ABC stock is trading at $100 per share. The stock price will either go up or go down by 25% in each of the next two years. The annual interest rate compounded continuously is 5%.

(i) Determine the price of a two-year European call option with the strike price X = $110.

(ii) Determine the price of a two-year European put option with the strike price X = $110.

(iii) Determine the price of a two-year American put option with the strike price X = $110.

Answer:

u = 1 + 25% = 1.25

d = 1 - 25% = 0.75

p = probability of an up state = (e^r - d) / (u - d) = (e^5% - 0.75) / (1.25 - 0.75) = 60.25%

Stock price tree, European call price tree and put price tree are shown below:

Value of call at any node = max (S - X, 0)

Value of put an any node = max (X - S, 0)

Su = u x S0 = 1.25 x 100 = 125; Sd = d x S0 = 0.75 x 100 = 75

Suu = u x Su = 1.25 x 125 = 156.25; Sud = d x Su = u x Sd = 0.75 x 125 = 93.75; Sdd = d x Sd = 0.75 x 75 = 56.25; X = 110

Part (i) European Call Option Tree:

Price of the European call option = PV of expected value of the call option

= (46.25 x 0.3630571 + 0 + 0)e^{-r x 2}

= 46.25 x 0.3630571 x e^{-2 x 5%}

= $ 15.19

Part (ii)

European Put option Tree:

Price of the European put option = PV of expected value of the put option

= (0 + 16.25 x 0.478970197 + 53.75 x 0.157972709)e^{-r x 2}

= (6.25 x 0.478970197 + 53.75 x 0.157972709) x 0.3630571 x e^{-2 x 5%}

= $ 14.73

Part (iii)

Expected price of put option at the node Su = Pu = e^-r[Puu x p + Pud x (1 - p)] = e^-5%[0 x 60.25% + 16.25 x (1 - 60.25%)] = 6.14 > 0 = payoff from put option if exercised at the node Su

Hence, there will be no exercise at the node Su

At Sd, if put option is exercised, payoff, Pd = max (X - Sd, 0) = max (110 - 75, 0) = 35

Expected value of put option at the node Sd = e^-r[Pud x p + Pdd x (1 - p)] = e^-5%[16.25 x 60.25% + 53.75 x (1 - 60.25%)] = 29.64 < 35

Hence, it makes sense to exercise the put option at the node Sd.

Hence, price of the American put option = e^-r[p x Pu + (1 - p) x Pd] = e^-5% x [60.25% x 6.14 + (1 - 60.25%) x 35] = $ 16.75

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