Arithmetic Brownian Motion

A easy-to-understand introduction to Arithmetic Brownian Motion and stock pricing, with simple calculations in Excel. Brownian motion is often described as a random walk with the following characteristics).

• initial value Z(0)=0
• mean of Z(0)=0
• standard deviation of σ
• normal distributed
• 50% chance of moving -1 and 50% chance of moving 1.
• variance is the sum of the squared of the time.

Note that that Brownian Motion is used to find the expected price movement, not the actual price

Problem 1

Let’s assume that the price of a stock can be described by arithmetic Brownian motion.  Its price at time t=5 is 56.  What is the probability that the price is more than 70 at t=13>

We want the probability that P{Z(13)>70} given that Z(5)=56.  This is written

Given that the variance is the sum of the square of the time, then the volatility is

Therefore

This problem can be easily solved in Excel using the NORM.DIST function, as given in the picture below.

But stock prices are volatile, and often have apparently random fluctuation.   Let’s propose that the expected stock price is the sum of

• a general rate of increase over time of β, known as the drift
• Brownian motion of magnitude σ Z(t), known as the volatility

Therefore the change in price of a stock is dX= βt + σ Z(t), with mean βt and standard deviation of σ t^0.5.

Problem 2

A stock has a drift of 1 and volatility of 0.15.  If the current price is 40, what is the probability that the price is less than 43 at t=4

The mean is the current price plus the drift component = X(0) + βt = 40 + 1 x 4 = 44.   Hence

Again, this calculation can be implemented in Excel as follows.