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.

Itō's Lemma

Itō's Lemma is a theorem of stochastic calculus that holds that within a closed integral, dz² can be replaced by dt, where dz is a stochastic variable with order of magnitude equal to the square root of dt. The Lemma is sometimes erroneously stated as "dz² equals dt," which is not generally true. Integration, with continuity, invokes the Law of Large Numbers. In the absence of continuity the variance of dz² is proportional to Δt, length of measurement intervals taken over the range.

Itō's Lemma is significant in finance because it provides the basis according to which a delta hedge is assumed to be riskless, an assumption that is essential to the Black-Scholes Equation.

The Lognormal Distribution and Risk-Neutral Pricing

I wrote previously that the ergodic property of the normal distribution is so useful that it often makes sense to assume a normal distribution even in the face of evidence to the contrary.

Before going to that extreme, however, it is sometimes possible to arrive at a normal distribution by looking at a function of an original variable rather than the variable itself. One example of this is the lognormal distribution, which is a distribution whose log is a normal distribution. In finance, the future price of a stock is often considered to have a lognormal distribution, which gives the rate of return on the stock a normal distribution.

In considering a lognormal distribution, we generally refer to aspects of its log. In particular, we usually define a particular lognormal distribution based upon the mean, μ, and variance, σ², of the log.

There are any number of things to be known about lognormal distributions, but the most important fact about them for my purposes is that the expected value of a lognormal distribution is exp(μ + ½σ²).

Getting back to finance, the future price of a stock at time t can be considered to have a lognormal distribution with log-mean lnS + μt and log-variance σ²t, where S is the price of the stock at time 0. (here I have effectively used μ and σ² as the mean and variance of the instantaneous rate of return of the stock, so that at any given point in time in the future t the rate of return on the stock will have mean μt and variance σ²t). This gives the expected value of the stock at time t as exp(lnS + μt + ½σ²t).

Under a risk-neutral pricing regime, however, the expected future price of the stock should be exp(lnS + rt), where r is the riskless interest rate. So if we want to preserve the lognormal distribution of the stock price, we somehow have to adapt our real world expectations into risk-neutral probabilities so that exp(lnS + μt + ½σ²t) equals exp(lnS + rt), or more simply so that μ + ½σ² equals r.

One method to accomplish this is simply to replace μ with r - ½σ². And this is precisely what the Black-Scholes options pricing model does, about which more later.