Revisiting the Classical Models : Black-Scholes and Heston With Stochastic Interest Rates and Term Structure of Volatilities

We consider the Black and Scholes (1973) and Heston (1993) models and we generalize them to stochastic interest rates and maturity-dependent volatilities. In the Black-Scholes case we solve the extended model and provide a concrete form for the term structure of volatilities. In the Heston case we prove that, under some conditions, the generalized model is equivalent to a hybrid model and we find semi-closed-form solutions in the Hull and White (1990) and Cox et al. (1985) cases.We address the problem of the consistency of the Black-Scholes model with the volatility surface and we show that, under general conditions, the Black-Scholes formula cannot be generalized to account for the volatility smile