Small-Time Asymptotics of Option Prices and First Absolute Moments

TL;DR

This paper analyzes the small-time behavior of at-the-money call option prices for general martingales, revealing how the leading term depends on the process's continuous or jump components and their variation.

Contribution

It provides a comprehensive characterization of the leading small-time asymptotics of option prices for processes with continuous parts and jumps, including stable-like jumps.

Findings

Leading term is of order √T if S has a continuous part.

Leading term is linear in T if S has finite variation.

Exact form of small jumps for stable-like processes derived.

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.