Question

A 2-year American put is written on a stock whose current price is $42. You expect that in each year the stock price either goes up by 15% or decreases by 5%. The one-period interest rate is 5%. The option’s exercise price is $45. Will you ever exercise the option early?

Answer 1

u: up factor = 1+15% = 1.15

d: down factor = 1-5% = 0.95

American put option is exercised as soon as it comes in money ie stock price < strike price

So: current stock price = 42

Strike price = 45

p: probability of up movement

r: risk free rate = 5%

dt: time step = 1 year

p = (e^(r*dt)-d)/(u-d) = 0.506

q: probability of down movement = 0.494

T=0	T=1	T=2	Probability	Put-option pay-off
		48.3*1.15 = 55.545	2-up movements = p*p = 0.5064	max(K-S,0) = max(45-55.545) = 0
	So*u = 42*1.15 = 48.3
So = 42		48.3*0.95 = 45.885	1up&1down = 2*p*q = 0.5	max(45-45.885,0) = 0
	So*d = 42*0.95 = 39.9
		39.9*0.95 = 37.905	2-down movements = q*q = 0.2437	max(45-37.905,0) = 7.095

Value of put option at (T=2) = Sum (probability*pay-off) = 0.2437*7.095 = 1.73

Value of put option at (T=0) = 1.73*e^(-5%*2) = 1.564 (price of put-option)

Pay-off from early exercise of put-option at (t=1) = max(45-39.9,0) = 5.1

Since 5.1>1.73, it is advisable to exercise put option early at T=1 if the stock price falls