Consider a binomial model with u = 1:2 and d = 0:9, the risk-free interest

rate is 10%, and the current stock price is 100.

a. What is the risk-neutral probability of the down state Sd = dS0?

b. What is the hedge ratio (delta) of a European put option with an exercise price

X = 100 and n = 1 periods left until expiry?

c. Calculate the price of a European call option with an exercise price X = 100 and

n = 2 periods left until expiry.

d. Calculate the price of a European put option with an exercise price X = 100 and

n = 2 periods left until expiry. (Hint: X needs to be discounted for 2 periods in the

put-call parity.)

e. A buttery spread is the purchase of one call at exercise price X1, the sale of two

calls at exercise price X2, and the purchase of one call at exercise price X3. X1 is less

than X2, and X2 is less than X3 by equal amounts, and all calls have the same expiration

date.

Suppose X1 = 80, X2 = 100, and X3 = 120. What is the payo of this butter

spread if ST = 95? What is the payo if ST = 115?

f. Use the Black-Scholes formula to nd the value of a put option on the following

stock:

Time to expiration = 1 year

Standard deviation = 20% per year

Exercise price = $20

Stock price = $20

Interest rate = 0%

Dividend = 0

You can use N(d1) = 0:54 and N(d2) = 0:46 in your calculation