European Option Pricing under Markov-Switching Two-Factor Heston Model with Stochastic Interest Rate: Model Calibration Using a Neural Network Optimized by Bat Algorithm

Abstract

In this paper, we study the pricing of European options under a Markov-switching double Heston model with a stochastic interest rate governed by the CIR process. We assume that the mean-reversion levels of both stochastic volatilities and the interest rate switch between bull and bear market states. The stability of the proposed Markov-switching model is analytically investigated, and a semi-analytical pricing of European options is developed based on its moment-generating function. After obtaining the semi-analytical option pricing formula, we propose a hybrid calibration framework based on a neural network structure. This framework uses as inputs the bull and bear prices, their corresponding returns and volatilities, along with standard option parameters. Then, the option prices generated by the Markov-switching model are incorporated into the neural network via a concatenation layer. To optimize the parameters of this hybrid calibration structure, the bat optimization algorithm is employed. Finally, numerical experiments and empirical applications are conducted to demonstrate the accuracy of the proposed pricing formula and calibration framework, as well as the effectiveness of the bat algorithm in solving the calibration problem. © 2026 Elsevier B.V.