Pricing of volatility derivatives in a Heston–CIR model with Markov-modulated jump diffusion

Abstract

This paper concerns the pricing of volatility and variance swaps with discrete sampling times using a hybrid Heston–CIR model with Markov-modulated jump–diffusion. We extend the regime-switching Heston stochastic volatility model by further considering the Cox–Ingersoll–Ross (CIR) stochastic interest rate with jump diffusion. The market parameters, including the jump–diffusion part, are modulated by a continuous-time Markov chain that represents different market states. The proposed hybrid model can capture the stochastic property of volatility rate, interest rate as well as the short-term and long-term effects on the financial market caused by unexpected events and structural changes of the macroeconomic environment respectively. The fair strike prices of variance swaps and volatility swaps based on four different formulae for the realized variance (volatility) are obtained by utilizing the characteristic function method. The accuracy and efficiency of our method are validated through the semi-Monte-Carlo simulation. Finally, numerical and sensitivity analysis are conducted to show the impact of the change of each market parameter. Overall, pricing results were greatly enhanced with a more realistic financial model, and more driving forces are considered to investigate the formation of the discrete variance and volatility swap prices more comprehensively.