Graduation Project Report - Nadhir Ben Rached

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.