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

  • *Corresponding author

    *Corresponding author

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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  • 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].

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


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

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

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

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

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

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

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

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

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

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

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

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

    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
     | Show Table
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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
     | Show Table
    DownLoad: CSV
  • [1] H. Albrecher, P. Mayer, W. Schoutens and J. Tistaert, The Little Heston Trap, KU Leuven Section of Statistics Technical Report, 2006.
    [2] L. Andersen and J. Andreasen, Volatility skews and extensions of the LIBOR market model, Applied Mathematical Finance, 7 (2000), 1-32. 
    [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. 
    [4] D. BauerA. 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. 
    [5] J. Bouchaud and  M. PottersTheory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management, Cambridge University Press, Cambridge, 2000. 
    [6] D. Brigo and F. Mercurio, Interest Rate Models-Theory and Practice: With Smile, Inflation and Credit, Springer Finance. Springer-Verlag, Berlin, 2006.
    [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.
    [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.
    [9] B. Choy, T. Dun and E. Schlögl, Correlating market models, RISK, (2004), 124–129. doi: 10.2139/ssrn.395640.
    [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.
    [11] C. CuchieroM. 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.
    [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.
    [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.
    [14] W. Feller, Introduction to the Theory of Probability and its Applications, John Wiley & Sons, Vol. 2, 1971.
    [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.
    [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.
    [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.
    [18] S. Heston and A. Rossi, A spanning series approach to options, The Review of Asset Pricing Studies, 6 (2016), 2-42. 
    [19] R. Jarrow and A. Rudd, Approximate option valuation for arbitrary stochastic processes, Financial Derivatives Pricing, (2008), 9–31. doi: 10.1142/9789812819222_0001.
    [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.
    [21] C. Kahl and P. Jäckel, Not-so-complex logarithms in the Heston model, Wilmott Magazine, 19 (2005), 94-103. 
    [22] C. Necula, G. Drimus and W. Farkas, A general closed form option pricing formula, Swiss Finance Institute Research Paper, (2017), 15–53.
    [23] M. PottersR. Cont and J. Bouchaud, Financial markets as adaptive systems, EPL (Europhysics Letters), 41 (1998), 239-239.  doi: 10.1209/epl/i1998-00136-9.
    [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.
    [25] J. Vedani and L. Devineau, Solvency assessment within the ORSA framework: issues and quantitative methodologies, Bulletin Français d'Actuariat, 13 (2013), 35–71.
    [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.
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