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An improved total variation regularized RPCA for moving object detection with dynamic background
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 [
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.
|
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
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.
|
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |












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 |
Method | CPU Time |
Heston | 425.1 |
Edgeworth | 8.2 |
Method | CPU Time |
Heston | 425.1 |
Edgeworth | 8.2 |
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