The Black-Scholes Risk-Neutral Pricing Probability Distribution Function

I wrote previously that the Black-Scholes options pricing model replaces the equity drift μ with r - ½σ²,  where r is the riskless interest rate and σ is the stochastic element of the equity's price movement. This substitution has the effect of eliminating the equity drift coefficient from the model, and obviating the need to decide what the drift coefficient actually is.

The reasoning given by Black and Scholes in their paper that introduced their model (as well as one of the very first uses of risk-neutral pricing generally) were (1) that, through continuous rebalancing, the second-order (dz²) stochastic terms in the return on a portfolio became riskless (this argument implicitly relies upon Itō's Lemma, although the paper did not explicitly refer to it), and (2) that even if they were not riskless due to discontinuity, the second-order terms should be subject to the riskless interest rate because they are uncorrelated with the market rate of return. Thus, they argued, second-order terms must be priced with the riskless interest rate directly, and any risk premium must apply only to any first-order stochastic terms.

This can only be achieved by adapting only the drift coefficient -- and not the stochastic coefficient -- in constructing a risk-neutral probability distribution. This means replacing the equity drift μ with r - ½σ².

And this is all roughly consistent with traditional utility theory.