential √gi ds and use above formulation to 6 arrive at the following decomposition, ds √gi √gi √gi ds = √gi dDi = dDi = + dDi by (A.21), dDi dDi Di d Di d− ds − i − i + Di di− Di di = − + − − dDi Di d Di d− − i − i 1 1 = + dDi, (A.23) Di d − Di d− − i − i which will now be used in integrating over time interval [t, Ti+1]:

Ti+1 √gi Ti+1 t = √gi ds − Zt Di(Ti+1,Ti+1) Di(Ti+1,Ti+1) dDi dDi = + by (A.23), D t,T Di d − D t,T Di d− Z i( i+1) − i Z i( i+1) − i + Di Ti+1, Ti+1 di Di Ti+1, Ti+1 di− = ln −+ ln − Di t, Ti+1 d − Di t, Ti+1 d− − i − i + Di Ti+1, Ti+1 di Di t, Ti+1 di− = ln − − + . (A.24) Di Ti+1, Ti+1 d− Di t, Ti+1 d − i − i From this result the solution Di t, Ti+1 is obtained. (A.24) is equivalent to D T , T d+ D t, T d i i+1 i+1 i i i+1 i− √gi Ti+1 t − − + = e − Di Ti+1, Ti+1 d− Di t, Ti+1 d − i − i D T , T d+ i i+1 i+1 i √gi Ti+1 t = Di t, Ti+1 1 − e− − ⇒ − Di Ti+1, Ti+1 d− − i D T , T d+ + i i+1 i+1 i √gi Ti+1 t = di di− − e− − − Di Ti+1, Ti+1 d− − i + Di Ti ,Ti d + +1 +1 − i √gi Ti+1 t di di− e− − − Di Ti+1,Ti+1 d− − i Di t, Ti+1 = , D T ,T d+ ⇐⇒ i i+1 i+1 i √gi Ti t 1 − e− +1− − Di Ti+1,Ti+1 d− − i + + √gi (Ti+1 t) d Di Ti+1, Ti+1 d− d− Di Ti+1, Ti+1 d e− − = i − i − i − i + √gi (Ti+1 t) Di Ti+1, Ti+1 d− Di Ti+1, Ti+1 d e − i − − i − − + + + + √gi (Ti+1 t) d Di Ti+1, Ti+1 d + √gi (d √gi) Di Ti+1, Ti+1 d e− − = i − i − i − − i + √gi (Ti+1 t) Di Ti+1, Ti+1 d 1 e− − +√gi − i − + + √gi √gi Ti+1 t + di Di Ti+1, Ti+1 di 1 1 + e− − + di √gi − − − di = , + √gi (Ti+1 t) Di Ti+1, Ti+1 d 1 e + √gi − i − − − (A.25) + since d d− = √gi, which is the solution for Di t, Ti+1 on interval [Ti, Ti+1] if gi = 0. i − i 6 65 A.5.1.2 The case gi = 0

+ In the case gi = 0, we have di = di− and (A.21) becomes

dDi + 2 t, Ti+1 = Di t, Ti+1 d . (A.26) dt − i Again Di t, Ti+1 is obtained by transforming a time integral into an integral with respect to D : Ti+1 Di(Ti+1,Ti+1) ds Ti+1 t = ds = dDi − dD Zt ZDi(t,Ti+1) i Di(Ti+1,Ti+1) dD = i by (A.26) + 2 Di(t,Ti+1) Di t, Ti+1 d Z − i 1 1 = + + Di t, Ti+1 d − Di Ti+1, Ti+1 d − i − i + 1 Ti+1 t Di Ti+1, Ti+1 di + 1 = + = − −+ , (A.27) ⇒ Di t, Ti+1 d Di Ti+1, Ti+1 d − i − i from where Di t, Ti+1 follows:

+ + Di Ti+1, Ti+1 di Di t, Ti+1 = di + − + 1+ Ti+1 t Di Ti+1, Ti+1 d − − i + + Di Ti+1, Ti+1 + diDiTi+1, Ti+1 di Ti+1 t = + − − , (A.28) 1+ Di Ti+1, Ti+1 d Ti+1 t − i − for times t [T , T ]. ∈ i i+1

A.5.2 An analytic solution for Ai t, Ti+1

As in the derivation of Di t, Ti+1 on time interval [Ti, Ti+1], the derivation of Ai t, Ti+1 for gi = 0 follows diﬀerent lines as the gi = 0 case which is why these two scenarios will be 6 treated separately again.

A.5.2.1 The case gi = 0 6

In solving (A.20a) for gi = 0 we ﬁrst introduce the function 6 D T , T d+ i i+1 i+1 i √gi Ti+1 t mi(t)= − e− − , (A.29) Di Ti+1, Ti+1 d− − i with which the solution for Di t, Ti+1 can be abbreviated as

+ di di−mi(t) Di t, Ti+1 = − (A.30) 1 mi(t) −

66 and which satisﬁes

dmi(t) = √g m (t). (A.31) dt i i

Now the total diﬀerential Di s, Ti+1 ds can be written as,

+ di di−mi(s) ds Di s, Ti+1 ds = − dmi 1 mi(s) dmi − + di di−mi(s) 1 = − dmi by (A.30) and (A.31) 1 mi(s) √gi mi(s) − + dm d d− = i i i √gi mi(s) 1 mi(s) − 1 mi(s) − − dmi + 1 mi(s)+ mi(s) di− = di − √gi mi(s) 1 mi(s) − 1 mi(s) − − + dmi + 1 di di− = di + − √gi mi(s) 1 mi(s) − dmi + 1 √gi = di + from (A.22), √gi mi(s) 1 mi(s) − which integrates to

Ti+1 + mi(Ti+1) mi(Ti+1) di dmi dmi Di s, Ti+1 ds = + √gi mi 1 mi Zt Zmi(t) Zmi(t) − + d mi(Ti+1) mi(Ti+1) 1 = i ln ln − . (A.32) √gi mi(t) − mi(t) 1 −

But this result leads us to the solution for Ai t, Ti+1 , since from (A.20a)

Ti+1 dAi s, Ti+1 Ai t, Ti+1 = Ai Ti+1, Ti+1 ds − ds Zt Ti+1 = Ai Ti+1, Ti+1 αi Di s, Ti+1 ds − Zt + di mi(Ti+1) mi(Ti+1) 1 = Ai Ti+1, Ti+1 αi ln ln − − √gi mi(t) − mi(t) 1 − + mi(Ti+1) 1 = Ai Ti+1, Ti+1 αi di Ti+1 t ln − , (A.33) − − − mi(t) 1 − m (T ) where in the last step the relation i i+1 = e√gi(Ti+1 t) was inserted, comp. (A.29). mi(t) −

67 Since

+ Di Ti ,Ti d +1 +1 − i 1 m (T ) 1 Di Ti+1,Ti+1 d− − i i+1 = − i − + mi(t) 1 Di Ti+1,Ti+1 d i √gi (Ti t) − − e− +1− 1 Di Ti+1,Ti+1 d− − − i + Di Ti+1, Ti+1 d Di Ti+1, Ti+1 d− = − i − − i + √gi (Ti+1 t) Di Ti+1, Ti+1 d e Di Ti+1, Ti+1 d− − i − − − − i + d d− = i − i √gi (Ti+1 t + √gi (Ti+1 t) + + Di Ti+1, Ti+1 1 e− − )+ d e d + d d− − i − − − i i − i √gi h i = , √gi (Ti+1 t) + √gi (Ti+1 t) Di Ti+1, Ti+1 1 e d 1 e + √gi − − − − i − − − and ln 1 = ln(x), (A.33) simpliﬁes to x − + Ai t, Ti+1 = Ai Ti+1, Ti+1 αi d Ti+1 t − i − 1 + √gi (Ti+1 t) + ln 1+ Di Ti+1, Ti+1 di 1 e− − (A.34) √gi − − A.5.2.2 The case gi = 0

