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Long-Time Trajectorial Large Deviations and Importance Sampling for Affine Stochastic Volatility Models

Part of: Limit theorems Mathematical finance

Published online by Cambridge University Press:  17 March 2021

Zorana Grbac ,
David Krief  and
Peter Tankov
Zorana Grbac*
Affiliation:
Université de Paris
David Krief*
Affiliation:
Université de Paris
Peter Tankov*
Affiliation:
ENSAE, Institut Polytechnique de Paris
*
*Postal address: 5 rue Thomas Mann, 75013Paris, France.
*Postal address: 5 rue Thomas Mann, 75013Paris, France.
****Postal address: 5 avenue Henry Le Chatelier, 91120Palaiseau, France. Email address: peter.tankov@ensae.fr
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Abstract

We establish a pathwise large deviation principle for affine stochastic volatility models introduced by Keller-Ressel (2011), and present an application to variance reduction for Monte Carlo computation of prices of path-dependent options in these models, extending the method developed by Genin and Tankov (2020) for exponential Lévy models. To this end, we apply an exponentially affine change of measure and use Varadhan’s lemma, in the fashion of Guasoni and Robertson (2008) and Robertson (2010), to approximate the problem of finding the measure that minimizes the variance of the Monte Carlo estimator. We test the method on the Heston model with and without jumps to demonstrate its numerical efficiency.

Keywords

Large deviationsMonte Carlo methodsimportance samplingaffine stochastic volatility

MSC classification

Primary: 60F10: Large deviations
Secondary: 91G60: Numerical methods (including Monte Carlo methods)
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 1 , March 2021 , pp. 220 - 250
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Barndorff-Nielsen, O. E. (1997). Processes of normal inverse Gaussian type. Finance Stoch. 2, 41–68.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-Based models and some of their uses in financial economics. J. R. Statist. Soc. B [Statist. Methodology] 63, 167–241.CrossRefGoogle Scholar
Bates, D. S. (1996). Jumps and stochastic volatility: exchange rate processes implicit in Deutsche mark options. Rev. Financial Studies 9, 69–107.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd edn. Springer, Berlin, Heidelberg.CrossRefGoogle Scholar
Duffie, D., Filipovic, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Prob. 13, 984–1053.Google Scholar
Dupuis, P. and Wang, H. (2004). Importance sampling, large deviations, and differential games. Stochastics 76, 481–508.Google Scholar
Dupuis, P. and Wang, H. (2007). Subsolutions of an Isaacs equation and efficient schemes for importance sampling. Math. Operat. Res. 32, 723–757.CrossRefGoogle Scholar
Ekeland, I. and Temam, R. (1999) Convex Analysis and Variational Problems. Society for Industrial and Applied Mathematics, Philadelphia, PA.CrossRefGoogle Scholar
Genin, A. and Tankov, P. (2020). Optimal importance sampling for Lévy processes. Stoch. Process. Appl. 130, 20–46.CrossRefGoogle Scholar
Glasserman, P., Heidelberger, P. and Shahabuddin, P. (1999). Asymptotically optimal importance sampling and stratification for pricing path-dependent options. Math. Finance 9, 117–152.CrossRefGoogle Scholar
Guasoni, P. and Robertson, S. (2008). Optimal importance sampling with explicit formulas in continuous time. Finance Stoch. 12, 1–19.CrossRefGoogle Scholar
Heston, S. (1993). A closed-form solutions for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327–343.CrossRefGoogle Scholar
Jacquier, A., Keller-Ressel, M. and Mijatović, A. (2013). Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models. Stochastics 85, 321–345.CrossRefGoogle Scholar
Keller-Ressel, M. (2011). Moment explosions and long-term behavior of affine stochastic volatility models. Math. Finance 21, 73–98.CrossRefGoogle Scholar
Léonard, C. (2000). Large deviations for Poisson random measures and processes with independent increments. Stoch. Process. Appl. 85, 93–121.CrossRefGoogle Scholar
Robertson, S. (2010). Sample path large deviations and optimal importance sampling for stochastic volatility models. Stoch. Process. Appl. 120, 66–83.CrossRefGoogle Scholar
Rockafellar, R. T. (1971). Integrals which are convex functionals. II. Pacific J. Math. 39, 439–469.CrossRefGoogle Scholar