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Option Valuation As an Expectation in The Complex Domain: The Black-Scholes Case

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Author Info
Hortensia Fontanals Albiol
Ramon Lacayo (Universitat de Barcelona)
Abstract

It is very well known that the first succesful valuation of a stock option was done by solving a deterministic partial differential equation (PDE) of the parabolic type with some complementary conditions specific for the option. In this approach, the randomness in the option value process is eliminated through a no-arbitrage argument. An alternative approach is to construct a replicating portfolio for the option. From this viewpoint the payoff function for the option is a random process which, under a new probabilistic measure, turns out to be of a special type, a martingale. Accordingly, the value of the replicating portfolio (equivalently, of the option) is calculated as an expectation, with respect to this new measure, of the discounted value of the payoff function. Since the expectation is, by definition, an integral, its calculation can be made simpler by resorting to powerful methods already available in the theory of analytic functions. In this paper we use precisely two of those techniques to find the well-known value of a European call.

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Paper provided by Universitat de Barcelona. Espai de Recerca en Economia in its series Working Papers in Economics with number 142.

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Length: 8 pages
Date of creation: 2005
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Handle: RePEc:bar:bedcje:2005142

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Postal: Espai de Recerca en Economia, Facultat de Ciències Econòmiques. Tinent Coronel Valenzuela, Num 1-11 08034 Barcelona. Spain.
Web page: http://www.ere.ub.es
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G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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