Question

please answer question 3

3. A stock has two possible ending prices six months from now: $ 45 or $60. A call option written on this stock has an exercise price of $48. The option expires in six months. The risk-free rate is 4% per year. The current price of the stock is SSO. What is the equilibrium price of the call option on this stock? Suppose you find this call option trading at $3.00, describe an arbitrage strategy you can use to take advantage of the mispricing and calculate your arbitrage profit per share used in the strategy. 

Answer 1

Q - 3

S0 = 50; Su = 60; Sd = 45; u = Su / S0 = 60 / 50 = 1.20 ; d = Sd / S0 = 45 / 50 = 0.9; K = 48; r = 4%

p = probability of an up state = (1 + r - d) / (u - d) = (1 + 4% - 0.9) / (1.2 - 0.9) =  0.4667

Cu = max (Su - K, 0) = max (60 - 48, 0) = 12; Cd = max (Sd - K, 0) = max (45 - 48, 0) = 0

time = t = 6 months = 0.5 years

Hence, equilibrium price of the call option = C = [p x Cu + (1 - p) x Cd] / (1 + r)^-t = [0.4667 x 12 + (1 - 0.4667) x 0] / (1 + 4%)^0.5 = $  5.49

If the call option is actually trading at a different price than this, there will be an arbitrage opportunity in the market. In this case, actual trading price = $ 3, which is lower than the calculated price. Hence, the arbitrage strategy should be to short the synthetic call and buy the actual call. Hence the trading arbitrage strategy should be as shown below:

Sl. No.	Arbitrage action	Cash flows at t= 0	Cash flows at maturity
			If S = Sd = 45	If S = Su = 60
1.	Short the stock	+ S0 = + 50	Buy the stock and cover the short position; Cash flows = - 45	Use the stock from exercise of call option to close the short position
2.	Buy the call	- 3	Don't exercise the call	Exercise the call; Cash flow = - 48
3.	Lend PV of at risk free rate for 6 months	- 47	+ 47 x (1 + 4%)0.5 = + 47.93	+ 47 x (1 + 4%)0.5 = + 47.93
	Total	0	$ 2.93	~0

Thus you end up in a situation where there is no initial investment but there is a nearly riskless return of 0 or $ 2.93. This is the arbitrage. And the arbitrage per share = $ 2.93