Answer:
The short sale proceeds in an arbitrage strategy is 1.2277
Step-by-step explanation:
From the question given,
The Possible outcome of stock price at end of 6 months (0.5 years)
The Outcome is:
The Stock price = 35
The Strike price = 45
The Payoff call = max(ST - K,0) = max(35-45,0) = 0
The Present value = PV = 0/(1+5%)^0.5 = 0
The possible Outcome 2:
The Stock price = 49
The Strike price = 45
The Payoff call = max{ST - K,0} = max{49-45,0} = 4
The Present value =
PV = 4/(1+5%)^0.5 = 3.903
Then,
The Probability of both outcomes = 0.5
Value of call option = 0.5*0 + 0.5 x 3.903 = 1.95
Therefore, the Short sale arbitrage opportunity is:
The Short the stock and buy a call option.
Invest the proceeds at 5% for 6 months:
Short stock = +41.6
long call = -1.95
Proceeds = 41.6 - 1.95 = 39.65
Amount after 6 months = 39.65*(1+5%)^0.5 = 40.629
The Case 1:
Stock price = 35
Payoff from long call = 0
Buy the stock at market price and close the short stock position = -35
The Total payoff = 40.629 - 35 = 5.629
For Case 2:
Stock price = 49
Payoff from long call = 49 - 45 = 4
Buy the stock from market price and close the short stock position = -49
Total payoff = 40.629 + 4 - 49 = -4.3708
The Present value of payoff from both cases = (0.5*5.629 + 0.5*(-4.3708))/(1+5%)^0.5
= 1.2581/1.0246 = 1.2277
Then the Arbitrage payoff = 1.2277