Small-Time Asymptotics for an Uncorrelated Local-Stochastic Volatility Model

We add some rigour to the work of Henry-Labordère (2009; Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing (London and New York: Chapman & Hall)), Lewis (2007; Geometries and Smile Asymptotics for a Class of Stochastic Volatility Models. Available at <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://www.optioncity.net">http://www.optioncity.net</ext-link > (accessed 28 May 2011)) and Paulot (2009; Asymptotic implied volatility at the second order with application to the SABR model, Working Paper, Available at papers.ssrn.com/sol3/papers.cfm?abstract_id=1413649 (accessed 11 June 2011)) on the small-time behaviour of a local-stochastic volatility model with zero correlation at leading order. We do this using the Freidlin—Wentzell (FW) theory of large deviations for stochastic differential equations (SDEs), 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 <italic>transversality</italic> 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 Hölder's inequality, and the implied volatility (using a general result in Roper and Rutkowski (forthcoming, A note on the behavior of the Black--Scholes implied volatility close to expiry, International Journal of Thoretical and Applied Finance). 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.