A Note on Relaxing the Black-Scholes Assumptions Without Changing the Price Formula

We provide explicit, simple price formulas for the European options under stochastic volatility and stochastic interest rate. The formulas are as simple as the classical Black-Scholes formula. Moreover, the formulas do not require the normality of the returns. We do not need to know the distribution of the returns/price. Furthermore, this approach enables us to avoid the incomplete markets problem. That is, we relax the key assumptions of the classical Black-Scholes model without changing their price formula.