Calibration of stochastic models for interest rate derivatives

For the pricing of interest rate derivatives various stochastic interest rate models are used. The shape of such a model can take very different forms, such as direct modelling of the probability distribution (e.g. a generalized beta function of second kind), a short-rate model (e.g. a Hull-White model) or a forward rate model (e.g. a LIBOR market model). This article describes the general structure of optimization in the context of interest rate derivatives. Optimization in finance finds its particular application within the context of calibration problems. In this case, calibration of the (vector-valued) state of a given stochastic model to some target state, which is determined by available relevant market data, implies a continuous optimization of the model parameters such that a global minimum of the distance between the target state and the model state is achieved. In this article, a novel numerical algorithm for the optimization of parameters of stochastic interest rate models is presented. The optimization algorithm operates within the model parameter space on an adaptive lattice with a number of lattice points per dimension which is both low and fixed. In this way, a considerable performance gain is achieved, as compared to algorithms working with non-adaptive lattices requiring increasing and/or large numbers of lattice points. As compared to standard algorithms, e.g. those of the Levenberg-Marquardt type, the presented adaptive lattice algorithm also reduces the danger of getting trapped near a wrong local minimum. As a numerical example, its application is demonstrated by optimizing volatility and mean reversion parameters of the Hull-White model, such that the latter becomes calibrated to the swaption volatility market relevant for a given OTC (over-the-counter, i.e. not exchange traded) bond option.