Asymptotics for short maturity Asian options in jump-diffusion models with local volatility

Abstract

We present a study of the short maturity asymptotics for Asian options in a jump-diffusion model with a local volatility component, where the jumps are modeled as a compound Poisson process. The analysis for out-of-the-money Asian options is extended to models with Lévy jumps, including the exponential Lévy model as a special case. Both fixed and floating strike Asian options are considered. Explicit results are obtained for the first-order asymptotics of the Asian options prices for a few popular models in the literature: the Merton jump-diffusion model, the double-exponential jump model, and the Variance Gamma model. We propose an analytical approximation for Asian option prices which satisfies the constraints from the short-maturity asymptotics, and test it against Monte Carlo simulations. The asymptotic results are in good agreement with numerical simulations for sufficiently small maturity.

Keywords:

• Asian options
• Short maturity
• Lévy jumps
• Local volatility

2010 Mathematics Subject Classifications:

• 91G20
• 91G80
• 60J75

Acknowledgments

The authors are grateful to the Associate Editor and two anonymous referees whose comments and suggestions have greatly helped improve the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The Lévy density $\nu \left(\mathrm{d}x\right)=\nu \left(x\right)\mathrm{d}x$ is said to be completely monotone if and only if, for all $k\ge 0$, one has $(-1{)}^k\mathrm{d}\nu (x)/\mathrm{d}{x}^k>0$ for x>0, and the same condition holds for x<0 with $\nu (-x)$.

Additional information

Funding

Lingjiong Zhu is partially supported by the National Science Foundation (NSF) [grant number DMS-2053454,NSF DMS-2208303].