Small time central limit theorems for semimartingales with applications

TL;DR

This paper establishes small-time Gaussian limit laws for semimartingales, including solutions to SDEs, and applies these results to financial option pricing and implied volatility analysis.

Contribution

It provides new conditions for small-time convergence of semimartingales to Gaussian laws, extending to functional limit theorems and practical financial applications.

Findings

Normalized distributions converge to Gaussian as time approaches zero

Results apply to solutions of SDEs with bounded, continuous coefficients

Applications include short-maturity digital options and implied volatility skews

Abstract

We give conditions under which the normalized marginal distribution of a semimartingale converges to a Gaussian limit law as time tends to zero. In particular, our result is applicable to solutions of stochastic differential equations with locally bounded and continuous coefficients. The limit theorems are subsequently extended to functional central limit theorems on the process level. We present two applications of the results in the field of mathematical finance: to the pricing of at-the-money digital options with short maturities and short time implied volatility skews.