We compute a  small-time expansion 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 BellaicheBel81 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 DaviesDav88 upper bound for the heat kernel on a manifold with bounded Ricci curvature to deal with the tail integrals. We also consider the case when the correlation \rho%DIFDELCMD < \le %%%  0 ; in this case 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 \lm\ne \lm \to 0pt%DIFAUXCMD , and we verify our results numerically for the SABR model using Monte Carlo simulation and the exact closed-form solution given in Antonov\&Spector\mbox{%DIFAUXCMD AS12 }\hspace{0pt}%DIFAUXCMD for the case \rho=} 0.
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 BellaicheBel81 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 DaviesDav88 upper bound for the heat kernel on a manifold with bounded Ricci curvature to deal with the tail integrals. %DIFDELCMD < \le %%% 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\mbox{%DIFAUXCMD LPP130pt%DIFAUXCMD , and we verify our results numerically for the SABR model using Monte Carlo simulation and the exact closed-form solution given in Antonov\&Spector\mbox{%DIFAUXCMD AS12 }\hspace{0pt}%DIFAUXCMD for the case \rho=} 0.
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sharp
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expansion
estimate
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We also consider the case when the correlation \rho
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For \rho <
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; in this case
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0. We later augment the model with a single jump-to-default with intensity
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, which produces qualitatively different behaviour for the short-maturity smile; in particular, for \rho=0, log-moneyness x
A
0, the implied volatility increases by
A
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
A
0. Finally, we compare our result with the general asymptotic expansion in Lorig,Pagliarani\&Pascucci\mbox{%DIFAUXCMD LPP13