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Fast strong approximation Monte Carlo schemes for stochastic volatility models

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  • Christian Kahl
  • Peter Jackel

Abstract

Numerical integration methods for stochastic volatility models in financial markets are discussed. We concentrate on two classes of stochastic volatility models where the volatility is either directly given by a mean-reverting CEV process or as a transformed Ornstein-Uhlenbeck process. For the latter, we introduce a new model based on a simple hyperbolic transformation. Various numerical methods for integrating mean-reverting CEV processes are analysed and compared with respect to positivity preservation and efficiency. Moreover, we develop a simple and robust integration scheme for the two-dimensional system using the strong convergence behaviour as an indicator for the approximation quality. This method, which we refer to as the IJK (137) scheme, is applicable to all types of stochastic volatility models and can be employed as a drop-in replacement for the standard log-Euler procedure.

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Bibliographic Info

Article provided by Taylor and Francis Journals in its journal Quantitative Finance.

Volume (Year): 6 (2006)
Issue (Month): 6 ()
Pages: 513-536

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Handle: RePEc:taf:quantf:v:6:y:2006:i:6:p:513-536

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Related research

Keywords: Stochastic volatility models; Stochastic numerical integration; Strong approximation error; Hyperbolic Ornstein-Uhlenbeck process; Hyperbolic volatility;

References

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  1. Wiggins, James B., 1987. "Option values under stochastic volatility: Theory and empirical estimates," Journal of Financial Economics, Elsevier, vol. 19(2), pages 351-372, December.
  2. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
  3. Brennan, Michael J. & Schwartz, Eduardo S., 1980. "Analyzing Convertible Bonds," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 15(04), pages 907-929, November.
  4. Beckers, Stan, 1980. " The Constant Elasticity of Variance Model and Its Implications for Option Pricing," Journal of Finance, American Finance Association, vol. 35(3), pages 661-73, June.
  5. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
  6. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-43.
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Citations

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Cited by:
  1. Denis Belomestny & Stanley Matthew & John Schoenmakers, 2007. "A stochastic volatility Libor model and its robust calibration," SFB 649 Discussion Papers SFB649DP2007-067, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  2. Masaaki Fujii & Akihiko Takahashi, 2012. "Perturbative Expansion of FBSDE in an Incomplete Market with Stochastic Volatility," CARF F-Series CARF-F-270, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Jun 2012.
  3. Andreas Neuenkirch & Lukasz Szpruch, 2012. "First order strong approximations of scalar SDEs with values in a domain," Papers 1209.0390, arXiv.org.
  4. Mordecai Avriel & Jens Hilscher & Alon Raviv, 2012. "Inflation Derivatives Under Inflation Target Regimes," Working Papers 43, Brandeis University, Department of Economics and International Businesss School.
  5. Liu, Peng & Tang, Ke, 2011. "The stochastic behavior of commodity prices with heteroskedasticity in the convenience yield," Journal of Empirical Finance, Elsevier, vol. 18(2), pages 211-224, March.
  6. Dell'Era, Mario, 2010. "Vanilla Option Pricing on Stochastic Volatility market models," MPRA Paper 25645, University Library of Munich, Germany.
  7. Dell'Era, Mario, 2010. "Geometrical Approximation method and stochastic volatility market models," MPRA Paper 22568, University Library of Munich, Germany.
  8. Benjamin Jourdain & Mohamed Sbai, 2009. "High order discretization schemes for stochastic volatility models," Working Papers hal-00409861, HAL.
  9. Fahim, Arash & Touzi, Nizar & Warin, Xavier, 2011. "A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs," Open Access publications from Université Paris-Dauphine urn:hdl:123456789/5524, Université Paris-Dauphine.
  10. Dell'Era, Mario, 2010. "Geometrical Considerations on Heston's Market Model," MPRA Paper 21523, University Library of Munich, Germany.
  11. Masaaki Fujii & Akihiko Takahashi, 2012. "Perturbative Expansion of FBSDE in an Incomplete Market with Stochastic Volatility," CIRJE F-Series CIRJE-F-840, CIRJE, Faculty of Economics, University of Tokyo.
  12. Paul Glasserman & Kyoung-Kuk Kim, 2011. "Gamma expansion of the Heston stochastic volatility model," Finance and Stochastics, Springer, vol. 15(2), pages 267-296, June.
  13. Masaaki Fujii & Akihiko Takahashi, 2012. "Perturbative Expansion of FBSDE in an Incomplete Market with Stochastic Volatility," Papers 1202.0608, arXiv.org, revised Sep 2012.

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