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