Solution of Derivative

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Unformatted text preview: 0181 -0.0257 -0.0599 37.00 3.9282 3.9403 3.9082 3.9263 3.8080 3.8519 -0.0121 -0.0181 -0.0438 37.50 3.6734 3.6809 3.6534 3.6655 3.5532 3.5834 -0.0075 -0.0121 -0.0302 38.00 3.4299 3.4341 3.4099 3.4174 3.3097 3.3285 -0.0042 -0.0075 -0.0187 38.50 3.1978 3.1998 3.1777 3.1820 3.0776 3.0870 -0.0020 -0.0043 -0.0094 39.00 2.9770 2.9778 2.9569 2.9590 2.8568 2.8587 -0.0008 -0.0020 -0.0019 39.50 2.7675 2.7677 2.7475 2.7481 2.6473 2.6433 -0.0002 -0.0006 0.0040 40.00 2.5694 2.5694 2.5493 2.5490 2.4492 2.4406 0.0000 0.0003 0.0085 40.50 2.3826 2.3824 2.3626 2.3615 2.2624 2.2502 0.0002 0.0011 0.0122 41.00 2.2071 2.2064 2.1871 2.1851 2.0869 2.0717 0.0007 0.0020 0.0152 41.50 2.0430 2.0411 2.0230 2.0195 1.9228 1.9047 0.0019 0.0035 0.0181 42.00 1.8903 1.8860 1.8702 1.8643 1.7701 1.7488 0.0042 0.0060 0.0213 42.50 1.7488 1.7408 1.7288 1.7190 1.6286 1.6034 0.0080 0.0098 0.0252 43.00 1.6187 1.6051 1.5987 1.5833 1.4985 1.4682 0.0137 0.0154 0.0304 43.50 1.5000 1.4783 1.4800 1.4567 1.3798 1.3426 0.0217 0.0233 0.0372 44.00 1.3926 1.3602 1.3726 1.3388 1.2724 1.2262 0.0324 0.0338 0.0461 44.50 1.2965 1.2502 1.2765 1.2291 1.1763 1.1186 0.0463 0.0474 0.0578 45.00 1.2118 1.1480 1.1917 1.1273 1.0916 1.0191 0.0638 0.0645 0.0725 45.50 1.1384 1.0531 1.1184 1.0328 1.0182 0.9273 0.0853 0.0855 0.0908 46.00 1.0763 0.9651 1.0563 0.9454 0.9561 0.8429 0.1112 0.1109 0.1133 46.50 1.0256 0.8837 1.0056 0.8645 0.9054 0.7652 0.1420 0.1411 0.1402 47.00 0.9862 0.8084 0.9662 0.7898 0.8660 0.6940 0.1779 0.1764 0.1721 47.50 0.9582 0.7388 0.9382 0.7209 0.8380 0.6286 0.2194 0.2173 0.2094 48.00 0.9415 0.6747 0.9215 0.6574 0.8213 0.5689 0.2668 0.2640 0.2524 48.50 0.9362 0.6156 0.9161 0.5990 0.8160 0.5143 0.3205 0.3171 0.3017 49.00 0.9421 0.5612 0.9221 0.5454 0.8219 0.4644 0.3809 0.3767 0.3575 49.50 0.9595 0.5113 0.9394 0.4961 0.8393 0.4190 0.4482 0.4433 0.4203 50.00 0.9881 0.4654 0.9681 0.4509 0.8679 0.3776 0.5227 0.5172 0.4903 50.50 1.0281 0.4233 1.0081 0.4095 0.9079 0.3400 0.6048 0.5986 0.5679 51.00 1.0795 0.3847 1.0594 0.3716 0.9593 0.3059 0.6947 0.6878 0.6534 51.50 1.1421 0.3494 1.1221 0.3370 1.0219 0.2750 0.7927 0.7851 0.7470 52.00 1.2162 0.3171 1.1961 0.3054 1.0960 0.2469 0.8990 0.8907 0.8490 52.50 1.3015 0.2876 1.2815 0.2765 1.1813 0.2216 1.0139 1.0049 0.9597 53.00 1.3982 0.2607 1.3782 0.2502 1.2780 0.1986 1.1375 1.1279 1.0794 53.50 1.5062 0.2362 1.4862 0.2263 1.3860 0.1780 1.2701 1.2599 1.2081 54.00 1.6256 0.2138 1.6056 0.2045 1.5054 0.1593 1.4118 1.4011 1.3461 54.50 1.7563 0.1934 1.7363 0.1847 1.6361 0.1425 1.5629 1.5516 1.4936 55.00 