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 PutCallParity 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 $1.00strike and show figures for all three strikes.
The future value of the $1.00strike 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
$1.00strike 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 $1strike call, it
is: $0.0376 × 1.062 = $0.04 . The following table shows the profit calculations of the $1.00strike
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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 Spring '14
 NguyenThiMaiLan
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