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July  2020, 16(4): 1699-1729. doi: 10.3934/jimo.2019025

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

 1 Université de Lyon, Université Lyon 1, Laboratoire de Science Actuarielle et Financière, ISFA, 50 avenue Tony Garnier, 69007, Lyon, France 2 Milliman R & D, 14 Avenue de la Grande Armée, 75017, Paris, France

*Corresponding author

Received  May 2017 Revised  November 2018 Published  May 2019

Fund Project: This research is supported by Milliman Paris. The authors thank Jean-Baptiste Garnier, Bernard Lapeyre, Abdallah Laraisse, Damien Louvet, Sophian Mehalla and Julien Vedani for their help

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 [26], 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 [17].

Citation: Laurent Devineau, Pierre-Edouard Arrouy, Paul Bonnefoy, Alexandre Boumezoued. Fast calibration of the Libor market model with stochastic volatility and displaced diffusion. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1699-1729. doi: 10.3934/jimo.2019025
##### References:

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##### References:
ATM Monte Carlo swaption volatilities for 5-years maturity
Monte Carlo swaption volatility skews for 5-years maturity
ATM swaption volatilities with given parameters for 5-years maturity
Swaption volatility skew with given parameters for 5-years maturity
ATM Monte Carlo swaption volatilities for 10-years maturity
Monte Carlo swaption volatility skews for 10-years maturity
ATM swaption volatilities with given parameters for 10-years maturity
Swaption volatility skews with given parameters for 10-years maturity
ATM Monte Carlo swaption volatilities for 20-years maturity
Monte Carlo swaption volatility skews for 20-years maturity
ATM swaption volatilities with given parameters for 20-years maturity
Swaption volatility skews with given parameters for 20-years maturity
Sum of squared differences between market volatilities and those calculated with each method after calibration
 Method Target function value Gram-Charlier (pricing) 3.59E-05 Edgeworth (pricing) 3.02E-05 Edgeworth (smile) 3.00E-05 Heston 2.03E-05
 Method Target function value Gram-Charlier (pricing) 3.59E-05 Edgeworth (pricing) 3.02E-05 Edgeworth (smile) 3.00E-05 Heston 2.03E-05
CPU time required for calibration using a 2500 optimization iterations budget
 Method CPU Time Heston 425.1 Edgeworth 8.2
 Method CPU Time Heston 425.1 Edgeworth 8.2
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