Multi-dimensional stochastic volatility for Interest Rates

The first part of this thesis is devoted to the study of an Affine Term Structure Model (ATSM) where we use Wishart-like processes to model the stochastic variance-covariance of interest rates. This work was initially motivated by some thoughts on calibration and model risk in hedging interest rates derivatives. The ambition of our work is to build a model which reduces as much as possible the noise coming from daily re-calibration of the model to the market. It is standard market practice to hedge interest rates derivatives using models with parameters that are calibrated on a daily basis to fit the market prices of a set of well chosen instruments (typically the instrument that will be used to hedge the derivative). The model assumes that the parameters are constant, and the model price is based on this assumption; however since these parameters are re-calibrated, they become in fact stochastic. Therefore, calibration introduces some additional terms in the price dynamics (precisely in the drift term of the dynamics) which can lead to poor P&L explain, and mishedging. The initial idea of our research work is to replace the parameters by factors, and assume a dynamics for these factors, and assume that all the parameters involved in the model are constant. Instead of calibrating the parameters to the market, we fit the value of the factors to the observed market prices. A large part of this work has been devoted to the development of an efficient numerical framework to implement the model. We study second order discretization schemes for Monte Carlo simulation of the model. We also study efficient methods for pricing vanilla instru- ments such as swaptions and caplets. In particular, we investigate expansion techniques for prices and volatility of caplets and swaptions. The arguments that we use to obtain the expansion rely on an expansion of the infinitesimal generator with respect to a perturbation factor. Finally we have studied the calibration problem. As mentioned before, the idea of the model we study in this thesis is to keep the parameters of the model constant, and calibrate the values of the factors to fit the market. In particular, we need to calibrate the initial values (or the variations) of the Wishart-like process to fit the market, which introduces a positive semidefinite constraint in the optimization problem. Semidefinite programming (SDP) gives a natural framework to handle this constraint. The second part of this thesis presents some of the work I have done on the hedging of interest rate risk in ALM. This work was motivated by the business at Cr edit Agricole S.A. and in particular by the Financial Division of the bank. The purpose of this part of the dis- sertation is twofold. First we want to communicate on a field of Finance which is less known by the mathematical finance community, and presents some interesting modeling challenges. Secondly we try to present an original approach to modeling and hedging interest rate risk. Chapter 6 is an attempt to formalize some of concepts that are used in practice in ALM. We recall some of the key concepts such as the schedule of an asset or a liability and the interest rate gap, and introduce a new concept : the notion of envelope. This concept will look familiar to people used to derivatives pricing, and the hedging of the interest rate risk of an asset or a liability (closing the gap using the language of ALM) is very similar to the hedging of an option. The remaining chapters present the results of the work we have done in three different projects.