Question

5. Consider the 3-period binomial model with So 100, u 2, dand r-1. (a) What is the current price of a lookback call option with a strike price of $100 that pays off (at time three) V3- max Sn - 100 Sn3 (b) What is the time-zero price of a lookback put option with a strike price of $100 that pays off (at time three) V 100-min Sn OSnK3 (c) What is the time-zero price of an Asian call option with a strike price of S100 whose payoff (at time three) is V- Ys - 100 4 - -0 Sk is the sum of stock prices between time 0 and time 3.

Answer 1

Let's draw the possible state of prices at the end of each period. The table below summarises different possible price of the underlying stock at the end of different period. The symbols have the same meaning as they have in the question.

Period 0	Period 1	Period 2	Period 3
S0 = 100.00	Su = u.S0 = 200.00	Suu = u.Su = 400.00	Suuu = u.Suu = 800.00
	Sd = d.S0 = 50.00	Sud = d.Su = u.Sd = 100.00	Suud = d.Suu = u.Sud = 200.00
		Sdd = d.Sd = 25.00	Sddu = u.Sdd = d.Sud = 50.00
			Sddd = d.Sdd =12.50

Maximum stock price over the life, max S_N = S_uuu = 800

Minimum stock price over the life, min S_N = S_ddd = 12.50

Sum of stock prices between t=0 to t=3 = sum of all the prices in the table above = Y₃ = 1,937.50

Discount rate, r = 1/4

Part (a)

V₃ = max S_N - 100 =  800 - 100 = 700

Hence, value of the call option today = PV (V₃) = V₃ / (1 + r)³ = 700 / (1 + 1/4)³ = 358.40

Part (b)

V₃ = 100 - min S_N = 100 - 12.50 = 87.50

Hence, value of the put option today = PV (V₃) = V₃ / (1 + r)³ = 87.50 / (1 + 1/4)³ = 44.80

Part (c)

V₃ = 1 / 4 x Y₃ - 100 = 1/4 x 1,937.50 - 100 = 384.38

Hence, value of the put option today = PV (V₃) = V₃ / (1 + r)³ = 384.38 / (1 + 1/4)³ = 196.80