SOCR EduMaterials Activities ApplicationsActivities StockSimulation - Revision history

IvoDinov at 19:10, 3 August 2008

2008-08-03T19:10:22Z

IvoDinov at 18:31, 3 August 2008

2008-08-03T18:31:01Z

IvoDinov at 18:30, 3 August 2008

2008-08-03T18:30:32Z

IvoDinov at 18:23, 3 August 2008

2008-08-03T18:23:45Z

IvoDinov at 18:23, 3 August 2008

2008-08-03T18:23:01Z

Nchristo at 16:18, 3 August 2008

2008-08-03T16:18:05Z

Nchristo at 16:15, 3 August 2008

2008-08-03T16:15:06Z

Nchristo at 16:14, 3 August 2008

2008-08-03T16:14:23Z

Nchristo: New page: == A Model for Stock prices == * '''Description''': You can access the stock simulation applet at http://www.socr.ucla.edu/htmls/app/ . * Process for Stock Prices: Assumed a drift rate...

2008-08-03T16:13:13Z

New page: == A Model for Stock prices == * '''Description''': You can access the stock simulation applet at http://www.socr.ucla.edu/htmls/app/ . * Process for Stock Prices: Assumed a drift rate...

New page

== A Model for Stock prices ==

* '''Description''': You can access the stock simulation applet at http://www.socr.ucla.edu/htmls/app/ .

* Process for Stock Prices: Assumed a drift rate equal to <math>\mu S</math> where <math>\mu</math> is the expected return of the stock, and variance <math>\sigma^2 S^2</math> where <math>\sigma^2</math> is the variance of the return of the stock. From Weiner process the model for stock prices is:
<math>
\Delta S = \mu S \Delta t + \sigma S \epsilon \sqrt{\Delta t}
</math>
or
<math>
\frac{\Delta S}{S} = \mu \Delta t + \sigma \epsilon \sqrt{\Delta t}.
</math>
Therefore
<math>
\frac{\Delta S}{S} \sim N(\mu \Delta t, \sigma \sqrt{\Delta t}).
</math>
 
<math>S</math> Price of the stock.
<math>\Delta S</math> Change in the stock price.
<math>\Delta t</math> Small interval of time.
<math>\epsilon</math> Follows <math>N(0,1)</math>.

 

*Example: The current price of a stock is <math>S_0=\$100</math>. The expected return is <math>\mu=0.10</math> per year, and the standard deviation of the return is <math>\sigma=0.20</math> (also per year).

* Find an expression for the process of the stock. 
<math>
\frac{\Delta S}{S}=0.14 \Delta t + 0.20 \epsilon \sqrt{\Delta t}
</math>
* Find the distribution of the change in <math>S<math> divided by <math>S<math> at the end of the first year. That is, find the distribution of <math>\frac{\Delta S}{S}</math>. 
<math>
\frac{\Delta S}{S} \sim N\left(0.10 \Delta t, 0.20 \sqrt{\Delta t}\right).
</math>
* Divide the year in weekly intervals and find the distribution of <math>\frac{\Delta S}{S}</math> at the end of each weekly interval. 
<math>
\frac{\Delta S}{S} \sim N\left(0.10 \frac{1}{52}, 0.20 \sqrt{\frac{1}{52}}\right).
</math>
 
* 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 <math>S_1 = S_0 + \Delta S_1</math>, where <math>\Delta S_1</math> 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 <math>\frac{1}{100}</math> of a year, when <math>S_0=\$20<math>, annual mean and standard deviation: <math>\mu=0.14, \sigma=0.20</math>.
 

* The applet will select a random sample of 100 observations from <math>N(0,1)</math> and will compute 
<math>
\frac{\Delta S}{S} = 0.14 (0.01) + 0.20 \epsilon \sqrt{0.01}.
</math>
Suppose that <math>\epsilon_1=0.58</math>. Then 
<math>
\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.
</math>
 
Therefore
<math>
\Delta S_1 = S_0 + \Delta S_1 = 20 + 0.26=20.26.
</math>
We continue in the same fashion until we reach the end of the year.

<center>[[Image: Christou_stock_simulation.jpg|600px]]</center>

@@ Line 2: / Line 2: @@
 ===Description===
-You can access the portfolio applet at [http://www.socr.ucla.edu/htmls/app/ the SOCR Applications Site], select ''Financial Applications'' --> ''StockSimulation''.
+You can access the Stock Prices applet at [http://www.socr.ucla.edu/htmls/app/ the SOCR Applications Site], select ''Financial Applications'' --> ''StockSimulation''.
-* Process for Stock Prices:  Assumed a drift rate equal to <math>\mu S</math> where <math>\mu</math> is the expected return of the stock, and variance <math>\sigma^2 S^2</math> where <math>\sigma^2</math> is the variance of the return of the stock.  From Weiner process the model for stock prices is:
+===Process for Stock Prices===
 : <math> \Delta S = \mu S \Delta t + \sigma S \epsilon \sqrt{\Delta t}
 </math>

← Older revision		Revision as of 18:31, 3 August 2008
Line 1:		Line 1:
−	== [[SOCR_EduMaterials_ApplicationsActivities \| SOCR Applications Activities]] -	+	== [[SOCR_EduMaterials_ApplicationsActivities \| SOCR Applications Activities]] - A Model for Stock Prices ==
−	A Model for Stock Prices ==

