2010/06/30 by Johannes Muhle-Karbe, Marcel Nutz · 1 citation
Economics, Econometrics and Finance · Mathematics · #q-fin.PR #math.PR #msc:91B25 #msc:60G44
paper · pdf · doi:10.1239/jap/1324046015
published as J. Appl. Probab. 48 (2011) 1003-1020 · 22 pages; forthcoming in 'Journal of Applied Probability'
arxiv created 2011/06/16 · arxiv updated 2019/07/10
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.