Nadhir Ben Rached, "Adaptive Monte Carlo Euler Method For Options With Low Regularity Payoffs", Graduation Project Report, Tunisia Polytechnic School, Hosting Institution: KAUST, June 2012.
In this report, an adaptive Monte Carlo Euler method for options with low regularity payoffs is developed. Two new smoothing methods for non regular payoffs are described. The first method is based on a cut off in the frequency space by applying a low pass filter. The second consists of using the Gaussian Kernel density and the Richardson extrapolation. Adaptive single level and multilevel algorithms with either deterministic or stochastic time steps are described. The main result is new applications of the adaptive algorithms using the approximate functions found by the two smoothing methods. In this work, we are interested essentially in smoothing the plain written European options like Call, Put, and the binary option for the one dimensional case the Basket and Spread Call for the multidimensional case. Numerical tests show the efficiency of the adaptive algorithm since the contribution of the error made by the smoothing methods on the global error is negligible compared to the one generated by the adaptive algorithm. The numerical results illustrate the efficiency of the multilevel adaptive algorithm compared to the single level one since it requires less computational work.