Figure 1.
ATM Monte Carlo swaption volatilities for 5-years maturity
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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 |
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 [
Keywords:
Libor market model,
stochastic volatility,
displaced diffusion,
swaption pricing,
model calibration,
Edgeworth expansions,
Gram-Charlier expansions.
Mathematics Subject Classification:
Primary: 60J60, 90-08; Secondary: 60G99.
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 and 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. |
| [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. 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.
|
| [5] |
J. Bouchaud and M. Potters, Theory 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. 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. |
| [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. 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. |
| [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. |
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. |
| [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. 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.
|
| [5] |
J. Bouchaud and M. Potters, Theory 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. 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. |
| [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. 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. |
| [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. |
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 |
| Method | Target function value |
| Gram-Charlier (pricing) | 3.59E-05 |
| Edgeworth (pricing) | 3.02E-05 |
| Edgeworth (smile) | 3.00E-05 |
| Heston | 2.03E-05 |
Table 2.
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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