# SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

### From Socr

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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</math>, \ E=\$29,\ R_f=0.05, \sigma=0.30,\ |

\mbox{Days to expiration}=40</math>. <br> | \mbox{Days to expiration}=40</math>. <br> | ||

* For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1). | * 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. | * The snapshot below from the SOCR Black Scholes Option Pricing model applet shows the path of the stock. |

## Revision as of 15:51, 3 August 2008

## 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.