SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing
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- | == Black-Scholes | + | == [[SOCR_EduMaterials_ApplicationsActivities | SOCR Applications Activities]] - Black-Scholes Option Pricing Model (with Convergence of Binomial) == |
- | + | ===Description=== | |
+ | 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''. | ||
- | + | ===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: | ||
- | <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> | </math> | ||
- | + | : <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> | + | |
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- | d_2=\frac{ln(\frac{S_0}{E})+(r-\frac{1}{2} \sigma^2)t} | + | |
- | {\sigma \sqrt{t}}=d_1-\sigma \sqrt{t} | + | |
- | </math> | + | |
- | + | ||
Where, <br> | 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=== | |
- | 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>, |
- | call using Black-Scholes remains the same regardless of <math>n</math>. The data used for this example are: | + | |
- | <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> | <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). | * 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> | <br> | ||
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<center>[[Image: Christou_black_scholes_binomial.jpg|600px]]</center> | <center>[[Image: Christou_black_scholes_binomial.jpg|600px]]</center> | ||
- | + | ===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 | + | * ''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. |
+ | * [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_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:
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:
- , , 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).
- The snapshot below from the SOCR Black Scholes Option Pricing model applet shows the path of the stock.
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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