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 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 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. For uncorrelated stochastic volatility models, our results also include a SABR-type model with \beta=1 and an affine mean-reverting drift, and the exponential Ornstein-Uhlenbeck model. 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 %DIFDELCMD < \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%DIFDELCMD < \to %%%  0. Finally, we compare our result with the general asymptotic expansion in Lorig, Pagliarani \& Pascucci \mbox{%DIFAUXCMD 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 AS12 for the case \rho=0.
We compute a sharp small-time estimate for implied volatility under a general uncorrelated local-stochastic volatility model . For this we use the Bellaiche 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 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, 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. For uncorrelated stochastic volatility models, our results also include a SABR-type model with \beta=1 and an affine mean-reverting drift, and the exponential Ornstein-Uhlenbeck model. 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 %DIFDELCMD < \ne %%% > 0, the implied volatility increases by \lm f(x) t +o(t) for some function f(x) which blows up %DIFDELCMD < \to %%% as x \searrow 0. Finally, we compare our result with the general asymptotic expansion in Lorig, Pagliarani \& Pascucci \mbox{%DIFAUXCMD 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 AS12 for the case \rho=0.
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\mbox{%DIFAUXCMD LPP15