Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models

TL;DR

This paper derives the asymptotic implied volatility smile for affine stochastic volatility models with jumps as maturity tends to infinity, using large deviation principles and convex duality.

Contribution

It provides a uniform limit formula for implied volatility in affine models with jumps, linking it to the convex dual of the cumulant generating function.

Findings

Explicit formula for the limiting implied volatility smile.

Application to models like Heston, Bates, and Barndorff-Nielsen-Shephard.

Connection between large deviations and implied volatility asymptotics.

Abstract

Let $\sigma_t(x)$ denote the implied volatility at maturity $t$ for a strike $K=S_0 e^{xt}$, where $x\in\bbR$ and $S_0$ is the current value of the underlying. We show that $\sigma_t(x)$ has a uniform (in $x$) limit as maturity $t$ tends to infinity, given by the formula $\sigma_\infty(x)=\sqrt{2}(h^*(x)^{1/2}+(h^*(x)-x)^{1/2})$, for $x$ in some compact neighbourhood of zero in the class of affine stochastic volatility models. The function $h^*$ is the convex dual of the limiting cumulant generating function $h$ of the scaled log-spot process. We express $h$ in terms of the functional characteristics of the underlying model. The proof of the limiting formula rests on the large deviation behaviour of the scaled log-spot process as time tends to infinity. We apply our results to obtain the limiting smile for several classes of stochastic volatility models with jumps used in applications (e.g. Heston with state-independent jumps, Bates with state-dependent jumps and Barndorff-Nielsen-Shephard model).