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

Hortensia Fontanals Albiol and Ramon Lacayo
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Ramon Lacayo: Universitat de Barcelona

No 142, Working Papers in Economics from Universitat de Barcelona. Espai de Recerca en Economia

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.

JEL-codes: G13 (search for similar items in EconPapers)
Pages: 8 pages
Date: 2005
New Economics Papers: this item is included in nep-cfn and nep-fin
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