Fast calibration of the LIBOR Market Model with Stochastic Volatility based on analytical gradient

TL;DR

This paper introduces an efficient gradient-based calibration method for the LIBOR Market Model with Stochastic Volatility, leveraging analytical expressions of swaption prices to improve accuracy and speed over traditional methods.

Contribution

It derives analytical gradients for swaption pricing in the DDSVLMM, enabling faster and more accurate calibration using gradient-based optimization.

Findings

Analytical gradients significantly reduce calibration time.

Gradient-based method improves calibration accuracy.

Compared to standard methods, the approach is more efficient.

Abstract

We propose to take advantage of the common knowledge of the characteristic function of the swap rate process as modelled in the LIBOR Market Model with Stochastic Volatility and Displaced Diffusion (DDSVLMM) to derive analytical expressions of the gradient of swaptions prices with respect to the model parameters. We use this result to derive an efficient calibration method for the DDSVLMM using gradient-based optimization algorithms. Our study relies on and extends the work by (Cui et al., 2017) that developed the analytical gradient for fast calibration of the Heston model, based on an alternative formulation of the Heston moment generating function proposed by (del Ba{\~n}o et al., 2010). Our main conclusion is that the analytical gradient-based calibration is highly competitive for the DDSVLMM, as it significantly limits the number of steps in the optimization algorithm while improving its accuracy. The efficiency of this novel approach is compared to classical standard optimization procedures.