Options, Futures, and Other Derivatives, 10th Edition

15.1 LOGNORMAL PROPERTY OF STOCK PRICES

The model of stock price behavior used by Black, Scholes, and Merton is the model we developed in Chapter 14. It assumes that percentage changes in the stock price in a very short period of time are normally distributed. Define

μ:	Expected return in a short period of time (annualized)
σ:	Volatility of the stock price.

The mean and standard deviation of the return in time Δt are approximately μ Δt and $\sigma \sqrt{\text{ Δ }t}$, so that

$$\frac{\text{ Δ }S}{S}\sim \varphi \left(\mu \text{ Δ}t,{\sigma }^2\text{ Δ }t\right)$$(15.1)

where ΔS is the change in the stock price S in time Δt, and ϕ(m, v) denotes a normal distribution with mean m and variance v. (This is equation (14.9).)

As shown in Section 14.7, the model implies that

$$\text{ln}{S}_T-\ln S_0\sim \varphi \left[\left(\mu -\frac{{\sigma }^2}{2}\right)T,{\sigma }^{2}T\right]$$

so that

$$\text{ln}\frac{S_T}{S_0}$$