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Chapter 13Market-Making and Delta-Hedging
Question 13.1.
The delta of the option is .2815. To delta hedge writing 100 options we must purchase 28.15 sharesfor a delta hedge. The total value of this position is 1028.9 which is the amount we will initiallyborrow. If the next day’s stock price is 39,
−28.15 + 26.56 − .23 = −1.82. (1)
If S rises to 40.50, the change in stock value and option value will be the total profit:
14.08 − 13.36 − .23 = .49. (2)
Question 13.2.
Using the Black Scholes formula we can solve for the put premium and the put’s delta: P = 1.9905and � = −0.4176. If we write this option, we will have a position that moves with the stock price.This implies our delta hedge will require shorting 41.76 shares (receiving $41.76 (40) = $1670.4).As before, we must look at the three components of the profit. There will now be interest earnedsince we are receiving both the option premium 199.05 as well as the 1670.40 on the short sale. This$1869.43 will earn (rounding to the nearest penny) 1869.43 ∗ e.08/365 − 1[1] = .41 in interest. If thestock falls to 39 we make 41.76 on our short sale and if the stock price rises to 40.5 we lose 20.88 onour short sale. If the stock prices falls to 39 or rises to 40.5 the price of the put option we wrote willbe (using T = 90/365) P (39) = 2.4331 or P (40.5) = 1.7808. This implies our option positionwill lose 243.31 − 199.05 = 44.26 if the stock falls by $1 and make 199.05 − 178.08 = 20.97 ifthe stock rises by $0.50. Combining these results, our profit will be
41.76 − 44.26 + .41 = −2.09 (3)
if the stock price falls to $39 and
−20.88 + 20.97 + .41 = .50. (4)
Notice that, as in the case of the call option, the large change implies a loss and the small changeinvolves a profit.
Question 13.3.
The unhedged delta will be 30.09 hence we have to short 30.09 shares of stock, receiv-ing $30.09 (40) = $1203.60. This implies will we earn interest (in one day) of
185
Part 3 Options
1203.6(e.08/365 − 1
) = .26383 ≈ .26. In the two scenarios, we have a profit of
30.09 − 30.04 + .26 = .31 (5)
if S falls to 39 and a profit of
−15.05 + 14.31 + .26 = −.48 (6)
if S rises to 40.5.
Question 13.4.
The 45-strike put has a premium of 5.0824 and a delta of−0.7185 and the 40-strike put has a premiumof 1.9905 and a delta of −0.4176. For the put ratio spread (assume on 100 shares), our total costis 508.24 − 200 (1.9905) = 110.14. The delta on this position is 100 (−0.7185 − 2 (−0.4176)) =11.67 hence our delta hedged requires shorting 11.67 shares (receiving 11.67 (40) = $466.80). Thisimplies that in one day we will receive 466.8
(e.08/365 − 1
) = 0.10232 ≈ .10 from our short saleproceeds. Our short sale of 11.67 shares will make 11.67 if S falls to 59 and will lose 5.89 if S risesto 60.5. If S falls to 39 in one day the 45-strike and 40-strike puts will be worth 5.8265 and 2.4331(respectively). This implies our put ratio spread will be worth 582.65 − (2) 243.31 = 96. 03 (welose 110.14 − 96.03 = 14.11). If S rises to 40.5 in one day the 45-strike and 40-strike puts willbe worth 4.7257 and 1.7808 (respectively) which implies put ratio spread will be worth 472.57 −2 (178.08) = 116.41 (we make 116.41 − 110.14 = 6.27). Combining these three components, ourprofit will be
11.67 − 14.11 + .10 = −2.34 (7)
if S falls to 39 and
−5.89 + 6.27 + .10 = .48 (8)
if S rises to 40.5. This suggests that the put ratio spread has a negative gamma at 40.
Question 13.5.
See Table One. Note the similarities with the delta hedged call.
