Probability distribution of returns in the Heston model with stochastic volatility
AbstractWe study the Heston model, where the stock price dynamics is governed by a geometrical (multiplicative) Brownian motion with stochastic variance. We solve the corresponding Fokker-Planck equation exactly and, after integrating out the variance, find an analytic formula for the time-dependent probability distribution of stock price changes (returns). The formula is in excellent agreement with the Dow-Jones index for time lags from 1 to 250 trading days. For large returns, the distribution is exponential in log-returns with a time-dependent exponent, whereas for small returns it is Gaussian. For time lags longer than the relaxation time of variance, the probability distribution can be expressed in a scaling form using a Bessel function. The Dow-Jones data for 1982-2001 follow the scaling function for seven orders of magnitude.
Download InfoIf you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
Bibliographic InfoArticle provided by Taylor and Francis Journals in its journal Quantitative Finance.
Volume (Year): 2 (2002)
Issue (Month): 6 ()
Contact details of provider:
Web page: http://taylorandfrancis.metapress.com/link.asp?target=journal&id=111405
You can help add them by filling out this form.
CitEc Project, subscribe to its RSS feed for this item.
- Agnieszka Janek & Tino Kluge & Rafal Weron & Uwe Wystup, 2010.
"FX Smile in the Heston Model,"
HSC Research Reports
HSC/10/02, Hugo Steinhaus Center, Wroclaw University of Technology.
- Janek, Agnieszka & Kluge, Tino & Weron, Rafal & Wystup, Uwe, 2010. "FX Smile in the Heston Model," MPRA Paper 25491, University Library of Munich, Germany.
- Agnieszka Janek & Tino Kluge & Rafał Weron & Uwe Wystup, 2010. "FX Smile in the Heston Model," SFB 649 Discussion Papers SFB649DP2010-047, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
- Agnieszka Janek & Tino Kluge & Rafal Weron & Uwe Wystup, 2010. "FX Smile in the Heston Model," Papers 1010.1617, arXiv.org.
- P. Friz & S. Gerhold & A. Gulisashvili & S. Sturm, 2010. "On refined volatility smile expansion in the Heston model," Papers 1001.3003, arXiv.org, revised Nov 2010.
- Buchbinder, G.L. & Chistilin, K.M., 2007. "Multiple time scales and the empirical models for stochastic volatility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 379(1), pages 168-178.
- Andreas Behr & Ulrich Pötter, 2009. "Alternatives to the normal model of stock returns: Gaussian mixture, generalised logF and generalised hyperbolic models," Annals of Finance, Springer, vol. 5(1), pages 49-68, January.
- Ballestra, Luca Vincenzo & Pacelli, Graziella & Zirilli, Francesco, 2007. "A numerical method to price exotic path-dependent options on an underlying described by the Heston stochastic volatility model," Journal of Banking & Finance, Elsevier, vol. 31(11), pages 3420-3437, November.
- Benoit Pochard & Jean-Philippe Bouchaud, 2003. "Option pricing and hedging with minimum expected shortfall," Science & Finance (CFM) working paper archive 500029, Science & Finance, Capital Fund Management.
- A. Monteiro & R. Tütüncü & L. Vicente, 2011. "Estimation of risk-neutral density surfaces," Computational Management Science, Springer, vol. 8(4), pages 387-414, November.
- del Baño Rollin, Sebastian & Ferreiro-Castilla, Albert & Utzet, Frederic, 2010. "On the density of log-spot in the Heston volatility model," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 2037-2063, September.
- Chiang, Thomas C. & Yu, Hai-Chin & Wu, Ming-Chya, 2009. "Statistical properties, dynamic conditional correlation and scaling analysis: Evidence from Dow Jones and Nasdaq high-frequency data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(8), pages 1555-1570.
- Anca Gheorghiu & Ion Spanulescu, 2009. "Macrostate Parameter, an Econophysics Approach for the Risk Analysis of the Stock Exchange Market Transactions," Papers 0907.5600, arXiv.org.
- Nakamura, Tomomichi & Small, Michael, 2007. "Tests of the random walk hypothesis for financial data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 377(2), pages 599-615.
- Giacomo Bormetti & Valentina Cazzola & Danilo Delpini, 2009. "Option pricing under Ornstein-Uhlenbeck stochastic volatility: a linear model," Papers 0905.1882, arXiv.org, revised May 2010.
- Anca Gheorghiu & Ion Spanulescu, 2012. "An Econophysics Model for the Migration Phenomena," Papers 1202.0996, arXiv.org.
- A. Gulisashvili & E. M. Stein, 2009. "Asymptotic Behavior of the Stock Price Distribution Density and Implied Volatility in Stochastic Volatility Models," Papers 0906.0392, arXiv.org.
- Bernardo Spagnolo & Davide Valenti, 2008. "Volatility Effects on the Escape Time in Financial Market Models," Papers 0810.1625, arXiv.org.
- López Martín, María del Mar & García, Catalina García & García Pérez, José, 2012. "Treatment of kurtosis in financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(5), pages 2032-2045.
- Oguzhan Cepni & Ahmet Goncu & Mehmet Oguz Karahan & Tolga Umut Kuzubas, 2013. "Goodness-of-fit of the Heston, Variance-Gamma and Normal-Inverse Gaussian Models," Working Papers 2013/16, Bogazici University, Department of Economics.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Michael McNulty).
If references are entirely missing, you can add them using this form.