Strong Convergence Rates for Euler Approximations to a Class of Stochastic Path-Dependent 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 process, 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 stochasticlocal 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.

Well-posedness and Stability Analysis of Two Classes of Generalized Stochastic Volatility Models

“…Apparently, (1.1) covers all the classical types of stochastic volatility models and path-dependent models, and it also covers the Heston-type stochastic path-dependent volatility model proposed in Cozma and Reisinger (2018) (as well as local maximum stochastic volatility model proposed in Bain et al (2019)) whose well-posedness is unknown, dS t = µ(t, S t , M t )S t dt + V t σ(t, S t , M t )S t dW t ,…”