TABLE ONE (Problem 13.5)Day 0 1 2 3 4 5
Stock ($) 40.00 40.50 39.25 38.75 40.00 40.00
Put ($) 199.05 178.08 230.55 254.05 195.49 194.58
Option Delta -0.417596 -0.385797 -0.468923 -0.504365 -0.419402 -0.41986Investment ($) -1869.43 -1740.56 -2071.07 -2208.46 -1873.10 -1874.02
Interest ($) 0.41 0.38 0.45 0.48 0.41Capital Gain ($) 0.09 -4.25 -0.05 -4.48 0.91
Daily Profit 0.50 -3.87 0.40 -4.00 1.32
186
Chapter 13 Market-Making and Delta-Hedging
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 -0.4594Investment ($) -1869.433 -1704.224 -1866.514 -2031.895 -2198.561 -2031.022
Interest ($) 0.41 0.37 0.41 0.45 0.48Capital Gain ($) -0.42 -0.35 -0.40 -0.45 -0.48
Daily Profit -0.01 0.02 0.01 0.00 0.00
Question 13.7.
See Table Three.
TABLE THREE (Problem 13.7)
Errors
Future S Approx Actual Approx Actual Approx Actual 1d 5d 25d
36.00 2.0108 2.0365 1.9571 1.9921 1.6883 1.7660 -0.0257 -0.0350 -0.0777
36.25 2.1206 2.1424 2.0669 2.0971 1.7981 1.8663 -0.0217 -0.0302 -0.0683
36.50 2.2333 2.2515 2.1796 2.2053 1.9108 1.9701 -0.0181 -0.0257 -0.0594
36.75 2.3488 2.3637 2.2951 2.3168 2.0263 2.0773 -0.0149 -0.0217 -0.0511
37.00 2.4672 2.4792 2.4134 2.4315 2.1446 2.1880 -0.0121 -0.0181 -0.0434
37.25 2.5883 2.5979 2.5346 2.5495 2.2658 2.3020 -0.0096 -0.0149 -0.0362
37.50 2.7123 2.7198 2.6586 2.6707 2.3898 2.4195 -0.0075 -0.0121 -0.0297
37.75 2.8392 2.8449 2.7854 2.7951 2.5166 2.5403 -0.0057 -0.0097 -0.0237
38.00 2.9688 2.9730 2.9151 2.9226 2.6463 2.6645 -0.0042 -0.0075 -0.0183
38.25 3.1014 3.1044 3.0476 3.0534 2.7788 2.7921 -0.0030 -0.0058 -0.0133
38.50 3.2367 3.2387 3.1829 3.1872 2.9141 2.9231 -0.0020 -0.0043 -0.0089
38.75 3.3749 3.3762 3.3211 3.3241 3.0523 3.0573 -0.0013 -0.0030 -0.0050
39.00 3.5159 3.5167 3.4621 3.4642 3.1933 3.1948 -0.0008 -0.0020 -0.0014
39.25 3.6597 3.6602 3.6060 3.6072 3.3372 3.3355 -0.0005 -0.0012 0.0017
39.50 3.8064 3.8067 3.7527 3.7533 3.4839 3.4794 -0.0002 -0.0006 0.0044
39.75 3.9560 3.9560 3.9022 3.9023 3.6334 3.6265 -0.0001 -0.0001 0.0069
40.00 4.1083 4.1083 4.0545 4.0542 3.7857 3.7767 0.0000 0.0003 0.0090
40.25 4.2635 4.2634 4.2097 4.2090 3.9409 3.9300 0.0001 0.0007 0.0109
40.50 4.4215 4.4213 4.3678 4.3667 4.0990 4.0863 0.0002 0.0011 0.0127
40.75 4.5824 4.5820 4.5286 4.5271 4.2598 4.2456 0.0004 0.0015 0.0142
41.00 4.7461 4.7454 4.6923 4.6903 4.4235 4.4078 0.0007 0.0020 0.0157
41.25 4.9126 4.9114 4.8588 4.8562 4.5900 4.5729 0.0012 0.0027 0.0172
41.50 5.0820 5.0800 5.0282 5.0247 4.7594 4.7408 0.0019 0.0035 0.0186
41.75 5.2542 5.2513 5.2004 5.1958 4.9316 4.9115 0.0029 0.0046 0.0201
42.00 5.4292 5.4250 5.3754 5.3695 5.1066 5.0848 0.0042 0.0060 0.0218
42.25 5.6071 5.6012 5.5533 5.5456 5.2845 5.2609 0.0059 0.0077 0.0236
42.50 5.7878 5.7798 5.7340 5.7242 5.4652 5.4395 0.0080 0.0098 0.0257
42.75 5.9713 5.9607 5.9175 5.9052 5.6487 5.6206 0.0106 0.0124 0.0281
43.00 6.1577 6.1440 6.1039 6.0885 5.8351 5.8043 0.0137 0.0154 0.0308
