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Markov Functional Interest Rate Models with Stochastic Volatility

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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 tractabil- ity. 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 de- vised 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 ap- proach 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 dis- count 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 func- tional 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.
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