The Calibration of Some Stochastic Volatility Models Used in Mathematical Finance

Affiliation(s)

Dipartimento di Matematica e Informatica, Università di Camerino, Camerino (MC), 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 (MC), 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

Stochastic volatility models are used in mathematical finance to describe the dynamics of asset prices. In these models, the asset price is modeled as a stochastic process depending on time implicitly defined by a stochastic differential Equation. The volatility of the asset price itself is modeled as a stochastic process depending on time whose dynamics is described by a stochastic differential Equation. The stochastic differential Equations for the asset price and for the volatility are coupled and together with the necessary initial conditions and correlation assumptions constitute the model. Note that the stochastic volatility is not observable in the financial markets. In order to use these models, for example, to evaluate prices of derivatives on the asset or to forecast asset prices, it is necessary to calibrate them. That is, it is necessary to estimate starting from a set of data the values of the initial volatility and of the unknown parameters that appear in the asset price/volatility dynamic Equations. These data usually are observations of the asset prices and/or of the prices of derivatives on the asset at some known times. We analyze some stochastic volatility models summarizing merits and weaknesses of each of them. We point out that these models are examples of stochastic state space models and present the main techniques used to calibrate them. A calibration problem for the Heston model is solved using the maximum likelihood method. Some numerical experiments about the calibration of the Heston model involving synthetic and real data are presented.

Stochastic volatility models are used in mathematical finance to describe the dynamics of asset prices. In these models, the asset price is modeled as a stochastic process depending on time implicitly defined by a stochastic differential Equation. The volatility of the asset price itself is modeled as a stochastic process depending on time whose dynamics is described by a stochastic differential Equation. The stochastic differential Equations for the asset price and for the volatility are coupled and together with the necessary initial conditions and correlation assumptions constitute the model. Note that the stochastic volatility is not observable in the financial markets. In order to use these models, for example, to evaluate prices of derivatives on the asset or to forecast asset prices, it is necessary to calibrate them. That is, it is necessary to estimate starting from a set of data the values of the initial volatility and of the unknown parameters that appear in the asset price/volatility dynamic Equations. These data usually are observations of the asset prices and/or of the prices of derivatives on the asset at some known times. We analyze some stochastic volatility models summarizing merits and weaknesses of each of them. We point out that these models are examples of stochastic state space models and present the main techniques used to calibrate them. A calibration problem for the Heston model is solved using the maximum likelihood method. Some numerical experiments about the calibration of the Heston model involving synthetic and real data are presented.

Cite this paper

L. Fatone, F. Mariani, M. Recchioni and F. Zirilli, "The Calibration of Some Stochastic Volatility Models Used in Mathematical Finance,"*Open Journal of Applied Sciences*, Vol. 4 No. 2, 2014, pp. 23-33. doi: 10.4236/ojapps.2014.42004.

L. Fatone, F. Mariani, M. Recchioni and F. Zirilli, "The Calibration of Some Stochastic Volatility Models Used in Mathematical Finance,"

References

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http://dx.doi.org/10.1002/andp.19053220806

[2] Keller, J.B. (1964) Wave propagation in random media. Proceedings of Symposia in Applied Mathematics, 13, 227-246.

[3] Black, F. and Scholes, M. (1973) The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637-659.

http://dx.doi.org/10.1086/260062

[4] Kalman, R.E. (1960) A new approach to linear filtering and prediction problems. Journal of Basic Engineering-Transactions of the ASME, Series D, 85, 35-45.

http://dx.doi.org/10.1115/1.3662552

[5] Bachelier, L. (1900) Théorie de la speculation. Annales Scientifiques de l’école Normale Supérieure, 3, 21-86.

[6] Merton, R.C. (1973) Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141-183.

http://dx.doi.org/10.2307/3003143

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http://dx.doi.org/10.1111/j.1540-6261.1987.tb02568.x

[8] Stein, E.M. and Stein, J.C. (1991) Stock price distribution with stochastic volatility: An analytic approach. Review of Financial Studies, 4, 727-752.

http://dx.doi.org/10.1093/rfs/4.4.727

[9] Heston, S.L. (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6, 327-343.

http://dx.doi.org/10.1093/rfs/6.2.327

[10] Cox, J.C., Ingersoll, J.E. and Ross, S.A. (1985) A theory of the term structure of interest rates. Econometrica, 53, 385-407.

http://dx.doi.org/10.2307/1911242

[11] Hagan, P.S., Kumar, D., Lesniewski, A.D. and Woodward, E. (2002) Managing smile risk. Wilmott Magazine, 2002, 84-108.

http://www.wilmott.com/pdfs/021118_smile.pdf

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[13] Harvey, A.C., Koopman, S.J. and Shephard, N. (2004) State space and unobserved component models: Theory and applications. Cambridge University Press, Cambridge.

