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m48-ch24 - Chapter 24 Interest Rate Models Question 24.1 a...

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Chapter 24 Interest Rate Models Question 24.1. a) F = P ( 0 , 2 ) /P ( 0 , 1 ) = . 8495 /. 9259 = . 91749. b) Using Black’s Formula, BSCall (. 8495 , . 9009 × . 9259 , . 1 , 0 , 1 , 0 ) = $0 . 0418 . (1) c) Using put call parity for futures options, p = c + KP ( 0 , 1 ) FP ( 0 , 1 ) = . 0418 + . 8341 . 8495 = $0 . 0264 (2) d) Since 1 + K R = 1 . 1, the caplet is worth 1.1 one year put options on the two year bond with strike price 1 / 1 . 1 = 0 . 9009 which is the same strike as before. Hence the caplet is worth 1 . 11 × . 0264 = . 0293. Question 24.2. a) The two year forward price is F = P ( 0 , 3 ) /P ( 0 , 2 ) = . 7722 /. 8495 = . 90901. b) Since FP ( 0 , 2 ) = P ( 0 , 3 ) the first input into the formula will be .7722. The present value of the strike price is . 9 P ( 0 , 2 ) = . 9 × . 8495 = . 76455. We can use this as the strike with no interest rate; we could also use a strike of .9 with an interest rate equal to the 2 year yield. Either way the option is worth BSCall (. 7722 , . 76455 , . 105 , 0 , 2 , 0 ) = $0 . 0494 (3) c) Using put call parity for futures options, p = c + KP ( 0 , 2 ) FP ( 0 , 2 ) = . 0494 + . 76455 . 7722 = $0 . 4175 . (4) d) The caplet is worth 1.11 two year put options with strike 1 / 1 . 11 = . 9009. The no interest formula will use (. 9009 ) (. 8495 ) = . 7653 as the strike. The caplet has a value of 1 . 11 BSP ut (. 7722 , . 7653 , . 105 , 0 , 2 , 0 ) = $0 . 0468 . (5) 304
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Chapter 24 Interest Rate Models Question 24.3. We must sum three caplets. The one year option has a value of . 0248, the two year option has a value . 0404, and the three year option has a value of . 0483. The three caplets have a combined value of 1 . 115 (. 0248 + . 0404 + . 0483 ) = $0 . 1266 . (6) Question 24.4. A flat yield curve implies the two bond prices are P 1 = e . 08 ( 3 ) = . 78663 and P 2 = e . 08 ( 6 ) = . 61878. If we have purchased the three year bond, the duration hedge is a position of N = − 1 2 P 1 P 2 = − 1 2 e 3 (. 08 ) = − . 63562 (7) in the six year bond. Notice the total cost of this strategy is V 8% = . 78663 . 63562 (. 61878 ) = . 39332 (8) which implies we will owe . 39332 e . 08 / 365 = . 39341 in one day. If yields rise to 8.25%, our portfolio will have a value V 8 . 25% = e . 0825 ( 3 1 / 365 ) . 63562 e . 0825 ( 6 1 / 365 ) = . 39338 . (9) If yields fall to 7.75%, the value will be V 7 . 75% = e . 0775 ( 3 1 / 365 ) . 63562 e . 0775 ( 6 1 / 365 ) = . 39338 . (10) Either way we lose . 00003. This is a binomial version of the impossibility of a no arbitrage flat (stochastic) yield curve. Question 24.5. For this question, let P 1 be the price of the 4 year, 5% coupon bond and let P 2 be the price of the 8 year, 7% coupon bond. Note we use a continuous yield and assume the coupon rates simple. We also use a continuous yield version of duration, i.e. D = − ∂P/∂y P . a) The bond prices are P 1 = 4 i = 1 . 05 e . 06 (i) + e . 06 ( 4 ) = . 95916 . (11) 305
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Part 5 Advanced Pricing Theory and P 2 = 8 i = 1 . 07 e . 06 (i) + e . 06 ( 8 ) = 1 . 0503 . (12) The (modified) durations are D 1 = 4 i = 1 . 05 ie . 06 (i) + 4 e . 06 ( 4 ) . 95916 = 3 . 7167 . (13) and D 2 = 8 i = 1 . 07 ie . 06 (i) + 8 e . 06 ( 8 ) 1 . 0503 = 6 . 4332 . (14) b) If we buy one 4-year bond the duration hedge involves a position of N = − D 1 D 2 P 1 P 2 = − . 5276 (15) of the 8-year bond. This has a total cost of . 9516 . 5276 ( 1 . 0503 ) = . 39746. The next day we will owe . 39746 e . 06 / 365 = . 39753. If yields rise in the next instant to 6 . 25 then bond prices will be P 1 = 4 i = 1 . 05 e . 0625 (i 1 / 365 ) + e . 0625 ( 4 1 / 365 ) = . 95045 (16) and P 2 = 8 i = 1 . 07 e . 0625 (i 1 / 365 ) + e . 0625 ( 8 1 / 365 ) = 1 . 0338 . (17) The duration hedge will have a value of . 95045 . 5276 ( 1 . 0338 ) = . 40502 < . 39753 and we will profit. If the yields fall to 5.75 then one can check P 1 = . 96827 and P 2 = 1 . 0675. The duration
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