Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T17:40:48.642Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic behaviour of randomised fractional volatility models

Part of: Stochastic processes Approximations and expansions Limit theorems

Published online by Cambridge University Press:  30 July 2019

Blanka Horvath ,
Antoine Jacquier  and
Chloé Lacombe
Blanka Horvath*
Affiliation:
King’s College London
Antoine Jacquier*
Affiliation:
Imperial College London and the Alan Turing Institute
Chloé Lacombe*
Affiliation:
King’s College London Imperial College London and the Alan Turing Institute
*
*Postal address: Department of Mathematics, King’s College London, London WC2R 2LS, UK.
***Postal address: Department of Mathematics, Imperial College London, London SW7 2AZ, UK.
***Postal address: Department of Mathematics, Imperial College London, London SW7 2AZ, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the asymptotic behaviour of a class of small-noise diffusions driven by fractional Brownian motion, with random starting points. Different scalings allow for different asymptotic properties of the process (small-time and tail behaviours in particular). In order to do so, we extend some results on sample path large deviations for such diffusions. As an application, we show how these results characterise the small-time and tail estimates of the implied volatility for rough volatility models, recently proposed in mathematical finance.

Keywords

Rough volatilitylarge deviationsimplied volatility asymptotics

MSC classification

Primary: 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Secondary: 60F10: Large deviations 60G15: Gaussian processes 60G22: Fractional processes, including fractional Brownian motion
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 2 , June 2019 , pp. 496 - 523
Copyright
© Applied Probability Trust 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. and Stegun, I.A. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York.Google Scholar
Aliprantis, C.D. and Border, K. C. (2005). Infinite Dimensional Analysis, a Hitchhiker’s Guide, 3rd edn. Springer, New York.Google Scholar
Alô’s, E., Leòn, J. 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.CrossRefGoogle Scholar
Azencott, R. (1980). Grandes déviations et applications. Lecture Notes Math. 774, 1–176.CrossRefGoogle Scholar
Baudoin, F. and Ouyang, C. (2011). Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions. Stoch. Process. Appl. 121, 759–792.CrossRefGoogle Scholar
Baudoin, F. and Ouyang, C. (2015). On small-time asymptotics for rough differential equations driven by fractional Brownian motions. In Large Deviations and Asymptotic Methods in Finance, Springer Proc. Math. Stats., Vol. 110.CrossRefGoogle Scholar
Bayer, C., Friz, P.K. and Gatheral, J. (2016). Pricing under rough volatility. Quant. Finance 16, 887– 904.CrossRefGoogle Scholar
Bayer, C. et al. Short-time near-the-money skew in rough fractional volatility models. To appear in Quant. Finance.Google Scholar
Ben Arous, G. (1988). Methods de Laplace et de la phase stationnaire sur l’espace de Wiener. Stochastics 25, 125–153.CrossRefGoogle Scholar
Bezuidenhout, C. (1987). A large deviations principle for small perturbations of random evolution equations. Ann. Prob. 15, 646–658.CrossRefGoogle Scholar
Bismut, J. M. (1984). Large Deviations and the Malliavin Calculus. Birkhauser, Basel.Google Scholar
Caravenna, F. and Corbetta, J. (2018). The asymptotic smile of a multiscaling stochastic volatility model. Stoch. Process. Appl. 128, 1034–1071.CrossRefGoogle Scholar
Carmona, R. and Tehranchi, M. R. (2006). Interest Rate Models: An Infinite-Dimensional Stochastic Analysis Perspective. Springer, New York.Google Scholar
Comte, F. and Renault, E. (1998). Long memory in continuous-time stochastic volatility models. Math. Finance 8, 291–323.CrossRefGoogle Scholar
Comte, F., Coutin, L. and Renault, E. (2012). Affine fractional stochastic volatility models. Ann. Finance 8, 337–378.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, Vol. 2. Springer, New York.CrossRefGoogle Scholar
Deuschel, J. D., Friz, P. K., Jacquier, A. and Violante, S. (2014). Marginal density expansions for diffusions and stochastic volatility, part I: Theoretical foundations. Commun. Pure Appl. Math. 67, 40–82.CrossRefGoogle Scholar
Deuschel, J. D., Friz, P. K., Jacquier, A. and Violante, S. (2014). Marginal density expansions for diffusions and stochastic volatility, part II: Applications. Commun. Pure Appl. Math. 67, 321–350.CrossRefGoogle Scholar
Deuschel, J.D. and Stroock, D. (2001). Large Deviations. Vol. 342, American Mathematical Society, Providence, RI.Google Scholar
