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

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