Question

eBook Problem 8-07 Binomial Model The current price of a stock is $16. In 6 months, the price will be either $20 or $12. The annual risk-free rate is 3%. Find the price of a call option on the stock that has an strike price of $15 and that expires in 6 months. (Hint: Use daily compounding.) Round your answer to the nearest cent. Assume a 365-day year. Do not round your intermediate calculations

Answer 1

P = $16

P(u) = $20

P(d) = $12

rRF = 3%

X = $15

Cu ending up option payoff = Max[(20 - 15), 0] = $5

Cd ending down option payoff = Max[(12 - 14), 0] = $0

Share of stock (Ns) = [Cu - Cd] / [P(u) - P(d)]

= [$5 - $0] / [$20 - $12] = 5 / 8 = 0.625

Hedge portofolio’s payoff if stock is up = [Ns * P(u)] - Cu

= [0.625 x ($20)] - $5= $7.5

Hedge portofolio’s payoff if stock is down = [Ns * P(d)] - Cd

= [0.625($12)] - $0 = $7.5

PV of riskless payoff = $7.5 / [{1 + (rRF / 365)}^365(t/n)]

= $7.5 / [{1 + (0.03 / 365)}^365(0.5/1) = $7.5 / 1.0151 = $7.39

Option’s Value (VC) = [Ns x P] - Present value of riskless payoff

= [0.625 x $16] - $7.39 = $2.61