# Difference between revisions of "SOCR EduMaterials Activities ApplicationsActivities StockSimulation"

## A Model for Stock prices

• Process for Stock Prices: Assumed a drift rate equal to $$\mu S$$ where $$\mu$$ is the expected return of the stock, and variance $$\sigma^2 S^2$$ where $$\sigma^2$$ 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. $$\Delta S$$ Change in the stock price. $$\Delta t$$ Small interval of time. $$\epsilon$$ Follows $$N(0,1)$$.

• Example: The current price of a stock is $$S_0=\100$$. The expected return is $$\mu=0.10$$ per year, and the standard deviation of the return is $$\sigma=0.20$$ (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 $$S_1 = S_0 + \Delta S_1$$, where $$\Delta S_1$$ 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 $$S_0=\20[itex], annual mean and standard deviation'"UNIQ-MathJax2-QINU"'. <br> * The applet will select a random sample of 100 observations from [itex]N(0,1)$$ and will compute

$$\frac{\Delta S}{S} = 0.14 (0.01) + 0.20 \epsilon \sqrt{0.01}.$$ Suppose that $$\epsilon_1=0.58$$. 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 $$\Delta S_1 = S_0 + \Delta S_1 = 20 + 0.26=20.26.$$ We continue in the same fashion until we reach the end of the year. Here is the SOCR applet.