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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  July 2020 Early access  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:
 [1] H. Albrecher, P. Mayer, W. Schoutens and J. Tistaert, The Little Heston Trap, KU Leuven Section of Statistics Technical Report, 2006. Google Scholar [2] L. Andersen and J. Andreasen, Volatility skews and extensions of the LIBOR market model, Applied Mathematical Finance, 7 (2000), 1-32.   Google Scholar [3] R. Balieiro Filho and R. Rosenfeld, Testing option pricing with the Edgeworth expansion, Physica A: Statistical Mechanics and its Applications, 344 (2004), 482-490.   Google Scholar [4] D. Bauer, A. Reuss and D. Singer, On the calculation of the solvency capital requirement based on nested simulations, ASTIN Bulletin: The Journal of the IAA, 42 (2012), 453-499.   Google Scholar [5] J. Bouchaud and M. Potters, Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management, Cambridge University Press, Cambridge, 2000.   Google Scholar [6] D. Brigo and F. Mercurio, Interest Rate Models-Theory and Practice: With Smile, Inflation and Credit, Springer Finance. Springer-Verlag, Berlin, 2006.  Google Scholar [7] P. Carr and D. Madan, Option valuation using the fast Fourier transform, Journal of Computational Finance, 2 (1999), 61-73.  doi: 10.21314/JCF.1999.043.  Google Scholar [8] J. Chateau, Valuing european put options under skewness and increasing [excess] kurtosis, Journal of Mathematical Finance, 4 (2014), Article ID: 45662, 17 pages. doi: 10.4236/jmf.2014.43015.  Google Scholar [9] B. Choy, T. Dun and E. Schlögl, Correlating market models, RISK, (2004), 124–129. doi: 10.2139/ssrn.395640.  Google Scholar [10] C. Corrado and T. Su, Skewness and kurtosis in S & P 500 index returns implied by option prices, Journal of Financial Research, 19 (1996), 175-192.  doi: 10.1111/j.1475-6803.1996.tb00592.x.  Google Scholar [11] C. Cuchiero, M. Keller-Ressel and J. Teichmann, Polynomial processes and their applications to mathematical finance, Finance and Stochastics, 16 (2012), 711-740.  doi: 10.1007/s00780-012-0188-x.  Google Scholar [12] L. De Leo, V. Vargas and S. Ciliberti and J. Bouchaud, We've walked a million miles for one of these smiles, preprint, 2012, arXiv: 1203.5703v2. Google Scholar [13] L. Devineau and S. Loisel, Construction d'un algorithme d'accélération de la méthode des "simulations dans les simulations" pour le calcul du capital économique Solvabilité Ⅱ, Bulletin Français d'Actuariat, 10 (2009), 188–221. Google Scholar [14] W. Feller, Introduction to the Theory of Probability and its Applications, John Wiley & Sons, Vol. 2, 1971.  Google Scholar [15] D. Filipović and M. Larsson, Polynomial diffusions and applications in finance, Finance and Stochastics, 20 (2016), 931-972.  doi: 10.1007/s00780-016-0304-4.  Google Scholar [16] P. Hall, The Bootstrap and Edgeworth Expansion, Springer Series in Statistics. Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4612-4384-7.  Google Scholar [17] S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.  Google Scholar [18] S. Heston and A. Rossi, A spanning series approach to options, The Review of Asset Pricing Studies, 6 (2016), 2-42.   Google Scholar [19] R. Jarrow and A. Rudd, Approximate option valuation for arbitrary stochastic processes, Financial Derivatives Pricing, (2008), 9–31. doi: 10.1142/9789812819222_0001.  Google Scholar [20] M. Joshi and R. Rebonato, A stochastic-volatility, displaced-diffusion extension of the LIBOR market model, Quantitative Finance, 3 (2003), 458-469.  doi: 10.1088/1469-7688/3/6/305.  Google Scholar [21] C. Kahl and P. Jäckel, Not-so-complex logarithms in the Heston model, Wilmott Magazine, 19 (2005), 94-103.   Google Scholar [22] C. Necula, G. Drimus and W. Farkas, A general closed form option pricing formula, Swiss Finance Institute Research Paper, (2017), 15–53. Google Scholar [23] M. Potters, R. Cont and J. Bouchaud, Financial markets as adaptive systems, EPL (Europhysics Letters), 41 (1998), 239-239.  doi: 10.1209/epl/i1998-00136-9.  Google Scholar [24] E. Schlögl, Option pricing where the underlying assets follow a Gram/Charlier density of arbitrary order, Journal of Economic Dynamics and Control, 37 (2013), 611-632.  doi: 10.1016/j.jedc.2012.10.001.  Google Scholar [25] J. Vedani and L. Devineau, Solvency assessment within the ORSA framework: issues and quantitative methodologies, Bulletin Français d'Actuariat, 13 (2013), 35–71. Google Scholar [26] L. Wu and F. Zhang, LIBOR market model with stochastic volatility, Journal of Industrial and Management Optimization, 2 (2006), 199-227.  doi: 10.3934/jimo.2006.2.199.  Google Scholar

