Title page for ETD etd-07072006-154259


Type of Document Master's Thesis
Author Strauss, Arne Karsten
Author's Email Address arne.strauss@vt.edu
URN etd-07072006-154259
Title Numerical Analysis of Jump-Diffusion Models for Option Pricing
Degree Master of Science
Department Mathematics
Advisory Committee
Advisor Name Title
Sachs, Ekkehard W. Committee Chair
Adjerid, Slimane Committee Member
Beattie, Christopher A. Committee Member
Keywords
  • Jump-diffusion processes
  • Option pricing
  • Finite differences
  • Fast Fourier Transform
  • Conjugate Gradient method
Date of Defense 2006-07-07
Availability unrestricted
Abstract
Jump-diffusion models can under certain assumptions be expressed as partial integro-differential equations (PIDE). Such a PIDE typically involves a convection term and a nonlocal integral like for the here considered models of Merton and Kou. We transform the PIDE to eliminate the convection term, discretize it implicitly using finite differences and the second order backward difference formula (BDF2) on a uniform grid. The arising dense linear system is solved by an iterative method, either a splitting technique or a circulant preconditioned conjugate gradient method. Exploiting the Fast Fourier Transform (FFT) yields the solution in only $O(n\log n)$ operations and just some vectors need to be stored. Second order accuracy is obtained on the whole computational domain for Merton's model whereas for Kou's model first order is obtained on the whole computational domain and second order locally around the strike price. The solution for the PIDE with convection term can oscillate in a neighborhood of the strike price depending on the choice of parameters, whereas the solution obtained from the transformed problem is stabilized.
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