1.8984 0.1749 1.8784 0.1667 1.7782 0.1274 1.7235 1.7116 1.6508 99 Part 3 Options Question 13.10. See Figures 2 & 3. Figure 2 (Problem 13.10) 16 14 12 10 8 6 Black Scholes ∆ Approx ∆ Γ Approx δ Γ θ Approx 4 2 0 -2 30 32 34 36 38 40 42 44 46 48 50 Stock Price in 5 Days Figure 3 (Problem 13.10) 0.5 ∆ Γ Error 0 ∆Γθ Error -0.5 -1 ∆ Error -1.5 -2 30 32 34 36 38 40 42 Stock Price in 5 Days 100 44 46 48 50 Chapter 13 Market-Making and Delta-Hedging Question 13.12. See Figures 5 & 6. Figure 5 (Problem 13.12) 9 8 7 Black Scholes ∆ Approx. ∆ Γ Approx. ∆ Γ θ Approx. 6 5 4 3 2 1 0 -1 30 32 34 36 38 40 42 44 46 48 50 Stock Price in 5 days Figure 6 (Problem 13.12) 0.5 ∆ Error ∆ Γ Error ∆ Γ θ Error. Zero Error 0 -0.5 -1 30 32 34 36 38 40 42 Stock Price in 5 days 101 44 46 48 50 Part 3 Options Question 13.14. Using the given parameters, a six month 45-strike put has a price and Greeks of P = 5.3659, = −.6028, = .045446, and per day = −.0025. Note that , as given in the software is a per day. Equation (13.9) uses annualized rates (i.e. h is in the equation. Hence for equation (13.9) we should use −.9139. For equation (13.9) we have a market-maker profit of .09 2 40 (.045446) − .9139 + .08 ((−.6028) 40 − 5.3659) h 2 = − (3.2721 − .9139 − 2.3582) h = 0 − (6) (7) Question 13.16. For our 45-strike put that we’ve written: P1 = 5.3596, 1 = −.6051, 1 = .0457. The 40-strike has C2 = 4.1217, 2 = .6151, 2 = .0454. Since we are “short” gamma (we wrote an option), we must buy 1 / 2 = .0457/.0454 = 1.007 40-strike calls. Our option position will have a total delta of .6051 + (1.007) .6151 = 1.2245 hence we have to short 1.2247 shares. Our total initial cash flow will be 5.3596 − 1.007 (4.1217) + 1.2247 (40) = 50.20. Using primes to denote next day prices, our one-day profit will be −P1 − 1.007C2 − 1.2247S + 50.20e.08/365 (8) We use Black Scholes with T − t = 179/365 to arrive at our profit in Figure 8. Figure 8 (Problem 13.16) 0.5 0.4 0.3 Overnight Profit ($) 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 30 32 34 36 38 40 42 Stock Price ($) 102 44 46 48 50 Chapter 13 Market-Making and Delta-Hedging Question 13.18. The relevant values are of the spread are: f = 5.0824 − 2 (1.9905) = 1.1014, = −.71845 − 2 (−.4176) = .11675, and = .05633 − 2 (.06516) = −.07399. Since we wrote the spread, to hedge we need to buy .07399/.04536 = 1.6312 options. The delta of the spread and the call will become .11675 + (1.6312) (.6151) = 1.120; t...
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This document was uploaded on 03/11/2014 for the course FIN 402 at FPT University.

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