A stock is selling for $41.60. The strike price on a call, maturing in 6 months, is $45. The possible stock prices at the end of 6 months are $35.00 and $49.00. Interest rates are 5.0%. Given an underpriced option, what are the short sale proceeds in an arbitrage strategy

The short sale proceeds in an arbitrage strategy is 1.2277

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

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