# SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

The value $$C$of a European call option at time [itex]t=0$$ 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 $$t$$ 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$.