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.