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The Partial Differential Equation Approach Under Geometric Brownian Motion

In: Derivative Security Pricing

Author

Listed:
  • Carl Chiarella

    (University of Technology Sydney)

  • Xue-Zhong He

    (University of Technology Sydney)

  • Christina Sklibosios Nikitopoulos

    (University of Technology Sydney)

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.

Suggested Citation

  • Carl Chiarella & Xue-Zhong He & Christina Sklibosios Nikitopoulos, 2015. "The Partial Differential Equation Approach Under Geometric Brownian Motion," Dynamic Modeling and Econometrics in Economics and Finance, in: Derivative Security Pricing, edition 127, chapter 0, pages 191-206, Springer.
    • Handle: RePEc:spr:dymchp:978-3-662-45906-5_9
    DOI: 10.1007/978-3-662-45906-5_9
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