Solution of Derivative

A symmetric butterfly spread cannot have a premium of

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Unformatted text preview: d mean that two calls (or puts) with different strikes have the same price – and we know by now that two instruments that have different payoff structures and the same underlying risk cannot have the same price without creating an arbitrage opportunity. A symmetric butterfly spread cannot have a premium of zero because it would violate the convexity condition of options. Question 3.18. The following three figures show the individual legs of each of the three suggested strategies. The last subplot shows the aggregate position. It is evident from the figures that you can indeed use all the suggested strategies to construct the same butterfly spread. Another method to show the claim of 3.18. mathematically would be to establish the equivalence by using the Put-CallParity on b) and c) and showing that you can write it in terms of the instruments of a). profit diagram part a) 20 Chapter 3 Insurance, Collars, and Other Strategies profit diagram part b) profit diagram part c) 21 Part 1 Insurance, Hedging, and Simple Strategies Question 3.20. Use separate cells for the strike price and the quantities you buy and sell for each strike (i.e., make use of the plus or minus sign). Then, use the maximum function to calculate payoffs and profits. The best way to solve this problem is probably to have the calculations necessary for the payoff and profit diagrams run in the background, e.g., in another auxiliary table that you are referencing to. Define the boundaries for the calculations dynamically and symmetrically around the current stock price. Then use the diagram function with the line style to draw the diagrams. 22 Chapter 4 Introduction to Risk Management Question 4.2. a) If the forward price were $0.80 instead of $1, we would get the following table: Copper price in one year $0.70 $0.80 $0.90 $1.00 $1.10 $1.20 Total Cost Unhedged profit $0.90 $0.90 $0.90 $0.90 $0.90 $0.90 -$0.20 -$0.10 0 $0.10 $0.20 $0.30 With a forward price of $0.45, we have: Copper price in Total Cost Unhedged profit one year $0.70 $0.90 -$0.20 $0.80 $0.90 -$0.10 $0.90 $0.90 0 $1.00 $0.90 $0.10 $1.10 $0.90 $0.20 $1.20 $0.90 $0.30 Profit on short forward $0.10 $0 -$0.10 -$0.20 -$0.30 -$0.40 Net income on hedged profit -$0.10 -$0.10 -$0.10 -$0.10 -$0.10 -$0.10 Profit on short forward -$0.25 -$0.35 -$0.45 -$0.55 -$0.65 -$0.75 Net income on hedged profit -$0.45 -$0.45 -$0.45 -$0.45 -$0.45 -$0.45 Although the copper forward price of $0.45 is below our total costs of $0.90, it is higher than the variable cost of $0.40. It still makes sense to produce copper, because even at a price of $0.45 in one year, we will be able to partially cover our fixed costs. Question 4.4. We will explicitly calculate the profit for the $1.00-strike and show figures for all three strikes. The future value of the $1.00-strike call premium amounts to: $0.0376 × 1.062 = $0.04 . 23 Part 1 Insurance, Hedging, and Simple Strategies Copper price in one year Total Cost Unhedged profit $0.70 $0.80 $0.90 $1.00 $1.10 $1.20 $0.90 $0.90 $0.90 $0.90 $0.90 $0.90 -$0.20 -$0.10 0 $0.10 $0.20 $0.30 Profit on short $1.00-strike call option 0 0 0 0 -$0.10 -$0.20 We obtain the following payoff graphs: 24 call premium received $0.04 $0.04 $0.04 $0.04 $0.04 $0.04 Net income on hedged profit -$0.16 -$0.06 $0.04 $0.14 $0.14 $0.14 Chapter 4 Introduction to Risk Management Question 4.6. a) Copper price in one year Total Cost $0.70 $0.80 $0.90 $1.00 $1.10 $1.20 $0.90 $0.90 $0.90 $0.90 $0.90 $0.90 Profit on short 1.025 put -$0.325 -$0.225 -$0.125 -$0.025 0 0 Profit on two long $0.975 puts Net premium Hedged profit $0.55 $0.35 $0.150 0 0 0 $0.0022 $0.0022 $0.0022 $0.0022 $0.0022 $0.0022 $0.0228 $0.0228 $0.0228 $0.0728 $0.1978 $0.2978 We can see from the following profit diagram (and the above table) that in the case of a favorable increase in copper prices, the hedged profit is almost identical to the unhedged profit. Profit diagram: 25 Part 1 Insurance, Hedging, and Simple Strategies b) Copper price in one year Total Cost $0.70 $0.80 $0.90 $1.00 $1.10 $1.20 $0.90 $0.90 $0.90 $0.90 $0.90 $0.90 Profit on two short 1.034 puts -$0.6680 -$0.4680 -$0.2680 -$0.0680 0 0 Profit on three long Net $1 puts premium Hedged profit $0.9 $0.6 $0.3 0 0 0 $0.0318 $0.0318 $0.0318 $0.0318 $0.1998 $0.2998 $0.0002 $0.0002 $0.0002 $0.0002 $0.0002 $0.0002 We can see from the following profit diagram (and the above table) that in the case of a favorable increase in copper prices, the hedged profit is almost identical to the unhedged profit. Profit diagram: Question 4.8. In this exercise, we need to first find the future value of the call premia. For the $1-strike call, it is: $0.0376 × 1.062 = $0.04 . The following table shows the profit calculations of the $1.00-strike call. The calculations for the two other calls are exactly similar. The figures on the next page compare the profit diagrams of all three possible hedging strategies. 26 Chapter 4 Introduction to Risk Management Copper price in one year $0.70 $0.80 $0.90 $1...
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