State Prices Implicit in Valuation Formulae for Derivative Securities∗

Derivative assets analysis usually takes a model of the underlying price process as given and attempts to value derivative securities relative to that model. This paper studies the following “inverse” problem: given a valuation formula for a derivative asset, what can be inferred about the underlying asset price process? Assuming continuous sample paths, we show that a sufficiently regular pricing formula for some derivative asset completely determines the risk-neutral law of the underlying price. In particular, such a valuation formula implies a unique set of state prices for payoffs contingent on the price path of the underlying security. As an illustration of our main result, we analyse certain pricing formulae for European options on zero-coupon bonds. ∗A previous version of this paper was circulated as Discussion Paper No. 181 of the LSE Financial Markets Group. I would like to thank Lucien Foldes for his encouragement and many helpful comments, and Phil Dybvig, Bruce Grundy and Stanley Pliska for very useful discussions. Thanks are also due to seminar participants at the FORC Warwick, Paris I (Sorbonne), and the Studienzentrum Gerzensee. Financial support by the German Academic Exchange Service (DAAD), the COLONIA Studienstiftung, the LSE Financial Markets Group and the ESRC in the UK is gratefully acknowledged. †Graduate School of Business, Stanford University, Stanford CA 94305-5015, USA; phone (+1) 650 723 5512, fax (+1) 650 725 0468, e-mail rady [email protected].