Abstract
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.
MSC (2010) Classification::
Acknowledgement
The author is grateful to Professor G. Peskir for fruitful discussions and hints concerning the reduction of the three-dimensional problem to two dimensions.