SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing
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
\(t<math> Time to expiration in years <br>
<math>\Phi(d_i)\) 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 \(n<math> is large. In the example below we value the call option using the binomial formula for different values of <math>n\) and also using the Black-Scholes formula. We then plot the value of the call (from binomial) against the number of periods \(n<math>. The value of the
call using Black-Scholes remains the same regardless of <math>n\). 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.
