Asymptotics for rough stochastic volatility models
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 $dS_t=S_tσ(Y_t) (\barρ dW_t +ρdB_t), \,dY_t=dB^H_t$ where $σ$ is $α$-Hölder continuous for some $α\in(0,1]$; 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: $dS_t= S_t^β |Y_t|^p dW_t,dY_t=dB^H_t$, 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 $\frac{1}{t^{2H}}(\log \int_0^t e^{2(μs+B^H_s)} ds-2 μt)$ 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 $μ>0$ as $t \to \infty$.