Lognormal Stock Prices

It would be useful at this point to stop for a moment and collect a few important results. We established the process of geometric Brownian motion whereby stock returns follow the process:

We then used Ito's lemma to prove that for , it must be that the logarithm of prices, ln(S_t), has the following form:

These are both models of stock return dynamics, but they are clearly not the same. The difference is subtle with the log form having a small correction factor. In fact, as the variance of the returns gets arbitrarily small, the two are equivalent. Recall from Chapter 1 that the difference between the arithmetic mean and the geometric mean goes to zero with the variance of the random return. That logic is applicable to explaining the difference here as well and as we shall see, the assumption that the mean of geometric Brownian motion is equal to μ overstates the mean and the correction factor is required. Let's try to sketch out an explanation as to why this is the case. I will do this in two steps.

The first step will derive the mean of a lognormally distributed return. To do this, let me begin with a very general model that governs stock price movements over time, starting with the stock price S_t at time t and letting υ_t represent a random shock at time t. Again, we assume that shocks are independent over time (independence means that the covariance of shocks is zero, for example, cov(υ_t,υ_(t + s)) = 0.) and we postulate that the difference between the current price and last period's price is due to the shock (which is caused by some event, anticipated or otherwise). As such, we model the basic price dynamic as:

This is not the deterministic growth model from the first equation. It is, however, related to the stochastic model , which reduces to if we restrict growth to zero. Implicitly, we assume that υ_t = e^(μΔt + ε√Δt). The two, therefore, are equivalent models of stochastic movement in stock prices. If we substitute for the lagged stock price, we get the following:

Successive substitution therefore generates the multiplicative model of stock prices as the product of the shocks:

This says that the price we currently see is the accumulated effect of the shocks, that is, the observed price is really the product of geometrically linking the individual shocks over time. It is important to note, if it is not already obvious, that for prices to remain positive, the shocks must also be positive. As we shall see further on, this property has implications concerning the probability distribution governing the shocks, υ. If we take natural logarithms on both sides, we can represent this as an additive model, to wit:

Thus, the discrete time return (which is defined as the difference in the log prices—see Chapter 1) is a sum of random shocks

With continuous compounding, P_1 = P_0 e^rt. Taking natural logs and noting that t = 1 for this example gives us the following: ln [(P_1) = ln [(P_0) + r]]. Equivalently, ln [(P_1/P_0) = r].

Let us denote the innovations to this returns process by . Equivalently, . It follows therefore that the logarithms of the shocks given by are normally distributed, which is consistent with the assumption that returns be normally distributed. The shocks must be strictly non-negative, since we are transforming them into logarithms and the log of a negative number is undefined (recall, ln(1) = 0, while the log of numbers larger than 1 are positive and the log of a fraction is a negative number, which approaches negative infinity as the fraction approaches zero). Denote the mean and variance of the logarithms of these shocks, respectively, by . For example, might be the annual expected return on a stock (for example, 15 percent in the examples in the previous section). It follows then that the shocks themselves (the υ_i) are lognormally distributed.

As lognormal random variables, the shocks do not have a mean described by ; rather, as lognormally distributed random variables, and variance . An example of a lognormal distribution is given in Figure 17.5 based on the assumption that and σ = 0.3. Prices are normalized to 100.

Figure 17.5 Lognormal Distribution

Notice that the lognormal distribution is skewed right (because prices must be positive). That is why the mean E(υ) contains the additive correction factor and this factor gives this distribution fatter tails than the normal distribution.

The second step involves showing that the mean in geometric Brownian motion is overstated and therefore requires the correction factor we see in Example 17.2. Again, let's begin with the same model of continuously compounded returns:

The expected value of the stock price in period t is therefore:

Taking logarithms, we solve for the mean return w:

If then we would conclude that . But, this is not the case. In fact, , indicating that the mean is overstated. As we showed in step 1, the mean is for the lognormal distribution and therefore, we must subtract the correction factor to establish the required equivalence, that is:

This result is in accordance with the result in Example 17.3.

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Example 17.3

Assume that the monthly return has mean µ = 1 percent and monthly volatility σ = 4 percent. Then the expected annual return would be (0.01 + 0.0008) × 12 which is close to 12 percent with annual volatility 4%(√12) = 13.86% under the assumption that returns are independent and lognormally distributed. Price dynamics over discrete time intervals given by Δt—for example, Δt could represent fractions of a year, say, for monthly, weekly, and daily frequencies—would suggest mean returns equal to μΔt and variances σ²Δt for a given interval defined on Δt.

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