In solving (A.20a) for gi = 0 we re–deﬁne the function mi(t),

+ mi(t)= Di Ti+1, Ti+1 d Ti+1 t , (A.35) − i − with which the solution for Di t, Ti+1 (A.28) can be abbreviated as

+ Di Ti+1, Ti+1 + d mi(t) D t, T = i (A.36) i i+1 m t 1+ i( ) and which satisﬁes

dmi(t) + = d Di Ti+1, Ti+1 . (A.37) dt i −

68 Therefore the integral of Di t, Ti+1 over time can be expressed as

Ti+1 Ti+1 + Di Ti+1, Ti+1 + d mi(s) D s, T ds = i ds i i+1 1+ m (s) Zt Zt i mi(Ti+1) + Di Ti+1, Ti+1 + d mi ds = i dm 1+ m dm i Zmi(t) i i mi(Ti+1) + Di Ti+1, Ti+1 + di mi dmi = + m t 1+ mi d Di Ti+1, Ti+1 Z i( ) i − mi(Ti+1) Di Ti+1, Ti+1 dmi = + d Di Ti+1, Ti+1 m t 1+ mi i − Z i( ) + mi(Ti+1) di 1+ mi 1 + + − dmi d Di Ti+1, Ti+1 m t 1+ mi i − Z i( ) Di Ti+1, Ti+1 1+ mi(Ti+1) = + ln d Di Ti+1, Ti+1 1+ mi(t) i − + di 1+ mi(Ti+1) + + mi Ti+1 mi(t) ln d Di Ti+1, Ti+1 − − 1+ mi(t) i − + Di Ti+1, Ti+1 di 1+ mi(Ti+1) = + − ln d Di Ti+1, Ti+1 1+ mi(t) i − =( 1) 1 − =ln , since mi(Ti+1) = 0 1+mi(t) | {z } | {z + } di + + mi(Ti+1) mi(t) di Di Ti+1, Ti+1 − − =0, from (A.35) + di | {z } = ln 1+ mi(t) + mi(t) − d Di Ti+1, Ti+1 i − + + = ln 1+ Di Ti+1, Ti+1 d Ti+1 t + d Ti+1 t , (A.38) − i − i − where in the last line the deﬁnition of mi(t) was inserted. From this Ai t, Ti+1 in the case gi = 0 follows by integration of (A.20a):

Ti+1 Ai t, Ti+1 = Ai Ti+1, Ti+1 αi Di s, Ti+1 ds − Zt + + = Ai Ti+1, Ti+1 αi ln 1+ Di Ti+1, Ti+1 d Ti+1 t + d Ti+1 t . − − i − i − (A.39)

69 A.5.3 Summary of the solution

The system of time dependent Riccati equations (A.20) is solved by

Ai t, Ti+1 = Ai Ti+1, Ti+1

+ 1 + √gi (Ti+1 t) d Ti+1 t + ln 1+ D Ti+1, Ti+1 d 1 e if g = 0, i √gi i i − − i αi − − − 6 −  + +  ln 1+ Di Ti+1, Ti+1 d Ti+1 t + d Ti+1 t if gi = 0,  − i − i − (A.40)  and

+ √gi √gi Ti+1 t Di Ti+1,Ti+1 d 1 1 e− − +√gi i d+ + − − − i di if gi = 0, + √g (T t) Di Ti ,T i d 1 e− i i+1− +√gi 6 Di t, Ti+1 =  +1 +1 − i − (A.41)  + +  Di Ti+1,Ti+1 +d Di Ti+1,Ti+1 d Ti+1 t  i − i − + if gi = 0, 1+ Di Ti+1,Ti+1 di Ti+1 t  − −  for times t [T , T ] with i = 1,...,N 1, and complex valued constants ∈ i i+1 − 2 gi = g Ti = Θ 4 f Ti , − 1 d± = Θ √gi , i 2e ± + = d d− = √gi. ⇒ i − i e

+ Θ If gi = 0, the relation di = di− = 2 holds. Furthermore, e

AN 1 TN , TN = DN 1 TN , TN = 0. − − A.6 Derivation of relation (3.28)

In the following we provide the derivation of the result

Tm ϕ µ Σt = E exp µ f(s) Σs ds Σt = exp Aµ,f t, Tm + Bµ,f t, Tm Σt − Zt for T t T where f(s) is a real valued function and the coeﬃcients A, B satisfy the 0 ≤ ≤ m Riccati system (3.29), which is a special case of a relation provided by Duﬃe et al. in [6] and for time t = T0 corresponds to expression (3.27).

We begin by taking the total derivative of ϕ µ Σ0 with respect to time. Observing that the derivative of the conditional probability distributio n of the process Σt, p Σt Σ0 , with

dp Σ Σ respect to time vanishes, i.e., t 0 = 0, the diﬀerential operator can be drawn into the dt

70 expectation and for T t T one obtains the ordinary diﬀerential equation (ODE) 0 ≤ ≤ m Tm dϕ µ Σt d = E exp µ f(s) Σs ds Σt dt dt − Zt Tm

= E µ f(t) Σt exp µ f(s) Σs ds Σt − − Zt = µ f(t) Σt ϕ µ Σt , − ϕ µ Σ T = 1. m Considering the process

dΣt =Θ Σ0 Σt dt + η Σt dVt, − p 2 it is obvious that the drift and variance coefficients, Θ Σ0 Σt and η Σt respectively, − are linear in Σt and thus affine. Furthermore, by Ito’s lemma the total differential of any function of Σt, F Σt , is given by

2 ∂F ∂F 1 ∂ F 2 dF = + dΣt + 2 dΣt ∂t ∂Σt 2 ∂Σt ∂F ∂F η2 ∂2F ∂F = +Θ Σ0 Σt + Σt dt + η Σt dVt. ∂t − ∂Σ 2 ∂Σ2 ∂Σ t t t p d( ) Applying the obtained diﬀerential operator dt· to above ODE, we see that it is equivalent to ∂ϕ ∂ϕ η2 ∂2ϕ +Θ Σ0 Σt + Σt 2 + µf(t)Σt ϕ = 0, ∂t − ∂Σt 2 ∂Σt the general solution of which can be written in the form

ϕ µ Σt = exp A t, Tm + B t, Tm Σt . (A.42) Upon inserting this ’ansatz’ we arrive at

2 dA dB η 2 + Σt +Θ Σ0 Σt B t, Tm + Σt B t, Tm + µf(t)Σt = 0, (A.43) dt dt − 2 which is equivalent to the Riccati system of ordinary diﬀerential equations dA t, T = Θ Σ B t, T , dt m − 0 m dB η2 t, T = µ f(t)+Θ B t, T B t, T 2, dt m m − 2 m

A Tm, Tm = B Tm, Tm = 0. (A.44) This system of ODEs is exactly the one stated in (3.29), where the dependence of A and B on µ and f(t) was made explicit.

