The Use of Second-Order Stochastic Dominance To Bound European Call Prices: Theory and Results

Summary

Introduction

Since the seminal article of Black and Scholes (1973) on option pricing, a vast amount of literature has been written concerning options. Most of these articles consider perfect and complete markets, where options can be replicated by self-financing strategies involving continual revisions in a portfolio containing the underlying security. The price is then shown to be the ‘riskneutral’ expected cash-flows discounted at the risk-free rate: see for instance the seminal articles of Harrison & Kreps (1979), Harrison & Pliska (1981, 1983), Karatzas (1988) and El Karoui & Rochet (1989).

However, very broadly speaking, in any financial model, the equilibrium price of any asset is found to aggregate the risk-aversions of the individuals and their demands for this asset taking into account its correlation with the state variables of the economy (see for instance the general equilibrium model of Cox, Ingersoll & Ross (1985)) or with a ‘well-defined’ variable (Breeden 1989). But expected utility is, in general, a function of all the moments of the distribution: rules and pricing involving only two moments or duplication are valid only for a limited class of utility functions, or for specific distributions. For several years, finance research has got rid of utility functions in option pricing through the inspired idea of duplication. If the option is simply, at each instant, a portfolio composed of two others assets, then of course its price is the simple sum of the prices of the two assets (the underlying and the risk-free assets), these two prices being determined elsewhere.