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

Journal article

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 converence 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.

Authors: Cozma, A; Reisinger, C

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Numerical Analysis
• Volume: 56
• Issue: 6
• Pages: 3430–3458
• Publication date: 2018-12-11
• Acceptance date: 2018-09-07
• DOI: 10.1137/17M1136754
• EISSN: 1095-7170
• ISSN: 0036-1429
• Keywords: path-dependent volatility; running maximum; Euler scheme; Cox{Ingersoll{Ross process; strong convergence order; Monte Carlo simulation