Algorithmic Adjoint Differentiation in Libor market Models

Specialeforsvar ved Troels Tang Karlsen

Titel: Algorithmic Adjoint Differentiation  in Libor Market Models

Abstract: Modelling of exotic interest rate derivatives has long been of great interest for banks around the world, due to the massive size of the market for interest rate products. Recent post financial crisis regulations such as Basel III and the Fundamental Review of the Trading Book (FRTB) has spawned renewed interest in these complex derivatives, this time focusing on effective methods for computing their sensitivities to market variables. Interest rate derivatives with early exercise features such as Bermuda swaptions are some of the most challenging and time consuming derivatives to price - and hence some of the most expensive ones to find sensitivities of, using the classic bump-and-revalue approach. In this thesis we model interest rates in the lognormal Forward-Libor Model to price Bermuda swaptions in a simulation scheme, by using the industry accepted practice of applying Least Squares regression to estimate early exercise boundaries. The entire simulation is then differentiated using backward Algorithmic Adjoint Differentiation (AAD), in what is known as the backward phase, to compute sensitivities of input parameters in constant time. Crucially, we implement the technique of check-pointing, to avoid the excessive memory consumption often associated with AAD. We further show how an efficient AAD library can be constructed by means of operator overloading and custom memory containers to maximize performance. We then explore how to differentiate numerical matrix procedures that are not directly differentiable in an overloaded AAD library, most notably the Cholesky decomposition used when sampling from a multivariate normal distribution. Several correlation matrices are found to be positive semi-definite and making their Cholesky decompositions unstable. We find that naive attempts at circumventing this lead to correct prices in the forward pricing phase of AAD, but unstable and wrong sensitivities in the backward phase of AAD. Lastly we find that all sensitivities of both European and Bermuda swaptions can be computed at the cost of only a few evaluations of the Monte Carlo procedure, an order of magnitude faster than bump-and-revalue. By applying AAD to simulations of long tenor Bermuda swaptions, we show that AAD can easily be 100x times faster than bump-and-revalue and even faster for more sophisticated volatility structures of the forward rates.

Vejleder: Rolf Poulsen
Censor:   Elisa Nicolato, AAU