Some Explicit Formulae for the Hull and White Stochastic Volatility Model

Author(s)
Lorella Fatone^{*},
Francesca Mariani^{*},
Maria Cristina Recchioni^{*},
Francesco Zirilli^{*}

Affiliation(s)

Dipartimento di Matematica e Informatica, Università di Camerino, Camerino, Italy.

Dipartimento di Scienze Economiche, Università degli Studi di Verona, Verona, Italy.

Dipartimento di Management, Università Politecnica delle Marche, Ancona, Italy.

Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Roma, Italy.

Dipartimento di Matematica e Informatica, Università di Camerino, Camerino, Italy.

Dipartimento di Scienze Economiche, Università degli Studi di Verona, Verona, Italy.

Dipartimento di Management, Università Politecnica delle Marche, Ancona, Italy.

Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Roma, Italy.

ABSTRACT

An explicit formula for the transition probability density function of the Hull and White stochastic volatility model in presence of nonzero correlation between the stochastic differentials of the Wiener processes on the right hand side of the model equations is presented. This formula gives the transition probability density function as a two dimensional integral of an explicitly known integrand. Previously an explicit formula for this probability density function was known only in the case of zero correlation. In the case of nonzero correlation from the formula for the transition probability density function we deduce formulae (expressed by integrals) for the price of European call and put options and closed form formulae (that do not involve integrals) for the moments of the asset price logarithm. These formulae are based on recent results on the Whittaker functions [1] and generalize similar formulae for the SABR and multiscale SABR models [2]. Using the option pricing formulae derived and the least squares method a calibration problem for the Hull and White model is formulated and solved numerically. The calibration problem uses as data a set of option prices. Experiments with real data are presented. The real data studied are those belonging to a time series of the USA S&P 500 index and of the prices of its European call and put options. The quality of the model and of the calibration procedure is established comparing the forecast option prices obtained using the calibrated model with the option prices actually observed in the financial market. The website:* http*://*www.econ.univpm.it*/*recchioni*/*finance*/*w*17 contains some auxiliary material including animations and interactive applications that helps the understanding of this paper. More general references to the work of the authors and of their coauthors in mathematical finance are available in the website: *http*://*www.econ.univpm.it*/*recchioni/finance*.

Cite this paper

L. Fatone, F. Mariani, M. Recchioni and F. Zirilli, "Some Explicit Formulae for the Hull and White Stochastic Volatility Model,"*International Journal of Modern Nonlinear Theory and Application*, Vol. 2 No. 1, 2013, pp. 14-33. doi: 10.4236/ijmnta.2013.21003.

L. Fatone, F. Mariani, M. Recchioni and F. Zirilli, "Some Explicit Formulae for the Hull and White Stochastic Volatility Model,"

References

[1] R. Szmytkowki and S. Bielski, “An Orthogonality Relation for the Whittaker Functions of the Second Kind of Imaginary Order,” Integral Transforms and Special Functions, Vol. 21, No. 10, 2010, pp. 739-744. doi:10.1080/10652461003643412

[2] L. Fatone F. Mariani, M. C. 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.

[3] 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

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

[5] C. O. Ewald, K. R. Schenk-Hoppé and Z. Yang, “Closed-Form Solutions for European and Digital Calls in the Hull and White Stochastic Volatility Model and Their Relation to Locally R-Minimizing and Delta Hedges,” Paper No. 07-11, Swiss Finance Institute Research, 2007. http://papers.ssrn.com/sol3/papers.cfm?abstract-id=957807

[6] C. Corrado and T. Su, “Empirical Test of the Hull and White Option Pricing Model,” The Journal of Futures Markets, Vol. 18, No. 4, 1998, pp. 363-378. doi:10.1002/(SICI)1096-9934(199806)18:4<363::AID-FUT1>3.0.CO;2-K

[7] B. A. Surya, “Two-Dimensional Hull-White Model for Stochastic Volatility and Its Nonlinear Filtering Estimation,” Procedia Computer Science, Vol. 4, 2011, pp. 1431-1440. doi:10.1016/j.procs.2011.04.154

