# SOCR EduMaterials Activities ApplicationsActivities StockSimulation

## A Model for Stock prices

• Process for Stock Prices: Assumed a drift rate equal to μS where μ is the expected return of the stock, and variance σ2S2 where σ2 is the variance of the return of the stock. From Weiner process the model for stock prices is:

$\Delta S = \mu S \Delta t + \sigma S \epsilon \sqrt{\Delta t}$ or $\frac{\Delta S}{S} = \mu \Delta t + \sigma \epsilon \sqrt{\Delta t}.$ Therefore $\frac{\Delta S}{S} \sim N(\mu \Delta t, \sigma \sqrt{\Delta t}).$
S Price of the stock. ΔS Change in the stock price. Δt Small interval of time. ε Follows N(0,1).

• Example: The current price of a stock is $S_0=\100$. The expected return is μ = 0.10 per year, and the standard deviation of the return is σ = 0.20 (also per year).
• Find an expression for the process of the stock.

$\frac{\Delta S}{S}=0.14 \Delta t + 0.20 \epsilon \sqrt{\Delta t}$

• Find the distribution of the change in S divided by S at the end of the first year. That is, find the distribution of $\frac{\Delta S}{S}$.

$\frac{\Delta S}{S} \sim N\left(0.10 \Delta t, 0.20 \sqrt{\Delta t}\right).$

• Divide the year in weekly intervals and find the distribution of $\frac{\Delta S}{S}$ at the end of each weekly interval.

$\frac{\Delta S}{S} \sim N\left(0.10 \frac{1}{52}, 0.20 \sqrt{\frac{1}{52}}\right).$

• Therefore, sampling from this distribution we can simulate the path of the stock. The price of the stock at the end of the first interval will be S1 = S0 + ΔS1, where ΔS1 is the change during the first time interval, etc.

• Using the SOCR applet we will simulate the stock's path by dividing one year into small intervals each one of length $\frac{1}{100}$ of a year, when $S_0=\20$, annual mean and standard deviation: μ = 0.14,σ = 0.20.

• The applet will select a random sample of 100 observations from N(0,1) and will compute

$\frac{\Delta S}{S} = 0.14 (0.01) + 0.20 \epsilon \sqrt{0.01}.$ Suppose that ε1 = 0.58. Then
$\frac{\Delta S}{S} = 0.14 (0.01) + 0.20 (0.58) \sqrt{0.01}= 0.013 \Rightarrow \Delta S_1= 20(0.013)=0.26.$
Therefore ΔS1 = S0 + ΔS1 = 20 + 0.26 = 20.26. We continue in the same fashion until we reach the end of the year. Here is the SOCR applet.