Valuing lookback options with barrier

In this paper, we introduce a new class of exotic options, termed lookback-barrier options, which literally combine lookback and barrier options by incorporating an activating barrier condition into the European lookback payoff. A prototype of lookback-barrier option was first proposed by Bermin (1998), where he intended to reduce the expensive cost of lookback option by considering lookback options with barrier. However, despite his novel trial, it has not attracted much attention yet. Thus, in this paper, we revisit the idea and extend the horizon of lookback-barrier option in order to enhance the marketability and applicability to equity-linked investments. Devising a variety of payoffs, this paper develops a complete valuation framework which allows for closed-form pricing formulas under the Black–Scholes model. Our closed-form pricing formulas provide a substantial advantage over the method of Monte Carlo simulation, because the extrema appearing in both of the lookback payoff and barrier condition would require a large number of simulations for exact calculation. Complexities involved in the derivation process would be resolved by the Esscher transform and the reflection principle of the Brownian motion. We illustrate our results with numerical examples.