Assign1correction - STAT/ACTSC 446/846...

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STAT/ACTSC 446/846 Assignment #1 (Solution) Exercise 2.5: Solution 1. The payoff to a short forward at expiration is equal to: Payoff to short forward = forward price - spot price at expiration Therefore, we can construct the following table: Price of asset in 6 months Agreed forward price Payoff to the short forward 40 50 10 45 50 5 50 50 0 55 50 - 5 60 50 - 10 2. The payoff to a purchased put option at expiration is: Payoff to put option = max[0 , strike price - spot price at expiration] The strike is given: It is $50. Therefore, we can construct the following table: Price of asset in 6 months Strike price Payoff to the call option 40 50 10 45 50 5 50 50 0 55 50 0 60 50 0 3. If we compare the 2 contracts, we see that the put option has a protection for increases in the price of the asset: If the spot price is above $50, the buyer of the put option can walk away, and need not incur a loss. The buyer of the short forward incurs a loss and must meet her obligations. However, she has the same payoff as the buyer of the put option if the spot price is below $50. Therefore, the put option should be more expensive. It is this attractive option to walk away if things are not as we want that we have to pay for. ——————————————————– 1
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Exercise 3.7: Solution Payoff equivalence Let us first consider the payoff of (a). If we short the index (let us name it S), we have to pay at the time of expiration T of the options: - S T . The payoffs of part (b) are more complicated. Let us look again at each region separately, and hope to be able to draw a conclusion when we look at the aggregate position. We have for the payoffs in (b): Instrument S T < K S T K Make repayment of loan - $1029 . 41 × 1 . 02 = - $1050 - $1029 . 41 × 1 . 02 = - $1050 Short Call Option - max( S T - 1050 , 0) = 0 - max( S T - 1050 , 0) = 1050 - S T Long Put Option max($1050 - S T , 0) 0 = $1050 - S T Total - S T - S T We see that the total aggregate position gives us - S T , no matter what the final index value is—but this is the same payoff as part (a). Our proof for the payoff equivalence is complete. Profit equivalence Now let us turn to the profits. If we sell the index today, we receive money that we can lend out. Therefore, we can lend $1,000, and be repaid $1,020 after one year. The profit for part (a) is thus: $1 , 020 - S T . The profits of the aggregate position in part (b) are the payoffs, less the future value of the put premium plus the future value of the call premium (because we sold the call), and less the future value of the loan we gave initially. Note that we know already that a risk-less bond is canceling out of the profit calculations. We can again tabulate: Instrument S T < K S T K Make repayment of loan - $1029 . 41 × 1 . 02 = - $1050 - $1029 . 41 × 1 . 02 = - $1050 Future value of borrowed $1050 $1050 money Short Call Option - max( S T - 1050 , 0) = 0 - max( S T - 1050 , 0) = 1050 - S T Future value of premium $71 . 802 × 1 . 02 = $73 . 24 $71 . 802 × 1 . 02 = $73 . 24 Long Put Option max($1050 - S T , 0) = 0 $1050 - S T Future value of premium - $101 . 214 × 1 . 02 = - $103 . 24 - $101 . 214 × 1 . 02 = - $103 . 24 Total $1 , 020 - S T $1 , 020 - S T Indeed, we see that the profits for part (a) and part (b) are identical as well.
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