Unbiased estimators for the Heston model with stochastic interest rates

TL;DR

This work tackles unbiased Monte Carlo estimation for pricing options under the Heston model with stochastic interest rates by developing a semi-exact log-Euler discretization that decouples the interest-rate driver from the Heston components. The authors establish an $L^2$ convergence rate of $O(h)$ for the scheme under mild assumptions, and show that, for both continuous and digital payoffs, the resulting price estimators achieve $O(h^2)$ accuracy in $L^2$ when integrated with Rhee–Glynn unbiased estimators. The methodology applies to a broad class of rate models, including CIR, Hull–White, and Black–Karasinski (and their Heston variants), and remains robust under correlation between rate and asset drives. Numerical experiments corroborate the theoretical rates and demonstrate practical, finite-time performance of the unbiased estimators in this setting.

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.

Figures (1)

• Figure 1: Convergence rate for the Heston model with stochastic interest rates. The model parameters are from Table \ref{['table1']}.

Theorems & Definitions (18)

• Remark 1
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Lemma 4.3