# Fast calibration of the Libor Market Model with Stochastic Volatility and Displaced Diffusion

## TL;DR

This paper introduces a fast calibration method for the Libor Market Model with Stochastic Volatility and Displaced Diffusion using Edgeworth and Gram-Charlier expansions, significantly reducing computational time.

## Contribution

It combines analytical moment-based density approximations with existing models to enhance calibration efficiency and interpretability.

## Key findings

- Achieves 98% reduction in calibration time
- Provides a smile formula relating volatility to moneyness
- Demonstrates effectiveness of Edgeworth and Gram-Charlier expansions

## 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).