Applying stochastic volatility models in the risk-neutral and real-world probability measures

Abstract

Stochastic volatility models have become immensely popular since their introduction in 1993 by Heston. This is because their dynamics are more consistent with market behaviour compared to the standard Black-Scholes model. More specifically, stochastic volatility models can somewhat capture the asymmetric distribution often observed in daily equity returns. Numerous extensions to the stochastic volatility model of Heston have since been proposed, including jumps and stochastic interest rates. Due to their complex dynamics, numerical methods such as Monte Carlo simulation, the fast Fourier transform (FFT), and the efficient method of moments (EMM) are often required to calibrate and implement stochastic volatility models. In this thesis, we explore the application of stochastic volatility models to a variety of problems for which research is still in its infancy phase. We consider the pricing of embedded derivatives in the South African life insurance industry given the illiquid derivatives market; the pricing of rainbow and spread options that depend on two underlying assets; the calibration of stochastic volatility models with jumps to historical equity returns; and the use of stochastic volatility models in static hedging. Our findings suggest that stochastic interest rates are the dominant risk driver when pricing long-dated contingent claims; the FFT significantly outperforms Monte Carlo simulation in terms of efficiency; jumps are an important factor required to explain daily equity returns; and static hedging is a simple and effective way to replicate vanilla and exotic options.