ROBUST REPLICATION UNDER MODEL UNCERTAINTY PAVEL V. GAPEEV, TOMMI SOTTINEN, AND ESKO VALKEILA Abstract. We consider the robust hedging problem in which an investor wants to super-hedge an option in the framework of uncertainty in a model of a stock price process. More specifically, the investor knows that the stock price process is H -self-similar with H ∈ (1/2, 1) , and that the log-returns are Gaussian. This leads to two natural but mutually exclusive hypotheses both being self-contained to fix the probabilistic model for the stock price. Namely, the investor may assume that either the market is efficient, i.e. the stock price process is a semimartingale, or that the centred log-returns are stationary. We show that to be able to super-hedge a convex European vanilla-type option robustly the investor must assume that the markets are efficient. If it turns out that if the other hypothesis of stationarity of the log-returns is true, then the investor can actually super-hedge the option as well as receive a net hedging profit. Mathematics Subject Classification 2000: Primary 91B28, 60G15, 60G18. Sec- ondary 60G44, 60H05, 91B70. JEL Classification: Primary G13. Secondary C52. Key words and phrases: Arbitrage option pricing and hedging, fractional Brownian motion, self-similarity, long-range dependence, fractional Black-Scholes market model, forward pathwise integral, model uncertainty, average risk-neutral measure, net hedging profit, Wick-Itˆo-Skorohod integral. 1. Introduction In the classical Black-Scholes model of financial market the logarithm of the stock price is modeled by a drifted Brownian motion. However, in some studies of real financial data it is concluded that the centred log-returns of the stock prices exhibit the so-called long-range dependency property (see, e.g., [12, Chapter IV]). This observation generates an intention to replace the driving Brownian motion with independent increments by another Gaussian process with long memory, or at least having the so-called H -self-similarity property, which is in many cases taken to be evidence for the long-range dependence (when H belongs to the interval (1/2, 1)). A natural candidate for the new driving process is the fractional Brownian motion, which is a Gaussian process characterized by being self-similar with stationary increments. This, what we call hypothesis (H1), will result in a market model that exhibits arbitrage opportunities (see, e.g., [5, 10, 13]). Another natural candidate for the replacement of the Gaussian driving process is an H -self-similar Gaussian martingale, which would still be in the realm of H -self-similarity but would not generate arbitrage. This, what we Date : October 1, 2007. This preprint is CDAM Research Report LSE-CDAM-2007-28. 1