The asymptotic smile of a multiscaling stochastic volatility model

We consider a stochastic volatility model which captures relevant stylized facts of financial series, including the multiscaling of moments. Using large deviations techniques, we determine the asymptotic shape of the implied volatility surface in any regime of small maturity $t \to 0$ or extreme log-strike $|\kappa| \to \infty$ (with bounded maturity). Even if the price has continuous paths, we show that out-of-the-money implied volatility diverges for small maturity, producing a very pronounced smile. When $|\kappa|$ is much larger than $t$, the implied volatility is asymptotically an explicit function of just the ratio $(\kappa / t)$.