# Difference between revisions of "SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing"

## SOCR Applications Activities - Black-Scholes Option Pricing Model (with Convergence of Binomial)

### Description

You can access the Black-Scholes Option Pricing Model applet at the SOCR Applications Site, select Financial Applications --> BlackScholesOptionPricing.

### Black-Scholes option pricing formula

The value $$C$$ of a European call option at time $$t=0$$ 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).

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### References

The materials above was partially taken from:

• Modern Portfolio Theory by Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, and William N. Goetzmann, Sixth Edition, Wiley, 2003.
• Options, Futues, and Other Derivatives by John C. Hull, Sixth Edition, Pearson Prentice Hall, 2006.
• SOCR Applications Site