SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing - Revision history

Nchristo: /* Binomial convergence to Black-Scholes option pricing formula */

2008-08-03T19:55:28Z

‎Binomial convergence to Black-Scholes option pricing formula

Nchristo: /* Binomial convergence to Black-Scholes option pricing formula */

2008-08-03T19:53:15Z

‎Binomial convergence to Black-Scholes option pricing formula

IvoDinov at 19:01, 3 August 2008

2008-08-03T19:01:17Z

Nchristo at 16:19, 3 August 2008

2008-08-03T16:19:45Z

Nchristo at 16:19, 3 August 2008

2008-08-03T16:19:15Z

Nchristo at 16:02, 3 August 2008

2008-08-03T16:02:34Z

Nchristo at 16:01, 3 August 2008

2008-08-03T16:01:58Z

Nchristo at 15:58, 3 August 2008

2008-08-03T15:58:53Z

Nchristo at 15:58, 3 August 2008

2008-08-03T15:58:26Z

Nchristo at 15:57, 3 August 2008

2008-08-03T15:57:07Z

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 * For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1).
-* The snapshot below from the [http://www.socr.ucla.edu/htmls/app/ SOCR Black Scholes Option Pricing model applet] shows the convergence of the call price calculated by the binomial option pricing model to the price of the call calculated using the Black-Schole model.
+* The snapshot below from the [http://www.socr.ucla.edu/htmls/app/ SOCR Black Scholes Option Pricing model applet] shows the convergence of the call price calculated by the binomial option pricing model to the price of the call calculated using the Black-Scholes model.
 <br>

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 * For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1).
-* The snapshot below from the [http://www.socr.ucla.edu/htmls/app/ SOCR Black Scholes Option Pricing model applet] shows the path of the stock.
+* The snapshot below from the [http://www.socr.ucla.edu/htmls/app/ SOCR Black Scholes Option Pricing model applet] shows the convergence of the call price calculated by the binomial option pricing model to the price of the call calculated using the Black-Schole model.
 <br>

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-== Black-Scholes option pricing model - Convergence of binomial ==
+== [[SOCR_EduMaterials_ApplicationsActivities | SOCR Applications Activities]] - Black-Scholes Option Pricing Model (with Convergence of Binomial) ==
-* '''Description''':  You can access the Black-Scholes applet at http://www.socr.ucla.edu/htmls/app/ .
+===Description===
-* Black-Scholes option pricing formula: <br>
+===Black-Scholes option pricing formula===
 The value <math>C</math> of a European call option at time <math>t=0</math> is:
-<math>
+: <math> C=S_0 \Phi (d_1) - \frac{E}{e^{rt}} \Phi(d_2) </math>
-C=S_0 \Phi (d_1) - \frac{E}{e^{rt}} \Phi(d_2)
+: <math> d_1=\frac{ln(\frac{S_0}{E})+(r+\frac{1}{2} \sigma^2)t} {\sigma \sqrt{t}}
 </math>
-<br>
+: <math> d_2=\frac{ln(\frac{S_0}{E})+(r-\frac{1}{2} \sigma^2)t} {\sigma \sqrt{t}}=d_1-\sigma \sqrt{t} </math>
-<math>
 Where, <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>      Continuously compounded risk-free interest <br>
+: <math>r</math>      Continuously compounded risk-free interest <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>
-<math>\Phi(d_i)</math>   Cumulative probability at <math>d_i</math> of the standard normal distribution <math>N(0,1)</math>  <br>
+: <math>\Phi(d_i)</math>   Cumulative probability at <math>d_i</math> of the standard normal distribution <math>N(0,1)</math>  <br>
-* Binomial convergence to Black-Scholes option pricing formula: <br>
+===Binomial convergence to Black-Scholes option pricing formula===
-The binomial formula converges to the Black-Scholes formula when
+The binomial formula converges to the Black-Scholes formula when the number of periods <math>n</math> is large.  In the example below we value the call option using the binomial formula for different values of <math>n</math> and also using the Black-Scholes formula.  We then plot the value of the call (from binomial) against the number of periods <math>n</math>.  The value of the call using Black-Scholes remains the same regardless of <math>n</math>.  The data used for this example are:
-the number of periods <math>n</math> is large.  In the example below we value the call option using the binomial formula for different values of <math>n</math> and also using the Black-Scholes formula.  We then plot the value of the call (from binomial) against the number of periods <math>n</math>.  The value of the
+: <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}=40</math>. <br>
 * For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1).
-* The snapshot below from the SOCR Black Scholes Option Pricing model applet shows the path of the stock.
+* The snapshot below from the [http://www.socr.ucla.edu/htmls/app/ SOCR Black Scholes Option Pricing model applet] shows the path of the stock.
 <br>
@@ Line 41: / Line 32: @@
 <center>[[Image: Christou_black_scholes_binomial.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.

@@ Line 4: / Line 4: @@
 * Black-Scholes option pricing formula: <br>
-The value <math>C<math> of a European call option at time <math>t=0</math> is:
+The value <math>C</math> of a European call option at time <math>t=0</math> is:
 <math>
 C=S_0 \Phi (d_1) - \frac{E}{e^{rt}} \Phi(d_2)

← Older revision		Revision as of 16:19, 3 August 2008
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	<center>[[Image: Christou_black_scholes_binomial.jpg\|600px]]</center>		<center>[[Image: Christou_black_scholes_binomial.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.

← Older revision		Revision as of 16:02, 3 August 2008
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	<br>		<br>

−	<center>[[Image: ~~Christou_black_scholes~~.jpg\|600px]]</center>	+	<center>[[Image: Christou_black_scholes_binomial.jpg\|600px]]</center>