The Small-Maturity Heston Forward Smile
In this paper we investigate the asymptotics of forward-start options and the forward implied volatility smile in the Heston model as the maturity approaches zero. We prove that the forward smile for out-of-the-money options explodes and compute a closed-form high-order expansion detailing the rate of the explosion. Furthermore the result shows that the square-root behaviour of the variance process induces a singularity such that for certain parameter configurations one cannot obtain high-order out-of-the-money forward smile asymptotics. In the at-the-money case a separate model-independent analysis shows that the small-maturity limit is well defined for any Ito diffusion. The proofs rely on the theory of sharp large deviations (and refinements) and incidentally we provide an example of degenerate large deviations behaviour.
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- Martin Forde & Antoine Jacquier, 2009. "Small-Time Asymptotics For Implied Volatility Under The Heston Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(06), pages 861-876.
- 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.
- Martin Forde & Antoine Jacquier & Aleksandar Mijatovic, 2009. "Asymptotic formulae for implied volatility in the Heston model," Papers 0911.2992, arXiv.org, revised May 2010.
- Aleksandar Mijatovi\'c & Peter Tankov, 2012. "A new look at short-term implied volatility in asset price models with jumps," Papers 1207.0843, arXiv.org, revised Jul 2012.
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