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. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of
the requirements for the degree of Doctor of Philosophy. / Degree Awarded: Spring Semester, 2008. / Date of Defense: November 15, 2007. / Rainbow Option, Basket Option, Jump Diffusion, Stochastic Volatility, Options, Convolution Integral, Spectral Element Method / Includes bibliographical references. / David A. Kopriva, Professor Directing Dissertation; Fred Huffer, Outside Committee Member; Bettye Anne Case, Committee Member; Alec N. Kercheval, Committee Member; Giray Okten, Committee Member; Xiaoming Wang, Committee Member.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_168669 |
| Contributors | Zhu, Wuming (authoraut), Kopriva, David A. (professor directing dissertation), Huffer, Fred (outside committee member), Case, Bettye Anne (committee member), Kercheval, Alec N. (committee member), Okten, Giray (committee member), Wang, Xiaoming (committee member), Department of Mathematics (degree granting department), Florida State University (degree granting institution) |
| Publisher | Florida State University |
| Source Sets | Florida State University |
| Language | English, English |
| Detected Language | English |
| Type | Text, text |
| Format | 1 online resource, computer, application/pdf |
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