SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

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== Black-Scholes option pricing model - Convergence of binomial ==
== Black-Scholes option pricing model - Convergence of binomial ==
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* '''Description''':  You can access the Black-Scholes applet at http://www.socr.ucla.edu/htmls/app/ .
* Black-Scholes option pricing formula: <br>
* Black-Scholes option pricing formula: <br>
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* The snapshot below from the SOCR Black Scholes Option Pricing model applet shows the path of the stock.
* The snapshot below from the SOCR Black Scholes Option Pricing model applet shows the path of the stock.
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<br>
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<center>[[Image: Christou_black_scholes.jpg|600px]]</center>

Revision as of 15:55, 3 August 2008

Black-Scholes option pricing model - Convergence of binomial

  • Black-Scholes option pricing formula:

The value C < math > ofaEuropeancalloptionattime < math > 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).
  • The snapshot below from the SOCR Black Scholes Option Pricing model applet shows the path of the stock.


File:Christou black scholes.jpg
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