The asymptotic smile of a multiscaling stochastic volatility model

TL;DR

This paper analyzes the asymptotic behavior of implied volatility surfaces in a multiscaling stochastic volatility model, revealing divergence in out-of-the-money implied volatility for small maturities using large deviations techniques.

Contribution

It introduces a model capturing multi-scaling of moments and derives the asymptotic shape of implied volatility in extreme regimes, extending understanding of volatility smiles.

Findings

Implied volatility diverges for out-of-the-money options as maturity approaches zero.

The model captures key stylized facts of financial time series, including multi-scaling.

Asymptotic analysis applies large deviations to characterize the volatility surface.

Abstract

We consider a stochastic volatility model which captures relevant stylized facts of financial series, including the multi-scaling of moments. The volatility evolves according to a generalized Ornstein-Uhlenbeck processes with super-linear mean reversion. Using large deviations techniques, we determine the asymptotic shape of the implied volatility surface in any regime of small maturity $t \to 0$ or extreme log-strike $|\kappa| \to \infty$ (with bounded maturity). Even if the price has continuous paths, out-of-the-money implied volatility diverges for small maturity, producing a very pronounced smile.