Difference between revisions of "SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing"
(New page: == Black-Scholes option pricing model - Convergence of binomial == * Black-Scholes option pricing formula: <br> The value <math>C<math> of a European call option at time <math>t=0</math> ...) |
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| Line 22: | Line 22: | ||
<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: <br> | ||
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 | + | 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: | call using Black-Scholes remains the same regardless of <math>n</math>. The data used for this example are: | ||
<math>S_0=\$30, \ E=\$29,\ R_f=0.05, \sigma=0.30,\ | <math>S_0=\$30, \ E=\$29,\ R_f=0.05, \sigma=0.30,\ | ||
Revision as of 10:56, 3 August 2008
Black-Scholes option pricing model - Convergence of binomial
- Black-Scholes option pricing formula:
The value \(C<math> of a European call option at time <math>t=0\) 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,
\(S_0\) Price of the stock at time \(t=0\)
\(E\) Exercise price at expiration
\(r\) Continuously compounded risk-free interest
\(\sigma\) Annual standard deviation of the returns of the stock
\(t\) Time to expiration in years
\(\Phi(d_i)\) Cumulative probability at \(d_i\) 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\[S_0=\$30, \ E=\$29,\ R_f=0.05, \sigma=0.30,\
\mbox{Days to expiration}=40\].
- 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.
