Risk-Free Internal Gains -- Black and Scholes Re-Examined

Abstract:     
In this paper, we first show that if a not-necessarily-self-financing portfolio has instantaneously riskless internal gains, then on an infinitesimal time-interval, the increase in the internal gains on the portfolio is the same as the change in the price of that amount of bonds which has the same wealth as the portfolio has. Then, using this result, we re-examine the original derivation of the Black-Scholes formula, and conclude that contrary to common belief, the argument of Black and Scholes can be made completely rigorous, employing the same delta-hedge portfolio that they used and keeping all their mathematical formulas; but the explanations they gave to support their formulas must be replaced by others.

Keywords: Mathematical finance, Black-Scholes formula, Wiener process, self-financing portfolio

JEL Classifications: G10, G13

Working Paper Series