Solution of Derivative

08 11641 we make 11641 11014 627 combining these

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Unformatted text preview: and a delta of −.4176. For the put ratio spread (assume on 100 shares), our total cost is 508.24 − 200 (1.9905) = 110.14. The delta on this position is 100 (−.7185 − 2 (−.4176)) = 11.67 hence our delta hedged requires shorting 11.67 shares (receiving 11.67 (40) = $466.80). This implies that in one day we will receive 466.8 e.08/365 − 1 = .10232 ≈ .10 from our short sale proceeds. Our short sale of 11.67 shares will make 11.67 if S falls to 59 and will lose 5.89 if S rises to 60.5. If S falls to 39 in one day the 45-strike and 40-strike puts will be worth 5.8265 and 2.4331 (respectively). This implies our put ratio spread will be worth 582.65 − (2) 243.31 = 96. 03 (we lose 110.14 − 96.03 = 14.11). If S rises to 40.5 in one day the 45-strike and 40-strike puts will be worth 4.7257 and 1.7808 (respectively) which implies put ratio spread will be worth 472.57 − 2 (178.08) = 116.41 (we make 116.41 − 110.14 = 6.27). Combining these three components, our 97 Part 3 Options profit will be 11.67 − 14.11 + .10 = −2.34 (3) −5.89 + 6.27 + .10 = .48 (4) if S falls to 39 and if S rises to 40.5. This suggests that the put ratio spread has a negative gamma at 40. Question 13.6. See Table Two. Once again, note the similarities with the delta hedged call. TABLE TWO (Problem 13.6) Day 0 1 2 3 4 5 Stock ($) 40 40.642 40.018 39.403 38.797 39.42 Put ($) 199.05 172.66 196.53 222.60 250.87 220.07 Option Delta -0.4176 -0.3768 -0.4173 -0.4592 -0.5020 -0.4594 Investment ($) -1869.433 -1704.224 -1866.514 -2031.895 -2198.561 -2031.022 Interest ($) 0.41 0.37 0.41 0.45 0.48 Capital Gain ($) -0.42 -0.35 -0.40 -0.45 -0.48 Daily Profit -0.01 0.02 0.01 0.00 0.00 Question 13.8. See Table Four. Note the errors are larger the farther out we go as the theta will be changing. With the one day the error is minimal at S = 40 due to no error due to changes in S (since it will not be changing) and little error due to our theta approximation for in doesn’t change much during the day. For 5 days, there is a theta error at S5/365 = 40 (of .0003) due to theta decreasing during the five days. Note that the error of .0003 is not constant across the range of prices. Besides the familiar delta-gamma error (i.e. ignoring third order changes of S ), there is the effect changes in S have on (technically the cross partial derivative ∂ 2 f (S, t) / (∂S∂t)). The delta gamma error is symmetric; however this cross partial error is not symmetric. To see this, we can use the fundamental theorem of calculus on the Black Scholes formula. By put call parity, the put and call will have the same second cross partial derivative which is equal to ∂ 2 f (S, t) =− ∂S∂t call r − σ 2 /2 . √ T −t (5) In this case this, r − σ 2 /2 > 0 and call > 0, hence the above term is negative; this implies our 2 f (S,t) approximation does not include terms like ∂ ∂S∂t ( S) ( t) which will be positive when ST < 40 and negative when ST > 40; hence our approximation will be underestimate the option value for low ST and overestimate it for large ST . 98 Chapter 13 Market-Making and Delta-Hedging TABLE FOUR (Problem 13.8) 1 day Future S Approx 5 days Actual Approx Errors 25 days Actual Approx Actual 1d 5d 25d 25.00 13.4458 13.5091 13.4257 13.5389 13.3256 13.6910 -0.0634 -0.1132 -0.3655 25.50 12.9188 13.0236 12.8988 13.0524 12.7986 13.2004 -0.1048 -0.1537 -0.4018 26.00 12.4032 12.5414 12.3831 12.5692 12.2830 12.7123 -0.1382 -0.1861 -0.4293 26.50 11.8989 12.0631 11.8789 12.0897 11.7787 12.2273 -0.1642 -0.2109 -0.4486 27.00 11.4059 11.5892 11.3859 11.6146 11.2857 11.7458 -0.1833 -0.2287 -0.4600 27.50 10.9243 11.1204 10.9043 11.1443 10.8041 11.2685 -0.1961 -0.2400 -0.4644 28.00 10.4541 10.6573 10.4340 10.6796 10.3339 10.7960 -0.2033 -0.2456 -0.4621 28.50 9.9951 10.2005 9.9751 10.2211 9.8749 10.3289 -0.2054 -0.2460 -0.4540 29.00 9.5475 9.7507 9.5275 9.7694 9.4273 9.8680 -0.2031 -0.2419 -0.4407 29.50 9.1113 9.3084 9.0913 9.3253 8.9911 9.4140 -0.1971 -0.2340 -0.4229 30.00 8.6864 8.8744 8.6663 8.8892 8.5662 8.9675 -0.1880 -0.2229 -0.4013 30.50 8.2728 8.4492 8.2528 8.4619 8.1526 8.5293 -0.1764 -0.2091 -0.3767 31.00 7.8706 8.0335 7.8505 8.0440 7.7504 8.1000 -0.1630 -0.1935 -0.3497 31.50 7.4797 7.6278 7.4596 7.6361 7.3595 7.6805 -0.1482 -0.1765 -0.3210 32.00 7.1001 7.2327 7.0801 7.2387 6.9799 7.2712 -0.1325 -0.1586 -0.2913 32.50 6.7319 6.8485 6.7119 6.8523 6.6117 6.8728 -0.1166 -0.1404 -0.2611 33.00 6.3750 6.4758 6.3550 6.4773 6.2548 6.4860 -0.1008 -0.1223 -0.2312 33.50 6.0295 6.1150 6.0095 6.1143 5.9093 6.1111 -0.0855 -0.1048 -0.2018 34.00 5.6953 5.7663 5.6753 5.7634 5.5751 5.7486 -0.0710 -0.0882 -0.1736 34.50 5.3724 5.4301 5.3524 5.4251 5.2522 5.3990 -0.0576 -0.0727 -0.1468 35.00 5.0609 5.1064 5.0409 5.0994 4.9407 5.0625 -0.0455 -0.0585 -0.1218 35.50 4.7607 4.7956 4.7407 4.7867 4.6405 4.7394 -0.0349 -0.0460 -0.0989 36.00 4.4719 4.4976 4.4519 4.4869 4.3517 4.4299 -0.0257 -0.0350 -0.0782 36.50 4.1944 4.2125 4.1744 4.2001 4.0742 4.1340 -0....
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