[14] Hansen, L.P. (1982) Large sample properties of generalized method of moments estimators. Ecometrica, 50, 1029-1054.

[15] Bollerslev, T. and Zhou, H. (2002) Estimating stochastic volatility diffusion using conditional moments of integrated volatility. Journal of Econometrics, 109, 33-65.

http://dx.doi.org/10.1016/S0304-4076(01)00141-5

[16] Jaynes, E.T. (1957) Information theory and statistical mechanics. Physical Review, 106, 620-630.

http://dx.doi.org/10.1103/PhysRev.106.620

[17] Shannon, C.E. (1948) A mathematical theory of communication. Bell System Technical Journal, 27, 379-423.

http://dx.doi.org/10.1002/j.1538-7305.1948.tb01338.x

[18] Avellaneda, M. (1998) Minimum-relative-entropy calibration of asset pricing models. International Journal of Theoretical and Applied Finance, 1, 447-472.

http://dx.doi.org/10.1142/S0219024998000242

[19] Ait-Sahalia, Y. and Kimmel, R. (2007) Maximum likelihood estimation of stochastic volatility models. Journal of Financial Economics, 83, 413-452.

http://dx.doi.org/10.1016/j.jfineco.2005.10.006

[20] Bates, D.S. (2006) Maximum likelihood estimation of latent affine processes. The Review of Financial Studies, 19, 909-965.

http://dx.doi.org/10.1016/j.jfineco.2005.10.006

[21] Mariani, F., Pacelli, G. and Zirilli, F. (2008) Maximum likelihood estimation of the Heston stochastic volatility model using asset and option prices: An application of nonlinear filtering theory. Optimization Letters, 2, 177-222.

http://dx.doi.org/10.1007/s11590-007-0052-7

[22] Fatone, L., Mariani, F., Recchioni, M.C. and Zirilli, F. (2012) The use of statistical tests to calibrate the Black-Scholes asset dynamics model applied to pricing options with uncertain volatility. Journal of Probability and Statistics, 2012, Article ID: 931609.

[23] Johnson, R.A. and Bhattacharyya, G.K. (2006) Statistics: Principles and methods. 5th Edition, John Wiley & Sons, New York.

[24] Fatone, L., Mariani, F., Recchioni, M.C. and Zirilli, F. (2013) The use of statistical tests to calibrate the normal SABR model. Journal of Inverse and Ill-Posed Problems, 21, 59-84.

http://dx.doi.org/10.1515/jip-2012-0093

[25] Christoffersen, P., Heston, S. and Jacobs, K. (2009) The shape and term structure of the index option smirk: Why multifactor stochastic volatility models work so well. Management Science, 55, 1914-1932.

http://dx.doi.org/10.1287/mnsc.1090.1065

[26] Fatone, L., Mariani, F., Recchioni, M.C. and Zirilli, F. (2009) An explicitly solvable multi-scale stochastic volatility model: Option pricing and calibration problems. Journal of Futures Markets, 29, 862-893.

http://dx.doi.org/10.1002/fut.20390

[27] Mergner, S. (2009) Application of state space models in finance: An empirical analysis of the time-varying relationship between macroeconomics, fundamentals and Pan-European industry portfolios. Universitätsverlag Göttingen, Göttingen.

[1] Einstein, A. (1905) über die von der molekularkinetischen theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik, 322, 549-560.

http://dx.doi.org/10.1002/andp.19053220806

[2] Keller, J.B. (1964) Wave propagation in random media. Proceedings of Symposia in Applied Mathematics, 13, 227-246.

[3] Black, F. and Scholes, M. (1973) The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637-659.

http://dx.doi.org/10.1086/260062

[4] Kalman, R.E. (1960) A new approach to linear filtering and prediction problems. Journal of Basic Engineering-Transactions of the ASME, Series D, 85, 35-45.

http://dx.doi.org/10.1115/1.3662552

[5] Bachelier, L. (1900) Théorie de la speculation. Annales Scientifiques de l’école Normale Supérieure, 3, 21-86.

