SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

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== Black-Scholes option pricing model - Convergence of binomial ==
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== [[SOCR_EduMaterials_ApplicationsActivities | SOCR Applications Activities]] - Black-Scholes Option Pricing Model (with Convergence of Binomial) ==
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* '''Description''':  You can access the Black-Scholes applet at http://www.socr.ucla.edu/htmls/app/ .
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===Description===
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You can access the Black-Scholes Option Pricing Model applet at [http://www.socr.ucla.edu/htmls/app/ the SOCR Applications Site], select ''Financial Applications'' --> ''BlackScholesOptionPricing''.
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* Black-Scholes option pricing formula: <br>
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===Black-Scholes option pricing formula===
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:
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<math>
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: <math> C=S_0 \Phi (d_1) - \frac{E}{e^{rt}} \Phi(d_2) </math>
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C=S_0 \Phi (d_1) - \frac{E}{e^{rt}} \Phi(d_2)
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: <math> d_1=\frac{ln(\frac{S_0}{E})+(r+\frac{1}{2} \sigma^2)t} {\sigma \sqrt{t}}
</math>
</math>
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<br>
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: <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>
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<math>
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d_1=\frac{ln(\frac{S_0}{E})+(r+\frac{1}{2} \sigma^2)t}
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{\sigma \sqrt{t}}
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</math>
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<br>
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<math>
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d_2=\frac{ln(\frac{S_0}{E})+(r-\frac{1}{2} \sigma^2)t}
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{\sigma \sqrt{t}}=d_1-\sigma \sqrt{t}
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</math>
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<br> 
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Where, <br>
Where, <br>
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<math>S_0</math>    Price of the stock at time <math>t=0</math> <br>
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: <math>S_0</math>    Price of the stock at time <math>t=0</math> <br>
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<math>E</math>      Exercise price at expiration <br>
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: <math>E</math>      Exercise price at expiration <br>
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<math>r</math>      Continuously compounded risk-free interest <br>
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: <math>r</math>      Continuously compounded risk-free interest <br>
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<math>\sigma</math> Annual standard deviation of the returns of the stock <br>
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: <math>\sigma</math> Annual standard deviation of the returns of the stock <br>
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<math>t</math>      Time to expiration in years <br>
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: <math>t</math>      Time to expiration in years <br>
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<math>\Phi(d_i)</math>  Cumulative probability at <math>d_i</math> of the standard normal distribution <math>N(0,1)</math>  <br>
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: <math>\Phi(d_i)</math>  Cumulative probability at <math>d_i</math> of the standard normal distribution <math>N(0,1)</math>  <br>
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* Binomial convergence to Black-Scholes option pricing formula: <br>
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===Binomial convergence to Black-Scholes option pricing formula===
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The binomial formula converges to the Black-Scholes formula when
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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:
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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
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: <math>S_0=\$30</math>, <math>E=\$29 </math>, <math>R_f=0.05</math>, <math>\sigma=0.30 </math>,  
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call using Black-Scholes remains the same regardless of <math>n</math>.  The data used for this example are:
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<math>S_0=\$30</math>, <math>E=\$29 </math>, <math>R_f=0.05</math>, <math>\sigma=0.30 </math>,  
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<math>\mbox{Days to expiration}=40</math>. <br>
<math>\mbox{Days to expiration}=40</math>. <br>
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* For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1).
* For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1).
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* The snapshot below from the SOCR Black Scholes Option Pricing model applet shows the path of the stock.
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* 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>
<br>
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<center>[[Image: Christou_black_scholes_binomial.jpg|600px]]</center>
<center>[[Image: Christou_black_scholes_binomial.jpg|600px]]</center>
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<br>
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===References===
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* The materials above was partially taken from <br>
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The materials above was partially taken from:
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''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003, and <br>
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* ''Modern Portfolio Theory'' by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003.
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''Options, Futues, and Other Derivatives'' by John C. Hull, Sixth Edition, Pearson Prentice Hall, 2006.
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* ''Options, Futues, and Other Derivatives'' by John C. Hull, Sixth Edition, Pearson Prentice Hall, 2006.
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* [http://www.socr.ucla.edu/htmls/app/ SOCR Applications Site]
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{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_EduMaterials_Activities_ApplicationsActivities_BlackScholesOptionPricing}}

Revision as of 18:55, 3 August 2008

Contents

SOCR Applications Activities - Black-Scholes Option Pricing Model (with Convergence of Binomial)

Description

You can access the Black-Scholes Option Pricing Model applet at the SOCR Applications Site, select Financial Applications --> BlackScholesOptionPricing.

Black-Scholes option pricing formula

The value C of a European call option at time t = 0 is:

C=S_0 \Phi (d_1) - \frac{E}{e^{rt}} \Phi(d_2)
d_1=\frac{ln(\frac{S_0}{E})+(r+\frac{1}{2} \sigma^2)t} {\sigma \sqrt{t}}
d_2=\frac{ln(\frac{S_0}{E})+(r-\frac{1}{2} \sigma^2)t} {\sigma \sqrt{t}}=d_1-\sigma \sqrt{t}

Where,

S0 Price of the stock at time t = 0
E Exercise price at expiration
r Continuously compounded risk-free interest
σ Annual standard deviation of the returns of the stock
t Time to expiration in years
Φ(di) Cumulative probability at di of the standard normal distribution N(0,1)

Binomial convergence to Black-Scholes option pricing formula

The binomial formula converges to the Black-Scholes formula when the number of periods n is large. In the example below we value the call option using the binomial formula for different values of n and also using the Black-Scholes formula. We then plot the value of the call (from binomial) against the number of periods n. The value of the call using Black-Scholes remains the same regardless of n. The data used for this example are:

S_0=\$30, E=\$29 , Rf = 0.05, σ = 0.30,

Days to expiration = 40.

  • For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1).


References

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.
  • Options, Futues, and Other Derivatives by John C. Hull, Sixth Edition, Pearson Prentice Hall, 2006.
  • SOCR Applications Site

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