Monte Carlo evaluation of sensitivities in computational finance

In computational finance, Monte Carlo simulation is used to compute the correct prices for financial options. More important, however, is the ability to compute the so-called "Greeks'', the first and second order derivatives of the prices with respect to input parameters such as the current asset price, interest rate and level of volatility.\ud \ud This paper discusses the three main approaches to computing Greeks: finite difference, likelihood ratio method (LRM) and pathwise sensitivity calculation. The last of these has an adjoint implementation with a computational cost which is independent of the number of first derivatives to be calculated. We explain how the practical development of adjoint codes is greatly assisted by using Algorithmic Differentiation, and in particular discuss the performance achieved by the FADBAD++ software package which is based on templates and operator overloading within C++.\ud \ud The pathwise approach is not applicable when the financial payoff function is not differentiable, and even when the payoff is differentiable, the use of scripting in real-world implementations means it can be very difficult in practice to evaluate the derivative of very complex financial products. A new idea is presented to address these limitations by combining the adjoint pathwise approach for the stochastic path evolution with LRM for the payoff evaluation