71 Obviously the system (A.44) corresponds to expression (A.19) with the replacements

TN Tm, −→ 2 2 α(t) ΘΣ = Θ, because Σ = 1, −→ η2 0 η2 0 Θ Θ, −→ η2 D(t, Tme) B(t, Tm), −→ − 2 η2 f(t) µf(t), −→ − 2 and can be discretised analogously to (A.20). Hence its solution on sub intervals [Ti, Ti+1], i = 1,...,m 1, is given by (A.40) and (A.41), where − 2 2 gi = g Ti =Θ + 2 η µf Ti , 1 d± = Θ √gi , i 2 ± + = d d− = √gi. ⇒ i − i A.7 2d–Markov functional integration

T T Deﬁning zTm := xTm , yTm with zT0 := 0, 0 , the two–dimensional integrals are evaluated. The inner expected value is hence given by 1 E(zTm ) := p zTm+1 zTm dzTm+1 2 D(T , T ; F¯ (T , z ), F (T ) ) ZR m+1 N+1 m+1 m+1 Tm+1 i m+1 i { } ∞ ∞ 1 = dxTm+1 px xTm+1 xTm py yTm+1 yTm dyTm+1 . | D(Tm+1, TN+1; xTm+1 , yTm+1 ) | Z−∞ Z−∞ (A.45) The outer expected value is then calculated as

p zTm zT0 E(zTm ) dzTm Zx∗ z∗ z∗ xTm − = 1 dxT px xT 0 E(xT , yT ) py yT 0 dyT , (A.46) − m m | m m m | m Z Z −∞ −∞ where the boundary value z∗ is deﬁned by

Tm x∗ µ¯m(s) ds ¯ 1 T0 z∗ = Fm(Tm; (xTm , 0))− = ln e− x∗ ¯ Fm(T0) R with the integrand of expression (4.11) as drift termµ ¯m(s). The conditional probability densities are given by Gaussian normal distributions: 2 1 1 (xT xT ) p x x m+1 − m , x Tm+1 Tm = exp 2 √2πσ (T , T ) −2 σx(Tm, Tm+1) x m m+1 2 1 1 (yTm+1 yTm ) p y y − , y Tm+1 Tm = exp 2 √2πσ (T , T ) −2 σy(Tm, Tm+1) y m m+1

72 with variances

Tm+1 2 2 2 σx(Tm, Tm+1) = Γ(s) (βm+1(s)λm+1(s)) ds, (A.47) ZTm Tm+1 2 2 2 σy(Tm, Tm ) = 1 Γ(s) (βm+1(s)λm+1(s)) ds (A.48) +1 − ZTm from expression (4.12).

73 74 Appendix B

The Heston Model

This chapter is dedicated to Hestons’s stochastic volatility model [11]. Although the model was originally introduced in the context of option pricing for equities, we present it as a model for forward swap rates. The main reason for this is that Piterbarg’s Libor Market Model (FL–TSS) is calibrated to market derived Heston parameters for european swaptions. Thus in the context of Piterbarg FL–TSS model calibration takes place in the parameter domain and for this the existence of a calibrated Heston model is a prerequisite.

B.1 Speciﬁcation of the model dynamics

Let S(t) denote the time t T par rate of a (forward) swap maturing at time T >t, the ≥ 0 ﬁnal time. Then the swap rate is modelled as a displaced diﬀusion process with stochastic volatility,

dS(t)= β S(t) + (1 β) S(T0) λ Σt dWt, (B.1a) −

dΣt =Θ Σ0 Σt dt + η Σt dVpt, (B.1b) − dWt, dVt = ρdt, p (B.1c) for T t T, where the constants β and λ denote the skew paramter and volatility 0 ≤ ≤ λ. The dynamics of the volatility level Σt is given by a CEV process with start value

Σ0 = Σ T0 = 1. Whereas above model speciﬁcaton allows ﬁve degrees of freedom, namely the swap rate skew β, the swap rate volatility λ, the mean reversion speed Θ, the volatility of variance η and the correlation between the Brownian drivers1 of the swap rate and variance processes ρ, we restrict the parameter space to the set of tuples (β, λ) . Thus in calibrating { } 1 Both Brownian motions dWt and dVt are considered under the swap measure induced by taking the present value of a basis point as numeraire.

75 the model to european swaption prices the mean reversion and volatility of variance as well as the correlation will be kept constant across swaption strikes and expiries2. The model dynamics (B.1) is equivalent to

dS(t)= d S(t)+ γ = S(t)+ γ βλ Σt dWt, (B.2a)

dΣt =Θ Σ0 Σt dt + η Σt dV t, p (B.2b) − dWt, dVt = ρdt, p (B.2c) 1 β γ = − S(T ), (B.2d) β 0 for T t T, where γ denotes the constant displacement parameter, comp. [23]. From 0 ≤ ≤ (B.2) it is obvious that the stochastic variable S(t) := S(t)+ γ follows a lognormal process with stochastic volatility βλ√Σt. e The value of any derivative product under this model depends on the conditional distribution of the stochastic process (S(t), Σt) t T0 . However, for valuation purposes it is { } ≥ more convenient to consider the stochastic process x(t) t T0 := (ln S(t), Σt) t T0 instead e { } ≥ { } ≥ of (S(t), Σt) t T0 itself. Since x(t) can be cast into the form { } ≥ e 1 2 2 0 2 (λβ) λβρ λβ 1 ρ dVt edx(t)= Σt dt + Σt − , (B.3) ΘΣ0 − Θ η 0 · dZt p p =:µ(xt) =:σ(xt) where dZt is| a Brownian{z motion orthogonal} | to dVt which{z facilitates} the decomposition 2 2 dWt = ρ dVt + 1 ρ dZt dW = dt and dWt, dVt = ρdt, since dVt, dZt = 0 , it is − → t T an aﬃne processp because µ(xt) and σ(xt) σ(x t) are aﬃne functions of xt. Expression (B.3) involves the dynamics of ln S(t) which is obtained by applying Ito’s lemma, e ∂ ln S(t) 1 ∂2 ln S(t) d ln S(t)= dS(t)+ dS(t)2 ∂S(t) 2 ∂S(t)2 e e e 1 e 1 1e 2 = S(t)βλ Σt dWt S(t)βλ Σt dWt S(te) e − 2 S(t)2 h p i h p i 1 e 2 e = (λβ) Σt dt + λβ Σt dWt, (B.4) −e2 e 2 p since dWt = dt. Furthermore we observe that

dΣt,d ln S(t) = (λβη) Σt ρdt, (B.5)

2 since dWt dt =0= dVt dt =0= dt . Thise will now be used to derive a diﬀerential equation for the characteristic function of the Heston model.

2The restriction to the mentioned parameter space corresponds to Piterbarg’s approach in [19] and [20]. There also a zero correlation between the driving Brownian motions is assumed. Nonetheless, we include the case of non–zero correlation for completeness of exposure.

