Stochastic Calculus and the Black-Scholes-Merton Model: A Simplified Approach

In the continuous-time finance literature, it is claimed that the expected rate of return of underlying asset does not affect the option pricing model. This paper has shown that with no arbitrage, i.e., under the Arbitrage (Gordan) theorem, different underlying asset price processes used in the Black-Scholes-Merton partial differential equation and the Black-Scholes-Merton option pricing formula require that risk-free interest rate be a linear function of underlying asset’s expected rate of return (alpha) and variance of return, or (as in the literature) risk-free interest rate equal underlying asset's alpha.