Monte Carlo methods for derivatives of options with discontinuous payoffs

Various Monte Carlo methods have been proposed to estimate the derivatives of contingent claims prices. The Monte Carlo approximate likelihood ratio estimator is studied. Recent convergence results are extended in order to show that the Monte Carlo approximate likelihood ratio derivative estimator is asymptotically equivalent, up to a second-order bias component, to an estimator based on a covariation approximation, the Monte Carlo Covariation estimator. Both converge slower than the Monte Carlo Malliavin derivative estimators. Theoretical convergence results are illustrated in a numerical experiment dealing with the risk management of digital options in a CEV model.