76 B.2 The characteristic function

The characteristic function is deﬁned as the fourier transform of the conditional probability density. For the stochastic process xt = (ln St, Σt) the conditional probability distribution is denoted as p(xu xt), T0 t

is xu is xu ϕtu(s xt)= e · p(xu xt) dxu = E e · xt , (B.6) | 2 | ZR where in the last step the interpretation of ϕtu(s xt) as conditional expected value of the | is xu function e · was established. An application of the tower law for conditional expectations shows that

is xu is xu ϕtu(s xt)= E e · xt = E E e · xv xt = E ϕvu(s xv) xt , T0 t v u T, | | ≤ ≤ ≤ ≤ (B.7) which indicates that ϕtu(s xt) as stochastic function of xt is a martingale. Therefore it has | dϕtu(s xt) to be driftless and the relation dt | = 0 must hold. From this requirement an ordinary diﬀerential equation for the characteristic function is derived by application of Ito’s lemma,

∂ϕtu ∂ϕtu ∂ϕtu dϕtu(s xt)= dt + dΣt + d(ln St) | ∂t ∂Σt ∂ ln St 2 2 2 1 ∂ ϕtu 2 ∂ ϕtu e ∂ ϕtu 2 + (dΣt) + 2 (dΣt)(d(ln St)) + (d(ln St)) 2 ∂Σ2 e 2 t ∂Σt∂(ln St) ∂(ln St) ∂ϕtu ∂ϕtu 1 2 ∂ϕtu e e = +Θ Σ0 Σt (λβ) Σt ∂t − ∂Σ − 2 e e t ∂ ln St 2 2 2 1 2 ∂ ϕtu ∂ ϕtu 1 2 ∂ ϕtu + η Σt + (λβη) Σt ρ + (λβ) Σt dt 2 ∂Σ2 e 2 2 t ∂Σt∂(ln St) ∂(ln St) + Brownian motion terms, e e where relations (B.4) and (B.5) were used, and from which the ODE follows:

∂ϕtu ∂ϕtu 1 2 ∂ϕtu 0= +Θ Σ0 Σt (λβ) Σt ∂t − ∂Σt − 2 ∂ ln St 2 2 2 1 2 ∂ ϕtu ∂ ϕtu 1 2 ∂ ϕtu + η Σt + (λβη) Σt ρ + (λβ) Σt T0 t T 2 e 2 2 ∂Σt ∂Σt ∂(ln St) 2 ∂(ln St) ≤ ≤ (B.8) e e From the analysis in section B.1 it is apparent that the process xt is aﬃne. The general form of the characteristic function for processes of this kind at ﬁnal time T is

ϕtT (s xt) = exp As(t, T )+ Ms(t, T ) xu , (B.9) | · with time dependent functions A : R2 [T , T ] R, (s,t) As(t, T ), and M : R2 [T , T ] × 0 → 7→ × 0 → R2, (s,t) Ms(t, T ), comp. [6], which will be further explored in the next section where 7→ the Heston ODE (B.8) is solved.

77 B.3 The solution of the Heston ODE

In accordance with (B.9) we deﬁne M(s)= Ms := Bs, Cs and make the ’ansatz’ ϕtT (s xt) = exp As(t, T )+ Ms(t, T ) xt = exp As(t, T )+ Bs(t, T ) ln St + Cs(t, T ) Σt , | · (B.10) e where the scalar product was expanded in the second step. Upon insertion into (B.8) we obtain

dAs dBs dCs 1 2 + ln St + Σt +Θ Σ0 Σt Cs(t, T ) (λβ) Σt Bs(t, T ) dt dt dt − − 2 1 2 2 1 2 2 + η Σt Ces(t, T ) + (λβη) Σt ρ Bs(t, T ) Cs(t, T )+ (λβ) Σt Bs(t, T ) = 0 2 2 ⇐⇒ dAs + ΘΣ Cs(t, T ) dt 0 2 dCs η 2 1 2 2 + Θ Cs(t, T )+ Cs(t, T ) + (λβ) Bs(t, T ) Bs(t, T ) dt − 2 2 − + (λβη)ρ Bs(t, T ) Cs(t, T ) Σt dBs + ln S = 0, dt t from which it is obviouse that above relation can only be fulﬁlled for all times t [T , T ] and ∈ 0 ln St, Σt if each row vanishes separately. This results in the following system of ordinary diﬀerential equations, e dAs = ΘΣ Cs(t, T ), (B.11a) dt − 0 2 dCs η 2 1 2 2 =Θ Cs(t, T ) Cs(t, T ) (λβ) Bs(t, T ) Bs(t, T ) dt − 2 − 2 − (λβη)ρ Bs(t, T )Cs(t, T ), (B.11b) − dBs = 0, (B.11c) dt for t [T , T ] and from (B.11c) it is clear that Bs(t, T ) const.. ∈ 0 ≡ B.3.1 Boundary conditions

From the deﬁnition of the characteristic function we know that at ﬁnal time T

is xT is xT is1 ln ST +is2ΣT T ϕTT (s xT )= E e · xT = e · = e , where s := (s , s ) , | 1 2 e and ϕ (s x ) = exp As( T, T)+ Bs(T, T ) ln S + Cs(T, T ) Σ by (B.10). TT | T T T e

78 Thus matching of exponents results in

As(T, T ) = 0, (B.12a)

Bs(T, T )= is1, (B.12b)

Cs(T, T )= is2, (B.12c) and since Bs(t, T ) const. on [T , T ] it follows from (B.12b) that ≡ 0

Bs(t, T )= is for all t [T , T ]. (B.13) 1 ∈ 0 B.3.2 A system of Riccati ODEs

Hence the characterisitc function of the Heston model at time t is given by

ϕtT (s1, s2 xt) = exp A (t, T )+ is1 ln St + C (t, T ) Σt , (B.14) | (s1,s2) (s1,s2) and the system (B.11) transforms into e

dA(s ,s ) 1 2 = ΘΣ C (t, T ), (B.15a) dt − 0 (s1,s2) 2 dC(s ,s ) η 1 1 2 = Θ is (λβη)ρ C (t, T ) C (t, T )2 + (λβ)2 s2 + is , (B.15b) dt − 1 (s1,s2) − 2 (s1,s2) 2 1 1

B(s1,0) = is1. (B.15c)

But this systems corresponds to the Riccati equations (A.19) discussed in appendix A with the following correspondents,

T T , → N η2 D t, T C (t, T ), →− 2 (s1,s2) 2 α(t) ΘΣ const., → η2 0 ≡ η2 f(t) (λβ)2 s2 + is const., →− 4 1 1 ≡ Θ Θ is (λβη )ρ, → − 1 for t [T , T ]. We therefore obtaine A t, T and C t, T for times T t T by ∈ 0 (s1,s2) (s1,s2) 0 ≤ ≤ reference to (A.40) and (A.41) and application of these discrete solutions to time interval

[T0, TN = T ]. As there are no sub intervals in this case, we have i = 0 and the replace- ment T T results in the correspondents A t, T A t, T , D t, T i+1 → N 0 N → (s1,s2) 0 N →

79 2 D t, T = η C t, T . Recalling that A(T, T ) = 0 and substituting parameters (s1,s2) − 2 (s1,s2) as described above we thus obtain,

A(s1,s2) t, T =

2 2 + 1 η + √g0(s1)(T t) η2 ΘΣ0 d0 (s1) T t + ln 1+ i 2 s2 d0 (s1) 1 e− − − − √g0(s1) − − −   if g0(s1) = 0,  6  2 η2 + +  2 ΘΣ0 ln 1+ i s2 d (s1) T t + d (s1) T t  − η − 2 − 0 − 0 −  if g0(s1)=0,   (B.16)  and

