Journal of Applied Probability

Small-time asymptotics of option prices and first absolute moments

Johannes Muhle-Karbe and Marcel Nutz

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We study the leading term in the small-time asymptotics of at-the-money call option prices when the stock price process S follows a general martingale. This is equivalent to studying the first centered absolute moment of S. We show that if S has a continuous part, the leading term is of order √T in time T and depends only on the initial value of the volatility. Furthermore, the term is linear in T if and only if S is of finite variation. The leading terms for pure-jump processes with infinite variation are between these two cases; we obtain their exact form for stable-like small jumps. To derive these results, we use a natural approximation of S so that calculations are necessary only for the class of Lévy processes.

Article information

J. Appl. Probab., Volume 48, Number 4 (2011), 1003-1020.

First available in Project Euclid: 16 December 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 91B25: Asset pricing models
Secondary: 60G44: Martingales with continuous parameter

Option price absolute moment small-time asymptotics approximation by Lévy processes


Muhle-Karbe, Johannes; Nutz, Marcel. Small-time asymptotics of option prices and first absolute moments. J. Appl. Probab. 48 (2011), no. 4, 1003--1020. doi:10.1239/jap/1324046015.

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