http://repub.eur.nl/res/aut/3603/ List of Publications en http://repub.eur.nl/static-eur/img/logo.png http://repub.eur.nl http://repub.eur.nl/res/pub/18571/ 2010-02-01T00:00:00Z Using an Euler discretization to simulate a mean-reverting CEV process gives rise to the problem that while the process itself is guaranteed to be nonnegative, the discretization is not. Although an exact and efficient simulation algorithm exists for this process, at present this is not the case for the CEV-SV stochastic volatility model, with the Heston model as a special case, where the variance is modelled as a mean-reverting CEV process. Consequently, when using an Euler discretization, one must carefully think about how to fix negative variances. Our contribution is threefold. Firstly, we unify all Euler fixes into a single general framework. Secondly, we introduce the new full truncation scheme, tailored to minimize the positive bias found when pricing European options. Thirdly and finally, we numerically compare all Euler fixes to recent quasi-second order schemes of Kahl and Jckel, and Ninomiya and Victoir, as well as to the exact scheme of Broadie and Kaya. The choice of fix is found to be extremely important. The full truncation scheme outperforms all considered biased schemes in terms of bias and root-mean-squared error http://repub.eur.nl/res/pub/13917/ 2008-11-21T00:00:00Z Since the Nobel-prize winning papers of Black and Scholes and Merton in 1973, the derivatives market has evolved into a multi-trillion dollar market. Structures which were once considered as exotic are now commonplace, appearing in retail products such as mortgages and investment notes. At the same time, new and more complex structures are invented on a regular basis. To price and risk manage such products, a financial engineer will typically: (1) choose a model which is both economically plausible and analytically tractable, (2) calibrate the model to the prices of traded options, and (3) price the exotic option with the calibrated model, using appropriate numerical techniques. This thesis mainly deals with the second and third steps in this process. For the analytically tractable class of affine models, containing among others the Black-Scholes model and Heston’s stochastic volatility model, it deals with topics such as the robust pricing of European options via Fourier inversion, the pricing of Bermudan options using convolution based methods, the simulation of stochastic volatility models and the pricing of Asian options. A separate chapter deals with a completely different topic, the mathematical properties of the principal components of term structure data. Roger Lord (1977) holds cum laude Master’s degrees in both Applied Mathematics (Eindhoven University of Technology) and Econometrics (Tilburg University). After graduating he joined Cardano Risk Management in 2001 as a financial engineer. Deciding to pursue a PhD degree, he joined Erasmus University Rotterdam as a PhD candidate in 2003. Throughout his PhD he held a part-time position as a quantitative analyst at the Derivatives Research & Validation team of Rabobank International. He has published articles in Applied Mathematical Finance, the Journal of Computational Finance, Mathematical Finance, Quantitative Finance and SIAM Journal on Scientific Computing, and presented his research at several international conferences. Since October 2006 he joined Rabobank International’s Financial Engineering team in London as a quantitative analyst, developing front-office pricing models for interest rate derivatives. http://repub.eur.nl/res/pub/7915/ 2006-07-18T00:00:00Z At the time of writing this article, Fourier inversion is the computational method of choice for a fast and accurate calculation of plain vanilla option prices in models with an analytically available characteristic function. Shifting the contour of integration along the complex plane allows for different representations of the inverse Fourier integral. In this article, we present the optimal contour of the Fourier integral, taking into account numerical issues such as cancellation and explosion of the characteristic function. This allows for robust and fast option pricing for almost all levels of strikes and maturities. http://repub.eur.nl/res/pub/7914/ 2006-07-17T00:00:00Z The characteristic functions of many affine jump-diffusion models, such as Heston’s stochastic volatility model and all of its extensions, involve multivalued functions such as the complex logarithm. If we restrict the logarithm to its principal branch, as is done in most software packages, the characteristic function can become discontinuous, leading to completely wrong option prices if options are priced by Fourier inversion. In this paper we prove under non-restrictive conditions on the parameters that the rotation count algorithm of Kahl and Jäckel chooses the correct branch of the complex logarithm. Under the same restrictions we prove that in an alternative formulation of the characteristic function the principal branch is the correct one. Seen as this formulation is easier to implement and numerically more stable than Heston’s formulation, it should be the preferred one. The remainder of this paper shows how complex discontinuities can be avoided in the Schöbel-Zhu model and the exact simulation algorithm of the Heston model, recently proposed by Broadie and Kaya. Finally, we show that Matytsin’s SVJJ model has a closed-form characteristic function, though the complex discontinuities that arise there due to the branch switching of the exponential integral cannot be avoided under all circumstances. http://repub.eur.nl/res/pub/7738/ 2006-05-17T00:00:00Z When using an Euler discretisation to simulate a mean-reverting square root process, one runs into the problem that while the process itself is guaranteed to be nonnegative, the discretisation is not. Although an exact and efficient simulation algorithm exists for this process, at present this is not the case for the Heston stochastic volatility model, where the variance is modelled as a square root process. Consequently, when using an Euler discretisation, one must carefully think about how to fix negative variances. Our contribution is threefold. Firstly, we unify all Euler fixes into a single general framework. Secondly, we introduce the new full truncation scheme, tailored to minimise the upward bias found when pricing European options. Thirdly and finally, we numerically compare all Euler fixes to a recent quasi-second order scheme of Kahl and Jäckel and the exact scheme of Broadie and Kaya. The choice of fix is found to be extremely important. The full truncation scheme by far outperforms all biased schemes in terms of bias, root-mean-squared error, and hence should be the preferred discretisation method for simulation of the Heston model and extensions thereof. http://repub.eur.nl/res/pub/6922/ 2005-09-06T00:00:00Z The first three factors resulting from a principal components analysis of term structure data are in the literature typically interpreted as driving the level, slope and curvature of the term structure. Using slight generalisations of theorems from total positivity, we present sufficient conditions under which level, slope and curvature are present. These conditions have the nice interpretation of restricting the level, slope and curvature of the correlation surface. It is proven that the Schoenmakers-Coffey correlation matrix also brings along such factors. Finally, we formulate and corroborate our conjecture that the order present in correlation matrices causes slope.