Solution of Derivative

12 a equations 915 of the textbook is violated we use

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Unformatted text preview: 5. 59 Part 3 Options Therefore, we use a call and put butterfly spread to profit from these arbitrage opportunities. Transaction Buy 1 50 strike call Sell 2 55 strike calls buy 1 60 strike call TOTAL t=0 -18 +28 -9.50 +0.50 ST < 50 0 0 0 0 Transaction Buy 1 50 strike put Sell 2 55 strike puts buy 1 60 strike put TOTAL t=0 -7 21.50 -14.45 +0.05 ST < 50 50 − ST 50 ≤ ST ≤ 55 0 2 × ST − 110 2 × ST − 110 60 − ST ST − 50 ≥ 0 50 ≤ ST ≤ 55 ST − 50 0 0 ST − 50 ≥ 0 60 − ST 0 55 ≤ ST ≤ 60 ST − 50 110 − 2 × ST 0 60 − ST ≥ 0 ST > 60 ST − 50 110 − 2 × ST ST − 60 0 55 ≤ ST ≤ 60 0 0 60 − ST 60 − ST ≥ 0 ST > 60 0 0 0 0 Please note that we initially receive money and have non-negative future payoffs. Therefore, we have found an arbitrage possibility, independent of the prevailing interest rate. Question 9.12. a) Equations (9.15) of the textbook is violated. We use a call bear spread to profit from this arbitrage opportunity. Transaction Sell 90 strike call Buy 95 strike call TOTAL t=0 +10 -4 +6 Expiration or Exercise ST < 90 90 ≤ ST ≤ 95 ST > 95 0 90 − ST 90 − ST 0 0 ST − 95 0 90 − ST > −5 - 5 Please note that we initially receive more money than our biggest possible exposure in the future. Therefore, we have found an arbitrage possibility, independent of the prevailing interest rate. b) Now, equation (9.15) is not violated anymore. However, we can still construct an arbitrage opportunity, given the information in the exercise. We continue to sell the 90-strike call and buy the 95-strike call, and we loan our initial positive net balance for two years until expiration. It is important that the options be European, because otherwise we would not be able to tell whether the 90-strike call could be exercised against us sometime (note that we do not have information regarding any dividends). 60 Chapter 9 Parity and Other Option Relationships We have the following arbitrage table: Transaction Sell 90 strike call Buy 95 strike call Loan 4.75 TOTAL Expiration t=T t=0 ST < 90 90 ≤ ST ≤ 95 +10 0 90 − ST -5.25 0 0 -4.75 5.80 5.80 0 5.80 95.8 − ST > 0 ST > 95 90 − ST ST − 95 5.8 + 0 .8 In all possible future states, we have a strictly positive payoff. We have created something out of nothing – we demonstrated arbitrage. c) We will first verify that equation (9.17) is violated. We have: C (K1 ) − C (K 2 ) 15 − 10 = = 0 .5 K 2 − K1 100 − 90 and C (K 2 ) − C (K 3 ) 10 − 6 = = 0 .8 , K3 − K2 105 − 100 which violates equation 9.17. We calculate lambda in order to know how many options to buy and sell when we construct the butterfly spread that exploits this form of mispricing. Using formula (9.19), we can calculate that lambda is equal to 1/3. To buy and sell round lots, we multiply all the option trades by 3. We use an asymmetric call and put butterfly spread to profit from these arbitrage opportunities. Transaction Buy 1 90 strike calls Sell 3 100 strike calls buy 2 105 strike calls TOTAL t=0 -15 +30 -12 +3 ST < 90 0 0 0 0 90 ≤ ST ≤ 100 ST − 90 0 0 100 ≤ ST ≤ 105 ST − 90 300 − 3 × ST 0 ST − 90 ≥ 0 210 − 2 × S T ≥ 0 ST > 105 ST − 90 300 − 3 × ST 2 × ST − 210 0 We indeed have an arbitrage opportunity. Question 9.14. This question is closely related to question 9.13. In this exercise, the strike is not cash anymore, but rather one share of Apple. In parts a) and b), there is no benefit in keeping Apple longer, because the dividend is zero. a) The underlying asset is the stock of Apple, which does not pay a dividend. Therefore, we have an American call option on a non-dividend-paying stock. It is never optimal to early exercise such an option. 61 Part 3 Options b) The underlying asset is the stock of Apple, and the strike consists of AOL. As AOL does not pay a dividend, the interest rate on AOL is zero. We will therefore never early exercise the put option, because we cannot receive earlier any benefits associated with holding Apple – there are none. If Apple is bankrupt, there is no loss from not early exercising, because the option is worth max[0, AOL – 0], which is equivalent to one share of AOL, because of the limited liability of stock. As AOL does not pay dividends, we are indifferent between holding the option and the stock. c) For the American call option, dividends on the stock are the reason why we want to receive the stock earlier, and now Apple pays a dividend. We usually benefit from waiting, because we can continue to earn interest on the strike. However, in this case, the dividend on AOL remains zero, so we do not have this benefit associated with waiting to exercise. Finally, we saw that there is a second benefit to waiting: the insurance protection, which will not be affected by the zero AOL dividend. Therefore, there now may be circumstances in which we will early exercise, but we will not always early exercise. For the American put option, there is no cost associated with w...
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This document was uploaded on 03/11/2014 for the course FIN 402 at FPT University.

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