Asymptotics for Rough Stochastic Volatility Models

Using the large deviation principle (LDP) for a rescaled 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)((rho) over bar dW(t) + rho dB(t)), dY(t) = dB(t)(H), where sigma is alpha-Holder continuous for some alpha is an element of (0, 1]; in particular, we show that t(H-1/2) log S-t satisfies the LDP as t -> 0 and the model has a well-defined implied volatility smile as t -> 0, when the log-moneyness k(t) = xt(1/2-H). Thus the smile steepens to infinity or flattens to zero depending on whether H is an element of (0, 1/2) or H is an element of(1/2,1). We also compute large-time asymptotics for a fractional local-stochastic volatility model of the form dS(t) = S-t(beta)vertical bar Y-t vertical bar(p)dW(t), dY(t) = dB(t)(H), and we generalize two identities in Matsumoto and Yor [Probab. Sarv., 2 (2005), pp. 312-347] to show that 1/t(2H) log 1/t integral(t)(0) e(2BsH) ds and 1/t(2H) (log fot integral(t)(0) e(2(mu s+BsH)) ds - mu t) converge in law to 2max(0 <= s <= 1)B(s)(H) and 2B(1), respectively, for H is an element of (0, 1/2) and mu > 0 as t -> infinity.