2 η + √g0(s1) √g0(s1) T t i s2 d (s1) 1 1 e− − +√g0(s1) 2 0 d+ s 2 + − − − − 0 ( 1) η2 d0 (s1)  η2 + √g (s )(T t) − i s2 d (s1) 1 e− 0 1 − +√g0(s1)  − 2 − 0 −  C(s ,s ) t, T =  if g0(s1) = 0, 1 2   η2 + η2 + 6  i s2+d (s1) i s2 d (s1) T t  2 − 2 0 − 2 − 0 − η2 η2 + − 1+ i 2 s2 d0 (s1) T t  − − −  if g0(s1)=0,   (B.17)  for times T t T and complex valued constants 0 ≤ ≤ 2 2 2 2 2 g0(s1)= gs T0 = Θ 4 fs T0 = Θ is1(λβη)ρ + η (λβ) s + is1 , (B.18) 1 s1 − 1 − 1 1 1 d±(s1)= Θ s g0(s1) = Θ is1(λβη)ρ g0(s1) . (B.19) 0 2 1 ± e 2 − ± p Θ p e + s1 If g0(s1) = 0, the relation d0 (s1)= d0−(s1)= 2 holds. With above solutions the characeristic functione of the Heston model is

ϕtT (s1, s2 xt) = exp A (t, T )+ is1 ln St + C (t, T ) Σt , (B.20) | (s1,s2) (s1,s2) for T0 t T. e ≤ ≤ B.4 Option pricing by transformation techniques

In this section we want to present the valuation of european swaptions within the Heston model. In turns out that prices of european options can be obtained by evaluation of an integral over the product of the characteristic function with the fourier transformed payoﬀ.

Now, if f(xT ) denotes a payoﬀ function at time T with respect to the stochastic variable x = (ln S , Σ ) its fourier transform f(s), s R2, is deﬁned by T T T ∈

i s xT e f(s)= e e · f(xT ) dxT , (B.21a) 2 ZR 1 s x e i T f(xT )= 2 e− · f(s) ds. (B.21b) (2π) 2 ZR e 80 Moreover, since the characteristic function (B.6) is the fourier transform of the probabiliy density function p(x x ), the relation T | t

1 i s xT p(xT xt)= 2 e− · ϕtT (s xt) ds (B.22) | (2π) 2 | ZR holds. With the necessary tools at hand we can proceed to calculate the expected value of the payoﬀ at time T,

E f(xT ) xt = f(xT ) p(xT xt) dxT 2 | ZR 1 s x i T = f(xT ) 2 e− · ϕtT (s xt) ds dxT 2 (2π) 2 | ZR ZR 1 i s xT = 2 e− · f(xT ) dxT ϕtT (s xt) ds 2 (2π) 2 | ZR ZR 1 = 2 f( s) ϕtT (s xt) ds, (B.23) (2π) 2 − | ZR where the right hand side can obviouslye be interpreted as fourier inversion of the function s x f( s) ϕ (s x ) ei T . − tT | t · We now turn to the valuation of european (payer) swaptions of strike K which certify e the right to enter a (payer) swap at time T whose par rate is then given by S(T ). Hence the swaption payoﬀ at time T is deﬁned as

max S(T ) K, 0 P (T ) S(T ) K + P (T ) − ≡ − = (S(T ) γ) (K γ) + P (T ) − − − =:K + = S(T ) K P|(T{z) by} deﬁnition of the displaced diﬀusion − e = eln S(T ) eln K + P (T ), (B.24) e −e e e where P (T ) denotes the present value of a basis point of the underlying swap at time T which will be used as numeraire. By the fundamental theorem of asset pricing, in a complete economy all numeraire rebased assets are martingales and therefore the value of a derivative V (t) with respect to the numeraire P (t) is given by, comp. [10], V (t) V (T ) = E t . P (t) P (T ) F

Since at option expiry T the derivative value equals the payoﬀ, application of the fundamental theorem of asset pricing to an european swaption results in the relation

ln S(T ) ln K V ln St, ln K := V (t)= P (t) E e e 1 ln S(t) , (B.25) − ln ST >ln K h e e i e e e e e 81 T since (B.2) is a Markov process and because the payoff function is f ln St, 0 = f(xT ) ≡ ln S(T ) ln K V (T )= e e P (T ). − e So in ordere to calculatee the derivative value one has to evaluate the expected value of the numeraire rebased payoff function which is calculated by integrating over the product of its fourier transform and the characteristic function according to (B.23). We therefore x ¯ f( T ) calculate the fourier transform f(s) of the numeraire rebased payoff function f(xT ) := P (T ) and in doing so we split the latter in an asset or nothing and a cash or nothing component, e

¯ ln ST ln K f(xT )= e e 1 − ln ST >ln K ln S e e ln K = e T 1 ee 1e ln ST >ln K − ln ST >ln K := f¯ e (x ) f¯ (x ). e AN T e− CNe T e e Starting with the asset or nothing component we have

s x i T ¯ fAN (s)= e · fAN (xT ) dxT 2 ZR e is1 ln ST +is2ΣT ln ST = dΣT e e 1ln S >ln K d ln ST R R T Z Z e e is2ΣT ∞ (is1+1) ln ST e e = e dΣT e d ln ST e R ln K Z Z e ln K is2ΣT ∞e i(s1 i) ln ST e (is1+1) ln ST = e dΣT e − d ln ST e d ln ST R − e Z Z e Z e −∞ −∞ =2π δ(s2) =2π δ(s1 i) − e e | {z } | e(is{z1+1) ln K } = 2πδ(s2) 2πδ(s1 i) , − − is + 1 e 1 since e(is1+1) ln ST 0 for ln S , and by deﬁnition of the delta distribution, δ(x) = → T → −∞ 1 eikx dk fore x C. 2π R ∈ e RNow we turn to the fourier transform of the cash or nothing component. Observing that

ln K ln K is1 ln St (is1+ǫ) ln St e d(ln St)= lim e d(ln St) e ǫ 0+ e Z∞ e → Z∞ e K e e(is1+ǫ) ln St ln e = lim ǫ 0+ is1 + ǫ e e → −∞ e(is1+ǫ) ln K e(is1+ǫ) ln St = lim lim + ǫ 0 is1 + ǫ e − ln St is1 + ǫ e → →−∞ e =0 eis1 ln K = , | {z } is1 e

82 we obtain

s x i T ¯ fCN (s)= e · fCN (xT ) dxT 2 ZR e ln K is1 ln ST +is2ΣT = e dΣT e 1ln S >ln K d ln ST R R T e Z Z e e e ln K ln K is2ΣT ∞ is1 ln ST e is1 ln ST = e e dΣT e d ln ST e d ln ST R − e e Z Z e Z e −∞ −∞ =2π δ(s2) =2π δ(s1) e e

is1 ln K ln|K {z } | e {z } = 2π e δ(s2) 2πδ(s1) . − is1 e e With the fourier transforms at hand we calculate the swaption value at time t according to (B.23):

Vt ln St, ln K; T = E eln S(T ) eln K 1 ln S(t) P t ln ST >ln K ( ) − e e h e e i 1 e e e = 2 fAN ( s) fCN ( s) ϕtT (s xt) ds (2π) R2 − − − | Z (1 is1) ln K 1 ∞ e ∞ e e − = ds1 ds2 δ( s2) 2πδ( s1 i) ϕtT (s1, s2 xt) 2π − − − − 1 is1 e | Z−∞ Z−∞ − is1 ln K 1 ∞ ∞ ln K e− ds1 ds2 e δ( s2) 2πδ( s1)+ ϕtT (s1, s2 xt) − 2π − − is1 e | Z−∞ Z−∞ e = 1) eln K ϕ (0, 0 x ) − tT | t e 1 ∞ is1 ln K 1 1 + ds1 e− + ϕtT (s1, 0 xt) 2π 1 is1 is1 | Z−∞ e − = 1) eln K ϕ (0, 0 x ) − tT | t e s 1 ∞ is1 ln K 1 1 + ds1 e− 2 i 2 2 ϕtT (s1, 0 xt) , 2π s1 + 1 − s1 1+ s1 | Z−∞ e (B.26) where the term involving the delta distribution δ( s i) vanishes because integration is − 1 − done over the real axis. From (B.18) we observe that

g (s )= Θ is (λβη)ρ 2 + η2(λβ)2 s2 + is 0 1 − 1 1 1 = Θ2 + (λβη)2 1 ρ2 s2 +i(λβη) (λβη ) 2Θρ s − 1 − 1 =:R(s1) =:I Is = R| (s )+ iIs{z= R(s )2}+ I2|eiϕ(s1) with{z ϕ(s }) = arctan 1 . 1 1 1 1 R(s ) 1 p

