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# Spectral Element Method for Pricing European Options and Their Greeks

 dc.contributor.advisor Liu, Qing H. dc.contributor.author Yue, Tianyao dc.date.accessioned 2013-01-16T20:28:52Z dc.date.available 2013-01-16T20:28:52Z dc.date.issued 2012 dc.identifier.uri https://hdl.handle.net/10161/6156 dc.description.abstract

Numerical methods such as Monte Carlo method (MCM), finite difference method (FDM) and finite element method (FEM) have been successfully implemented to solve financial partial differential equations (PDEs). Sophisticated computational algorithms are strongly desired to further improve accuracy and efficiency.

The relatively new spectral element method (SEM) combines the exponential convergence of spectral method and the geometric flexibility of FEM. This dissertation carefully investigates SEM on the pricing of European options and their Greeks (Delta, Gamma and Theta). The essential techniques, Gauss quadrature rules, are thoroughly discussed and developed. The spectral element method and its error analysis are briefly introduced first and expanded in details afterwards.

Multi-element spectral element method (ME-SEM) for the Black-Scholes PDE is derived on European put options with and without dividend and on a condor option with a more complicated payoff. Under the same Crank-Nicolson approach for the time integration, the SEM shows significant accuracy increase and time cost reduction over the FDM. A novel discontinuous payoff spectral element method (DP-SEM) is invented and numerically validated on a European binary put option. The SEM is also applied to the constant elasticity of variance (CEV) model and verified with the MCM and the valuation formula. The Stochastic Alpha Beta Rho (SABR) model is solved with multi-dimensional spectral element method (MD-SEM) on a European put option. Error convergence for option prices and Greeks with respect to the number of grid points and the time step is analyzed and illustrated.

dc.subject Electrical engineering dc.subject Applied mathematics dc.subject Finance dc.subject Binary Options dc.subject Black-Scholes dc.subject Gauss Quadrature dc.subject Option Greeks dc.subject Spectral Element Method dc.subject Stochastic Volatility dc.title Spectral Element Method for Pricing European Options and Their Greeks dc.type Dissertation dc.department Electrical and Computer Engineering dcterms.provenance PDF updated on 26 January 2022 by request of the author. The final page of the original PDF was removed, since it contained a biographical statement with sensitive information that should not be shown publicly.
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