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Small-Time Asymptotics of Option Prices and First Absolute Moments

By Johannes Muhle-Karbe and Marcel Nutz

Abstract

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 $\sqrt{T}$ 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 L\'evy processes.

OAI identifier: oai:RePEc:arx:papers:1006.2294
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