83 Since R(s )= R( s ) and arctan ( x)= arctan (x) for all x R, it follows that 1 − 1 − − ∈

g (s )∗ = g ( s ), 0 1 0 − 1 ϕ(s ) ϕ( s ) ∗ 2 2 i 1 2 2 i − 1 and g (s ) = R(s ) + I e− 2 = R( s ) + I e 2 = g ( s ). 0 1 1 − 1 0 − 1 p p p p Similarly by reference to (B.19), 1 d±(s )= Θ is (λβη)ρ g (s ) 0 1 2 − 1 ± 0 1 1 p ϕ(s ) 1 ϕ(s ) = Θ R(s )2 + I2 cos 1 i (λβη)ρ s R(s )2 + I2 sin 1 , 2 ± 1 2 − 2 1 ∓ 1 2 h p i h p i ϕ( s1) ϕ(s1) ϕ(s1) and because R(s )= R( s ), cos ( s ) = cos (s ), sin − = sin = sin , 1 − 1 − 1 1 2 − 2 − 2 we have

d±(s ) = d±( s ) , d±(s ) = d±( s ) , ℜ 0 1 ℜ 0 − 1 ℑ 0 1 −ℑ 0 − 1 d±(s )∗ = d±( s ). (B.27) 0 1 0 − 1

Thus g0(s1), g0(s1) and d0±(s1) all have symmetric real and antisymmetric imaginary parts. Furthermorep for any pair of complex numbers with symmetric real and antisymmetric imaginary parts, E(s )= E(s ) eiϕ1(s1) and F (s )= F (s ) eiϕ2(s1), the relations 1 | 1 | 1 | 1 |

E(s )∗ = E( s ), F (s )∗ = F ( s ), 1 − 1 1 − 1 E s ∗ E s E s E s ( 1) ( 1) i(ϕ1(s1) ϕ2(s1)) ( 1) i(ϕ1( s1) ϕ2( s1)) ( 1) = | | e− − = | − | e − − − = − , F (s ) F (s ) F ( s ) F ( s ) 1 | 1 | | − 1 | − 1 ln E(s ) ∗ = ln E(s ) iϕ (s )=ln E( s ) + iϕ ( s )=ln E( s ) 1 | 1 | − 1 1 | − 1 | 1 − 1 − 1 hold. Thus

+ + d (s ) ∗ d ( s ) 0 1 = 0 − 1 , g (s ) g ( s ) 0 1 0 − 1 ∗ √g0(s1)(T t) √g0(s1) (T t) √g0( s1)(T t) and e− p − ∗ = ep− − = e− − − . With these relations at hand we observe that the coeﬃcients (B.16) and (B.16) evaluated at s2 = 0,

A(s1,0) t, T =

2 + 1 + √g0(s1)(T t) η2 ΘΣ0 d0 (s1) T t + ln 1 d0 (s1) 1 e− − − − − √g0(s1) −   if g0(s1) = 0,  6  2 + +  2 ΘΣ0 ln 1 d (s1)(T t) + d (s1)(T t)  − η − 0 − 0 −  if g0(s1)=0,    84 and

+ √g0(s1) √g0(s1) T t d (s1) 1 1 e− − √g0(s1) 0 − − d+(s ) − 2 d+ s 0 1  η2 0 ( 1) − d+(s) 1 e √g0(s1)(T t) g (s )  0 1 − − √ 0 1  − − C(s ,0) t, T =  , 1  if g0(s1) = 0,  + 2 6  2 d (s1) (T t)  0 − η2 + 1 d0 (s1)(T t)  − −  if g0(s1)=0,   for times T0 t T and complex valued constants ≤ ≤ 2 2 2 2 2 g0(s1)= gs T0 = Θ 4 fs T0 = Θ is1(λβη)ρ + η (λβ) s + is1 , (B.28) 1 s1 − 1 − 1 1 1 d±(s1)= Θ s g0(s1) = Θ is1(λβη)ρ g0(s1) , (B.29) 0 2 1 ± e 2 − ± p p satisfy e

∗ ∗ A(s1,0) t, T = A( s1,0) t, T , C(s1,0) t, T = C( s1,0) t, T , − − which is a behaviour that transfers to the characteristic function (B.20) evaluated at s2 = 0,

ϕtT (s1, 0 xt)∗ = exp A (t, T )+ is1 ln St + C (t, T ) Σt ∗ | (s1,0) (s1,0)

= expA (t, T )∗ is1 ln St + C (t, T )∗ Σt (s1,0) − e (s1,0)

= expA( s1,0)(t, T )+ i( s1) ln St + C( s1,0)(t, T) Σt − − e − = ϕ ( s , 0 x ). tT − 1 | t e

As a consequence, for any complex valued function f(s1) with symmetric real and antisymmetric imaginary part, i.e., f(s ) = f( s ), 1 ∗ − 1 ∞ ∞ ϕtT (s1, 0 xt)f(s1) ds1 = ϕtT (s1, 0 xt)∗f(s1)∗ + ϕtT (s1, 0 xt)f(s1) ds1 | 0 | | Z−∞ Z ∞ = 2 ϕtT (s1, 0 xt)f(s1) ds1. 0 ℜ | Z Applying this result to equation (B.26) for the numeraire rebased swaption value one obtains

Vt ln St, ln K; T = 1) eln K ϕ (0, 0 x ) P (t) − tT | t e e e s 1 ∞ is1 ln K 1 1 + e− 2 i 2 2 ϕtT (s1, 0 xt) ds1 , 2π s1 + 1 − s1 1+ s1 | Z−∞ e ln K = 1) e ϕtT (0, 0 xt) − | 1e 1 s ln K ∞ is1 ln K 1 e e− 2 i 2 2 ϕtT (s1, 0 xt) ds1. − π 0 ℜ s1 + 1 − s 1+ s | e Z e 1 1 (B.30) This equation establishes the link between the characteristic function and swaption prices. In computations the integral in (B.30) has to be evaluated numerically.

