A Hybrid Model for Equity Indices and Stochastic Interest Rates

The growth optimal portfolio (GOP) plays an important role in finance, in particular in derivative pricing, where it is employed as a num\'eraire portfolio, allowing to price contingent claims directly under the real world probability measure. This paper derives an extension of a time dependent constant elasticity of variance (TCEV) model which takes into account stochastic interest rate risk. This results in a hybrid framework that models the stochastic dynamics of the GOP and the short rate simultaneously. We estimate and compare a variety of continuous-time models for short-term interest rates using non-parametric kernel-based estimation. Taking interest rate dynamics into account, we show that the hybrid model remains highly tractable and fits well the observed dynamics of diversified equity indices and interest rates. Our results are important for pricing and hedging of various derivative products, allowing to derive closed-form solutions for standard derivatives. Across all models under consideration we find that the hybrid model with 3/2 dynamics for the interest rate provides the best fit to the data. It leads to the lowest prices and the least expensive hedges