Interest Rate Derivative Pricing with Stochastic Volatility PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus Prof.ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties, in het openbaar te verdedigen op dinsdag 25 september 2012 om 15:00 uur door Bin CHEN Master of Science in Engineering and Policy Analysis geboren te Hangzhou, China Dit proefschrift is goedgekeurd door de promotor: Prof.dr.ir. C.W. Oosterlee Samenstelling promotiecommissie: Rector Magnificus, voorzitter Prof.dr.ir. C.W. Oosterlee, TU Delft, promotor Prof.dr. C. V´azquezCend´on, University of Coru~na,Spain Prof.dr.ir. B. Koren, TU Eindhoven Prof.dr. M.H. Vellekoop, University of Amsterdam Drs. S. van Weeren, Rabobank International Prof.dr.ir. A.W. Heemink, TU Delft Prof.dr.ir. C. Vuik, TU Delft Prof. dr. F.H.J. Redig, TU Delft, reservelid Interest Rate Derivative Pricing with Stochastic Volatility. Dissertation at Delft University of Technology. The work described in this thesis was financially supported by Rabobank Inter- national and CWI Amsterdam. ISBN 978-94-6203-052-7 Copyright c 2012 by B. Chen Printed in The Netherlands by: W¨ohrmannPrint Service Acknowledgements This thesis concludes my four years of research in Centrum Wiskunde Informat- ica (CWI) and the Derivative Research & Validation (DR&V) team of Rabobank International. In the past four years, many people have helped me go through the painstaking process towards the final completion of this thesis. At this mo- ment, I would like to express my thanks to all those who contributed in many ways to the success of this research. First and foremost, I would like to thank Prof. dr. ir. Cornelis Oosterlee for his academic support, project management skills, and kind patience. Without his continuing advice, mathematically as well as personally, this thesis would not have been possible. I was especially amazed by his readiness to work at any time. Occasionally I would send emails at 11 o'clock in the evening or 6:30 in the morning, but I always received response within a very short period of time, even on weekends. His hard-working attitude was a great motivation for me. My special words of thanks should also go to Drs. Sacha van Weeren for his sharp opinion, deep knowledge of financial products and no-nonsense attitude, which were a great inspiration driving my research. Most of the time, his no- nonsense attitude was combined with a high level of directness. A discussion with him could be tough if I presented an idea which I had not yet fully thought through or if my report did not provide strong enough evidence to support the argument. Thus I would have to constantly push further for a deeper under- standing, better arguments and more thoughtful solutions, which was reflected in the content of my research papers. Sincere gratitude goes to Dr. Lech Grzelak who acted as the co-author of one of my four research papers, but our cooperation went far beyond that paper. He is an amazingly fast learner with great technical skills. It was great pleasure to work with him closely and to have stimulating discussions with him on a lot of things inside and outside our research. I am also grateful to Dr. Natalia Borovykh and Dr. Tim Dijkstra, who are scientists turned finance professionals. With them I could discuss many details of my work and they gave me many useful comments, which helped me to improve the scientific quality of my research. I should not forget my friendly colleagues in CWI, Linda Plantagie, Benjamin Sanderse, Willem Haverkort, Shashi Jain, Marjon Ruijter and Bram van Es, iii iv with whom I attend weekly scientific meetings and we enjoyed regular sport activities together. They made my stay in CWI so much more enjoyable. My special words of appreciation also go to my other colleagues in DR&V Rabobank International, Erik van Raaij, Erik Hennink, Thomas Zelders, Marcel Wijnen and Maurice Lutterot, among others. It has been great to work with you! Finally and most importantly, I would like to thank my family for their support, especially my wife Fei. It is her love, encouragement and devotion that have enabled me to go this far. For all her sacrifices and tolerance of my occasionally vulgar moods, I cannot express my gratitude enough through words. Thank you, Fei, for everything. Bin Chen Utrecht, August 2012 Summary Interest Rate Derivative Pricing with Stochastic Volatility Bin Chen One purpose of exotic derivative pricing models is to enable financial institu- tions to quantify and manage their financial risk, arising from large books of portfolios. These portfolios consist of many non-standard exotic financial prod- ucts. Risk is managed by means of the evaluation of sensitivity parameters, i.e. the so-called Greeks, the deltas, vegas, gammas and also volgas, vannas, and others. In practice, practitioners do not expect an exotic derivative pricing