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, with mild linear growth conditions on the drift and vol-of-vol. 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. For $\rho < 0$, our approach still works if the drift of the volatility takes a specific functional form and there is no local volatility component, and our results include the SABR model for $\beta=1, \rho \le 0$. We later augment the model with a single jump-to-default with intensity $\lm$, which produces qualitatively different behaviour for the short-maturity smile; in particular, for $\rho=0$, log-moneyness $x \ne 0$, the implied volatility increases by $\lm f(x) t +o(t) $ for some symmetric function $f(x)$ which blows up at $x=0$, and we see that the jump affects the smile convexity but not the skew at leading order as $t\to 0$. Finally, we compare our result with the general asymptotic expansion in Lorig,Pagliarani\&Pascucci\cite{LPP13}, 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 $\rho=0$.