Asymptotic Properties of Monte Carlo Estimators of Derivatives

We study the convergence of Monte Carlo estimators of derivatives when the transition density of the underlying state variables is unknown. Three types of estimators are compared. These are respectively based on Malliavin derivatives, on the covariation with the driving Wiener process, and on finite difference approximations of the derivative. We analyze two different estimators based on Malliavin derivatives. The first one, the Malliavin path estimator, extends the path derivative estimator of Broadie and Glasserman (1996) to general diffusion models. The second one, the Malliavin weight estimator, proposed by Fournié et. al. (1999), is based on an integration by parts argument and generalizes the likelihood ratio derivative estimator. It is shown that for discontinuous payoff functions only the estimators based on Malliavin derivatives attain the optimal convergence rate for Monte Carlo schemes. Estimators based on the covariation or on finite difference approximations are found to converge at slower rates. Their asymptotic distributions are shown to depend on additional second order biases even for smooth payoff functions. ——————————— Detemple is affiliated with Boston University School of Management and CIRANO, Garcia with the Economics Department, Université de Montréal, CIREQ and CIRANO and Rindisbacher with the Rotman School of Management, University of Toronto and CIRANO. Financial support from MITACS is gratefully acknowledged. The second author also thanks Hydro-Québec and the Bank of Canada for financial support. Address for correspondence: Jérôme Detemple, Boston University School of Management, 595 Commonwealth Avenue, Boston, MA, 02215.