Euler Scheme and Tempered Distributuions

TL;DR

This paper investigates the convergence rate of the Euler scheme for smooth diffusions, showing it converges at rate 1/n for a broad class of test functions including tempered distributions, with applications to option pricing.

Contribution

It establishes the convergence speed of the Euler scheme for tempered distributions and functions with exponential growth, extending previous results.

Findings

Convergence rate of 1/n for tempered distributions.

Applicability to option pricing and hedging.

Numerical rates for prices, deltas, and gammas.

Abstract

Given a smooth R^d-valued diffusion, we study how fast the Euler scheme with time step 1/n converges in law. To be precise, we look for which class of test functions f the approximate expectation E[f(X^{n,x}_1)] converges with speed 1/n to E[f(X^x_1)]. If X is uniformly elliptic, we show that this class contains all tempered distributions, and all measurable functions with exponential growth. We give applications to option pricing and hedging, proving numerical convergence rates for prices, deltas and gammas.