Asymptotics for rough stochastic volatility models

TL;DR

This paper derives small- and large-time asymptotics for fractional stochastic volatility models using large deviation principles, revealing how the implied volatility smile behaves depending on the Hurst parameter.

Contribution

It provides new asymptotic results for rough volatility models, including the behavior of implied volatility smiles and extensions of known identities for fractional Brownian motion.

Findings

Implied volatility smile steepens or flattens depending on Hurst parameter H.

Small-time asymptotics show the log-price scaled by t^{H-1/2} satisfies an LDP.

Large-time asymptotics extend identities for fractional Brownian motion to new convergence results.

Abstract

Using the large deviation principle (LDP) for a re-scaled fractional Brownian motion $B^H_t$ where the rate function is defined via the reproducing kernel Hilbert space, we compute small-time asymptotics for a correlated fractional stochastic volatility model of the form $dS_t=S_t\sigma(Y_t) (\bar{\rho} dW_t +\rho dB_t), \,dY_t=dB^H_t$ where $\sigma$ is $\alpha$-H\"{o}lder continuous for some $\alpha\in(0,1]$; in particular, we show that $t^{H-\frac{1}{2}} \log S_t $ satisfies the LDP as $t\to0$ and the model has a well-defined implied volatility smile as $t \to 0$, when the log-moneyness $k(t)=x t^{\frac{1}{2}-H}$. Thus the smile steepens to infinity or flattens to zero depending on whether $H\in(0,\frac{1}{2})$ or $H\in(\frac{1}{2},1)$. We also compute large-time asymptotics for a fractional local-stochastic volatility model of the form: $dS_t= S_t^{\beta} |Y_t|^p dW_t,dY_t=dB^H_t$, and we generalize two identities in Matsumoto&Yor05 to show that $\frac{1}{t^{2H}}\log \frac{1}{t}\int_0^t e^{2 B^H_s} ds$ and $\frac{1}{t^{2H}}(\log \int_0^t e^{2(\mu s+B^H_s)} ds-2 \mu t)$ converge in law to $ 2\mathrm{max}_{0 \le s \le 1} B^H_{s}$ and $2B_1$ respectively for $H \in (0,\frac{1}{2})$ and $\mu>0$ as $t \to \infty$.