TY - JOUR
T1 - Asymptotics of the Time-Discretized Log-Normal Sabr Model
T2 - The Implied Volatility Surface
AU - Pirjol, Dan
AU - Zhu, Lingjiong
N1 - Publisher Copyright:
© 2020 The Author(s). Published by Cambridge University Press.
PY - 2021/10/1
Y1 - 2021/10/1
N2 - 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.
AB - 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.
KW - applied probability
KW - mathematical finance
KW - stochastic modelling
UR - http://www.scopus.com/inward/record.url?scp=85085393885&partnerID=8YFLogxK
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U2 - 10.1017/S0269964820000248
DO - 10.1017/S0269964820000248
M3 - Article
AN - SCOPUS:85085393885
SN - 0269-9648
VL - 35
SP - 942
EP - 974
JO - Probability in the Engineering and Informational Sciences
JF - Probability in the Engineering and Informational Sciences
IS - 4
ER -