model to be a high precision predictive model. What is important is a high precision replication of the hedging instruments, as well as efficient computation with the model. Plain vanilla interest rate options like swaptions and caps are liquidly traded instruments, serving as fundamental building blocks of hedging portfolios for ex- otic products. In the early twenty-first century, the so-called implied volatility skew and smile in the market became pronounced in the interest rate plain vanilla market. The stochastic alpha beta rho (SABR) model [46] then be- came widely accepted as the market standard to model this implied volatility skew/smile. The model's popularity is due to the existence of an accurate analytic ap- proximation for the implied volatilities, presented by Hagan et al., in [46]. This approximation formula is often used by practitioners to inter- and extrapolate the implied volatility surface. The application of the SABR model is so prevalent that one can even observe SABR-type implied volatility curves in the market nowadays (which means that the SABR model can perfectly resemble one set of market implied volatilities with different strike prices). This PhD thesis considers the SABR model as its basis for further extension, and focuses on the various problems arising from the application of the SABR model in both plain vanilla and exotic option pricing, from a modelling as well as numerical point of view. v vi In Chapter 2, we present an analytic approximation to the convexity cor- rection of Constant Maturity Swap (CMS) products under a two-factor SABR model by means of small time asymptotic expansion technique. In Chapter 3, we apply the small time asymptotic expansion differently, to a problem of approximating the first and second moments of the integrated vari- ance of the log-normal volatility process in the context of defining a low-bias discretization scheme for the SABR model. With the approximated moment in- formation, we can approximate the density of the integrated variance by means of a log-normal distribution with the first two moments matched to that informa- tion. The conditional SABR process turns out to be a squared Bessel process, given the terminal volatility level and the integrated variance. Based on the idea of mixing conditional distributions and a direct inversion of the noncentral chi-square distributions, we propose the low-bias SABR Monte Carlo scheme. The low-bias scheme can handle the asset price process in the vicinity of the zero boundary well. The scheme is stable and exhibits a superior convergence behaviour compared to the truncated Euler scheme. In Chapter 4, we extend the discretization scheme proposed in Chapter 3 towards a SABR model with stochastic interest rate in the form of a Hull-White short rate model, the SABR-HW model. The hybrid model is meant for pricing long-dated equity-interest-rate linked exotic options with exposure to both the interest rate and the equity price risk. To facilitate the calibration of the SABR- HW model, we propose a projection formula, mapping the SABR-HW model parameters onto the parameters of the nearest SABR model. The numerical inversion of the projection formula can be used to calibrate the model. In Chapter 5, we focus on a version of the stochastic volatility LIBOR Mar- ket Model with time-dependent skew and volatility parameters. As a result of choosing time-dependent parameters, the model has the flexibility to match to the market quotes of an entire swaption cube (in terms of various combinations of expiry, tenor and strike), as observed in the current interest rate market. Thus, this model is in principle well-suited for managing the risk of a complete exotic option trading book in a financial institution, consisting of both exotic options and its plain vanilla hedge instruments. The calibration of the model to the swaption quotes relies on a model- mapping procedure, which relates the model parameters (most often time- dependent) in a high-dimensional LMM model to swaption prices. The model- mapping procedure maps the high-dimensional swap rate dynamics implied by the model onto a one-dimensional displaced diffusion process with time- dependent coefficients. Those time-dependent parameters are subsequently av- eraged to obtain the effective constant parameters of the projected model. Two known projection methods that are available in the literature, the freezing pro- jection and the more involved Markov projection, have been compared within the calibration process. The basic freezing projection achieves a good accuracy at significantly less computational cost in our tests, and it is thus applied within the calibration purpose. A second advantage of the freezing projection formula is that it enables us to formulate the time-dependent skew calibration problem as a convex opti- mization problem.