# SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

## 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,\ R_f=0.05, \sigma=0.30,\ \mbox{Days to expiration}=40[/itex].

• 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.