Robust hedging of the lookback option

The aim of this article is to nd bounds on the prices of exotic derivatives , and in particular the lookback option, in terms of the (market) prices of call options. This is achieved without making explicit assumptions about the dynamics of the price process of the underlying asset, but rather by inferring information about the potential distribution of asset prices from the call prices. Thus the bounds we obtain and the associated hedging strategies are model independent. The appeal and signiicance of the hedging strategies arises from their universality and robustness to model mis-speciication. Trading in call options is now so liquid that some authors (including Dupire 5, 6]) have argued that calls should no longer be treated as derivative assets to be priced using a Black-Scholes model (or otherwise). Instead calls should be treated as primary assets, whose prices are xed exogenously by market sentiments. It is only more complicated contingent claims which should be treated as derivative securities. These latter claims should be priced in a fashion consistent with call prices so as to preclude arbitrage. The development of this theory reeects the new perspective that term structure models have brought to the pricing of bonds and interest rate derivatives. In the traditional viewpoint of, for example, Vasi cek 21], there was a single state variable (typically a short term interest rate) from which all bond prices could be calculated. Subsequently Heath, Jarrow and Morton 10] argued that the whole of the current yield curve is needed to summarise current information, so that bond prices must be inputs to, and not outputs from, an interest rate model. Similarly here we are progressing from a model in which the asset price is the sole state variable, to one which incorporates the prices of call options. If the dynamics of the asset price process are speciied (for example if the asset price is assumed to follow the solution of a stochastic diierential equation), then it is possible, at least in principle, to calculate the nite-dimensional distributions of the price process. If moreover the market is complete then all derivatives can be replicated. Standard arguments then imply that the prices of contingent claims can be expressed as the (discounted) expected payoo of the claim under the equivalent martingale measure (EMM). Thus prices of many derivative securities can be calculated via a two-stage process: rstly calculate the marginal distributions of the …