Approximating Parabolic Partial Differential Equations with Applications to Financial Option Pricing

Open Access
Author:
Singh, Simranjeet
Area of Honors:
Interdisciplinary in Finance and Mathematics
Degree:
Bachelor of Science
Document Type:
Thesis
Thesis Supervisors:
  • Anna Mazzucato, Thesis Supervisor
  • Leonid Vaserstein, Honors Advisor
  • Brian Davis, Thesis Supervisor
Keywords:
  • Green's function
  • partial differential equations
  • option pricing
  • commutator
Abstract:
Herein, we first consider the basic assumptions and derivation of the original Black-Scholes model via a generalized portfolio replication argument. Thereafter, we analyze the Dyson-Taylor commutator method for short-time expansions of the heat kernel (typically referred to as the Green's function), and apply it to the Constant Elasticity of Variance (CEV) local volatility model, in order to find closed-form approximate solutions for the pricing kernel.