Affine Recursion Problem and a General Framework for Adjoint Methods for Calculating Sensitivities for Financial Instruments

Abstract

Options with embedded early exercise features are of fundamental importance in finance. A simple example is the hedge of a multi-callable bond. This instrument is hedged using a Bermudan swaption.

Bermudan swaptions also play a key role when pricing callable constant maturity swaps or flexible swaps and are thus relevant for risk management of Pension funds and retail banks. It is market practice to consider multi-factor models such as the Libor Market model to price and hedge such instruments.

For hedging the key figures are the sensitivities to market and model parameters. For instance the Delta is the sensitivity of changing the underlying rate. To this end it is necessary to calculate such sensitivities in a fast and robust fashion. In this paper we propose a general framework for dealing with adjoints. We call it the Affine Recursion Problem (ARP). There are two solutions to the ARP. One is the forward method and the other one is the adjoint method. After a short review of approximating diffusion processes by numerical schemes we approximate the sensitivities Delta, Gamma and Vega using the general procedure. We treat the application of the general framework to Bermudan Swaptions and Trigger Swaps in Libor Market Model. For all steps of the approach we provide source code in Matlab such that not only the general approach is outlined and can be adapted to computer programs but also the reader can use this approach to set up adjoint methods.

Keywords: Libor Marekt Model, Adjoint Method, Affine Recursion, Bermudan Swaption, Trigger Swaps

JEL Classification: C15, C63