Type of Document	Dissertation
Author	Zhu, Wuming
Author's Email Address	zhuwuming@gmail.com
URN	etd-11302007-145119
Title	A Spectral Element Method to Price Single and Multi-Asset European Options
Degree	Doctor of Philosophy
Department	Mathematics, Department of
Advisory Committee	Advisor Name Title David A. Kopriva Committee Chair Alec N. Kercheval Committee Member Bettye Anne Case Committee Member Fred Huffer Committee Member Giray Okten Committee Member Xiaoming Wang Committee Member
Keywords	Rainbow Option Basket Option Jump Diffusion Stochastic Volatility Options Convolution Integral Spectral Element Method
Date of Defense	2007-11-15
Availability	unrestricted
Abstract We develop a spectral element method to price European options under the Black-Scholes model, Merton’s jump diffusion model, and Heston’s stochastic volatility model with one or two assets. The method uses piecewise high order Legendre polynomial expansions to approximate the option price represented pointwise on a Gauss-Lobatto mesh within each element. This piecewise polynomial approximation allows an exact representation of the non-smooth initial condition. For options with one asset under the jump diffusion model, the convolution integral is approximated by high order Gauss-Lobatto quadratures. A second order implicit/explicit (IMEX) approximation is used to integrate in time, with the convolution integral integrated explicitly. The use of the IMEX approximation in time means that only a block diagonal, rather than full, system of equations needs to be solved at each time step. For options with two variables, i.e., two assets under the Black-Scholes model or one asset under the stochastic volatility model, the domain is subdivided into quadrilateral elements. Within each element, the expansion basis functions are chosen to be tensor products of the Legendre polynomials. Three iterative methods are investigated to solve the system of equations at each time step with the corresponding second order time integration schemes, i.e., IMEX and Crank-Nicholson. Also, the boundary conditions are carefully studied for the stochastic volatility model. The method is spectrally accurate (exponentially convergent) in space and second order accurate in time for European options under all the three models. Spectral accuracy is observed in not only the solution, but also in the Greeks.
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