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2015 | OriginalPaper | Chapter

8. The Martingale Approach

Authors : Carl Chiarella, Xue-Zhong He, Christina Sklibosios Nikitopoulos

Published in: Derivative Security Pricing

Publisher: Springer Berlin Heidelberg

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Abstract

The martingale approach is widely used in the literature on contingent claim analysis. Following the definition of a martingale process, we give some examples, including the Wiener process, stochastic integral, and exponential martingale. We then present the Girsanov’s theorem on a change of measure. As an application, we derive the Black–Scholes formula under risk neutral measure. A brief discussion on the pricing kernel representation and the Feynman–Kac formula is also included.

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Note that we use ξ(t, T) to denote ξ(T)∕ξ(t) where ξ(t) is defined in Eq. (8.38).

The notation δ(x −X) should be interpreted as

$$\displaystyle{\delta (x_{1} - X_{1})\delta (x_{2} - X_{2})\cdots \delta (x_{n} - X_{n}).}$$

For the purposes of the discussion in this section it is useful to have a notation for the expectation operator that indicates both the time, t, as well as the initial value, x, of the underlying stochastic process when expectations are formed. We shall not use this notation elsewhere.

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Title: The Martingale Approach
Authors: Carl Chiarella
Xue-Zhong He
Christina Sklibosios Nikitopoulos
Publisher: Springer Berlin Heidelberg
Book: Derivative Security Pricing
Print ISBN: 978-3-662-45905-8

Electronic ISBN: 978-3-662-45906-5

Copyright Year: 2015
DOI: https://doi.org/10.1007/978-3-662-45906-5_8