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

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Listed:
  • John Armstrong
  • Martin Forde
  • Matthew Lorig
  • Hongzhong Zhang

Abstract

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 $\rho=0$.

Suggested Citation

  • John Armstrong & Martin Forde & Matthew Lorig & Hongzhong Zhang, 2013. "Small-time asymptotics for a general local-stochastic volatility model with a jump-to-default: curvature and the heat kernel expansion," Papers 1312.2281, arXiv.org, revised Sep 2016.
    • Handle: RePEc:arx:papers:1312.2281
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    3. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(1), pages 29-50, January.
    4. J. D. Deuschel & P. K. Friz & A. Jacquier & S. Violante, 2011. "Marginal density expansions for diffusions and stochastic volatility, part I: Theoretical Foundations," Papers 1111.2462, arXiv.org, revised May 2013.
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    Cited by:

    1. Masaaki Fukasawa, 2015. "Short-time at-the-money skew and rough fractional volatility," Papers 1501.06980, arXiv.org.
    2. Martin Forde & Hongzhong Zhang, 2016. "Small-time asymptotics for basket options -- the bi-variate SABR model and the hyperbolic heat kernel on $\mathbb{H}^3$," Papers 1603.02896, arXiv.org, revised Apr 2016.

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