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Small-Time Asymptotics of Option Prices and First Absolute Moments

Part of: Mathematical economics Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Johannes Muhle-Karbe  and
Marcel Nutz
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: johannes.muhle-karbe@math.ethz.ch
∗∗ Current address: Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA. Email address: mnutz@math.columbia.edu
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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.

Keywords

Option priceabsolute momentsmall-time asymptoticsapproximation by Lévy processes

MSC classification

Secondary: 91B25: Asset pricing models 60G44: Martingales with continuous parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 4 , December 2011 , pp. 1003 - 1020
Copyright
Copyright © Applied Probability Trust 2011 

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