Computing Greeks Using Multilevel Path Simulation

Abstract

We investigate the extension of the multilevel Monte Carlo method (M.B. Giles, Improved multilevel Monte Carlo convergence using the Milstein scheme, In A. Keller, S. Heinrich, and H. Niederreiter, editors, Monte Carlo and Quasi-Monte Carlo Methods 2006, 343–358, Springer-Verlag, 2007; M.B. Giles, Oper Res 56(3):607–617, 2008) to the calculation of Greeks. The pathwise sensitivity analysis (P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, New York, 2004) differentiates the path evolution and effectively reduces the smoothness of the payoff. This leads to new challenges: the use of naive algorithms is often impossible because of the inapplicability of pathwise sensitivities to discontinuous payoffs.These challenges can be addressed in three different ways: payoff smoothing using conditional expectations of the payoff before maturity (P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, New York, 2004); an approximation of the above technique using path splitting for the final timestep (S. Asmussen and P. Glynn, Stochastic Simulation, Springer, New York, 2007); the use of a hybrid combination of pathwise sensitivity and the Likelihood Ratio Method (M.B. Giles, Vibrato Monte Carlo sensitivities, In P. L’Ecuyer and A. Owen, editors, Monte Carlo and Quasi-Monte Carlo Methods 2008, 369–382, Springer, 2009). We discuss the strengths and weaknesses of these alternatives in different multilevel Monte Carlo settings.