Question

A stock that does not pay dividend is trading at $100. The stock price will increase by 10% or decrease by 10% in one year. After that, the stock price will increase or decrease by 20% in the second year. The risk-free interest rate is 5% per annum with continuous compounding. Value a two-year American put option with strike price of $102. Note that risk-neutral probabilities or replicating portfolios may differ across two periods. You must also check for early exercise.

Answer 1

American Put
Current Stock Price (S0)	100	Given
U1	1.1	Given
D1	0.9	Given
U2	1.2	Given
D2	0.8	Given
Risk-Free Rate(r)	0.05	Given
Strike Price (X)	102	Given
Stock Price at (S1)	110	=S0*U1
Stock Price at (S2)	90	=S0*D1
Stock Price at (S3)	132	=S1*U2
Stock Price at (S4)	88	=S1*D2
Stock Price at (S5)	108	=S2*U2
Stock Price at (S6)	72	=S2*D2
Risk Neutral Probabilities:
q2 (for Second Period)	0.625	=((1+r)-D2)/(U2-D2)
1-q2 (for Second Period)	0.375	=1-q2
q1 (for First Period)	0.75	=((1+r)-D1)/(U1-D1)
1-q1 (for First Period)	0.25	=1-q1
Payoffs:
Payoff at S3 (s3)	0	=MAX(0,X-S3)
Payoff at S4 (s4)	14	=MAX(0,X-S4)
Payoff at S5 (s5)	0	=MAX(0,X-S5)
Payoff at S6 (s6)	30	=MAX(0,X-S6)
Payoff at S1 (s1)	5	=MAX((X-S1),((q2*s3)+({1-q2}*s4)/(1+r)))
Payoff at S2 (s2)	12	=MAX((X-S2),((q2*s5)+({1-q2}*s6)/(1+r)))
Payoff at S0 (s0)	6.61	=((q1*s1)+({1-q1}*s2)/(1+r))
Option Price at S0	6.61	=s0