Question

Binominal Option Pricing in One Period

The simple riskfree rate is 10%. The current stock price is 60. In one period, the underlying stock price may go up to 90 or down to 54.

What are the payoffs of the European put option with K=72 in the up and down states?

What are the risk-neutral probabilities for the two states in one period?

What is the price of the European put today?

Would the value of the corresponding American put option be the same?

Answer 1

What are the payoffs of the European put option with K=72 in the up and down states?

Payoff in the up state, Pu = max (K - Su, 0) = max (72 - 90, 0) = 0

Payoff in the down state, Pd = max (K - Sd, 0) = max (72 - 54, 0) = 18

What are the risk-neutral probabilities for the two states in one period?

Probability of an up state, p = (S0e^rt - Sd) / (Su - Sd) = (60e^{10% x 1} - 54) / (90 - 54) = 34.20%

Probability of a dwon state = 1 - p = 1- 34.20 = 65.80%

What is the price of the European put today?

Price of the put optiion today = [p x Pu + (1 - p) x Pd]e^-rt = [34.20% x 0 + 65.80% x 18]e^{-10% x 1} = 10.72

Would the value of the corresponding American put option be the same?

No, the price of thr American Put option has to be different from this. An American put option has an additional flexibity if being exercised earlier than its maturity date, and hence it must carry a premium for this flexibility. The price of the American put option should therefore be different from that of a European option.