85 B.5 Calibration of the Heston Model

As mentioned at the beginning we restrict the parameter space of the time dependent Heston model to the set of tuples (β, λ) while keeping the mean reversion Θ, the volatility of { } variance η and the correlation ρ of the Brownian drivers at ﬁxed values. Hence for each maturity T the swaption values given by (B.30) depend on the tuples (β , λ ) , namely { T T } + the swaption skew and volatility level through the variables g0(s1) and d0 (s1), which both enter the characteristic function. Thus the value of a swaption expiring at Ti with strike K can be expressed as

Vt ln St, ln K; Ti Vt ln St, ln K; Ti, (βT , λT ) ≡ i i = V ln(S + γ ), ln(K + γ ); T , (β , λ ) , e e t e t eTi Ti i Ti Ti and the tuple (βTi , λTi ) can be obtained by ﬁtting expression (B.30) with T = Ti to swaption i i market prices for a series of strikes Kj, VMarket(Kj; Ti). For a number of N market quotes i i corresponding to strikes K1,...,KN this is done by minimising

N 2 i i i Mi = w(K ) V (K ; Ti) Vt ln(St + γT ), ln(K + γT ); Ti, (βT , λT ) , (B.31) j Market j − i j i i i j=1 X i where the weight function w(Kj) is chosen as a monotone decreasing function of the moneyness. We will use an exponential weight function of the form

i i 2 w(K ) = exp ξ (St K ) , ξ> 0, j − − j i from which it is obvious that strikes Kj which are further away from the ATM-point St = i S(t)= Kj are assigned lower weights. Thereby it becomes apparent that for each swaption maturity Ti the tuple (βTi , λTi ) captures the Black76–volatility smile which is encoded in i the strikes Kj, j = 1,...,N. Based on this ﬁtting procedure the calibration of the model is performed for the single swaption expiries Ti according to a bootstrapping algorithm, comp. [8]. Starting with swaptions expiring at T1 the tuple (βT1 , λT1 ) is determined by minimisation of M1 according to (B.31). These values are then used in the next time step T2 where they enter the recursion scheme for the calculation of the characteristic function ϕtT2 on time interval

[T0, T2] according to (A.40) and (A.41). There market data of swaptions expiring at T2 2 for various strikes Kj , j = 1,...,N, are needed, and the tuple (βT2 , λT2 ) is obtained by minimisation of M2. Continuing along these lines for swaption expiries T3,...,TN results in a calibrated Heston model on time interval [T0, TN ].

86 Appendix C

Tables and ﬁgures

Table C.1: EUR term structure at T0 = May 7th, 2008 for maturities Ti = i y.

Maturity Discount Factor

T1 0.95958686

T2 0.92140904

T3 0.88521045

T4 0.85033692

T5 0.81669553

T6 0.78418277

T7 0.75260437

T8 0.72171881

T9 0.69149311

T10 0.66207251

T11 0.63345619

Table C.2: Euribor–1y forward rates Fi(T0; Ti, Ti+1) at T0 = May 7th, 2008 for reset times Ti = i y and i = 1,..., 10.

Forward rate Forward rate value

F1 0.04194355

F2 0.04169943

F3 0.04146102

F4 0.0412333

F5 0.04105982

F6 0.04095103

F7 0.04089707

F8 0.04118552

F9 0.0409216

F10 0.04087711

87 Table C.3: Piecewise constant skew and volatility term structure of forward rates Fi(t) for times Tj with T0 =0y Tj < Ti = i y with i = 1,..., 10 (rounded to three decimal places). ≤ Initially the skew/volatility was set to 0.5/0.02.

Skew/Time T0 T1 T2 T3 T4 T5 T6 T7 T8 T9

β1 0.503 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

β2 0.456 0.515 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

β3 0.396 0.439 0.545 0.5 0.5 0.5 0.5 0.5 0.5 0.5

β4 0.369 0.346 0.436 0.577 0.5 0.5 0.5 0.5 0.5 0.5

β5 0.315 0.298 0.301 0.448 0.653 0.5 0.5 0.5 0.5 0.5

β6 0.275 0.203 0.226 0.284 0.457 0.659 0.5 0.5 0.5 0.5

β7 0.232 0.235 0.094 0.147 0.244 0.549 0.66 0.5 0.5 0.5

β8 0.262 0.191 0.206 -0.06 0.194 0.311 0.501 0.744 0.5 0.5

β9 0.288 0.211 0.145 0.132 -0.314 0.26 0.372 0.511 0.73 0.5

β10 0.167 0.197 0.137 0.066 0.137 -0.417 0.262 0.436 0.521 0.752

Volatility/Time T0 T1 T2 T3 T4 T5 T6 T7 T8 T9

λ1 0.113 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02

λ2 0.132 0.119 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02

λ3 0.133 0.138 0.12 0.02 0.02 0.02 0.02 0.02 0.02 0.02

λ4 0.138 0.133 0.14 0.123 0.02 0.02 0.02 0.02 0.02 0.02

λ5 0.144 0.133 0.132 0.142 0.13 0.02 0.02 0.02 0.02 0.02

λ6 0.123 0.158 0.129 0.133 0.14 0.127 0.02 0.02 0.02 0.02

λ7 0.124 0.124 0.16 0.126 0.13 0.13 0.128 0.02 0.02 0.02

λ8 0.11 0.129 0.131 0.157 0.119 0.128 0.128 0.126 0.02 0.02

λ9 0.115 0.121 0.123 0.132 0.143 0.123 0.124 0.13 0.125 0.02

λ10 0.112 0.122 0.12 0.122 0.129 0.135 0.117 0.121 0.135 0.124

88 1.0

0.8 )) T ( 0.6 ¯ F ; 1 1 1 T, 1 1 T ( D 0.4

0.2

0.0 0.042 0.043 0.044 0.045 0.046 0.047 ¯ F1(T 1)

Figure C.1: The numeraire discount bond as ¯ ¯ functional of F1(T1, zT 1): D(T1, T11; F1(T1, zT 1)).

1.0

0.8 )) T ( 0.6 ¯ F ; 1 3 3 T, 3 1 T ( D 0.4

0.2

0.0 0.042 0.044 0.046 0.048 0.050 0.052 0.054 ¯ F3(T 3)

Figure C.2: The numeraire discount bond as ¯ ¯ functional of F3(T3, zT 3): D(T3, T11; F3(T3, zT 3)).

89 1.0

0.8 )) T ( 0.6 ¯ F ; 1 5 5 T, 5 1 T ( D 0.4

0.2

0.0 0.042 0.044 0.046 0.048 0.050 ¯ F5(T 5)

Figure C.3: The numeraire discount bond as ¯ ¯ functional of F5(T5, zT 5): D(T5, T11; F5(T5, zT 5)).

1.0

0.8 )) T ( 0.6 ¯ F ; 1 7 7 T, 7 1 T (

D 0.4

0.2

0.0 0.040 0.045 0.050 0.055 ¯ F7(T 7)

Figure C.4: The numeraire discount ¯ bond as functional of F10(T7, zT 7): ¯ D(T7, T11; F7(T7, zT 7)).

90 Table C.4: Digital caplet (on Euribor–1y) implied volatilities and (derived) numeraire re- V (T0;Ti,K) based prices for expiries Ti = i y and absolute moneyness Mi = Fi(T0; Ti, Ti+1) K D(T0,T11) − for i = 1,..., 10. Implied volatilities were observed on T0 = May 7th, 2008.

