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Solution of Derivative

F short put we have no control over the exercise

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Unformatted text preview: ess the future value of the premium and it occurs at a terminal stock price of zero. f) Short Put We have no control over the exercise decision when we write a put. The buyer of the put option decides whether to exercise or not, and he will only exercise if he makes a profit. As we have the opposite side, we will never make any money at the expiration of the put option. Our profit is restricted to the future value of the premium, and we make this maximum profit whenever the stock price at expiration is greater than the strike price. However, we lose money whenever the stock price is smaller than the strike, hence the largest loss occurs when the stock price attains its smallest possible value, zero. We lose the strike price because somebody sells us an asset for the strike that is worth nothing. We are only compensated by the future value of the premium we received. 9 Part 1 Insurance, Hedging, and Simple Strategies Question 12.14. a) In order to be able to draw profit diagrams, we need to find the future values of the put premia. They are: i) ii) iii) 35-strike put: $1.53 × (1 + 0.08) = $1.6524 40-strike put: $3.26 × (1 + 0.08) = $3.5208 45-strike put: $5.75 × (1 + 0.08) = $6.21 We get the following payoff diagrams: 10 Chapter 2 An Introduction to Forwards and Options We get the profit diagram by deducting the option premia from the payoff graphs. The profit diagram looks as follows: b) Intuitively, whenever the 35-strike put option pays off (i.e., has a payoff bigger than zero), the 40-strike and the 35-strike options also pay off. However, there are some instances in which the 40-strike option pays off and the 35-strike options does not. Similarly, there are some instances in which the 45-strike option pays off, and neither the 40-strike nor the 35-strike pay off. Therefore, the 45-strike offers more potential than the 40- and 35-strike, and the 40-strike offers more potential than the 35-strike. We pay for these additional payoff possibilities by initially paying a higher premium. It makes sense that the premium is increasing in the strike price. 11 Part 1 Insurance, Hedging, and Simple Strategies Question 2.16. The following is a copy of an Excel spreadsheet that solves the problem: 12 Chapter 3 Insurance, Collars, and Other Strategies Question 3.2. This question constructs a position that is the opposite to the position of Table 3.1. Therefore, we should get the exact opposite results from Table 3.1. and the associated figures. Mimicking Table 3.1., we indeed have: S&R Index S&R Put Payoff -(Cost +Interest) Profit -900.00 -100.00 -1000.00 1095.68 95.68 -950.00 -50.00 -1000.00 1095.68 95.68 -1000.00 0.00 -1000.00 1095.68 95.68 -1050.00 0.00 -1050.00 1095.68 45.68 -1100.00 0.00 -1100.00 1095.68 -4.32 -1150.00 0.00 -1150.00 1095.68 -54.32 -1200.00 0.00 -1200.00 1095.68 -104.32 On the next page, we see the corresponding payoff and profit diagrams. Please note that they match the combined payoff and profit diagrams of Figure 3.5. Only the axes have different scales. Payoff-diagram: 13 Part 1 Insurance, Hedging, and Simple Strategies Profit diagram: Question 3.4. This question is another application of Put-Call-Parity. Initially, we have the following cost to enter into the combined position: We receive $1,000 from the short sale of the index, and we have to pay the call premium. Therefore, the future value of our cost is: ($120.405 − $1,000) × (1 + 0.02) = −$897.19 . Note that a negative cost means that we initially have an inflow of money. 14 Chapter 3 Insurance, Collars, and Other Strategies Now, we can directly proceed to draw the payoff diagram: We can clearly see from the figure that the payoff graph of the short index and the long call looks like a fixed obligation of $950, which is alleviated by a long put position with a strike price of 950. The following profit diagram, including the cost for the combined position, confirms this: 15 Part 1 Insurance, Hedging, and Simple Strategies Question 3.6. We now move from a graphical representation and verification of the Put-Call-Parity to a mathematical representation. Let us first consider the payoff of (a). If we buy the index (let us name it S), we receive at the time of expiration T of the options simply ST . The payoffs of part (b) are a little bit more complicated. If we deal with options and the maximum function, it is convenient to split the future values of the index into two regions: one where ST < K and another one where ST ≥ K . We then look 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 Get repayment of loan Long Call Option Short Put Option ST < K = 950 $931.37 × 1.02 = $950 max (ST − 950, 0) = 0 − max ($950 − S T , 0) ST ≥ K = 950 $931.37 × 1.02 = $950 ST − 950 0 Total = S T − $950 ST ST We now see that the total aggregate position only gives us ST , no matter what the final index value is – but this is the same...
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