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

Laurent Devineau; Pierre-Edouard Arrouy; Paul Bonnefoy; Alexandre Boumezoued

doi:10.3934/jimo.2019025

Article Contents

2020, Volume 16, Issue 4: 1699-1729. Doi: 10.3934/jimo.2019025

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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
^*Corresponding author

Received: April 30, 2017

Revised: October 31, 2018

Early access: May 2019

Published: May 2020

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

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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 [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].

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Mathematics Subject Classification: Primary: 60J60, 90-08; Secondary: 60G99.

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Figure 1. ATM Monte Carlo swaption volatilities for 5-years maturity

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Figure 2. Monte Carlo swaption volatility skews for 5-years maturity

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Figure 3. ATM swaption volatilities with given parameters for 5-years maturity

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Figure 4. Swaption volatility skew with given parameters for 5-years maturity

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Figure 5. ATM Monte Carlo swaption volatilities for 10-years maturity

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Figure 6. Monte Carlo swaption volatility skews for 10-years maturity

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Figure 7. ATM swaption volatilities with given parameters for 10-years maturity

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Figure 8. Swaption volatility skews with given parameters for 10-years maturity

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Figure 9. ATM Monte Carlo swaption volatilities for 20-years maturity

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Figure 10. Monte Carlo swaption volatility skews for 20-years maturity

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Figure 11. ATM swaption volatilities with given parameters for 20-years maturity

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Figure 12. Swaption volatility skews with given parameters for 20-years maturity

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Table 1. 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

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Table 2. CPU time required for calibration using a 2500 optimization iterations budget

Method	CPU Time
Heston	425.1
Edgeworth	8.2

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