Approximating the Heston-Hull-White Model

Abstract

The hybrid Heston-Hull-White (HHW) model combines the Heston (1993) stochastic volatility and Hull and White (1990) short rate models. Compared to stochastic volatility models, hybrid models improve upon the pricing and hedging of longdated options and equity-interest rate hybrid claims. When the Heston and HullWhite components are uncorrelated, an exact characteristic function for the HHW model can be derived. In contrast, when the components are correlated, the more useful case for the pricing of hybrid claims, an exact characteristic function cannot be obtained. Grzelak and Oosterlee (2011) developed two approximations for this correlated case, such that the characteristics functions are available. Within this dissertation, the approximations, referred to as the determinist and stochastic approximations, were implemented to price vanilla options. This involved extending the Carr and Madan (1999) method to a stochastic interest rate setting. The approximations were then assessed for accuracy and efficiency. In determining an appropriate benchmark for assessing the accuracy of the approximations, the full truncation Milstein and Quadratic Exponential (QE) schemes, which are popular Monte Carlo discretisation schemes for the Heston model, were extended to the HHW model. These schemes were then compared against the characteristic function for the uncorrelated case, and the QE scheme was found to be more accurate than the Milstein-based scheme. With the differences in performance becoming increasingly noticeable when the Feller (1951) condition was not satisfied and the maturity and volatility of the Hull-White model (⌘) was large. In assessing the accuracy of the approximations against the QE scheme, both approximations were similarly accurate when ⌘ was small. In contrast, when ⌘ was large, the stochastic approximation was more accurate than the deterministic approximation. However, the deterministic approximation was significantly faster than the stochastic approximation and the stochastic approximation displayed signs of potential instability. When ⌘ is small, the deterministic approximation is therefore recommended for use in applications such as calibration. With its shortcomings, the stochastic approximation could not be recommended. However, it did show promising signs of accuracy that warrants further investigation into its efficiency and stability.