# SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

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

• Black-Scholes option pricing formula:

The value $$C[itex] of a European call option at time [itex]t=0$$ 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
$$t[itex] Time to expiration in years <br> [itex]\Phi(d_i)$$ 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 $$n[itex] is large. In the example below we value the call option using the binomial formula for different values of [itex]n$$ and also using the Black-Scholes formula. We then plot the value of the call (from binomial) against the number of periods $$n[itex]. The value of the call using Black-Scholes remains the same regardless of [itex]n$$. 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.