Approximate Solutions to Second Order Parabolic Equations with Applications to Option Pricing

Open Access
Author:
Cheng, Wen
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
September 15, 2011
Committee Members:
  • Victor Nistor, Dissertation Advisor
  • Victor Nistor, Committee Chair
  • Anna L Mazzucato, Committee Chair
  • John C Liechty, Committee Member
  • Alexei Novikov, Committee Member
Keywords:
  • Green function
  • Fokker Planck equation
  • parabolic equation
  • Option Pricing
Abstract:
In this thesis, we consider second order parabolic equations with coefficients that vary both in space and in time (non-autonomous). We derive closed-form approx- imations to the associated fundamental solution by extending the Dyson-Taylor commutator method recently developed by Costantinescu, Costanzino, Mazzucato and Nistor for autonomous equations. We establish error bounds in Sobolev spaces and show that, by including enough terms, our approximation can be proven to be accurate to arbitrary high order in the short-time limit. We show how our method can give an approximation of the solution for any fixed time and within any given tolerance. To extend our results to large time, we introduce the so called boot- strap scheme, and show that the total error is still under control in this scheme, but the short time limitation can be removed. For applications, we adapt our ideas to Kolmogorov backward equations that appear in various research fields, such as option pricing. We also numerically compare our results with many other methods in the literature and show that our Dyson-Taylor Commutator method is algorithmically more elementary, it works for more general PDEs, and it gives fairly accurate approximations that are good enough in practice.