[6] Merton, R.C. (1973) Theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141-183.

http://dx.doi.org/10.2307/3003143

[7] Hull, J. and White, A. (1987) The pricing of options on assets with stochastic volatilities. Journal of Finance, 42, 281-300.

http://dx.doi.org/10.1111/j.1540-6261.1987.tb02568.x

[8] Stein, E.M. and Stein, J.C. (1991) Stock price distribution with stochastic volatility: An analytic approach. Review of Financial Studies, 4, 727-752.

http://dx.doi.org/10.1093/rfs/4.4.727

[9] Heston, S.L. (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6, 327-343.

http://dx.doi.org/10.1093/rfs/6.2.327

[10] Cox, J.C., Ingersoll, J.E. and Ross, S.A. (1985) A theory of the term structure of interest rates. Econometrica, 53, 385-407.

http://dx.doi.org/10.2307/1911242

[11] Hagan, P.S., Kumar, D., Lesniewski, A.D. and Woodward, E. (2002) Managing smile risk. Wilmott Magazine, 2002, 84-108.

http://www.wilmott.com/pdfs/021118_smile.pdf

[12] Cox, J. (1975) Notes on option pricing I: Constant elasticity of diffusions. Unpublished Draft, Stanford University.

[13] Harvey, A.C., Koopman, S.J. and Shephard, N. (2004) State space and unobserved component models: Theory and applications. Cambridge University Press, Cambridge.

[14] Hansen, L.P. (1982) Large sample properties of generalized method of moments estimators. Ecometrica, 50, 1029-1054.

[15] Bollerslev, T. and Zhou, H. (2002) Estimating stochastic volatility diffusion using conditional moments of integrated volatility. Journal of Econometrics, 109, 33-65.

http://dx.doi.org/10.1016/S0304-4076(01)00141-5

[16] Jaynes, E.T. (1957) Information theory and statistical mechanics. Physical Review, 106, 620-630.

http://dx.doi.org/10.1103/PhysRev.106.620

[17] Shannon, C.E. (1948) A mathematical theory of communication. Bell System Technical Journal, 27, 379-423.

http://dx.doi.org/10.1002/j.1538-7305.1948.tb01338.x

[18] Avellaneda, M. (1998) Minimum-relative-entropy calibration of asset pricing models. International Journal of Theoretical and Applied Finance, 1, 447-472.

http://dx.doi.org/10.1142/S0219024998000242

[19] Ait-Sahalia, Y. and Kimmel, R. (2007) Maximum likelihood estimation of stochastic volatility models. Journal of Financial Economics, 83, 413-452.

http://dx.doi.org/10.1016/j.jfineco.2005.10.006

[20] Bates, D.S. (2006) Maximum likelihood estimation of latent affine processes. The Review of Financial Studies, 19, 909-965.

http://dx.doi.org/10.1016/j.jfineco.2005.10.006

[21] Mariani, F., Pacelli, G. and Zirilli, F. (2008) Maximum likelihood estimation of the Heston stochastic volatility model using asset and option prices: An application of nonlinear filtering theory. Optimization Letters, 2, 177-222.

http://dx.doi.org/10.1007/s11590-007-0052-7

[22] Fatone, L., Mariani, F., Recchioni, M.C. and Zirilli, F. (2012) The use of statistical tests to calibrate the Black-Scholes asset dynamics model applied to pricing options with uncertain volatility. Journal of Probability and Statistics, 2012, Article ID: 931609.

[23] Johnson, R.A. and Bhattacharyya, G.K. (2006) Statistics: Principles and methods. 5th Edition, John Wiley & Sons, New York.

[24] Fatone, L., Mariani, F., Recchioni, M.C. and Zirilli, F. (2013) The use of statistical tests to calibrate the normal SABR model. Journal of Inverse and Ill-Posed Problems, 21, 59-84.

http://dx.doi.org/10.1515/jip-2012-0093

[25] Christoffersen, P., Heston, S. and Jacobs, K. (2009) The shape and term structure of the index option smirk: Why multifactor stochastic volatility models work so well. Management Science, 55, 1914-1932.

http://dx.doi.org/10.1287/mnsc.1090.1065

[26] Fatone, L., Mariani, F., Recchioni, M.C. and Zirilli, F. (2009) An explicitly solvable multi-scale stochastic volatility model: Option pricing and calibration problems. Journal of Futures Markets, 29, 862-893.

http://dx.doi.org/10.1002/fut.20390

[27] Mergner, S. (2009) Application of state space models in finance: An empirical analysis of the time-varying relationship between macroeconomics, fundamentals and Pan-European industry portfolios. Universitätsverlag Göttingen, Göttingen.