On the calibration of stochastic volatility models to estimate the real-world measure used in option pricing

Abstract

It is widely noted that the Heston stochastic volatility model fails to capture the fat tails often observed in daily equity returns. Adding random jumps improves the model’s ability to capture extreme events. This extension is known as the Bates stochastic volatility jump (SVJ) model. The model parameters for the Heston and Bates SVJ models are generally calibrated to option prices inducing the so-called risk-neutral measure. However, in the absence of a sufficiently liquid options market, one has to resort to calibration under the realworld measure. In this paper, we calibrate the Heston and Bates SVJ models to historical equity returns in the United States and South Africa using the efficient method of moments(EMM). We then show how a real-world stochastic volatility model can be used in practice to test a simple volatility targeting strategy. Our findings suggest that stochastic volatility and jumps are both required to characterise equity returns in South Africa. Furthermore, volatility targeting is an effective strategy that allows investors to manage the downside risk of a portfolio.