A note on portfolios with risk-free internal gains

In this paper we 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. As an application of this result, we derive the Black-Scholes PDE by using the original derivation of Black and Scholes, and we show that it can be made completely rigorous.