Efficient Sensitivity Analysis using Algorithmic  Differentiation in Financial Applications

Abstract: One of the most essential tasks of a financial institution is to keep the financial risk the institution is facing down to an acceptable level. This risk can for example be incurred due to bought or sold financial contracts, however, it can usually be dealt with using some kind of hedging technique. Certain quantities refereed to as "the Greeks" are often used to manage risk. The Greeks are usually determined using Monte Carlo simulation in combination with a finite difference approach, this can in some cases be very demanding considering the computational cost. Because of this, alternative methods for determining the Greeks are of interest. In this report a method called Algorithmic differentiation is evaluated. As will be described, there are two different settings of Algorithmic differentiation, namely, forward and adjoint mode. The evaluation will be done by firstly introducing the theory of the method and applying it to a simple, non financial, example. Then the method is applied to three different situations often arising in financial applications. The first example covers the case where a grid of local volatilities is given and sensitivities of an option price with respect to all grid points are sought. The second example deals with the case of a basket option. Here sensitivities of the option with respect to all of the underlying assets are desired. The last example covers the case where sensitivities of a caplet with respect to all initial LIBOR rates, under the assumption of a LIBOR Market Model, are sought.  It is shown that both forward and adjoint mode produces results aligning with the ones determined using a finite difference approach. Also, it is shown that using the adjoint method, in all these three cases, large savings in computational cost can be made compared to using forward mode or finite difference.