show all references

##### References:
 [1] H. Albrecher, P. Mayer, W. Schoutens and J. Tistaert, The Little Heston Trap, KU Leuven Section of Statistics Technical Report, 2006. Google Scholar [2] L. Andersen and J. Andreasen, Volatility skews and extensions of the LIBOR market model, Applied Mathematical Finance, 7 (2000), 1-32.   Google Scholar [3] R. Balieiro Filho and R. Rosenfeld, Testing option pricing with the Edgeworth expansion, Physica A: Statistical Mechanics and its Applications, 344 (2004), 482-490.   Google Scholar [4] D. Bauer, A. Reuss and D. Singer, On the calculation of the solvency capital requirement based on nested simulations, ASTIN Bulletin: The Journal of the IAA, 42 (2012), 453-499.   Google Scholar [5] J. Bouchaud and M. Potters, Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management, Cambridge University Press, Cambridge, 2000.   Google Scholar [6] D. Brigo and F. Mercurio, Interest Rate Models-Theory and Practice: With Smile, Inflation and Credit, Springer Finance. Springer-Verlag, Berlin, 2006.  Google Scholar [7] P. Carr and D. Madan, Option valuation using the fast Fourier transform, Journal of Computational Finance, 2 (1999), 61-73.  doi: 10.21314/JCF.1999.043.  Google Scholar [8] J. Chateau, Valuing european put options under skewness and increasing [excess] kurtosis, Journal of Mathematical Finance, 4 (2014), Article ID: 45662, 17 pages. doi: 10.4236/jmf.2014.43015.  Google Scholar [9] B. Choy, T. Dun and E. Schlögl, Correlating market models, RISK, (2004), 124–129. doi: 10.2139/ssrn.395640.  Google Scholar [10] C. Corrado and T. Su, Skewness and kurtosis in S & P 500 index returns implied by option prices, Journal of Financial Research, 19 (1996), 175-192.  doi: 10.1111/j.1475-6803.1996.tb00592.x.  Google Scholar [11] C. Cuchiero, M. Keller-Ressel and J. Teichmann, Polynomial processes and their applications to mathematical finance, Finance and Stochastics, 16 (2012), 711-740.  doi: 10.1007/s00780-012-0188-x.  Google Scholar [12] L. De Leo, V. Vargas and S. Ciliberti and J. Bouchaud, We've walked a million miles for one of these smiles, preprint, 2012, arXiv: 1203.5703v2. Google Scholar [13] L. Devineau and S. Loisel, Construction d'un algorithme d'accélération de la méthode des "simulations dans les simulations" pour le calcul du capital économique Solvabilité Ⅱ, Bulletin Français d'Actuariat, 10 (2009), 188–221. Google Scholar [14] W. Feller, Introduction to the Theory of Probability and its Applications, John Wiley & Sons, Vol. 2, 1971.  Google Scholar [15] D. Filipović and M. Larsson, Polynomial diffusions and applications in finance, Finance and Stochastics, 20 (2016), 931-972.  doi: 10.1007/s00780-016-0304-4.  Google Scholar [16] P. Hall, The Bootstrap and Edgeworth Expansion, Springer Series in Statistics. Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4612-4384-7.  Google Scholar [17] S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.  Google Scholar [18] S. Heston and A. Rossi, A spanning series approach to options, The Review of Asset Pricing Studies, 6 (2016), 2-42.   Google Scholar [19] R. Jarrow and A. Rudd, Approximate option valuation for arbitrary stochastic processes, Financial Derivatives Pricing, (2008), 9–31. doi: 10.1142/9789812819222_0001.  Google Scholar [20] M. Joshi and R. Rebonato, A stochastic-volatility, displaced-diffusion extension of the LIBOR market model, Quantitative Finance, 3 (2003), 458-469.  doi: 10.1088/1469-7688/3/6/305.  Google Scholar [21] C. Kahl and P. Jäckel, Not-so-complex logarithms in the Heston model, Wilmott Magazine, 19 (2005), 94-103.   Google Scholar [22] C. Necula, G. Drimus and W. Farkas, A general closed form option pricing formula, Swiss Finance Institute Research Paper, (2017), 15–53. Google Scholar [23] M. Potters, R. Cont and J. Bouchaud, Financial markets as adaptive systems, EPL (Europhysics Letters), 41 (1998), 239-239.  doi: 10.1209/epl/i1998-00136-9.  Google Scholar [24] E. Schlögl, Option pricing where the underlying assets follow a Gram/Charlier density of arbitrary order, Journal of Economic Dynamics and Control, 37 (2013), 611-632.  doi: 10.1016/j.jedc.2012.10.001.  Google Scholar [25] J. Vedani and L. Devineau, Solvency assessment within the ORSA framework: issues and quantitative methodologies, Bulletin Français d'Actuariat, 13 (2013), 35–71. Google Scholar [26] L. Wu and F. Zhang, LIBOR market model with stochastic volatility, Journal of Industrial and Management Optimization, 2 (2006), 199-227.  doi: 10.3934/jimo.2006.2.199.  Google Scholar
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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