Fast calibration of the Libor market model with stochastic volatility and displaced diffusion

Abstract

This paper demonstrates the efficiency of using Edgeworth and Gram-Charlier expansions in the calibration of the Libor Market Model with Stochastic Volatility and Displaced Diffusion (DD-SV-LMM). Our approach brings together two research areas; first, the results regarding the SV-LMM since the work of Wu and Zhang (2006), especially on the moment generating function, and second the approximation of density distributions based on Edgeworth or Gram-Charlier expansions. By exploring the analytical tractability of moments up to fourth order, we are able to perform an adjustment of the reference Bachelier model with normal volatilities for skewness and kurtosis, and as a by-product to derive a smile formula relating the volatility to the moneyness with interpretable parameters. As a main conclusion, our numerical results show a 98% reduction in computational time for the DD-SV-LMM calibration process compared to the classical numerical integration method developed by Heston (1993).

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

“…As a main result, this paper shows that the analytical gradient method is highly competitive both from a computational standpoint and calibration accuracy, as it achieves to significantly limit the number of steps in the optimization algorithm while still offering accurate data replication. Our calibration experiments on real data also show that expansion methodologies that reduce the computational cost of an objective function call (see for instance (Devineau et al, 2020) and ) is a complementary efficient alternative; the combination of the two strategies (expansion approach and computation of the related analytical gradient) appears as a promising direction which is left for further research.…”

“…As a matter of fact, this method requires a lot of evaluations of the objective function and the computation of the swaption prices is very expensive due to the computation of integrals in the complex field and the recursive definition of the characteristic function. This has already been pointed out by (Devineau et al, 2020). Consequently, an optimization algorithm that does not require a lot of objective functions evaluations is preferred in order to achieve a fast calibration.…”

“…In (Wu and Zhang, 2006), pricing under the SVLMM is performed based on both the classical (Heston, 1993) numerical integration method and the famous Fast Fourier Transform (FFT) approach of (Carr and Madan, 1999), which has become a standard for option valuation for models with known characteristic function, as it is particularly the case for affine diffusion processes. As an alternative to moment generating function calculation, (Devineau et al, 2020) have shown the efficiency of using Edgeworth and Gram-Charlier expansions in the calibration of the DDSVLMM. Further work on this type of strategies that reduce the computational cost of the objective function is provided in a companion paper, see .…”

“…In such calibration problem, the regularity of the objective function is unknown. This latter issue can be tackled using optimization algorithms unconcerned about the regularity of the target function; it has been done in (Devineau et al, 2020) in which authors used the Nelder-Mead method.…”

“…Finally, the efficiency of our approach is compared to the following methods: the classical Heston-type pricing method (Heston, 1993) or the Edgeworth expansion of swaption prices as developed in (Devineau et al, 2020), both combined with the Nelder-Mead algorithm as well as the Levenberg-Marquart algorithm with numerical gradient. As a main result, this paper shows that the analytical gradient method is highly competitive both from a computational standpoint and calibration accuracy, as it achieves to significantly limit the number of steps in the optimization algorithm while still offering accurate data replication.…”