Mathematical Finance, Vol. 8, No. 2 (April 1998), 93–126 ROBUSTNESS OF THE BLACK AND SCHOLES FORMULA NICOLE EL KAROUI Laboratoire de Probabilit´ es, Universit´ e Pierre et Marie Curie MONIQUE JEANBLANC-PICQU ´ E Equipe d’Analyse et Probabilit´ es, Universit´ e d’Evry STEVEN E. SHREVE Department of Mathematical Sciences, Carnegie Mellon University Consider an option on a stock whose volatility is unknown and stochastic. An agent assumes this volatility to be a specific function of time and the stock price, knowing that this assumption may result in a misspecification of the volatility. However, if the misspecified volatility dominates the true volatility, then the misspecified price of the option dominates its true price. Moreover, the option hedging strategy computed under the assumption of the misspecified volatility provides an almost sure one-sided hedge for the option under the true volatility. Analogous results hold if the true volatility dominates the misspecified volatility. These comparisons can fail, however, if the misspecified volatility is not assumed to be a function of time and the stock price. The positive results, which apply to both European and American options, are used to obtain a bound and hedge for Asian options. KEY WORDS: option pricing, hedging strategies, stochastic volatility 1. INTRODUCTION Since the development of the Black–Scholes option pricing formula (Black and Scholes 1973), practitioners have used it extensively, even to evaluate options whose underlying asset (hereafter called the “stock”) is known to not satisfy the Black–Scholes hypothesis of a deterministic volatility. In this paper, we provide conditions under which the Black– Scholes formula is robust with respect to a misspecification of volatility. We extend the well-known property of the option price being an increasing function of a deterministic volatility to a comparison of the price of contingent claims associated with two different stochastic volatilities. Our principal assumptions are that the contingent claims have convex payoffs and the only source of randomness in the misspecified volatility is a dependence on the current price of the stock. Under these assumptions, if the misspecified volatility dominates (respectively, is dominated by) the true volatility, then the contingent claim price corresponding to the misspecified volatility dominates (respectively, is dominated by) the true contingent claim price. A counterexample, based on ideas by M. Yor and reported in The authors thank Hans F ¨ ollmer, H´ elyette Geman, Terry Lyons, and Marc Yor for helpful comments on earlier versions of this paper, and also thank the Isaac Newton Institute for Mathematical Sciences for hosting visits by the authors while this work was underway. The work of S. E. Shreve was supported by the United States National Science Foundation under grants DMS-9203360 and DMS-9500626. Manuscript received June 1994; final revision received January 1997. Address correspondence to S. E. Shreve at Dept. of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213; e-mail: shreve@cmu.edu c  1998 Blackwell Publishers, 350 Main St., Malden, MA 02148, USA, and 108 Cowley Road, Oxford, OX4 1JF, UK. 93