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

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