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Small-Time Asymptotics of Option Prices and First Absolute Moments

Published online by Cambridge University Press:  14 July 2016

Johannes Muhle-Karbe*
Affiliation:
Universität Wien
Marcel Nutz*
Affiliation:
ETH Zürich
*
Current address: Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland. Email address: [email protected]
∗∗ Current address: Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA. Email address: [email protected]
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Abstract

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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.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2011 

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