Question

A stock is currently selling at $100, with a 75% chance of increasing by 25% and a 25% chance of decreasing by 20% each year. The risk-free interest rate is 5% per year (with annual compounding). Assume that the stock will not pay any dividend for the next two years. Consider a put option on this stock, with an exercise price of $105 and two years to maturity. Use a binomial model with two time periods – year one and year two.

a. What is the value of the put if it is European?

b. What is the value of the put if it is American?

Explain and show your work

Answer 1

K strike price = 105

r: risk free rate = 5% = 0.05

dt: lenght of time step = 1 year

u: up factor = 1 +25% = 1.25

d: down factor = 1 - 20% = 0.8

d = 1/1.424 = 0.702

p: probability of up movement = 75% = 0.75

q: probability of down movement

q = 25% = 0.25

Put pay-off = Max (K - S,0)

T = 0	T = 1	T = 2	Put Payoff	Probability
		125*u = 156.25	Max (105 - 156.25,0) = 0	p*p (2 consecutive ups) = 0.75^2 = 0.5625
	100 * u = 100*1.25 = 125
Stock price = 100		125*d = 100	Max (105 - 100,0) = 5	2*p*q (1 up 1 down in any sequence) = 0.375
	100 * 0.8 = 100*0.8 = 80
		80*d = 64	Max (105 - 64,0) = 41	q*q (2 consecutive downs) = 0.0625

Payoff put option at (T =2) = Sum[Probability*Payoff] = 0.5625*0 + 0.375*5 + 0.0625*41 = 4.4375

Present value of payoff (by discounting for 2 year) = e^(-0.05*2) * 4.4375 = 4.015 (value of european put option)

American Put Options are excersised as soon as they become in the money

When T = 1

If stock price = 80

Put pay-off = max(105 - 80, 0) = 25

When T = 1

If stock price = 125 (put option will not be exercised)

Value of put option = Present value of future pay-off

Value of put option = e^(-0.05*1)*(75%*0 + 25%*5)

**in the above equation: the exponential terms is discount factor for 1 year. The second term: 75% chance that option pay-off is zero & 25% chance option payoff is 5)

Value of put option = 1.189

Finding value of put option at T = 0

Discounted value of put options value off at T = 1

Put value = e^(-0.05*1)*(75%*1.189 + 25%*25)

**in the above equation: the exponential terms is discount factor for 1 year. The second term: 75% chance that option value is 1.189 & 25% chance option payoff is 25)

Put value = 6.793 (value of american put option)