# SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

### From Socr

## Black-Scholes option pricing model - Convergence of binomial

- Black-Scholes option pricing formula:

The value *C* < *m**a**t**h* > *o**f**a**E**u**r**o**p**e**a**n**c**a**l**l**o**p**t**i**o**n**a**t**t**i**m**e* < *m**a**t**h* > *t* = 0 is:

Where,

*S*_{0} 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(*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:
, \ E=\$29,\ R_f=0.05, \sigma=0.30,\
\mbox{Days to expiration}=40</math>.

- 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.