El Euch, O., Fukasawa, M. and Rosenbaum, M. (2018). The microstructural foundations of leverage effect and rough volatility. Finance Stoch. 22, 241–280.CrossRefGoogle Scholar
Forde, M. and Jacquier, A. (2009). Small-time asymptotics for implied volatility under the Heston model. IJTAF 12, 861–876.Google Scholar
Forde, M., Jacquier, A. and Lee, R. (2012). The small-time smile and term structure of implied volatility under the Heston model. SIAM J. Financial Math. 3, 690–708.CrossRefGoogle Scholar
Forde, M. and Zhang, H. (2017). Asymptotics for rough stochastic volatility models. SIAM J. Financial Math. 8, 114–145.CrossRefGoogle Scholar
Friedman, A. (1975). Stochastic Differential Equations and Applications, Vol. 1. Academic Press, New York.Google Scholar
Freidlin, M. and Wentzell, A. (1984). Random Perturbations of Dynamical Systems. Vol. 260, Springer, New York.Google Scholar
Fukasawa, M. (2017). Short-time at-the-money skew and rough fractional volatility. Quant. Finance 17, 189–198.CrossRefGoogle Scholar
Gao, K. and Lee, R. (2014). Asymptotics of implied volatility to arbitrary order. Finance Stoch. 18, 349–392.CrossRefGoogle Scholar
Garcia, J. (2008). A large deviation principle for stochastic integrals. J. Theoret. Prob. 21, 476–501.CrossRefGoogle Scholar
Gatheral, J. The Volatility Surface: A Practitioner’s Guide. John Wiley, New York.CrossRefGoogle Scholar
Gatheral, J., Jaisson, T. and Rosenbaum, M. (2018). Volatility is rough. Quant. Finance 18, 933–949.CrossRefGoogle Scholar
Genin, A. and Tankov, P. (2019). Optimal importance sampling for Lévy processes. To appear in Stoch. Process. Appl.Google Scholar
Guasoni, P. and Robertson, S. (2008). Optimal importance sampling with explicit formulas in continuous time. Finance Stoch. 12, 1–19.CrossRefGoogle Scholar
Gulisashvili, A., Viens, F. and Zhang, X. (2018). Small-time asymptotics for Gaussian self-similar stochastic volatility models. Appl. Math. Optimization, 1–41.Google 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
Jacquier, A., Martini, C. and Muguruza, A. (2018). On VIX futures in the rough Bergomi model. Quant. Finance 18, 45–61.CrossRefGoogle Scholar
Jacquier, A., Pakkanen, M. and Stone, H. (2018). Pathwise large deviations for the rough Bergomi model. J. Appl. Prob. 55, 1078–1092.CrossRefGoogle Scholar
Jacquier, A. and Roome, P. (2013). Asymptotics of forward implied volatility. SIAM J. Financial Math. 6, 307–351.CrossRefGoogle Scholar
Jacquier, A. and Roome, P. (2018). Black–Scholes in a CEV random environment: A new approach to smile modelling. Math. Financial Econom. 12, 445–474.CrossRefGoogle Scholar
Jacquier, A. and Shi, F. (2019). The randomised Heston model. SIAM J. Financial Math. 10, 89–129.CrossRefGoogle Scholar
Mayer-Wolf, E., Nualart, D. and PéRez-Abreu, V. (1992). Large deviations for multiple Wiener–Itô integral processes. Séminaire de Probabilités 26, 11–31.CrossRefGoogle Scholar
Mechkov, S. (2016). ‘Hot-start’ initialization of the Heston model. Risk, November.Google Scholar
Mellouk, M. (2000). A large-deviation principle for random evolution equations. Bernoulli 6, 977–999.CrossRefGoogle Scholar
Millet, A., Nualart, D. and Sanz, M. (1992). Large deviations for a class of anticipating stochastic differential equations. Ann. Prob. 20, 1902–1931.CrossRefGoogle Scholar
Mishura, Y. S. (2008). Stochastic Calculus for Fractional Brownian Motion and Related Processes. Springer, New York.CrossRefGoogle Scholar
Norros, I., Valkeika, E. and Virtamo, J. (1999). An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5, 571–587.CrossRefGoogle Scholar
Pham, H. (2010). Large deviations in finance. In Proc. 3rd SMAI European Summer School in Financial Mathematics.Google Scholar
Peithmann, D. (2007). Large Deviations and Exit Time Asymptotics for Diffusions and Stochastic Resonance. PhD thesis, Humboldt University, Berlin.Google Scholar
Robertson, S. (2010). Sample path large deviations and optimal importance sampling for stochastic volatility models. Stoch. Process. Appl. 120, 66–83.CrossRefGoogle Scholar
Schilder, M. (1966). Asymptotic formulas for Wiener integrals. Trans. Amer. Math. Soc. 125, 63–85.CrossRefGoogle Scholar
Schöbel, R. and Zhu, J. (1999). Stochastic volatility with an Ornstein–Uhlenbeck process: An extension. Rev. Finance 3, 23–46.CrossRefGoogle Scholar
Stein, E. and Stein, J. (1991). Stock-price distributions with stochastic volatility – an analytic approach. Rev. Financial Studies 4, 727–752.CrossRefGoogle Scholar
Varadhan, S. R. S. (1967). On the behaviour of the fundamental solution of the heat equation with variable coefficients. Commun. Pure Appl. Math. 20 431–455.CrossRefGoogle Scholar
Yan, L., Lu, Y. and Xu, Z. (2008). Some properties of the fractional Ornstein–Uhlenbeck process. J. Phys. A: Math. and Theor. 41, 1–17.CrossRefGoogle Scholar