[8] E. Alòs, “A Generalization of the Hull and White Formula with Applications to Option Pricing Approximation,” Finance and Stochastics, Vol. 10, No. 3, 2006, pp. 353-365. doi:10.1007/s00780-006-0013-5

[9] E. Alòs, J. A. León, M. Pontier and J. Vives, “A Hull and White Formula for a General Stochastic Volatility Jump-Diffusion Model with Applications to the Study of the Short-Time Behavior of the Implied Volatility,” Journal of Applied Mathematics and Stochastic Analysis, 2008, Article ID: 359142, 17p.

[10] L. A. Grzelakab, L. A. Oosterleeac and S. Van Weeren, “Extension of Stochastic Volatility Equity Models with the Hull-White Interest Rate Process,” Quantitative Finance, Vol. 12, No. 1, 2012, pp. 89-105. doi:10.1080/14697680903170809

[11] E. Benhamou, E. Gobet and M. Miri, “Analytical Formulas for a Local Volatility Model with Stochastic Rates,” Quantitative Finance, Vol. 12, No. 2, 2012, pp. 185-198. doi:10.1080/14697688.2010.523011

[12] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “The Use of Statistical Tests to Calibrate the Normal Sabr Model,” Journal of Inverse and III Posed Problems, Vol. 21, No. 1, 2013, pp. 59-84.

[13] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “Closed Form Formulae for the Moments of the Lognormal SABR Model Variables and Their Use to Solve Two Calibration Problems,” Inverse Problems in Science and Engineering, 2012.

[14] B. Wong and C. C. Heyde, “On Changes of Measure in Stochastic Volatility Models,” Journal of Applied Mathematics and Stochastic Analysis, 2006, Article ID: 18130, 13p. doi:10.1155/JAMSA/2006/18130

[15] S. B. Yakubovich and M. M. Rodrigues, “Heat Kernel in Terms of Whittaker’s Functions,” International Symposium on Orthogonal Polynomials and Special Functions—A Complex Analytic Perspective, Copenhagen, 11-15 June 2012. http://cmup.fc.up.pt/cmup/v2/include/filedb.php?id=398&table=publicacoes&field=file.

[16] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions,” Dover, New York, 1970.

[17] P. A. Becker, “On the Integration of Products of Whittaker Functions with Respect to the Second Index,” Journal of Mathematical Physics, Vol. 45, No. 2, 2004, pp. 761-773. doi:10.1063/1.1634351

[18] 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

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

[20] R. Szmytkowki 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

[21] R. Beals and Y. Kannai, “Inverse Laplace Transforms of Products of Whittaker Functions,” Proceedings of the Royal Society A, Vol. 464, No. 2092, 2008, pp. 795-806. doi:10.1098/rspa.2007.0248

[22] F. Oberhettinger, “Tables of Bessel Transform,” Springer-Verlag, Berlin, 1972. doi:10.1007/978-3-642-65462-6

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

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

[25] P. L. Lions and M. Musiela, “Correlation and Bounds for Stochastic Volatility Models,” Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Vol. 24, No. 1, 2007, pp. 1-16.

[26] W. Schoutens, “Lévy Processes in Finance, John Wiley & Sons, Chichester, 2003. doi:10.1002/0470870230

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

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

[1] R. Szmytkowki and S. Bielski, “An Orthogonality Relation for the Whittaker Functions of the Second Kind of Imaginary Order,” Integral Transforms and Special Functions, Vol. 21, No. 10, 2010, pp. 739-744. doi:10.1080/10652461003643412

[2] L. Fatone F. Mariani, M. C. 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.

