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.