Abstract
We derive an explicit representation of the transitions of the Heston stochastic volatility model and use it for fast and accurate simulation of the model. Of particular interest is the integral of the variance process over an interval, conditional on the level of the variance at the endpoints. We give an explicit representation of this quantity in terms of infinite sums and mixtures of gamma random variables. The increments of the variance process are themselves mixtures of gamma random variables. The representation of the integrated conditional variance applies the Pitman–Yor decomposition of Bessel bridges. We combine this representation with the Broadie–Kaya exact simulation method and use it to circumvent the most time-consuming step in that method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abate, J., Whitt, W.: The Fourier-series method for inverting transforms of probability distributions. Queueing Syst. 10, 5–88 (1992)
Ahlfors, L.V.: Complex Analysis, 3rd and international edn. McGraw–Hill, New York (1979)
Alfonsi, A.: On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl. 11, 355–384 (2005)
Alfonsi, A.: High order discretization schemes for the CIR process: Application to affine term structure and Heston models. Math. Comput. 79, 209–237 (2010)
Andersen, L.: Simple and efficient simulation of the Heston stochastic volatility model. J. Comput. Finance 11(3), 1–42 (2008)
Bondesson, L.: On simulation from infinitely divisible distributions. Adv. Appl. Probab. 14, 855–869 (1982)
Berkaoui, A., Bossy, M., Diop, A.: Euler scheme for SDEs with non-Lipschitz diffusion coefficient: Strong convergence. ESAIM: Probab. Stat. 12, 1–11 (2008)
Broadie, M., Kaya, Ö.: Exact simulation of stochastic volatility and other affine jump diffusion processes. Oper. Res. 54, 217–231 (2006)
Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall/CRC Press, London/Boca Raton (2004)
Cox, J.C., Ingersoll, J.E., Ross, S.A.: A theory of the term structure of interest rates. Econometrica 53, 1943–1978 (1985)
Deelstra, G., Delbaen, F.: Convergence of discretized stochastic (interest rate) processes with stochastic drift term. Appl. Stoch. Models Data Anal. 14, 77–84 (1998)
Delbaen, F., Shirakawa, H.: No arbitrage condition for positive diffusion price processes. Asia-Pac. Financ. Mark. 9, 159–168 (2002)
Devroye, L.: Simulating Bessel random variables. Stat. Probab. Lett. 57, 249–257 (2002)
Duffie, D., Glynn, P.: Efficient Monte Carlo simulation of security prices. Ann. Appl. Probab. 5, 897–905 (1995)
Duffie, D., Pan, J., Singleton, K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376 (2000)
Dutt, S.K., Welke, G.M.: Just-in-time Monte Carlo for path-dependent American options. J. Deriv. 15, 29–47 (2008)
Fishman, G.S.: Monte Carlo: Concepts, Algorithms, and Applications. Springer, New York (1996)
Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 13, 585–625 (1993)
Higham, D.J., Mao, X.: Convergence of Monte Carlo simulations involving the mean-reverting square root process. J. Comput. Finance 8, 35–62 (2005)
Iliopoulos, G., Karlis, D.: Simulation from the Bessel distribution with applications. J. Stat. Comput. Simul. 73, 491–506 (2003)
Kahl, C., Jäckel, P.: Fast strong approximation Monte Carlo schemes for stochastic volatility models. Quant. Finance 6, 513–536 (2006)
Kallenberg, O.: Foundations of Modern Probability, 2nd ed. Springer, Berlin (2002)
Kim, K.-K.: Affine processes in finance: Numerical approximation, simulation and model properties. Ph.D. thesis, Columbia University (2008). http://pqdtopen.proquest.com
Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. 3rd ed. Springer, Berlin (1999)
Longstaff, F.A., Schwartz, E.S.: Valuing American options by simulation: A simple least-squares approach. Rev. Financ. Stud. 14, 113–147 (2001)
Lord, R., Koekkoek, R., van Dijk, D.: A comparison of biased simulation schemes for stochastic volatility models. Quant. Finance (2009, forthcoming). http://www.rogerlord.com
Pitman, J., Yor, M.: A decomposition of Bessel bridges. Z. Wahrscheinlichkeitstheor. Verw. Geb. 59, 425–457 (1982)
Pitman, J., Yor, M.: Infinitely divisible laws associated with hyperbolic functions. Can. J. Math. 55, 292–330 (2003)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, New York (1999)
Scott, L.O.: Simulating a multi-factor term structure model over relatively long discrete time periods. In: Proceedings of the IAFE First Annual Computational Finance Conference. Graduate School of Business, Stanford University (1996)
van Haastrecht, A., Pelsser, A.: Efficient, almost exact simulation of the Heston stochastic volatility model. Working paper, University of Amsterdam (2008). http://papers.ssrn.com
Yuan, L., Kalbfleisch, J.D.: On the Bessel distribution and related problems. Ann. Inst. Stat. Math. 52, 438–447 (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Glasserman, P., Kim, KK. Gamma expansion of the Heston stochastic volatility model. Finance Stoch 15, 267–296 (2011). https://doi.org/10.1007/s00780-009-0115-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00780-009-0115-y