A closed-form pricing formula for European options under a new stochastic volatility model with a stochastic long-term mean

Abstract

Based upon the fact that a constant long-term mean could not provide a good description of the term structure of the implied volatility and variance swap curve, as suggested by Byelkina and Levin (in: Sixth world congress of the Bachelier Finance Society, Toronto, 2010) and Forde and Jacquier (Appl Math Finance 17(3):241–259, 2010), this paper presents a new stochastic volatility model, by assuming the long-term mean of the volatility in the Heston model be stochastic. An important feature of our model is that it still preserves the essential advantage of the Heston model, i.e., the analytic tractability, because a closed-form pricing formula for European options can be derived, which could not only facilitate the risk management process but also help save plenty of time in terms of model calibration. The effect of the newly introduced stochastic long-term mean is demonstrated through the numerical comparison with the Heston model. It is also shown that the current model can overall lead to more accurate option prices than the Heston model, through a carefully designed empirical study.