On the lookback option with fixed strike

The lookback option with fixed strike in the case of finite horizon was examined with help of the solution to the optimal stopping problem for a three-dimensional Markov process in [P. Gapeev, Discounted optimal stopping for maxima in diffusion models with finite horizon, Electron. J. Probab. 11 (2006), pp. 1031-1048]. The purpose of this paper was to illustrate another derivation of the solution in [P. Gapeev, Discounted optimal stopping for maxima in diffusion models with finite horizon, Electron. J. Probab. 11 (2006), pp. 1031-1048]. The key idea is to use the Girsanov change-of-measure theorem which allows to reduce the three-dimensional optimal stopping problem to a two-dimensional optimal stopping problem with a scaling strike. This approach simplifies the discussion and expressions for the arbitrage-free price and the rational exercise boundary. We derive a closed-form expression for the value function of the two-dimensional problem in terms of the optimal stopping boundary and show that the optimal stopping boundary itself can be characterized as the unique solution to a nonlinear integral equation. Using these results we obtain the arbitrage-free price and the rational exercise boundary of the option.