	===Description===		===Description===

← Older revision		Revision as of 18:30, 3 August 2008
Line 1:		Line 1:
−	== A Model for Stock ~~prices~~ ==	+	== [[SOCR_EduMaterials_ApplicationsActivities \| SOCR Applications Activities]] -
		+	A Model for Stock Prices ==

	===Description===		===Description===

@@ Line 59: / Line 59: @@
 * The materials above was partially taken from <br>
 ''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003, and ''Options, Futues, and Other Derivatives'' by John C. Hull, Sixth Edition, Pearson Prentice Hall, 2006.
-* [http://www.socr.ucla.edu/htmls/app/ the SOCR Applications Site]
+* [http://www.socr.ucla.edu/htmls/app/ SOCR Applications Site]
 {{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_ApplicationsActivities_StockSimulation}}

← Older revision		Revision as of 16:18, 3 August 2008
Line 59:		Line 59:

	<center>[[Image: Christou_stock_simulation.jpg\|600px]]</center>		<center>[[Image: Christou_stock_simulation.jpg\|600px]]</center>
		+
		+	<br>
		+	* The materials above was partially taken from <br>
		+	''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003, and <br>
		+	''Options, Futues, and Other Derivatives'' by John C. Hull, Sixth Edition, Pearson Prentice Hall, 2006.

@@ Line 1: / Line 1: @@
 == A Model for Stock prices ==
-* '''Description''':  You can access the stock simulation applet at http://www.socr.ucla.edu/htmls/app/ .
+===Description===
 * Process for Stock Prices:  Assumed a drift rate equal to <math>\mu S</math> where <math>\mu</math> is the expected return of the stock, and variance <math>\sigma^2 S^2</math> where <math>\sigma^2</math> is the variance of the return of the stock.  From Weiner process the model for stock prices is:
-<math>
+: <math> \Delta S = \mu S \Delta t + \sigma S \epsilon \sqrt{\Delta t}
 </math>
 or
-<math>
+: <math>
 \frac{\Delta S}{S} = \mu \Delta t + \sigma \epsilon \sqrt{\Delta t}.
 </math>
 Therefore
-<math>
+: <math>
 \frac{\Delta S}{S} \sim N(\mu \Delta t, \sigma \sqrt{\Delta t}).
 </math>
 <br>
-<math>S</math>         Price of the stock.
+: <math>S</math>         Price of the stock.
-<math>\Delta S</math>  Change in the stock price.
+: <math>\Delta S</math>  Change in the stock price.
-<math>\Delta t</math>  Small interval of time.
+: <math>\Delta t</math>  Small interval of time.
-<math>\epsilon</math>  Follows <math>N(0,1)</math>.
+: <math>\epsilon</math>  Follows <math>N(0,1)</math>.
 <br>
-*Example:  The current price of a stock is <math>S_0=\$100</math>.  The expected return is <math>\mu=0.10</math> per year, and the standard deviation of the return is <math>\sigma=0.20</math> (also per year).
+===Example===
 * Find an expression for the process of the stock. <br>
@@ Line 30: / Line 31: @@
 </math>
 * Find the distribution of the change in <math>S</math> divided by <math>S</math> at the end of the first year.  That is, find the distribution of <math>\frac{\Delta S}{S}</math>. <br>
-<math>
+: <math>
 \frac{\Delta S}{S} \sim N\left(0.10 \Delta t, 0.20 \sqrt{\Delta t}\right).
 </math>
 * Divide the year in weekly intervals and find the distribution of <math>\frac{\Delta S}{S}</math> at the end of each weekly interval.  <br>
-<math>
+: <math>
 \frac{\Delta S}{S} \sim N\left(0.10 \frac{1}{52}, 0.20 \sqrt{\frac{1}{52}}\right).
 </math>
@@ Line 40: / Line 41: @@
 * 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 <math>S_1 = S_0 + \Delta S_1</math>, where <math>\Delta S_1</math> is the change during the first time interval, etc.
 <br>
-* Using the SOCR applet we will simulate the stock's path by dividing one year into small intervals each one of length <math>\frac{1}{100}</math> of a year, when <math>S_0=\$20</math>, annual mean and standard deviation: <math>\mu=0.14, \sigma=0.20</math>.
+* Using the [http://www.socr.ucla.edu/htmls/app/ SOCR applet] we will simulate the stock's path by dividing one year into small intervals each one of length <math>\frac{1}{100}</math> of a year, when <math>S_0=\$20</math>, annual mean and standard deviation: <math>\mu=0.14, \sigma=0.20</math>.
 <br>
 * The applet will select a random sample of 100 observations from <math>N(0,1)</math> and will compute <br>
-<math>
+: <math> \frac{\Delta S}{S} =  0.14 (0.01) + 0.20 \epsilon \sqrt{0.01}.
 </math>
 Suppose that <math>\epsilon_1=0.58</math>.  Then <br>
-<math>
+: <math> \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. </math>
 <br>
 Therefore
-<math>
+: <math> \Delta S_1 = S_0 +  \Delta S_1 = 20 + 0.26=20.26. </math>
 We continue in the same fashion until we reach the end of the year.  Here is the SOCR applet.
@@ Line 62: / Line 58: @@
 <br>
 * The materials above was partially taken from <br>
-''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003, and <br>
+''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003, and ''Options, Futues, and Other Derivatives'' by John C. Hull, Sixth Edition, Pearson Prentice Hall, 2006.
-''Options, Futues, and Other Derivatives'' by John C. Hull, Sixth Edition, Pearson Prentice Hall, 2006.
+* [http://www.socr.ucla.edu/htmls/app/ the SOCR Applications Site]