Direct Inversion for the Squared Bessel Process and Applications

ArXiv ID: 2412.16655 “View on arXiv”

Authors: Unknown

Abstract

In this paper we derive a new direct inversion method to simulate squared Bessel processes. Since the transition probability of these processes can be represented by a non-central chi-square distribution, we construct an efficient and accurate algorithm to simulate non-central chi-square variables. In this method, the dimension of the squared Bessel process, equivalently the degrees of freedom of the chi-square distribution, is treated as a variable. We therefore use a two-dimensional Chebyshev expansion to approximate the inverse function of the central chi-square distribution with one variable being the degrees of freedom. The method is accurate and efficient for any value of degrees of freedom including the computationally challenging case of small values. One advantage of the method is that noncentral chi-square samples can be generated for a whole range of values of degrees of freedom using the same Chebyshev coefficients. The squared Bessel process is a building block for the well-known Cox-Ingersoll-Ross (CIR) processes, which can be generated from squared Bessel processes through time change and linear transformation. Our direct inversion method thus allows the efficient and accurate simulation of these processes, which are used as models in a wide variety of applications.

Keywords: Squared Bessel Process, CIR Model, Monte Carlo Simulation, Non-central Chi-square, Chebyshev Expansion, Fixed Income

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper presents advanced mathematical techniques involving two-dimensional Chebyshev expansions and direct inversion of non-central chi-square distributions, indicating high complexity. It includes empirical comparisons of errors and pricing applications for financial instruments, demonstrating significant implementation and backtest-ready elements.

  flowchart TD
    A["Research Goal: Develop efficient, direct inversion<br>method for simulating Squared Bessel processes"] --> B["Key Methodology: 2D Chebyshev Expansion"]
    B --> C["Computational Process: Approximate inverse CDF<br>of central chi-square (d.o.f. & variable)"]
    C --> D["Outcome: Accurate Non-central<br>Chi-square sampling"]
    D --> E["Outcome: Efficient Monte Carlo<br>simulation of Squared Bessel Processes"]
    E --> F["Application: Direct simulation of<br>Cox-Ingersoll-Ross (CIR) models"]