Expiry T1 T2 T3 T4 T5 Mi Impl. vol. Price Impl. vol. Price Impl. vol. Price Impl. vol. Price Impl. vol. Price -0.025 0.189 1.455 0.199 1.396 0.203 1.333 0.206 1.263 0.21 1.19 -0.023 0.175 1.455 0.185 1.394 0.189 1.325 0.192 1.249 0.196 1.169 -0.02 0.163 1.455 0.173 1.39 0.177 1.312 0.18 1.227 0.183 1.142 -0.018 0.153 1.454 0.162 1.381 0.166 1.29 0.169 1.197 0.172 1.107 -0.015 0.143 1.453 0.152 1.362 0.156 1.257 0.159 1.155 0.162 1.062 -0.013 0.135 1.447 0.144 1.328 0.148 1.207 0.151 1.099 0.154 1.005 -0.01 0.128 1.426 0.137 1.267 0.14 1.134 0.143 1.026 0.146 0.935 -0.008 0.122 1.366 0.131 1.168 0.134 1.035 0.137 0.934 0.14 0.851 -0.005 0.118 1.231 0.126 1.027 0.129 0.91 0.132 0.825 0.135 0.756 -0.003 0.114 0.996 0.122 0.848 0.125 0.766 0.128 0.704 0.131 0.653 0.0 0.112 0.695 0.12 0.651 0.123 0.614 0.125 0.58 0.128 0.549 0.003 0.111 0.41 0.119 0.466 0.121 0.471 0.124 0.463 0.126 0.449 0.005 0.111 0.206 0.118 0.312 0.121 0.346 0.123 0.358 0.125 0.36 0.008 0.111 0.09 0.118 0.198 0.121 0.247 0.123 0.272 0.125 0.284 0.01 0.112 0.036 0.119 0.121 0.121 0.172 0.123 0.203 0.125 0.222 0.013 0.113 0.013 0.12 0.073 0.122 0.119 0.124 0.151 0.126 0.173 0.015 0.115 0.005 0.122 0.043 0.124 0.082 0.126 0.112 0.127 0.134 0.018 0.117 0.002 0.124 0.025 0.125 0.056 0.127 0.083 0.129 0.104 0.02 0.119 0.001 0.125 0.015 0.127 0.039 0.129 0.062 0.131 0.082 0.023 0.121 0.0 0.127 0.009 0.129 0.027 0.131 0.047 0.132 0.064 0.025 0.123 0.0 0.129 0.005 0.131 0.019 0.133 0.035 0.134 0.051

Expiry T6 T7 T8 T9 T10 Mi Impl. vol. Price Impl. vol. Price Impl. vol. Price Impl. vol. Price Impl. vol. Price -0.025 0.209 1.121 0.208 1.054 0.205 0.99 0.203 0.934 0.201 0.879 -0.023 0.195 1.098 0.194 1.029 0.191 0.964 0.19 0.908 0.188 0.853 -0.02 0.182 1.068 0.182 0.999 0.179 0.934 0.177 0.878 0.176 0.824 -0.018 0.171 1.032 0.171 0.962 0.168 0.898 0.167 0.843 0.165 0.791 -0.015 0.161 0.987 0.161 0.918 0.158 0.857 0.157 0.804 0.155 0.753 -0.013 0.153 0.932 0.152 0.866 0.149 0.808 0.148 0.758 0.147 0.711 -0.01 0.145 0.867 0.144 0.805 0.142 0.752 0.14 0.706 0.139 0.662 -0.008 0.139 0.79 0.138 0.736 0.135 0.688 0.134 0.647 0.133 0.608 -0.005 0.133 0.705 0.132 0.659 0.13 0.619 0.129 0.583 0.127 0.549 -0.003 0.129 0.613 0.128 0.577 0.126 0.545 0.125 0.515 0.123 0.487 0.0 0.126 0.521 0.125 0.495 0.123 0.47 0.122 0.447 0.12 0.425 0.003 0.124 0.433 0.123 0.416 0.121 0.399 0.12 0.382 0.118 0.365 0.005 0.123 0.353 0.122 0.344 0.12 0.334 0.119 0.322 0.117 0.31 0.008 0.123 0.285 0.122 0.282 0.12 0.277 0.119 0.269 0.117 0.261 0.01 0.124 0.227 0.123 0.229 0.12 0.229 0.119 0.224 0.118 0.219 0.013 0.125 0.181 0.123 0.186 0.121 0.188 0.12 0.187 0.119 0.184 0.015 0.126 0.145 0.125 0.152 0.123 0.156 0.121 0.156 0.12 0.155 0.018 0.127 0.116 0.126 0.124 0.124 0.129 0.123 0.13 0.121 0.131 0.02 0.129 0.093 0.128 0.101 0.126 0.107 0.124 0.11 0.123 0.112 0.023 0.131 0.075 0.13 0.084 0.127 0.09 0.126 0.093 0.125 0.095 0.025 0.132 0.061 0.131 0.069 0.129 0.076 0.128 0.079 0.127 0.082

91 92 Bibliography

[1] Leif B.G. Andersen and Jesper Andreasen. Volatility skews and extensions of the Libor Market Model. Journal of Applied Mathematical Finance, 7, 2000.

[2] Leif B.G. Andersen and Jesper Andreasen. Volatile volatilities. Risk, 15(12), December 2002.

[3] Fischer Black. The pricing of commodity contracts. Journal of Financial Economics, 3:167–179, 1976.

[4] A. Brace, D. Gatarek, and M. Musiela. The market model of interest rate dynamics. Mathematical Finance, 7:127–154, 1996.

[5] J. C. Cox and S. A. Ross. The valuation of options for alternate stochastic processes. Journal of Financial Economics, 3:145–166, 1976.

[6] Darrell Duffie, Jun Pan, and Kenneth Singleton. Transform analysis and asset–pricing for affine jump–diffusions. Econometrica, 68(6):1343–1376, 2000.

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

[8] Alberto Elices. Aﬃne Concatenation. Wilmott Journal, 1(3):155–162, June 2009.

[9] Z. Hu et al. Cutting edges using domain integration. Risk, 19(11):95–99, 2006.

[10] J.M. Harrison and D. Kreps. Martingales and Arbitrage in Multiperiod Securities Markets. Journal of Economic Theory, 20:381–408, 1979.

[11] S. Heston. A closed form solution for options with stochastic volatilities with applica- tions to bond and currency options. Review of Financial Studies, 6:327–343, 1993.

[12] Phil Hunt. Explorations into stochastic volatility modelling. Unpublished lecture notes, 2007.

[13] Phil Hunt and Joanne Kennedy. Financial Derivatives in Theory and Practice. John Wiley & Sons, Chichester, 2004.

93 [14] Phil Hunt and Joanne Kennedy. Longstaﬀ–Schwartz, Eﬀective Model Dimensionality and Reducible Markov–Functional Models. Social Science Research Network, February 2005.

[15] Phil Hunt, Joanne Kennedy, and Antoon Pelsser. Markov–Functional Interest Rate Models. Finance and Stochastics, 4(4):391–408, 2000.

[16] F. Jamshidian. Libor and swap market models and measures. Finance and Stochastics, 1(4), 1997.

[17] Linus Kaisajuntti and Joanne Kennedy. An N–Dimensional Markov–Functional Inter- est Rate Model. Social Science Research Network, July 2008.

[18] Antoon Pelsser. Eﬃcient Methods for Valuing Interest Rate Derivatives. Springer Finance, London, 2000.

[19] Vladimir V. Piterbarg. Stochastic volatility model with time–dependent skew. Journal of Applied Mathematical Finance, 12(2):147–185, 2005.

[20] Vladimir V. Piterbarg. Time to smile. Risk, 18(5), May 2005.

[21] Riccardo Rebonato. Modern Pricing of Interest Rate Derivatives. Princeton University Press, 2002.

[22] Riccardo Rebonato. Volatility and Correlation. John Wiley & Sons, Chichester, 2004.

[23] Mark Rubinstein. Displaced Diﬀusion Option Pricing. Journal of Finance, 38(1):213– 217, March 1983.