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22 October, 16:44

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

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  1. 22 October, 16:47
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    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
  2. 22 October, 16:51
    0
    Possible outcome of stock price at end of 6 months (0.5 years)

    Outcome 1:

    Stock price = 35

    Strike price = 45

    Payoff call = max{ST - K, 0} = max{35-45,0} = 0

    Present value =

    PV = 0 / (1+5%) ^0.5 = 0

    Outcome 2:

    Stock price = 49

    Strike price = 45

    Payoff call = max{ST - K, 0} = max{49-45,0} = 4

    Present value =

    PV = 4 / (1+5%) ^0.5 = 3.903

    Probability of both outcomes = 0.5

    Value of call option = 0.5*0 + 0.5*3.903 = 1.95

    Short sale arbitrage opportunity:

    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

    Case 1:

    Stock price = 35

    Payoff from long call = 0

    Buy the stock at market price and close the short stock position = - 35

    Total payoff = 40.629 - 35 = 5.629

    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

    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

    Arbitrage payoff = 1.2277
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