Asymptotics for Rough Stochastic Volatility Models

Using the large deviation principle (LDP) for a rescaled fractional Brownian motion BHₜ, 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 dSₜ=Sₜσ(Yₜ) (ρ̄ dWₜ +ρ dBₜ), \,dYₜ=dBHₜ, where σ is α-Hölder continuous for some α∈(0,1]; in particular, we show that tH-1/2 log Sₜ 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)=x t1/2-H. Thus the smile steepens to infinity or flattens to zero depending on whether H∈(0,1/2) or H∈(1/2,1). We also compute large-time asymptotics for a fractional local-stochastic volatility model of the form dSₜ= Sₜβ |Yₜ|ᵖ dWₜ,dYₜ=dBHₜ, and we generalize two identities in Matsumoto and Yor [Probab. Surv., 2 (2005), pp. 312--347] to show that 1t2Hlog 1/t∫₀ᵗ e2 BHₛ ds and 1t2H(log ∫₀ᵗ e2(μ s+BHₛ) ds-2 μ t) converge in law to 2 max0 ≤ s ≤ 1 BHₛ and 2B₁, respectively, for H ∈ (0,1/2) and μ>0 as t → ∞.