Conquering The Greeks In Monte Carlo: Efficient Calculation Of The Market Sensitivities And Hedge-Ratios Of Financial Assets By Direct Numerical Simulation

The calculation of price-sensitivities of contingent claims is formulated in the framework of Monte Carlo simulation. In this article, we compute the sensitivities of prices of contingent claims by varying the probabilities, or weights assigned to the different paths in the simulation. This contrasts with the standard approach which consists of perturbing the parameters of the model and computing the price corresponding to new scenarios. The procedure is related to work of Broadie and Glasserman (1996) and to recent proposals based on Malliavin Calculus (Fournié, E. ; J.-M. Lasry, J. Lebuchoux, P.-L. Lions and N. Touzi (1998)). The hedging technique that we propose here is based on calculating regression coefficients of cashflows onto the cashflows of benchmark securities and taking advantage, whenever possible, of closed-form solution for the sensitivities of benchmark instruments. We show numerically that the sensitivities computed in this way are in excellent agreement with analytic closed-form solutions whenever the latter are available, e.g. with the Greeks of the Black-Scholes model, and with approximate analytic solutions for Basket Options in multi-asset models. The advantage of the new sensitivities is that they are “universal” (non-parametric) and simple to compute: they do not require performing multiple MC simulations, discrete-differentiation, or re-calibration of the simulation.