Solution of Derivative

Therefore we must use the value of 20404 in all

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Unformatted text preview: 0 0 0 Using these values at the previous node and at the initial node yields: delta B Call premium value of early exercise t=0 0.7400 -55.7190 18.2826 5 node d 0.4870 -35.3748 6.6897 0 node u 0.9528 -83.2073 33.1493 27.1250 Please note that in all instances the value of immediate exercise is smaller than the continuation value, the (European) call premium. Therefore, the value of the European call and the American call are identical. 66 Chapter 10 Binomial Option Pricing: 1 b) We can calculate similarly the binomial prices at each node in the tree. We can calculate for the different nodes of the tree: delta B Put premium value of early exercise node uu node ud = du 0 -0.1034 0 12.968 0 2.0624 0 0 node dd -1 92.5001 17.904 20.404 Using these values at the previous node and at the initial node yields: delta B Put premium value of early exercise t=0 -0.26 31.977 5.979 0 node d -0.513 54.691 10.387 8.6307 node u -0.047 6.859 1.091 0 c) From the previous tables, we can see that at the node dd, it is optimal to early exercise the American put option, because the value of early exercise exceeds the continuation value. Therefore, we must use the value of 20.404 in all relevant previous nodes when we back out the prices of the American put option. We have for nodes d and 0 (the other nodes remain unchanged): delta B Put premium value of early exercise t=0 -0.297 36.374 6.678 0 node d -0.594 63.005 11.709 8.6307 The price of the American put option is indeed 6.678. Question 10.12. a) We can calculate u and d as follows: u = e (r −δ )h +σ d = e (r −δ )h −σ h h = e (0.08 )×0.25+0.3× = e (0.08 )×0.25−0.3× 0.25 0.25 = 1.1853 = 0.8781 67 Part 3 Options b) We need to calculate the values at the relevant nodes in order to price the European call option: delta B Call premium t=0 0.6074 -20.187 4.110 node d 0.1513 -4.5736 0.7402 node u 1 -39.208 8.204 c) We can calculate at the relevant nodes (or, equivalently, you can use put-call-parity for the European put option): European put delta B Put premium t=0 -0.3926 18.245 2.5414 node d -0.8487 34.634 4.8243 node u 0 0 0 For the American put option, we have to compare the premia at each node with the value of early exercise. We see from the following table that at the node d, it is advantageous to exercise the option early; consequently, we use the value of early exercise when we calculate the value of the put option. American put delta B Put premium value of early exercise t=0 -0.3968 18.441 2.5687 0 node d -0.8487 34.634 4.8243 4.8762 node u 0 0 0 0 Question 10.14. a) We can calculate the price of the call currency option in a very similar way to our previous calculations. We simply replace the dividend yield with the foreign interest rate in our formulas. Thus, we have: delta B Call premium node uu node ud = du 0.9925 0.9925 -0.8415 -0.8415 0.4734 0.1446 node dd 0.1964 -0.1314 0.0150 Using these call premia at all previous nodes yields: delta B Call premium t=0 0.7038 -0.5232 0.1243 node d 0.5181 -0.3703 0.0587 68 node u 0.9851 -0.8332 0.2544 Chapter 10 Binomial Option Pricing: 1 The price of the European call option is $0.1243. b) For the American call option, the binomial approach yields: delta B Call premium value of early exercise node uu node ud = du 0.9925 0.9925 -0.8415 -0.8415 0.4734 0.1446 0.4748 0.1436 node dd 0.1964 -0.1314 0.0150 0 Using the maximum of the call premium and the value of early exercise at the previous nodes and at the initial node yields: t=0 node d node u delta 0.7056 0.5181 0.9894 B -0.5247 -0.3703 -0.8374 Call premium 0.1245 0.0587 0.2549 value of early exercise 0.07 0 0.2540 The price of the American call option is: $0.1245. Question 10.16. aa) We now have to inverse the interest rates: We have a Yen-denominated option, therefore, the dollar interest rate becomes the foreign interest rate. With these changes, and equipped with an exchange rate of Y120/$ and a strike of Y120, we can proceed with our standard binomial procedure. delta B Call premium node uu node ud = du 0.9835 0.1585 -119.6007 -17.4839 9.3756 1.0391 node dd 0 0 0 Using these call premia at all previous nodes yields: delta B Call premium t=0 0.3283 -36.6885 2.7116 node d 0.0802 -8.4614 0.5029 node u 0.5733 -66.8456 5.0702 The price of the European Yen-denominated call option is $2.7116. 69 Part 3 Options ab) For the American call option, the binomial approach yields: node uu node ud = du delta 0.9835 0.1585 B -119.6007 -17.4839 Call premium 9.3756 1.0391 Value of early exercise 11.1439 0 node dd 0 0 0 0 Using the maximum of the call premium and the value of early exercise at the previous nodes and at the initial node yields: delta B Call premium value of early exercise b) t=0 0.3899 -43.6568 3.1257 0 node d 0.0802 -8.4614 0.5029 0 node u 0.6949 -81.2441 5.9259 5.4483 For the Yen-denominated put option, we have: delta B Put premium value of early exercise node uu node ud = du node dd 0 -0.8249 -0.9835 0 102.1168 119.6007 0 5.7287 17.2210 0 3.1577 15.8997 We can clearly see that early exercise is never optimal at those stages. We can therefore c...
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