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

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.

Short Maturity Forward Start Asian Options in Local Volatility Models

“…where it was obtained by solving the Euler-Lagrange equation for the variational problem (46). The solution given here appears also in Section 3.2 in [35] and is more convenient for practical applications.…”

“…Let us compare this with the rate function for the short maturity asymptotics of European options in the log-normal SABR model given in equation ( 6.2) of [35]. Expressed in the notations of the current paper, the SDE of the model dS t = σ t S t dW t , dσ t = ωσ t dZ t becomes dS t = √ V t S t dW t , dV t = 2ωV t dZ t , which corresponds to σ(v) ≡ 2ω.…”

“…Expressed in the notations of the current paper, the SDE of the model dS t = σ t S t dW t , dσ t = ωσ t dZ t becomes dS t = √ V t S t dW t , dV t = 2ωV t dZ t , which corresponds to σ(v) ≡ 2ω. The rate function from [35] is…”

Asymptotics for the Laplace transform of the time integral of the geometric Brownian motion

“…The function R(b, 0) appears also in the short maturity expansion of the at-the-money (ATM) implied volatility in the β = 1 SABR model in the combined small vol-of-vol and large volatility limit, see Proposition 23 in [PZ21]. An examination of the singularities of the function R(b, 0) in the b complex plane shows that the series expansion (2.13) has a finite convergence radius.…”

“…Finally, the theoretical analysis in this paper relies on the large deviations method [DZ98], which has been used in similar contexts in the previous literature [PZ16,PZ17,PZ21]. The rate function for the large deviations in our context can be expressed as a variational problem which does not have a simple closed form in general.…”