Small-Time Asymptotics for Implied Volatility Under a General Local-Stochastic Volatility Model

We build on of the work of Henry-Labordµere and Lewis on the small-time behaviour of the return distribution under a general local-stochastic volatility model with zero correlation. We do this using the Freidlin-Wentzell theory of large deviations for stochastic differential equations, and then converting to a differential geometry problem of computing the shortest geodesic from a point to a vertical line on a Riemmanian manifold, whose metric is induced by the inverse of the diffusion coefficient. The solution to this variable endpoint problem is obtained using a transversality condition, where the geodesic is perpendicular to the vertical line under the aforementioned metric. We then establish the corresponding small-time asymptotic behaviour for call options using Holder's inequality, and the implied volatility. This avoids the use of possibly non-differentiable viscosity solutions, and we show that the distance function is actually a classical solution to the non-linear eikonal equation, which is analysed at length in Busca et al. We also derive a series expansion for the implied volatility in the small-maturity limit, in powers of the log-moneyness, and we show how to calibrate such a model to the observed implied volatility smile in the small-maturity limit. Finally, we introduce the hyperbolic stochastic volatility model, which satisfies the boundedness and Lipschitz conditions required to be able to apply the Freidlin-Wentzell theory, and we give numerical examples