State Prices Implicit in Valuation Formulae for Derivative Securities

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 to 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 Litzenberger's 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