[3] 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

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

[5] C. O. Ewald, K. R. Schenk-Hoppé and Z. Yang, “Closed-Form Solutions for European and Digital Calls in the Hull and White Stochastic Volatility Model and Their Relation to Locally R-Minimizing and Delta Hedges,” Paper No. 07-11, Swiss Finance Institute Research, 2007. http://papers.ssrn.com/sol3/papers.cfm?abstract-id=957807

[6] C. Corrado and T. Su, “Empirical Test of the Hull and White Option Pricing Model,” The Journal of Futures Markets, Vol. 18, No. 4, 1998, pp. 363-378. doi:10.1002/(SICI)1096-9934(199806)18:4<363::AID-FUT1>3.0.CO;2-K

[7] B. A. Surya, “Two-Dimensional Hull-White Model for Stochastic Volatility and Its Nonlinear Filtering Estimation,” Procedia Computer Science, Vol. 4, 2011, pp. 1431-1440. doi:10.1016/j.procs.2011.04.154

[8] E. Alòs, “A Generalization of the Hull and White Formula with Applications to Option Pricing Approximation,” Finance and Stochastics, Vol. 10, No. 3, 2006, pp. 353-365. doi:10.1007/s00780-006-0013-5

[9] E. Alòs, J. A. León, M. Pontier and J. Vives, “A Hull and White Formula for a General Stochastic Volatility Jump-Diffusion Model with Applications to the Study of the Short-Time Behavior of the Implied Volatility,” Journal of Applied Mathematics and Stochastic Analysis, 2008, Article ID: 359142, 17p.

[10] L. A. Grzelakab, L. A. Oosterleeac and S. Van Weeren, “Extension of Stochastic Volatility Equity Models with the Hull-White Interest Rate Process,” Quantitative Finance, Vol. 12, No. 1, 2012, pp. 89-105. doi:10.1080/14697680903170809

[11] E. Benhamou, E. Gobet and M. Miri, “Analytical Formulas for a Local Volatility Model with Stochastic Rates,” Quantitative Finance, Vol. 12, No. 2, 2012, pp. 185-198. doi:10.1080/14697688.2010.523011

[12] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “The Use of Statistical Tests to Calibrate the Normal Sabr Model,” Journal of Inverse and III Posed Problems, Vol. 21, No. 1, 2013, pp. 59-84.

[13] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “Closed Form Formulae for the Moments of the Lognormal SABR Model Variables and Their Use to Solve Two Calibration Problems,” Inverse Problems in Science and Engineering, 2012.

[14] B. Wong and C. C. Heyde, “On Changes of Measure in Stochastic Volatility Models,” Journal of Applied Mathematics and Stochastic Analysis, 2006, Article ID: 18130, 13p. doi:10.1155/JAMSA/2006/18130

[15] S. B. Yakubovich and M. M. Rodrigues, “Heat Kernel in Terms of Whittaker’s Functions,” International Symposium on Orthogonal Polynomials and Special Functions—A Complex Analytic Perspective, Copenhagen, 11-15 June 2012. http://cmup.fc.up.pt/cmup/v2/include/filedb.php?id=398&table=publicacoes&field=file.

[16] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions,” Dover, New York, 1970.

[17] P. A. Becker, “On the Integration of Products of Whittaker Functions with Respect to the Second Index,” Journal of Mathematical Physics, Vol. 45, No. 2, 2004, pp. 761-773. doi:10.1063/1.1634351

[18] 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

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

[20] R. Szmytkowki 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

[21] R. Beals and Y. Kannai, “Inverse Laplace Transforms of Products of Whittaker Functions,” Proceedings of the Royal Society A, Vol. 464, No. 2092, 2008, pp. 795-806. doi:10.1098/rspa.2007.0248

[22] F. Oberhettinger, “Tables of Bessel Transform,” Springer-Verlag, Berlin, 1972. doi:10.1007/978-3-642-65462-6

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

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

[25] P. L. Lions and M. Musiela, “Correlation and Bounds for Stochastic Volatility Models,” Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Vol. 24, No. 1, 2007, pp. 1-16.

[26] W. Schoutens, “Lévy Processes in Finance, John Wiley & Sons, Chichester, 2003. doi:10.1002/0470870230

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

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