Small-time asymptotics for an uncorrelated local-stochastic volatility model - Jacquier,A; Forde,M (Applied Mathematical Finance, 2011)
We refine the work of Henry-Labordere, Lewis and Paulot on the small-time behaviour of a local-stochastic volatility model with zero correlation at leading order. We do this using the Freidlin-Wentzell theory of large deviations for SDEs, and then converting to a di®erential 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 diffsion 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. 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.