Conquering the Greeks: Efficient Calculation of Market Sensitivities and Hedge Ratios of Financial Instruments by Direct Numerical Simulation
Marco Avellaneda and Roberta Gamba

The calculation of price-sensitivities  of contingent  claims  is formulated in the framework of Weighted Monte Carlo simulation. Rather than perturbing the  parameters  that drive the economic stata-variables of the model, we perturb the vector of probabilities of simulated paths in a neighborhood of the uniform distribution. The resulting  hedge-ratios  (sensitivities  with respect  to  input prices) are characterized in terms of higher-order moments of  simulated cash-flows. The computed sensitivities display 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'' and simple to compute: they do not require performing multiple MC simulations, discrete differentiation, or re-calibration of the simulation.