Think Value

---

A common model for the price movement process of non-dividend-paying stocks is the Geometric Brownian Motion (Robert Brown (1773–1858) was a Scottish botanist). Initially, Brown observed that dust particles suspended in water moved in a jittery, random manner. Later, Einstein explained that this was caused by molecular collisions, Wiener provided the mathematical formulation, and Itô refined it to the specific form known as Geometric Brownian Motion.

Directly applying a generalized Wiener process to model stock price movements is not appropriate, because investors perceive expected stock price changes in percentage rather than absolute terms.

The generalized Wiener process is given by:
dx = a·dt + b·dz  (find here)

If we were to substitute the stock price directly into this formula, the expected drift rate of the variable would be an absolute number, which would be incorrect. That’s because for an investor, it doesn’t matter whether a stock costs $10 or $1,000 — what matters is the expected return in percentage terms.

Thus, if we assume there are no random fluctuations and we remove the second component from the generalized Wiener process, we get:

ΔS = μS·Δt, where μ is the expected rate of return.

For infinitesimally small time intervals (as Δt → 0), we get:
dS = μS·dt, or equivalently dS / S = μ·dt.

If we integrate (sum) over a time period T, we obtain:
Sₜ = S₀·e^(μT) — where μ represents the continuously compounded rate of return per unit of time.

However, the real world also includes random fluctuations. Common sense suggests that the range of volatility should be proportional, regardless of the stock’s price level. That is, if the investor’s required expected return remains constant and stems from a certain level of uncertainty, then the perceived uncertainty should also be constant in percentage terms, regardless of the stock’s price.

Accordingly, we get:
dS = μS·dt + σS·dz,
where σ is the standard deviation, and therefore σ² is the variance of the stock price.

This leads us to the Itôs process known as Geometric Brownian Motion, which can also be written as:
dS / S = μ·dt + σ·dz.

Its discrete version (for finite time intervals) is:
ΔS = μS·Δt + σS·ε√Δt or ΔS / S = μ·Δt + σ·ε√Δt,
where ε is a random variable with a standard normal distribution.

---

Example and Monte Carlo Simulation

The chart below illustrates one possible path of how a stock price may evolve according to the Geometric Brownian Motion formula.

Given parameters:
μ = 15%, σ = 30%, S = $100, Δt = 1 week, ε = random number between 0 and 1

This is just one possible evolution of the stock price. However, we can perform a Monte Carlo simulation in Excel and observe the results for 1,000 different ε values.

Since ε is a random variable following a normal distribution, we can use this Excel formula:
=NORM.S.INV(RAND())

After running the simulation for 10 weeks, the expected price range was:
an average of $113, with 50% of outcomes falling between $91 and $132,
though the uncertainty range was wide — from a pessimistic $46 to an optimistic $267.

---

Excel – Stock Price Motion

Adapted from:
Options, Futures, and Other Derivatives — John C. Hull

---