On arbitrarily slow convergence rates for strong numerical approximations of Cox–Ingersoll–Ross processes and squared Bessel processes

Abstract

Cox-Ingersoll-Ross (CIR) processes are extensively used in state-ofthe-art models for the approximative pricing of financial derivatives. In particular, CIR processes are day after day employed to model instantaneous variances (squared volatilities) of foreign exchange rates and stock prices in Heston-type models and they are also intensively used to model short-rate interest rates. The prices of the financial derivatives in the above mentioned models are very often approximately computed by means of explicit or implicit Euler-or Milstein-type discretization methods based on equidistant evaluations of the driving noise processes. In this article we study the strong convergence speeds of all such discretization methods. More specifically, the main result of this article reveals that each such discretization method achieves at most a strong convergence order of δ/2, where 0 < δ < 2 is the dimension of the squared Bessel process associated to the considered CIR process. In particular, we thereby reveal that discretization methods currently employed in the financial industry may converge with arbitrarily slow strong convergence rates to the solution of the considered CIR process. We thereby lay open the need of the development of other more sophisticated approximation methods which are capable to solve CIR processes in the strong sense in a reasonable computational time and which thus can not belong to the class of algorithms which use equidistant evaluations of the driving noise processes.

On the performance of the Euler–Maruyama scheme for SDEs with discontinuous drift coefficient

“…Note that the recent literature on numerical approximation of SDEs contains a number of examples of SDEs with coefficients that are not Lipschitz continuous and such that the Euler-Maruyama scheme at the final time does not achieve an L p -error rate of 1/2, see [2,6,9,11,12,22,29]. Furthermore, in [3] numerical studies are carried out for a number of SDEs (1) with a discontinuous µ satisfying (A1) and σ = 1, and for several of these SDEs an empirical L 2 -error rate significantly smaller than 1/2 is observed for the Euler-Maruyama scheme at the final time.…”

Lower error bounds for strong approximation of scalar SDEs with non-Lipschitzian coefficients

“…If δ < 1 then combining the upper bound of Hefter and Herzwurm [18, Theorem 2] with the lower bound of Hefter and Jentzen [19,Theorem 1] shows that there exist constants c, C ∈ (0, ∞) such that for all n ∈ N it holds that…”

“…One of them consists in establishing sub-polynomial lower error bounds for particular equations with smooth coefficients in order to come closer to a characterization of polynomial convergence in that case, see Gerencsér, Jentzen, and Salimova [13], Hairer, Hutzenthaler, and Jentzen [16], Jentzen, Müller-Gronbach, and Yaroslavtseva [28], Müller-Gronbach and Yaroslavtseva [39], Yaroslavtseva [50]. The other one aims at a thorough analysis of strong approximation of Cox-Ingersoll-Ross processes as a prototype of SDEs with a diffusion coefficient that is Hölder continuous in the state variable with a Hölder exponent strictly between zero and one, see Hefter and Herzwurm [17,18], Hefter and Jentzen [19].…”

Strong Convergence Rates for Euler Approximations to a Class of Stochastic Path-Dependent Volatility Models

“…In Theorem 1 in [27], a lower error bound was established for all discretization schemes for the CIR process based on equidistant evaluations of the Brownian motion in the accessible boundary regime. As a consequence of this result, the FTE scheme achieves at most a strong convergence order of ν when ν < 1/2.…”