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Sample path Large Deviations and optimal importance sampling for stochastic volatility models

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  • Robertson, Scott

Abstract

Sample path Large Deviation Principles (LDP) of the Freidlin-Wentzell type are derived for a class of diffusions, which govern the price dynamics in common stochastic volatility models from Mathematical Finance. LDP are obtained by relaxing the non-degeneracy requirement on the diffusion matrix in the standard theory of Freidlin and Wentzell. As an application, a sample path LDP is proved for the price process in the Heston stochastic volatility model. Using the sample path LDP for the Heston model, the problem is considered of selecting an importance sampling change of drift, for both the price and volatility, which minimize the variance of Monte Carlo estimators for path-dependent option prices. An asymptotically optimal change of drift is identified as a solution to a two-dimensional variational problem. The case of the arithmetic average Asian put option is solved in detail.

Suggested Citation

  • Robertson, Scott, 2010. "Sample path Large Deviations and optimal importance sampling for stochastic volatility models," Stochastic Processes and their Applications, Elsevier, vol. 120(1), pages 66-83, January.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:1:p:66-83
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    References listed on IDEAS

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    1. Huyen Pham, 2007. "Some applications and methods of large deviations in finance and insurance," Papers math/0702473, arXiv.org, revised Feb 2007.
    2. Paolo Guasoni & Scott Robertson, 2008. "Optimal importance sampling with explicit formulas in continuous time," Finance and Stochastics, Springer, vol. 12(1), pages 1-19, January.
    3. Paul Glasserman & Philip Heidelberger & Perwez Shahabuddin, 1999. "Asymptotically Optimal Importance Sampling and Stratification for Pricing Path‐Dependent Options," Mathematical Finance, Wiley Blackwell, vol. 9(2), pages 117-152, April.
    4. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    5. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
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