Asymptotics of the Time-Discretized Log-Normal Sabr Model: The Implied Volatility Surface

Dan Pirjol, Lingjiong Zhu

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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.

Original languageEnglish
Pages (from-to)942-974
Number of pages33
JournalProbability in the Engineering and Informational Sciences
Volume35
Issue number4
DOIs
StatePublished - 1 Oct 2021

Keywords

  • applied probability
  • mathematical finance
  • stochastic modelling

Fingerprint

Dive into the research topics of 'Asymptotics of the Time-Discretized Log-Normal Sabr Model: The Implied Volatility Surface'. Together they form a unique fingerprint.

Cite this