Some control variates for exotic options

There are no known exact formulas for the valuation of a number of exotic options, and this is particularly true for options under discrete monitoring and for American style options. Therefore, one usually recourses to a Monte Carlo Simulation approach, amongst other numerical methods, to estimate the value of these options. The problem which then arises with this method is one of variance reduction. Control variates are often used, and we present some results for the optimization of these control variables, for the valuation of Asian and lookback options. An inequality on functions of correlations useful for comparing estimators in variance reduction procedures is also provided.