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On the structure of proper Black-Scholes formulae

Part of: Stochastic systems and control

Published online by Cambridge University Press:  14 July 2016

Peter Whittle
Peter Whittle*
Affiliation:
University of Cambridge
*
1Postal address: Centre for Mathematical Studies, University of Cambridge, Cambridge CB3 0WB, UK.
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Abstract

The thinking behind the original Black-Scholes formula is criticised on the grounds that it holds out the quite unrealistic prospect of risk-free operation, that it can sacrifice asset maximisation to exact meeting of the contract, and that it restricts investment to those stocks on which an option is being sold. An alternative approach is given, based on a model of risk-averse asset maximisation, which, while meeting these criticisms, gives the option price in the familiar form of a discounted and weighted conditional expectation of the seller's liability at maturity. This evaluation is extended to a completely general stochastic model of stock price evolution; consideration is also given to the possibility of seller's ruin.

Keywords

Option pricingBlack-Scholes formularisk-averseness

MSC classification

Secondary: 93E20: Optimal stochastic control
Type
Other stochastic models
Information
Journal of Applied Probability , Volume 38 , Issue A: Probability, Statistics and Seismology , 2001 , pp. 243 - 248
Copyright
Copyright © Applied Probability Trust 2001 

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References

Baxter, M. and Rennie, A. (1996). Financial Calculus. Cambridge University Press.CrossRefGoogle Scholar
Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. J. Political Economy 81, 637–654.CrossRefGoogle Scholar
Harrison, M. and Pliska, S. (1981). Martingales and stochastic integrals in the theory of continuous trading. Stoch. Proc. Applic. 11, 215–260.CrossRefGoogle Scholar
Merton, R. C. (1973). Theory of rational option pricing. Bell J. Econ. Mgmt. Sci. 4, 141–183.Google Scholar
Whittle, P. (2000). Probability via Expectation , 4th edn. Springer, New York.CrossRefGoogle Scholar