SOCR EduMaterials Activities ApplicationsActivities BinomialOptionPricing - Revision history

IvoDinov: /* Description */

2008-08-03T19:06:41Z

‎Description

IvoDinov: /* Definitions */

2008-08-03T18:54:32Z

‎Definitions

IvoDinov at 18:48, 3 August 2008

2008-08-03T18:48:18Z

Nchristo at 16:36, 3 August 2008

2008-08-03T16:36:03Z

Nchristo: New page: == Binomial Option Pricing Model = * '''Description''': You can access the portfolio applet at http://www.socr.ucla.edu/htmls/app/ . Define:
$S_0$ Stock price at
2008-08-03T16:35:09Z

New page: == Binomial Option Pricing Model = * '''Description''': You can access the portfolio applet at http://www.socr.ucla.edu/htmls/app/ . Define: <math>S_0</math> Stock price at <mat...

New page
== Binomial Option Pricing Model =

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

Define: 

<math>S_0</math> Stock price at <math>t=0</math> 
<math>S_1</math> Stock price at <math>t=1<math> 
<math>E</math> Exercise price of the call option 
<math>u</math> <math>1+ \%</math> change in stock price from <math>t=0</math> to <math>t=1</math> if stock price increases (<math>u>1</math>) 
<math>d</math> <math>1+ \%</math> change in stock price from <math>t=0</math> to <math>t=1</math> if stock price decreases (<math>d<1</math>) 
<math>C</math> The call price 
<math>\alpha</math> The number of shares of stocks purchased per one call (hedge ratio) 
<math>C_u</math> Price of call at <math>t=1</math> if stock price increases: <math>max(S_1-E,0)</math> or <math>max(uS_0-E,0)</math> 
<math>C_d</math> Price of call at <math>t=1</math> if stock price decreases: <math>max(S_1-E,0)</math> or <math>max(dS_0-E,0)</math> 
<math>r</math> Continuous risk-free interest rate 

The value <math>C</math> of a European call option at time <math>t=0</math> is: 
<math>
C=S_0 \sum_{j=k}^{n} {n \choose j} p'^{j}(1-p')^{n-j} -
\frac{E}{e^{rt}} \sum_{j=k}^{n} {n \choose j} p^{j}(1-p)^{n-j}
</math>
 
Where, 
<math>
u=e^{+\sigma \sqrt{\frac{t}{n}}}, \ \
d=e^{-\sigma \sqrt{\frac{t}{n}}}=\frac{1}{u}
</math>
 
<math>
p=\frac{e^{rt}-d}{u-d}, \ \ p'=\frac{up}{e^{rt}}
</math>

 
<math>S_0</math> Price of the stock at time <math>t=0</math> 
<math>E</math> Exercise price at expiration 
<math>r</math> Risk-free interest rate per period 
<math>n</math> Number of periods 
<math>\sigma</math> Annual standard deviation of the returns of the stock 
<math>t</math> &Time to expiration in years 

The SOCR Binomial Option Pricing applet provides the price of the stock and the price of the call at each node. Note that at expiration the nodes for which the call is in the money (<math>S > E</math> are colored green, while the nodes for which the call is out of the money (<math>S \le E)</math> are colored blue. The example below uses the following data: 
<math>S_0=\$30</math>, <math>E=\$29</math>, <math>r_f=0.05</math>,
<math>\sigma=0.30</math>, <math>\mbox{days to expiration}=73</math>,
<math>\mbox{number of steps}=5</math>.

<center>[[Image: Christou_binomial_applet1.jpg|600px]]</center>

<center>[[Image: Christou_binomial_applet2.jpg|600px]]</center>

 
* The materials above was partially taken from 
''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.

@@ 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'' --> ''BinomialOptionPricing''.
+You can access the Binomial Option Pricing Model applet at [http://www.socr.ucla.edu/htmls/app/ the SOCR Applications Site], select ''Financial Applications'' --> ''BinomialOptionPricing''.
 ===Definitions===

@@ Line 6: / Line 6: @@
 ===Definitions===
 : <math>S_0</math>   Stock price at <math>t=0</math> <br>
-: <math>S_1</math>   Stock price at <math>t=1<math> <br>
+: <math>S_1</math>   Stock price at <math>t=1</math> <br>
 : <math>E</math>       Exercise price of the call option <br>
 : <math>u</math>     <math>1+ \%</math> change in stock price from

@@ Line 1: / Line 1: @@
-== Binomial Option Pricing Model =
+== [[SOCR_EduMaterials_ApplicationsActivities | SOCR Applications Activities]] - Binomial Option Pricing Model ==
-* '''Description''':  You can access the portfolio applet at http://www.socr.ucla.edu/htmls/app/ .
+===Description===
-Define: <br>
+===Definitions===
-<math>S_0</math>   Stock price at <math>t=0</math> <br>
+*The value <math>C</math> of a European call option at time <math>t=0</math> is:
-<math>S_1</math>   Stock price at <math>t=1<math> <br>
+: <math> C=S_0 \sum_{j=k}^{n} {n \choose j} p'^{j}(1-p')^{n-j} -
 \frac{E}{e^{rt}} \sum_{j=k}^{n} {n \choose j} p^{j}(1-p)^{n-j}
 </math>
 <br>
 Where, <br>
-<math>
+: <math> u=e^{+\sigma \sqrt{\frac{t}{n}}}, d=e^{-\sigma \sqrt{\frac{t}{n}}}=\frac{1}{u} </math>
 <br>
-<math>
+: <math> p=\frac{e^{rt}-d}{u-d}, \ \ p'=\frac{up}{e^{rt}}
 </math>
 <br>
-<math>S_0</math>    Price of the stock at time <math>t=0</math> <br>
+: <math>S_0</math>    Price of the stock at time <math>t=0</math> <br>
-<math>E</math>      Exercise price at expiration <br>
+: <math>E</math>      Exercise price at expiration <br>
-<math>r</math>      Risk-free interest rate per period <br>
+: <math>r</math>      Risk-free interest rate per period <br>
-<math>n</math>      Number of periods <br>
+: <math>n</math>      Number of periods <br>
-<math>\sigma</math> Annual standard deviation of the returns of the stock <br>
+: <math>\sigma</math> Annual standard deviation of the returns of the stock <br>
-<math>t</math>      &Time to expiration in years <br>
+: <math>t</math>      &Time to expiration in years <br>
-* The SOCR Binomial Option Pricing applet provides the price of the stock and the price of the call at each node.  Note that at expiration the nodes for which the call is in the money (<math>S > E</math> are colored green, while  the nodes for which the call is out of the money (<math>S \le E)</math> are colored blue.  The example below uses the following data: <br>
+* The [http://www.socr.ucla.edu/htmls/app/ SOCR Binomial Option Pricing applet] provides the price of the stock and the price of the call at each node.  Note that at expiration the nodes for which the call is in the money (<math>S > E</math> are colored green, while  the nodes for which the call is out of the money (<math>S \le E)</math> are colored blue.  The example below uses the following data: <br>
-<math>S_0=\$30</math>, <math>E=\$29</math>, <math>r_f=0.05</math>,
+: <math>S_0=\$30</math>, <math>E=\$29</math>, <math>r_f=0.05</math>,
 <math>\sigma=0.30</math>, <math>\mbox{days to expiration}=73</math>,
 <math>\mbox{number of steps}=5</math>.
@@ Line 52: / Line 50: @@
 <center>[[Image: Christou_binomial_applet2.jpg|600px]]</center>
-<br>
+===References===
-* The materials above was partially taken from <br>
+The materials above was partially taken from:
-''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.
-''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.