On arbitrarily slow convergence rates for strong numerical approximations of Cox-Ingersoll-Ross processes and squared Bessel processes
For practitioners in quantitative finance who rely on CIR processes for derivative pricing, this paper shows that commonly used discretization methods may be unreliable, motivating the need for new approximation algorithms.
The paper proves that standard Euler- and Milstein-type discretization methods for Cox-Ingersoll-Ross (CIR) processes achieve at most a strong convergence order of δ/2, where δ is the dimension of the associated squared Bessel process, implying that these methods can converge arbitrarily slowly. This reveals a fundamental limitation of current industry-standard numerical methods for CIR processes.
Cox-Ingersoll-Ross (CIR) processes are extensively used in state-of-the-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.