Question

Suppose a stock is priced today at 40 and can go in one year to either 50 or 30. Assume the one-year risk free rate is 0%.

What is the fair value of a one-year call option on the stock with a strike of 42?
Assuming the expected rate of return on the stock is 10%, what is the "real world" probability of an up move in the stock?
What is the expected return on the call option?

Answer 1

a)

Payoff from the call option if the stock ends up at 50 = 50-42 = 8

Payoff from the call option if the stock ends up at 30 = 0

Expected payoff at maturity = 0.5*8 + 0.5*0 = 4

Expected payoff today = 4*e^(-0*1) = 4

Fair value of a one-year call option on the stock with a strike of 42 = 4

b)

The size of the up-move U = 50/40 = 1.25

The size of the down-move D = 30/40 = 0.75

Real world up-move probaility = (e^(q*t) -D) / (U-D) = (e^(0.1*1) -0.75) / (1.25-0.75)

Real world up-move probaility = 0.71034

c) Value of the call option today = 4

Expected payoff = Probability of up-move* Payoff on up-move = 0.71034*8 = 5.68272

Expected return = (5.68272-4)/4 = 0.42068 = 42%