Small-time asymptotics for a general local-stochastic volatility model with a jump-to-default: curvature and the heat kernel expansion

We compute a sharp small-time estimate for implied volatility under a general uncorrelated local-stochastic volatility model. For this we use the Bellaiche \cite{Bel81} heat kernel expansion combined with Laplace's method to integrate over the volatility variable on a compact set, and (after a gauge transformation) we use the Davies \cite{Dav88} upper bound for the heat kernel on a manifold with bounded Ricci curvature to deal with the tail integrals. If the correlation $\rho 0$, the implied volatility increases by $\lm f(x) t +o(t) $ for some function $f(x)$ which blows up as $x \searrow 0$. Finally, we compare our result with the general asymptotic expansion in Lorig, Pagliarani \& Pascucci \cite{LPP15}, and we verify our results numerically for the SABR model using Monte Carlo simulation and the exact closed-form solution given in Antonov \& Spector \cite{AS12} for the case $\rho=0$.