Type of Document Dissertation Author Zhu, Wuming Author's Email Address firstname.lastname@example.org 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
- Convolution Integral
- Spectral Element Method
Date of Defense 2007-11-15 Availability unrestricted AbstractWe develop a spectral element method to price European options under the Black-Scholes
model, Mertonís jump diffusion model, and Hestonís 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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