ABSTRACT
This chapter studies lookback options. The payoff of a lookback option depends on either the maximum price of XY (t) or the minimum price of XY (t). We describe the contract on the maximum price. In order to collect the payoff that depends on the maximum, all the intermediate price levels between zero and the maximum have to be reached during the lifetime of the option, and thus one can think of the lookback option as a combination of knock-in option contracts. Once a particular level K is reached, the price may end up either above or below this level. The first case corresponds to a plain vanilla European option; the second case corresponds to a knock-in barrier option. Using the results from the previous sections, the second contract can also be expressed as a plain vanilla European option. Thus we obtain a representation of the lookback option price in terms of two plain vanilla European call options. We also give partial differential equations that correspond to the pricing of the lookback option together with the characterization of the hedging portfolio. The last section introduces the maximum drawdown, a widely used portfolio performance measure.
