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ASYMPTOTICS OF THE TIME-DISCRETIZED LOG-NORMAL SABR MODEL: THE IMPLIED VOLATILITY SURFACE

Published online by Cambridge University Press:  22 May 2020

Dan Pirjol Open the ORCID record for Dan Pirjol [Opens in a new window]  and
Lingjiong Zhu
Dan Pirjol
Affiliation:
School of Business, Stevens Institute of Technology, Hoboken, NJ07030, USA E-mail: dpirjol@gmail.com
Lingjiong Zhu
Affiliation:
Department of Mathematics, Florida State University, 1017 Academic Way, Tallahassee, FL32306, USA
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Abstract

We propose a novel time discretization for the log-normal SABR model which is a popular stochastic volatility model that is widely used in financial practice. Our time discretization is a variant of the Euler–Maruyama scheme. We study its asymptotic properties in the limit of a large number of time steps under a certain asymptotic regime which includes the case of finite maturity, small vol-of-vol and large initial volatility with fixed product of vol-of-vol and initial volatility. We derive an almost sure limit and a large deviations result for the log-asset price in the limit of a large number of time steps. We derive an exact representation of the implied volatility surface for arbitrary maturity and strike in this regime. Using this representation, we obtain analytical expansions of the implied volatility for small maturity and extreme strikes, which reproduce at leading order known asymptotic results for the continuous time model.

Keywords

applied probabilitymathematical financestochastic modelling
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Issue 4 , October 2021 , pp. 942 - 974
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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