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Small-time moderate deviations for the randomised Heston model

Part of: Mathematical economics Mathematical finance Limit theorems

Published online by Cambridge University Press:  04 May 2020

Antoine Jacquier  and
Fangwei Shi
Antoine Jacquier*
Affiliation:
Imperial College London and Alan Turing Institute
Fangwei Shi*
Affiliation:
Imperial College London
*
*Postal address: Department of Mathematics and Alan Turing Institute, Imperial College London, London SW7 2AZ, UK.
*Postal address: Department of Mathematics and Alan Turing Institute, Imperial College London, London SW7 2AZ, UK.
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Abstract

We extend previous large deviations results for the randomised Heston model to the case of moderate deviations. The proofs involve the Gärtner–Ellis theorem and sharp large deviations tools.

Keywords

Stochastic volatilitymoderate deviationsHeston modelimplied volatilityasymptotic expansion

MSC classification

Primary: 60F10: Large deviations
Secondary: 91G20: Derivative securities 91B70: Stochastic models
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 19 - 28
Copyright
© Applied Probability Trust 2020

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References

Albrecher, H., Mayer, P., Schoutens, W. and Tistaert, J. (2007). The little Heston trap. Wilmott Magazine Jan, 83–92.Google Scholar
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.CrossRefGoogle Scholar
Babu, J. G. and Deo, C. M. (1981). Probabilities of moderate deviations in a Banach space. Proc. AMS 24, 392–397.Google Scholar
Bayer, C., Friz, P. and Gatheral, J. (2016). Pricing under rough volatility. Quant. Finance 16, 887–904.CrossRefGoogle Scholar
Bayer, C., Friz, P., Gulisashvili, A., Horvath, B. and Stemper, B. (2019). Short-time near-the-money skew in rough fractional volatility models. Quant. Finance 19, 779–798.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation, Cambridge University Press.Google Scholar
Caravenna, F. and Corbetta, J. (2016). General smile asymptotics with bounded maturity. SIAM J. Financial Math. 7, 720–759.CrossRefGoogle Scholar
De Acosta, A. (1992). Moderate deviations and associated Laplace approximations for sums of independent random vectors. Trans. AMS 329, 357–375.CrossRefGoogle Scholar
Dembo, A. (1996). Moderate deviations for martingales with bounded jumps. Electron. Commun. Prob. 1, 11–17.10.1214/ECP.v1-973CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, Springer, Berlin.CrossRefGoogle Scholar
El Euch, O. and Rosenbaum, M. (2019). The characteristic function of rough Heston models. Math. Finance 29, 3–38.10.1111/mafi.12173CrossRefGoogle Scholar
Forde, M. and Jacquier, A. (2009). Small-time asymptotics for implied volatility under the Heston model. Internat. J. Theoret. Appl. Finance 12, 861–876.CrossRefGoogle Scholar
Friz, P., Gerhold, S. and Pinter, A. (2018). Option pricing in the moderate deviations regime. Math. Finance 28, 962–988.CrossRefGoogle ScholarPubMed
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, 342–392.CrossRefGoogle Scholar
Gatheral, J., Jaisson, T. and Rosenbaum, M. (2018). Volatility is rough. Quant. Finance 18, 933–949.CrossRefGoogle Scholar
Guennoun, H., Jacquier, A., Roome, P. and Shi, F. (2018). Asymptotic behaviour of the fractional Heston model. SIAM J. Financial Math. 9, 1017–1045.CrossRefGoogle Scholar
Guillin, A. (2003). Averaging principle of SDE with small diffusion: Moderate deviations. Ann. Prob. 31, 413–443.Google Scholar
Guillin, A. and Liptser, R. (2006). Examples of moderate deviation principle for diffusion processes. Discrete Cont. Dynam. Syst. B 6, 803–828.CrossRefGoogle Scholar
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327–343.CrossRefGoogle Scholar
Jacquier, A. and Shi, F. (2019). The randomised Heston model. SIAM J. Financial Math. 10, 89–129.CrossRefGoogle Scholar
Jacquier, A. and Spiliopoulos, K. Pathwise moderate deviations for option pricing. To appear in Math. Finance.Google Scholar
Jacquier, A., Martini, C. and Muguruza, A. (2018). On VIX futures in the rough Bergomi model. Quant. Finance 18, 45–61.CrossRefGoogle Scholar
Liming, W. (1995). Moderate deviations of dependent random variables related to CLT. Ann. Prob. 23, 420–445.CrossRefGoogle Scholar
Mechkov, S. (2016). ‘Hot-start’ initialization of the Heston model. Risk Nov.Google Scholar
Mijatović, A. and Tankov, P. (2016). A new look at short-term implied volatility in asset price models with jumps. Math. Finance 26, 149–183.CrossRefGoogle Scholar
Rubin, H. and Sethuraman, J. (1965). Probabilities of moderate deviations. Sankhyā A 27, 325–346.Google Scholar