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Martingale Approach to Pricing Perpetual American Options

Published online by Cambridge University Press:  29 August 2014

Hans U. Gerber  and
Elias S.W. Shiu
Hans U. Gerber*
Affiliation:
Université de Lausanne, Switzerland
Elias S.W. Shiu*
Affiliation:
The University of Iowa, U.S.A.
*
Ecole des hautes études commerciales, Université de Lausanne, CH-1015 Lausanne, Switzerland.
Department of Statistics and Actuarial Science, The University of Iowa, Iowa City, Iowa 52242-1419, U.S.A.
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Abstract

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The method of Esscher transforms is a tool for valuing options on a stock, if the logarithm of the stock price is governed by a stochastic process with stationary and independent increments. The price of a derivative security is calculated as the expectation, with respect to the risk-neutral Esscher measure, of the discounted payoffs. Applying the optional sampling theorem we derive a simple, yet general formula for the price of a perpetual American put option on a stock whose downward movements are skip-free. Similarly, we obtain a formula for the price of a perpetual American call option on a stock whose upward movements are skip-free. Under the classical assumption that the stock price is a geometric Brownian motion, the general perpetual American contingent claim is analysed, and formulas for the perpetual down-and-out call option and Russian option are obtained. The martingale approach avoids the use of differential equations and provides additional insight. We also explain the relationship between Samuelson's high contact condition and the first order condition for optimality.

Keywords

Black-Scholes formulaoption-pricing theoryequivalent martingale measureEsscher transformperpetual American call optionperpetual American put optionperpetual down-and-out American call optionperpetual American strangleperpetual American straddleRussian optionoptional sampling theoremoptimal stoppinghigh contact conditionsmooth pasting condition
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 24 , Issue 2 , November 1994 , pp. 195 - 220
Copyright
Copyright © International Actuarial Association 1994

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