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Small-Time Asymptotics of Option Prices and First Absolute Moments

  • Johannes Muhle-Karbe (a1) and Marcel Nutz (a2)
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 √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.

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Copyright
Corresponding author
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
References
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  • ISSN: 0021-9002
  • EISSN: 1475-6072
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