Asymptotics for rough stochastic volatility models

Abstract

Using the large deviation principle (LDP) for a re-scaled fractional Brownian motion $B_t^H$ 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 $d{S}_t=S_t\sigma (Y_t)(\bar{\rho }d{W}_t+\rho d{B}_t),d{Y}_t=d{B}_t^H$ where $\sigma$ is $\alpha$-Hölder continuous for some $\alpha \in (0,1]$; in particular, we show that $t^{H-\frac{1}{2}}\log S_t$ satisfies the LDP as $t\to 0$ 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: $d{S}_t=S_t^{\beta }|Y_t{|}^{p}d{W}_t,d{Y}_t=d{B}_t^H$, and we generalize two identities in Matsumoto&Yor05 to show that $\frac{1}{t^{2H}}\log \frac{1}{t}{\int }_0^t{e}^{2B_s^H}ds$ and $\frac{1}{t^{2H}}(\log {\int }_0^t{e}^{2(\mu s+B_s^H)}ds-2\mu t)$ converge in law to $2{\max }_{0\le s\le 1}{B}_s^H$ and $2B_1$ respectively for $H\in (0,\frac{1}{2})$ and $\mu >0$ as $t\to \infty$.

The argument for the case of unbounded volatility was incorrect because Prob(Lambda_H(eps^H B^H)>c) = 1 i.e. the probability that the rate function of the realized re-scaled fBM path is infinite is 1

References in corpus (1)

• Large deviations for rough paths of the fractional Brownian motion

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