Some Explicitly Solvable SABR and Multiscale SABR Models: Option Pricing and Calibration

Affiliation(s)

Department of Mathematics and Information Technology, University of Camerino, Camerino, Italy.

Department of Economics, University of Verona, Verona, Italy.

Department of Management, University Politecnica Marche, Ancona, Italy.

Department of Mathematics, “G. Castelnuovo”, University of Rome “La Sapienza”, Rome, Italy.

Department of Mathematics and Information Technology, University of Camerino, Camerino, Italy.

Department of Economics, University of Verona, Verona, Italy.

Department of Management, University Politecnica Marche, Ancona, Italy.

Department of Mathematics, “G. Castelnuovo”, University of Rome “La Sapienza”, Rome, Italy.

ABSTRACT

A multiscale SABR model that describes the dynamics of forward prices/rates is presented. New closed form formulae for the transition probability density functions of the normal and lognormal SABR and multiscale SABR models and for the prices of the corresponding European call and put options are deduced. The technique used to obtain these formulae is rather general and can be used to study other stochastic volatility models. A calibration problem for these models is formulated and solved. Numerical experiments with real data are presented.

Cite this paper

L. Fatone, F. Mariani, M. Recchioni and F. Zirilli, "Some Explicitly Solvable SABR and Multiscale SABR Models: Option Pricing and Calibration,"*Journal of Mathematical Finance*, Vol. 3 No. 1, 2013, pp. 10-32. doi: 10.4236/jmf.2013.31002.

L. Fatone, F. Mariani, M. Recchioni and F. Zirilli, "Some Explicitly Solvable SABR and Multiscale SABR Models: Option Pricing and Calibration,"

References

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[2] G. Fiorentini, A. Leon and G. Rubio, “Estimation and Empirical Performance of Heston’s Stochastic Volatility Model: The Case of Thinly Traded Market,” Journal of Empirical Finance, Vol. 9, No 2, 2002, pp. 225-255. doi:10.1016/S0927-5398(01)00052-4

[3] B. Chen, C. W. Oosterlee and S. van Weeren, “Analytical Approximation to Constant Maturity Swap Convexity Corrections in a Multi-Factor SABR Model,” International Journal of Theoretical and Applied Finance, Vol 13, No. 7, 2010, pp. 1019-1046. doi:10.1142/S0219024910006091

[4] F. Mercurio and N. Moreni, “Inflation Modelling with SABR Dynamics,” Risk Magazine, 1 June 2009, pp. 106111.

[5] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “Calibration of a Multiscale Stochastic Volatility Model Using as Data European Option Prices,” Mathematical Methods in Economics and Finance, Vol. 3, No. 1, 2008, pp. 49-61.

[6] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “An Explicitly Solvable Multi-Scale Stochastic Volatility Model: Option Pricing and Calibration,” Journal of Futures Markets, Vol. 29, No. 9, 2009, pp. 862-893. doi:10.1002/fut.20390

[7] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “The Analysis of Real Data Using a Multiscale Stochastic Volatility Model,” European Financial Management, Vol. 19, No. 1, 2013, pp. 153-179. doi:10.1111/j.1468-036X.2010.00584.x

[8] O. Islah, “Solving SABR in Exact Form and Unifying It with LIBOR Market Model,” SSRN eLibrary, 2009. http://papers.ssrn.com/sol3/papers.cfm?astract-id=1489428

[9] P. S. Hagan, A. S. Lesniewski and D. E. Woodward, “Probability Distribution in the SABR Model of Stochastic Volatility,” 2005. http://lesniewski.us/papers/working/ProbDistrForSABR.eps

[10] J. Hull and A. White, “The Pricing of Options on Assets with Stochastic Volatilities,” The Journal of Finance, Vol. 42, No. 2, 1987, pp. 281-300. doi:10.1111/j.1540-6261.1987.tb02568.x

[11] B. A. Surya, “Two-Dimensional Hull-White Model for Stochastic Volatility and Its Nonlinear Filtering Estimation,” International Conference on Computational Science, ICCS 2011, Procedia Computer Science, Vol. 4, 2011, pp. 14311440.

[12] S. B. Yakubovich, “The Heat Kernel and Heisenberg Inequalities Related to the Kontorovich-Lebedev Transform,” Communications on Pure and Applied Analysis, Vol. 10, No. 2, 2011, pp. 745-760. doi:10.3934/cpaa.2011.10.745

[13] T. Bjork and C. Landen, “On the Term Structure of Futures and forward Prices,” In: H. Geman, D. Madan, S. Pliska and T. Vorst, Eds., Mathematical Finance—Bachelier Congress 2000, Springer Verlag, Berlin, 2002, pp. 111-150.

[14] M. Musiela and M. Rutkowski, “Martingale Methods in Financial Modelling,” Springer-Verlag, Berlin, 2005.

[15] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, “Higher Trascendental Functions,” McGraw-Hill Book Company, New York, 1953.

[16] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, “Tables of Integral Transforms,” McGraw-Hill Book Company, New York, 1954.

[17] A. Erdelyi, W. Magnus, F. Oberhettinger and F. Tricomi, “Tables of Integral Transforms,” McGraw-Hill Book Company, New York, 1954.

