2. Suppose some stock currently selling for $80 will either increase in value over the next year to $100, or decrease in value to $64. The risk free rate over the period is 10% given annual compounding. [Let* r* denote the continuously compounded rate per year. Thus* er×*1 = 1*.*1.] A **European** call option on the stock with an exercise price of $75 matures in one period (1 year). If you want to price the option with a **one-step** binomial tree.

**1.** **What are ****u**** and ****d****?**

*u *= $100/$80 = 1.25

*d = *$64/%80 = 0.80

**2.** **What are the payoffs from the call in each state of the world?**

If stock rises to $100 next year, the payoff will be $25 ($100 – $75). If stock decreases to $64 next year, the payoff will be $0 ($64 – $75).

**3.** **What is the European call price at time 0?**

= (c1+r)-D/(U-D)

= (1.1 – 0.8) / (1.25 – 0.8)

= .6666 (probability of up)

Call price at t0 = (.6666 * 25 + 0) / 1.1 = **$15.15**

**4.** **What are the pseudoprobabilities of the up and down movements in the stock price?**

= (c1+r)-D/(U-D)

= (1.1 – 0.8) / (1.25 – 0.8)

**= 66.66% probability of up**

= (1 – .6666)

**= 33.33% probability of down**