Asymptotics for rough stochastic volatility models

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 $dS_t=S_t\sigma(Y_t) (\bar{\rho} dW_t +\rho dB_t), ,dY_t=dB^H_t$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 $satisfiestheLDPas$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: $d{S}_t=S_t^{\beta }\mathrm{\mid }{Y}_t{\mathrm{\mid }}^{p}d{W}_t,d{Y}_t=d{B}_t^H$, and we generalize two identities in Matsumoto&Yor05 to show that $\frac{1}{t^{2H}}\log \frac{1}{t}\int_0^t e^{2 B^H_s} ds$and$\frac{1}{t^{2H}}(\log \int_0^t e^{2(\mu s+B^H_s)} ds-2 \mu t)$convergeinlawto$ 2$\mathrm{m}\mathrm{a}\mathrm{x}${0 \le s \le 1} B^H{s}$and$2B_1$respectivelyfor$H \in (0,\frac{1}{2})$and$\mu>0$as$t \to \infty$.