Fin 441 Chapter 4 Homework

Daniel Kalmikov

FIN 441

HW #2

May 7, 2019

Homework (#2)

1.        Consider a stock currently trading at $25 that can go up or down by 15% per period. The risk-free rate is 10 percent. Use one-period binomial model.

a.        Determine the two possible stock prices for the next period. 

        Possibility 1: stock price goes up 15%: Su = 25 × 1.15 = 28.75

        Possibility 2: stock price goes down 15%:Sd = 25 × 0.85 = 21.25

b.         Determine the intrinsic values at expiration of a European call

        with an exercise price of $25.

        Possibility 1: stock goes up 15%          Cu = Max(0, Su – X) = 28.75 – 25 = 3.75

        Possibility 2: stock goes down 15%         Cd = Max(0, Sd – X) = 21.25 – 25 = 0

c.        Find the value of the option today. 

        p = [(1 + r) – d] ÷ (u – d) =(1.10 – 0.85) ÷ (1.15 – 0.85) = 0.8333

        🡪 1 – p = 1 - 0.8333 = 0.1667

        C = {(p × Cu) + [(1 – p) × Cd]} ÷ (1 + r) 

    = [(0.8333 × 3.75) + (0.1667 × 0)] ÷ 1.10 = 2.84

d.         Construct a hedge by combining a position in stock with a position

in the call. Calculate the hedge ratio and show that the return on the hedge portfolio is the risk-free rate regardless of the outcome, assuming that the call trading at the price obtained in part c.

h = (Cu – Cd) ÷ (Su – Sd) =(3.75 – 0) ÷ (28.75 – 21.25) = 0.50

Hedge: buy 500 shares & sell 1000 calls

        V = (h × S) – C =(500 × 25) – (1000 × 2.84) = 9660

        Vu = (h × Su) – Cu =(500 × 28.75) – (1000 × 3.75) = 10,625

        Vd = (h × Sd) – Cd =(500 × 21.75) – (1000 × 0) = 10,625

        Rh = (Vu ÷ V) – 1 =(10,625 ÷ 9660) – 1 = 0.099896 ≈ 10% risk-free rate