American Monte Carlo option pricing under pure jump levy models

[摘要] ENGLISH ABSTRACT: We study Monte Carlo methods for pricing American options where the stock price dynamics followexponential pure jump L évy models. Only stock price dynamics for a single underlying are considered.The thesis begins with a general introduction to American Monte Carlo methods. We then consider twoclasses of these methods. The fi rst class involves regression - we briefly consider the regression method ofTsitsiklis and Van Roy [2001] and analyse in detail the least squares Monte Carlo method of LongstaandSchwartz [2001]. The variance reduction techniques of Rasmussen [2005] applicable to the least squaresMonte Carlo method, are also considered. The stochastic mesh method of Broadie and Glasserman [2004]falls into the second class we study. Furthermore, we consider the dual method, independently studiedby Andersen and Broadie [2004], Rogers [2002] and Haugh and Kogan [March 2004] which generates ahigh bias estimate from a stopping rule. The rules we consider are estimates of the boundary between thecontinuation and exercise regions of the option. We analyse in detail how to obtain such an estimate inthe least squares Monte Carlo and stochastic mesh methods.These models are implemented using both a pseudo-random number generator, and the preferred choiceof a quasi-random number generator with bridge sampling. As a base case, these methods are implementedwhere the stock price process follows geometric Brownian motion.However the focus of the thesis is to implement the Monte Carlo methods for two pure jump L évymodels, namely the variance gamma and the normal inverse Gaussian models. Wefirst provide a broaddiscussion on some of the properties of L évy processes, followed by a study of the variance gamma modelof Madan et al. [1998] and the normal inverse Gaussian model of Barndor -Nielsen [1995]. We also providean implementation of a variation of the calibration procedure of Cont and Tankov [2004b] for these models.We conclude with an analysis of results obtained from pricing American options using these models.

[发布机构] Stellenbosch University