State Prices Implicit in Valuation Formulae for Derivative Securities: A Martingale Approach

A derivative asset is a security whose payoff is entirely determined by the prices of one or more underlying securities. Call and put options on stocks are simple examples. Since 1973, when Black and Scholes published their path-breaking option price formula, a rapidly growing literature has dealt with the valuation of derivatives for various models of the underlying price processes. Some researchers have studied the converse problem. They seek in infer properties of the underlying asset price from given prices of derivatives. The properties of the underlying price which are relevant for valuation purposes can be summarised in what are referred to as Arrow- Debreu state prices. These are the prices of elementary securities that pay one unit if the realisation of the underlying price path belongs to some specified set, and nothing otherwise. Breeden and Litzenberger showed in 1978 that a subset of these state prices can indeed be inferred from a sufficiently large collection of option prices. In a similar spirit, the present paper investigates the restrictions which a pricing formula for a derivative asset imposes on the underlying price processes. The valuation formulae considered satisfy a partial differential equation which is common in the literature on derivatives. It is shown that such formulae uniquely determine the full set of state prices. In contrast to Breeden and Litzenbergers work, the approach chosen does not rely on the particular payoff profiles of standard options, but allows for arbitrary derivative assets. In the last part of the paper, the general result is used to analyse two types of valuation formulae for options on pure discount bonds. The analysis yields a characterisation of the interest rate behaviour implicit in the valuation formulae and highlights the shortcomings of either type of formula.