Solution of Derivative

If we buy the index today we need to finance it

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Unformatted text preview: payoff as in part (a). Our proof for the payoff equivalence is complete. Now let us turn to the profits. If we buy the index today, we need to finance it. Therefore, we borrow $1,000, and have to repay $1,020 after one year. The profit for part (a) is thus: S T − $1,020 . The profits of the aggregate position in part (b) are the payoffs, less the future value of the call premium plus the future value of the put premium (because we have sold the put), and less the future value of the loan we gave initially. Note that we already know that a risk-less bond is canceling out of the profit calculations. We can again tabulate: Instrument Get repayment of loan Future value of given loan Long Call Option Future value call premium Short Put Option ST < K $931.37 × 1.02 = $950 -$950 max (ST − 950, 0) = 0 − $120.405 × 1.02 = −$122.81 − max ($950 − S T , 0) ST ≥ K $931.37 × 1.02 = $950 -$950 ST − 950 − $120.405 × 1.02 = −$122.81 0 = S T − $950 Future value put premium $51.777 × 1.02 = $52.81 $51.777 × 1.02 = $52.81 Total S T − 1020 ST − 1020 Indeed, we see that the profits for part (a) and part (b) are identical as well. 16 Chapter 3 Insurance, Collars, and Other Strategies Question 3.8. This question is a direct application of the Put-Call-Parity. We will use equation (3.1) in the following, and input the given variables: Call (K , t ) − Put (K , t ) = PV (F0,t − K ) ⇔ Call (K , t ) − Put (K , t ) − PV (F0,t ) = − PV (K ) ⇔ Call (K , t ) − Put (K , t ) − S 0 = − PV (K ) ⇔ $109.20 − $60.18 − $1,000 = − ⇔ K = $970.00 K 1.02 Question 3.10. The strategy of selling a call (or put) and buying a call (or put) at a higher strike is called call (put) bear spread. In order to draw the profit diagrams, we need to find the future value of the cost of entering in the bull spread positions. We have: Cost of call bear spread: ($71.802 − $120.405) × 1.02 = −$49.575 Cost of put bear spread: ($101.214 − $51.777 ) × 1.02 = $50.426 The payoff diagram shows that the payoff to the call bear spread is uniformly less than the payoffs to the put bear spread. The difference is exactly $100, equal to the difference in strikes and as well equal to the difference in the future value of the costs of the spreads. There is a difference, because the call bear spread has a negative initial cost, i.e., we are receiving money if we enter into it. The higher initial cost for the put bear spread is exactly offset by the higher payoff so that the profits of both strategies turn out to be the same. It is easy to show this with equation (3.1), the put-call-parity. Payoff-Diagram: 17 Part 1 Insurance, Hedging, and Simple Strategies Profit Diagram: 18 Chapter 3 Insurance, Collars, and Other Strategies Question 3.12. Our initial cash required to put on the collar, i.e. the net option premium, is as follows: − $51.873 + $51.777 = −$0.096 . Therefore, we receive only 10 cents if we enter into this collar. The position is very close to a zero-cost collar. The profit diagram looks as follows: If we compare this profit diagram with the profit diagram of the previous exercise (3.11.), we see that we traded in the additional call premium (that limited our losses after index decreases) against more participation on the upside. Please note that both maximum loss and gain are higher than in question 13.11. Question 3.14. a) This question deals with the option trading strategy known as Box spread. We saw earlier that if we deal with options and the maximum function, it is convenient to split the future values of the index into different regions. Let us name the final value of the S&R index ST . We have two strike prices, therefore we will use three regions: One in which S T < 950 , one in which 950 ≤ S T < 1,000 and another one in which ST ≥ 1,000 . We then look at each region separately, and hope to be able to see that indeed when we aggregate, there is no S&R risk when we look at the aggregate position. 19 Part 1 Insurance, Hedging, and Simple Strategies Instrument long 950 call short 1000 call short 950 put long 1000 put Total S T < 950 0 0 S T − $950 $1,000 − ST $50 950 ≤ S T < 1,000 S T − $950 0 0 $1,000 − ST $50 ST ≥ 1,000 ST − $950 $1,000 − ST 0 0 $50 We see that there is no occurrence of the final index value in the row labeled total. The option position does not contain S&R price risk. b) The initial cost is the sum of the long option premia less the premia we receive for the sold options. We have: Cost = $120.405 − $93.809 − $51.77 + $74.201 = $49.027 c) The payoff of the position after 6 months is $50, as we can see from the above table. d) The implicit interest rate of the cash flows is: $50.00 ÷ $49.027 = 1.019 ≅ 1.02 . The implicit interest rate is indeed 2 percent. Question 3.16. A bull spread or a bear spread can never have an initial premium of zero, because we are buying the same number of calls (or puts) that we are selling and the two legs of the bull and bear spreads have different strikes. A zero initial premium woul...
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This document was uploaded on 03/11/2014 for the course FIN 402 at FPT University.

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