Risk-Management Methods for the Libor Market Model Using Semidefinite Programming

TL;DR

This paper introduces a novel approach using semidefinite programming to enhance risk management in the Libor Market Model, enabling better hedging and sensitivity analysis.

Contribution

It demonstrates how duality in the calibration program can be used for risk management and portfolio hedging in the Libor Market Model.

Findings

Optimal dual variables correspond to hedging portfolios.

Local sensitivities of covariance matrices can be computed from dual solutions.

Semidefinite programming effectively manages Gamma exposure.

Abstract

When interest rate dynamics are described by the Libor Market Model as in BGM97, we show how some essential risk-management results can be obtained from the dual of the calibration program. In particular, if the objetive is to maximize another swaption's price, we show that the optimal dual variables describe a hedging portfolio in the sense of \cite{Avel96}. In the general case, the local sensitivity of the covariance matrix to all market movement scenarios can be directly computed from the optimal dual solution. We also show how semidefinite programming can be used to manage the Gamma exposure of a portfolio.