Black–Scholes and Heston Models with Stochastic Interest Rates and Term Structure of Volatilities

Abstract

This article considers the Black–Scholes and Heston models and generalize them to stochastic interest rates and maturity-dependent volatilities. In the Black–Scholes case, the author solves the extended model and provides a concrete form for the term structure of volatilities. In the Heston case, he proves that, under some conditions, the generalized model is equivalent to a hybrid model and finds semi-closed-form solutions in the Hull and White and CIR cases. TOPICS: Options, statistical methods, fixed income and structured finance