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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.
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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 

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