Mini-symposium on automatic differentiation and its applications in the financial industry
Automatic differentiation has been involved for long in applied mathematics as an alternative to finite difference to improve the accuracy of numerical computation of derivatives. Each time a numerical minimisation is involved, automatic differentiation can be used. In between formal derivation and standard numerical schemes, this approach is based on software solutions applying mechanically the chain rule formula to obtain an exact value for the desired derivative. It has a cost in memory and cpu consumption.
For participants of financial markets (banks, insurances, financial intermediaries, etc), computing derivatives is needed to obtain the sensitivity of their exposure to well-defined potential market moves. It is a way to understand variations of their balance sheets in specific cases. Since the 2008 crisis, regulation demands to compute this kind of exposure to many different cases, to be sure that market participants are aware and ready to face a wide spectrum of market configurations.
This paper shows how automatic differentiation provides a partial answer to this recent explosion of computations to be performed. One part of the answer is a straightforward application of Adjoint Algorithmic Differentiation (AAD), but it is not enough. Since financial sensitivities involve specific functions and mix differentiation with Monte-Carlo simulations, dedicated tools and associated theoretical results are needed. We give here short introductions to typical cases arising when one uses AAD on financial markets.