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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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    Citations

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    Cited by:

    1. Archil Gulisashvili, 2020. "Time-inhomogeneous Gaussian stochastic volatility models: Large deviations and super roughness," Papers 2002.05143, arXiv.org, revised Dec 2020.
    2. Antoine Jacquier & Konstantinos Spiliopoulos, 2018. "Pathwise moderate deviations for option pricing," Papers 1803.04483, arXiv.org, revised Dec 2018.
    3. Peter K. Friz & Paul Gassiat & Paolo Pigato, 2018. "Precise asymptotics: robust stochastic volatility models," Papers 1811.00267, arXiv.org, revised Nov 2020.
    4. Zorana Grbac & David Krief & Peter Tankov, 2021. "Long-Time Trajectorial Large Deviations and Importance Sampling for Affine Stochastic Volatility Models," Post-Print hal-03899237, HAL.
    5. Genin, Adrien & Tankov, Peter, 2020. "Optimal importance sampling for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 20-46.
    6. Jacquier, Antoine & Pannier, Alexandre, 2022. "Large and moderate deviations for stochastic Volterra systems," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 142-187.
    7. Archil Gulisashvili, 2020. "Large deviation principles for stochastic volatility models with reflection and three faces of the Stein and Stein model," Papers 2006.15431, arXiv.org.
    8. Archil Gulisashvili, 2018. "Gaussian stochastic volatility models: Scaling regimes, large deviations, and moment explosions," Papers 1808.00421, arXiv.org, revised Jun 2019.
    9. Archil Gulisashvili, 2022. "Multivariate Stochastic Volatility Models and Large Deviation Principles," Papers 2203.09015, arXiv.org, revised Nov 2022.
    10. Adrien Genin & Peter Tankov, 2016. "Optimal importance sampling for L\'evy Processes," Papers 1608.04621, arXiv.org.
    11. dos Reis, Gonçalo & Smith, Greig & Tankov, Peter, 2023. "Importance sampling for McKean-Vlasov SDEs," Applied Mathematics and Computation, Elsevier, vol. 453(C).
    12. Aur'elien Alfonsi & David Krief & Peter Tankov, 2018. "Long-time large deviations for the multi-asset Wishart stochastic volatility model and option pricing," Papers 1806.06883, arXiv.org.
    13. Baldi, P. & Caramellino, L., 2011. "General Freidlin-Wentzell Large Deviations and positive diffusions," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 1218-1229, August.
    14. Antoine Jacquier & Alexandre Pannier, 2020. "Large and moderate deviations for stochastic Volterra systems," Papers 2004.10571, arXiv.org, revised Apr 2022.
    15. Martin Forde & Antoine Jacquier, 2011. "The large-maturity smile for the Heston model," Finance and Stochastics, Springer, vol. 15(4), pages 755-780, December.
    16. Djellout, Hacène & Guillin, Arnaud & Samoura, Yacouba, 2017. "Estimation of the realized (co-)volatility vector: Large deviations approach," Stochastic Processes and their Applications, Elsevier, vol. 127(9), pages 2926-2960.
    17. Zorana Grbac & David Krief & Peter Tankov, 2018. "Long-time trajectorial large deviations for affine stochastic volatility models and application to variance reduction for option pricing," Papers 1809.06153, arXiv.org.
    18. Gulisashvili, Archil, 2020. "Gaussian stochastic volatility models: Scaling regimes, large deviations, and moment explosions," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3648-3686.
    19. Gulisashvili, Archil, 2021. "Time-inhomogeneous Gaussian stochastic volatility models: Large deviations and super roughness," Stochastic Processes and their Applications, Elsevier, vol. 139(C), pages 37-79.
    20. Marc Geha & Antoine Jacquier & Zan Zuric, 2021. "Large and moderate deviations for importance sampling in the Heston model," Papers 2111.00348, arXiv.org.

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