Small-Time Asymptotics of Option Prices and First Absolute Moments

By Johannes Muhle‐Karbe, Marcel Nutz. Journal of Applied Probability, 2010

52 citations recorded by the source corpus.

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.

Subfields: Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/d223bb65ca1485ee92a46181323633de92d3688b

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Citations in the corpus

A NEW LOOK AT SHORT‐TERM IMPLIED VOLATILITY IN ASSET PRICE MODELS WITH JUMPS (2012, background): “It follows from the results in [17] that the ATM option price satisfies E[(et − 1)] ∼ t1/αE[(Z∗)+], where Z∗ is a stable random variable with the Lévy density c |x|1+α .”
Rough volatility: Evidence from option prices (2017, background): “This idea can be justified by the fact that in most models, the at-the-money implied volatility tends to the spot volatility as maturity goes to zero, see for example [20].”
Asymptotics for Exponential Levy Processes and Their Volatility Smile: Survey and New Results (2012, unknown): “Other relevant papers about short-time option prices in models with jumps include [2], [91], and [96].”