Consider the performance of an options writer who misspecifies the dynamics of the price process of the underlying asset by overestimating asset price volatility. When does he overprice the option? If he follows the hedging strategy suggested by his model, when does the terminal value of his strategy dominate the option payout? We show that both these events happen if the option payoff is a convex function of the price of the underlying at maturity. The proofs involve the simple, powerful and intuitive techniques of coupling. 1. Robust hedging and superreplication. The standard approach in mathematical finance, and particularly in the pricing of derivative securities, is to begin by writing down a stochastic model, which is assumed, without comment, to correctly and precisely specify the dynamics of the underlying asset. Arbitrage theory, backed by change of measure techniques and martingale representation theorems, then allows options to be priced and hedged. The fairness of the price and the success of the hedge depend crucially on the truth of the underlying model. The purpose of this article is to consider the implications of a misspecification of the dynamics of the asset price process. In particular, if the options writer uses an incorrect model, when does he overcharge for the option? Further, if he attempts to hedge using this incorrect model, can he still replicate (or rather superreplicate) the option payoff? (A superreplicating strategy is a dynamic hedging strategy which generates a terminal wealth which stochastically dominates the option payout.) It is well known that the price of a call option in the Black-Scholes model is an increasing function of the volatility parameter and that this monotonicity property extends to all European options with convex payoff profiles. We generalize this result to prove a price comparison theorem for pairs of models with stochastic volatilities. We have in mind the scenario where one model is used by the options writer to price and hedge options, and the second model represents the (unknown) truth.
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