Asymptotic smiles for an affine jump-diffusion model

Abstract

In this paper, we study the asymptotic behaviors of implied volatility in an affine jump-diffusion model. By assuming that log stock prices under the risk-neutral measure follow an affine jump-diffusion model, we show that an explicit form of the moment-generating function for log stock price can be obtained by solving a set of ordinary differential equations. A large-time large deviation principle for log stock prices is derived by applying the Gärtner-Ellis theorem. We characterize the asymptotic behaviors of implied volatility in the large-maturity and large-strike regimes using the rate function in the large deviation principle. The asymptotics of the implied volatility for fixed-maturity, large-strike and small-strike regimes are also studied. Numerical results are provided to validate the theoretical work.

Keywords

Affine jump-diffusion model / Large deviations / Implied volatility / Asymptotics

Acknowledgements

Nian Yao was supported in part by the Natural Science Foundation of China(Grant No. 12071361), the Natural Science Foundation of Guangdong Province(Grant No. 2020A1515010822) and Shenzhen Natural Science Fund (the Stable Support Plan Program 20220810152104001).

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