Monte Carlo Method for Pricing Lookback Type Options in Lévy Models

We construct a universal Monte Carlo method for pricing the options whose payout function depends on the final position of the extremum of the Lévy process. The proposed method is capable of evaluating the prices of floating and fixed strike lookback options not only at the initial time but also during the entire period when the current position of the Lévy process may be different from its extremum. Our algorithm involves three stages: approximation of the cumulative distribution function (c.d.f.) of the extremum process, evaluation of its inversion, and simulation of the final position of the extremum of the Lévy process. We obtain new approximate formulas for the c.d.f.'s of the supremum and infimum processes for Lévy models via Wiener--Hopf factorization. We also describe the principles of developing a hybrid Monte Carlo method, which combines classical numerical methods for construction of the c.d.f. of the final position of the extremum process and machine learning methods for inverting the c.d.f. with the help of tensor neural networks. The efficiency of the universal Monte Carlo method for lookback option pricing is supported by numerical experiments.