The order barrier for the $L^1$-approximation of the log-Heston SDE at a single point

TL;DR

This paper investigates the limitations of $L^1$-approximation methods for the log-Heston SDE at a single point, revealing an order barrier linked to the Feller index and establishing the optimality of Euler schemes in certain regimes.

Contribution

It establishes a fundamental order barrier for $L^1$-approximation of the log-Heston SDE and shows Euler schemes are optimal when the Feller index is at least 1.

Findings

Methods cannot surpass order $ u$ or 1/2, whichever is smaller.

Euler schemes achieve near 1/2 order for $ u \,\geq\, 1$.

Optimality of Euler schemes in the regime $ u \,\geq\, 1$.

Abstract

We study the $L^1$-approximation of the log-Heston SDE at the terminal time point by arbitrary methods that use an equidistant discretization of the driving Brownian motion. We show that such methods can achieve at most order $ \min \{ \nu, \tfrac{1}{2} \}$, where $\nu$ is the Feller index of the underlying CIR process. As a consequence Euler-type schemes are optimal for $\nu \geq 1$, since they have convergence order $\tfrac{1}{2}-\epsilon$ for $\epsilon >0$ arbitrarily small in this regime.