Large deviations and Stochastic Volatility with jumps: asymptotic implied volatility for affine models
Please always quote using this URN:urn:nbn:de:0296-matheon-11559
- Let $\sigma_� t(x)$ denote the implied volatility at maturity t for a strike $K = S_0 e^{x t}$, where $x \in R$ and $S_0$ is the current value of the underlying. We show that � $\sigma_� t(x)$ has a uniform (in $x$) limit as maturity t tends to infinity, given by the formula � \sigma_{\infty}(x) = \sqrt2 (h^*(x)^{1/2} + (h^*(x) − x)^{1/2}� , for $x$ in some compact neighbourhood of zero in the class of affine stochastic volatility models. The function $h^*$ is the convex dual of the limiting cumulant generating function $h$ of the scaled log-spot process. We express $h$ in terms of the functional characteristics of the underlying model. The proof of the limiting formula rests on the large deviation behaviour of the scaled log-spot process as time tends to infinity. We apply our results to obtain the limiting smile for several classes of stochastic volatility models with jumps used in applications (e.g. Heston with state-independent jumps, Bates with state-dependent jumps and Barndorff-Nielsen-Shephard model).
Author: | Antoine Jacquier, Martin Keller-Ressel, Aleksandar Mijatovic |
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URN: | urn:nbn:de:0296-matheon-11559 |
Referee: | Volker Mehrmann |
Document Type: | Preprint, Research Center Matheon |
Language: | English |
Date of first Publication: | 2015/12/23 |
Release Date: | 2015/12/23 |
Tag: | Affine processes; Implied volatility in the large maturity limit.; Large deviation principle; Stochastic volatility with jumps |
Institute: | Technische Universität Berlin |
MSC-Classification: | 60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Fxx Limit theorems [See also 28Dxx, 60B12] / 60F10 Large deviations |
60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Gxx Stochastic processes / 60G44 Martingales with continuous parameter | |
91-XX GAME THEORY, ECONOMICS, SOCIAL AND BEHAVIORAL SCIENCES / 91Gxx Mathematical finance / 91G20 Derivative securities | |
Preprint Number: | 1093 |