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 er1 = 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
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