Unbiased estimators for the Heston model with stochastic interest rates

Paper: arXiv 2301.12072

Authors: Chao Zheng, Jiangtao Pan

Abstract

We combine the unbiased estimators in Rhee and Glynn (Operations Research: 63(5), 1026-1043, 2015) and the Heston model with stochastic interest rates. Specifically, we first develop a semi-exact log-Euler scheme for the Heston model with stochastic interest rates. Then, under mild assumptions, we show that the convergence rate in the $L^2$ norm is $O(h)$, where $h$ is the step size. The result applies to a large class of models, such as the Heston-Hull-While model, the Heston-CIR model and the Heston-Black-Karasinski model. Numerical experiments support our theoretical convergence rate.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 4.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents strong theoretical mathematics by developing a semi-exact log-Euler scheme and proving L2 convergence rates for complex Heston models with stochastic interest rates. However, empirical rigor is limited to theoretical numerical experiments rather than backtests or market data analysis.

Research Flowchart

  flowchart TD
    A[Research Goal: Develop unbiased estimators for<br/>Heston model with stochastic interest rates] --> B[Inputs: Heston SV + Stochastic Rate Models<br/>Heston-HW, Heston-CIR, Heston-BK]
    B --> C[Methodology: Develop Semi-Exact Log-Euler Scheme]
    C --> D[Methodology: Apply Rhee & Glynn<br/>Unbiased Estimator Framework]
    D --> E[Theoretical Proof: Prove L2-norm<br/>convergence rate of O h ]
    E --> F[Numerical Experiments & Validations]
    F --> G[Outcome: Unbiased Option Pricing with<br/>Validated O h  Convergence Rate]