# Strong convergence rates for Euler approximations to a class of stochastic path-dependent volatility models

## TL;DR

This paper proves strong convergence rates of order 1/2 (up to a log factor) for Euler schemes approximating a class of path-dependent stochastic volatility models, including Heston-type models, under mild conditions.

## Contribution

It establishes the first positive convergence rate for Euler schemes applied to these complex path-dependent stochastic volatility models.

## Key findings

- Convergence order of 1/2 (up to a log factor) is achieved.
- Euler schemes are proven to be effective for path-dependent models.
- The results include Heston-type and other stochastic-local volatility models.

## Abstract

We consider a class of stochastic path-dependent volatility models where the stochastic volatility, whose square follows the Cox-Ingersoll-Ross model, is multiplied by a (leverage) function of the spot price, its running maximum, and time. We propose a Monte Carlo simulation scheme which combines a log-Euler scheme for the spot process with the full truncation Euler scheme or the backward Euler-Maruyama scheme for the squared stochastic volatility component. Under some mild regularity assumptions and a condition on the Feller ratio, we establish the strong convergence with order 1/2 (up to a logarithmic factor) of the approximation process up to a critical time. The model studied in this paper contains as special cases Heston-type stochastic-local volatility models, the state-of-the-art in derivative pricing, and a relatively new class of path-dependent volatility models. The present paper is the first to prove the convergence of the popular Euler schemes with a positive rate, which is moreover consistent with that for Lipschitz coefficients and hence optimal.