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Martingales versus PDEs in finance: an equivalence result with examples

Part of: Applications Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

David Heath  and
Martin Schweizer
David Heath*
Affiliation:
University of Technology, Sydney
Martin Schweizer*
Affiliation:
Technische Universität Berlin
*
∗ Postal address: University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia.
∗∗ Postal address: Technische Universität Berlin, Fachbereich Mathematik, MA 7–4, Str. des 17. Juni 136, D-10623 Berlin, Germany. Email address: [email protected]
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Abstract

We provide a set of verifiable sufficient conditions for proving in a number of practical examples the equivalence of the martingale and the PDE approaches to the valuation of derivatives. The key idea is to use a combination of analytic and probabilistic assumptions that covers typical models in finance falling outside the range of standard results from the literature. Applications include Heston's stochastic volatility model and the Black-Karasinski term structure model.

Keywords

Option valuationmartingale approachpartial differential equationsfinanceFeynman-Kac formula

MSC classification

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 62P05: Applications to actuarial sciences and financial mathematics
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 4 , December 2000 , pp. 947 - 957
Copyright
Copyright © by the Applied Probability Trust 2000 

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