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Pathwise large deviations for the rough Bergomi model

Part of: Limit theorems Mathematical finance Stochastic processes

Published online by Cambridge University Press:  16 January 2019

Antoine Jacquier ,
Mikko S. Pakkanen  and
Henry Stone
Antoine Jacquier*
Affiliation:
Imperial College London
Mikko S. Pakkanen*
Affiliation:
Imperial College London CREATES
Henry Stone*
Affiliation:
Imperial College London
*
* Postal address: Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK.
* Postal address: Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK.
* Postal address: Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK.
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Abstract

Introduced recently in mathematical finance by Bayer et al. (2016), the rough Bergomi model has proved particularly efficient to calibrate option markets. We investigate some of its probabilistic properties, in particular proving a pathwise large deviations principle for a small-noise version of the model. The exponential function (continuous but superlinear) as well as the drift appearing in the volatility process fall beyond the scope of existing results, and a dedicated analysis is needed.

Keywords

Rough volatilitylarge deviationssmall-time asymptoticsGaussian measurereproducing kernel Hilbert space

MSC classification

Primary: 60F10: Large deviations
Secondary: 60G22: Fractional processes, including fractional Brownian motion 91G20: Derivative securities 60G15: Gaussian processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 4 , December 2018 , pp. 1078 - 1092
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1]Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.Google Scholar
[2]Aliprantis, C. D. and Border, K. C. (2006). Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd edn. Springer, Berlin.Google Scholar
[3]Alòs, E., León, J. A. and Vives, J. (2007). On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility. Finance Stoch. 11, 571–589.Google Scholar
[4]Bajgrowicz, P., Scaillet, O. and Treccani, A. (2015). Jumps in high-frequency data: spurious detections, dynamics, and news. Manag. Sci. 62, 2198–2217.Google Scholar
[5]Bayer, C., Friz, P. and Gatheral, J. (2016). Pricing under rough volatility. Quant. Finance 16, 887–904.Google Scholar
[6]Bayer, C. et al. (2017). Short-time near-the-money skew in rough fractional volatility models. Preprint. Available at https://arxiv.org/pdf/1703.05132.pdf.Google Scholar
[7]Bennedsen, M., Lunds, A. and Pakkanen, M. S. (2016). Decoupling the short- and long-term behavior of stochastic volatility. Preprint. Available at https://arxiv.org/pdf/1610.00332.pdf.Google Scholar
[8]Bennedsen, M., Lunds, A. and Pakkanen, M. S. (2017). Hybrid scheme for Brownian semistationary processes. Finance Stoch. 21, 931–965.Google Scholar
[9]Biagini, F., Hu, Y.,Øksendal, B. and Zhang, T. (2008). Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, London.Google Scholar
[10]Carmona, R. A. and Tehranchi, M. R. (2006). Interest Rate Models: An Infinite Dimensional Stochastic Analysis Perspective. Springer, Berlin.Google Scholar
[11]Cherny, A. (2008). Brownian moving averages have conditional full support. Ann. Appl. Prob. 18, 1825–1830.Google Scholar
[12]Christensen, K., Oomen, R. C. A. and Podolskij, M. (2014). Fact or friction: jumps at ultra high frequency. J. Financial Econom. 114, 576–599.Google Scholar
[13]Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Springer, Berlin.Google Scholar
[14]Deuschel, J. D., Friz, P. K., Jacquier, A. and Violante, S. (2014). Marginal density expansions for diffusions and stochastic volatility I: theoretical foundations. Commun. Pure Appl. Math. 67, 40–82.Google Scholar
[15]Deuschel, J.-D. and Stroock, D. W. (1989). Large Deviations. Academic Press, Boston.Google Scholar
[16]Fernique, X. (1970). Intégrabilité des vecteurs gaussiens. C. R. Acad. Sci. Paris A-B 270, A1698–A1699.Google Scholar
[17]Figueroa-López, J. E. and Forde, M. (2012). The small-maturity smile for exponential Lévy models. SIAM J. Financial Math. 3, 33–65.Google Scholar
[18]Forde, M. and Jacquier, A. (2009). Small-time asymptotics for implied volatility under the Heston model. Int. J. Theoret. Appl. Finance 12, 861–876.Google Scholar
[19]Forde, M. and Zhang, H. (2017). Asymptotics for rough stochastic volatility models. SIAM J. Financial Math. 8, 114–145.Google Scholar
[20]Friz, P. K. et al. (eds) (2015). Large Deviations and Asymptotic Methods in Finance. Springer, Cham.Google Scholar
[21]Fukasawa, M. (2011). Asymptotic analysis for stochastic volatility: martingale expansion. Finance Stoch. 15, 635–654.Google Scholar
[22]Fukasawa, M. (2017). Short-time at-the-money skew and rough fractional volatility. Quant. Finance 17, 189–198.Google Scholar
[23]Garcia, J. (2008). A large deviation principle for stochastic integrals. J. Theoret. Prob. 21, 476–501.Google Scholar
[24]Gatheral, J., Jaisson, T. and Rosenbaum, M. (2018). Volatility is rough. Quant. Finance 18, 933–949.Google Scholar
[25]Guillin, A. (2003). Averaging principle of SDE with small diffusion: moderate deviations. Ann. Prob. 31, 413–443.Google Scholar
[26]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.Google Scholar
[27]Jacquier, A., Martini, C. and Muguruza, A. (2018). On VIX futures in the rough Bergomi model. Quant. Finance 18, 45–61.Google Scholar
[28]Jacquier, A. and Roome, P. (2015). Asymptotics of forward implied volatility. SIAM J. Financial Math. 6, 307–351.Google Scholar
[29]Jakubowski, A., Mémin, J. and Pagès, G. (1989). Convergence en loi des suites d’intégrales stochastiques sur l’espace D1 de Skorokhod. Prob. Theory Relat. Fields 81, 111–137.Google Scholar
[30]Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.Google Scholar
[31]Kurtz, T. G. and Protter, P. E. (1996). Weak convergence of stochastic integrals and differential equations. II. Infinite-dimensional case. In Probabilistic Models for Nonlinear Partial Differential Equations (Lecture Notes Math. 1627). Springer, Berlin, pp. 197–285.Google Scholar
[32]Lévy, P. (1953). Random functions: general theory with special reference to Laplacian random functions. Univ. California Publ. Statist. 1, 331–390.Google Scholar
[33]Mandelbrot, B. B.and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437.Google Scholar
[34]Morse, M. and Spiliopoulos, K. (2018). Importance sampling for slow-fast diffusions based on moderate deviations. Preprint. Available at https://arxiv.org/pdf/1805.10229.pdf.Google Scholar
[35]Olver, F. W. J. (1997). Asymptotics and Special Functions. A K Peters/CRC Press, Boca Raton, FL.Google Scholar
[36]Robertson, S. (2010). Sample path large deviations and optimal importance sampling for stochastic volatility models. Stoch. Process. Appl. 120, 66–83.Google Scholar
[37]Spiliopoulos, K. (2013). Large deviations and importance sampling for systems of slow-fast motion. Appl. Math. Optimization 67, 123–161.Google Scholar
[38]Titchmarsh, E. C. (1926). The zeros of certain integral functions. Proc. London Math. Soc. 25, 283–302.Google Scholar