[18] R. Szmytkowski and S. Bielski, “Comment on the Orthogonality of the Macdonald Functions of Imaginary Order,” Journal of Mathematical Analysis and Applications, Vol. 365, No. 1, 2010, pp. 195-197. doi:10.1016/j.jmaa.2009.10.035

[19] S. B. Yakubovich, “Beurling’s Theorems and Inversion Formulas for Certain Index Trasforms,” Opuscula Mathematica, Vol. 29, No. 1, 2009, pp. 93-110.

[20] S. Rabinowitz, “How to Find the Square Root of a Complex Number,” Mathematics and Informatics Quarterly, Vol. 3, 1993, pp. 54-56.

[21] A. Mordecai, “Nonlinear Programming: Analysis and Methods,” Dover Publishing, New York, 2003.

[1] P. S. Hagan, D. Kumar, A. S. Lesniewski and D. E. Woodward, “Managing Smile Risk,” Wilmott Magazine, September 2002, pp. 84-108. http://www.wilmott.com/pdfs/021118-smile.eps

[2] G. Fiorentini, A. Leon and G. Rubio, “Estimation and Empirical Performance of Heston’s Stochastic Volatility Model: The Case of Thinly Traded Market,” Journal of Empirical Finance, Vol. 9, No 2, 2002, pp. 225-255. doi:10.1016/S0927-5398(01)00052-4

[3] B. Chen, C. W. Oosterlee and S. van Weeren, “Analytical Approximation to Constant Maturity Swap Convexity Corrections in a Multi-Factor SABR Model,” International Journal of Theoretical and Applied Finance, Vol 13, No. 7, 2010, pp. 1019-1046. doi:10.1142/S0219024910006091

[4] F. Mercurio and N. Moreni, “Inflation Modelling with SABR Dynamics,” Risk Magazine, 1 June 2009, pp. 106111.

[5] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “Calibration of a Multiscale Stochastic Volatility Model Using as Data European Option Prices,” Mathematical Methods in Economics and Finance, Vol. 3, No. 1, 2008, pp. 49-61.

[6] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “An Explicitly Solvable Multi-Scale Stochastic Volatility Model: Option Pricing and Calibration,” Journal of Futures Markets, Vol. 29, No. 9, 2009, pp. 862-893. doi:10.1002/fut.20390

[7] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “The Analysis of Real Data Using a Multiscale Stochastic Volatility Model,” European Financial Management, Vol. 19, No. 1, 2013, pp. 153-179. doi:10.1111/j.1468-036X.2010.00584.x

[8] O. Islah, “Solving SABR in Exact Form and Unifying It with LIBOR Market Model,” SSRN eLibrary, 2009. http://papers.ssrn.com/sol3/papers.cfm?astract-id=1489428

[9] P. S. Hagan, A. S. Lesniewski and D. E. Woodward, “Probability Distribution in the SABR Model of Stochastic Volatility,” 2005. http://lesniewski.us/papers/working/ProbDistrForSABR.eps

[10] J. Hull and A. White, “The Pricing of Options on Assets with Stochastic Volatilities,” The Journal of Finance, Vol. 42, No. 2, 1987, pp. 281-300. doi:10.1111/j.1540-6261.1987.tb02568.x

[11] B. A. Surya, “Two-Dimensional Hull-White Model for Stochastic Volatility and Its Nonlinear Filtering Estimation,” International Conference on Computational Science, ICCS 2011, Procedia Computer Science, Vol. 4, 2011, pp. 14311440.

[12] S. B. Yakubovich, “The Heat Kernel and Heisenberg Inequalities Related to the Kontorovich-Lebedev Transform,” Communications on Pure and Applied Analysis, Vol. 10, No. 2, 2011, pp. 745-760. doi:10.3934/cpaa.2011.10.745

[13] T. Bjork and C. Landen, “On the Term Structure of Futures and forward Prices,” In: H. Geman, D. Madan, S. Pliska and T. Vorst, Eds., Mathematical Finance—Bachelier Congress 2000, Springer Verlag, Berlin, 2002, pp. 111-150.

[14] M. Musiela and M. Rutkowski, “Martingale Methods in Financial Modelling,” Springer-Verlag, Berlin, 2005.

[15] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, “Higher Trascendental Functions,” McGraw-Hill Book Company, New York, 1953.

[16] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, “Tables of Integral Transforms,” McGraw-Hill Book Company, New York, 1954.

[17] A. Erdelyi, W. Magnus, F. Oberhettinger and F. Tricomi, “Tables of Integral Transforms,” McGraw-Hill Book Company, New York, 1954.

[18] R. Szmytkowski and S. Bielski, “Comment on the Orthogonality of the Macdonald Functions of Imaginary Order,” Journal of Mathematical Analysis and Applications, Vol. 365, No. 1, 2010, pp. 195-197. doi:10.1016/j.jmaa.2009.10.035

[19] S. B. Yakubovich, “Beurling’s Theorems and Inversion Formulas for Certain Index Trasforms,” Opuscula Mathematica, Vol. 29, No. 1, 2009, pp. 93-110.

[20] S. Rabinowitz, “How to Find the Square Root of a Complex Number,” Mathematics and Informatics Quarterly, Vol. 3, 1993, pp. 54-56.

[21] A. Mordecai, “Nonlinear Programming: Analysis and Methods,” Dover Publishing, New York, 2003.