Approximate solution to second order parabolic equations, with application to financial modeling

Open Access
Author:
Liang, Chao
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
April 18, 2014
Committee Members:
  • Victor Nistor, Dissertation Advisor
  • Xiantao Li, Dissertation Advisor
  • Anna L Mazzucato, Committee Member
  • John C Liechty, Committee Member
  • Qiang Du, Committee Member
  • Yuxi Zheng, Special Member
Keywords:
  • Partial Differential Equations
  • Financial Modeling
  • Option Pricing
  • Approximate Solutions
  • Symbolic Computation
Abstract:
In this dissertation, we consider second order parabolic equations with variable coefficients. We derive the closed-form approximations to the associated fundamental solution, as well as general solutions for certain initial conditions. The starting point is the Dyson-Taylor commutator method which was recently developed. We derive the approximate formula in a new form, which is easy to be calculated by computer programming. We next show that the Dyson series can be computed without Taylor expansion, which leads to more elementary computations, as well as cleaner results. In the section on theory, we have proven that the approximate solution is asymptotic series under certain regularity conditions; in the section for practical application, we have successfully implemented symbolic computation to several popular models in finance: CEV model, SABR model and Heston model. We not only approximate the fundamental solution (transition kernel), but also the option price (which is important for calibration and pricing), the implied volatility (which is quoted by traders), as well as the characteristic function of the underlying stochastic process (which is important for understanding the volatility skew/smile). The computation is carried out by MATLAB codes, and they are available upon request to {\it charlesleung2009@gmail.com} or {\it cul192@psu.edu}