The Partial Differential Equation Approach Under Geometric Brownian Motion

Carl Chiarella*, Xue Zhong He, Christina Sklibosios Nikitopoulos

*Corresponding author for this work

Research output: Chapter in Book or Report/Conference proceedingChapterpeer-review

Abstract

The Partial Differential Equation (PDE) Approach is one of the techniques in solving the pricing equations for financial instruments. The solution technique of the PDE approach is the Fourier transform, which reduces the problem of solving the PDE to one of solving an ordinary differential equation (ODE). The Fourier transform provides quite a general framework for solving the PDEs of financial instruments when the underlying asset follows a jump-diffusion process and also when we deal with American options. This chapter illustrates that in the case of geometric Brownian motion, the ODE determining the transform can be solved explicitly. It shows how the PDE approach is related to pricing derivatives in terms of integration and expectations under the risk-neutral measure.

Original languageEnglish
Title of host publicationDynamic Modeling and Econometrics in Economics and Finance
PublisherSpringer Science and Business Media Deutschland GmbH
Pages191-206
Number of pages16
DOIs
Publication statusPublished - 2015
Externally publishedYes

Publication series

NameDynamic Modeling and Econometrics in Economics and Finance
Volume21
ISSN (Print)1566-0419
ISSN (Electronic)2363-8370

Keywords

  • Exercise Price
  • Geometric Brownian Motion
  • Kolmogorov Equation
  • Payoff Function
  • Underlying Asset

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