43.25 6.3469 6.3295 6.2931 6.2741 6.0243 5.9903 0.0174 0.0191 0.0340
43.50 6.5389 6.5173 6.4852 6.4619 6.2164 6.1787 0.0217 0.0233 0.0377
43.75 6.7338 6.7071 6.6800 6.6519 6.4112 6.3694 0.0267 0.0282 0.0418
44.00 6.9315 6.8991 6.8778 6.8440 6.6090 6.5623 0.0324 0.0338 0.0466
1 day 5 days 25 days
Question 13.8.
See Table Four on the next page. Note the errors are larger the farther out we go as the theta will bechanging. With the one day the error is minimal at S = 40 due to no error due to changes in S (since it
187
Part 3 Options
TABLE FOUR (Problem 13.8)
Errors
Future S Approx Actual Approx 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 -0.0181 -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
1 day 5 days 25 days
188
Chapter 13 Market-Making and Delta-Hedging
will not be changing) and little error due to our theta approximation for in doesn’t change muchduring the day. For 5 days, there is a theta error at S5/365 = 40 (of .0003) due to theta decreasingduring the five days. Note that the error of .0003 is not constant across the range of prices. Besidesthe familiar delta-gamma error (i.e. ignoring third order changes of S), there is the effect changes inS have on (technically the cross partial derivative ∂2f (S, t) / (∂S∂t)). The delta gamma error issymmetric; however this cross partial error is not symmetric. To see this, we can use the fundamentaltheorem of calculus on the Black Scholes formula. By put call parity, the put and call will have thesame second cross partial derivative which is equal to
∂2f (S, t)
∂S∂t= −�call
r − σ 2/2√T − t
. (9)
In this case this, r − σ 2/2 > 0 and �call > 0, hence the above term is negative; this implies our
approximation does not include terms like ∂2f (S,t)∂S∂t
(�S) (�t) which will be positive when ST < 40and negative when ST > 40; hence our approximation will underestimate the option value for lowST and overestimate it for large ST .
Question 13.9.
See Figure 1. Note there is no visible difference between the � − � approximation and the � −� − approximation; however there is a quantitative difference for if S = 40 including willhelp capture time decay.
30 32 34 36 38 40 42 44 46 48 50-4
-2
0
2
4
6
8
10
12
Stock Price in One Day
Figure 1 (Problem 13.9)
ActualDeltaDelta-GammaDelta-Gamma-Theta
189
Part 3 Options
Question 13.10.
See Figures 2 & 3.
30 32 34 36 38 40 42 44 46 48 50-2
0
2
4
6
8
10
12
14
16Figure 2 (Problem 13.10)
Stock Price in 5 Days
Black Scholes
∆ Approx
∆ Γ Approx
δ Γ θ Approx
30 32 34 36 38 40 42 44 46 48 50-2
-1.5
-1
-0.5
0
0.5Figure 3 (Problem 13.10)
Stock Price in 5 Days
∆ Γ Error
∆ Error
∆ Γ θ Error
190
Chapter 13 Market-Making and Delta-Hedging
Question 13.11.
See Figure 4.
30 32 34 36 38 40 42 44 46 48 50-4
-2
0
2
4
6
8
10
Stock Price in One Day
Figure 4 Problem 13.11)
Black Scholes
∆ Approx.
∆ Γ Approx.
∆ Γ θ Approx.
Question 13.12.
See Figures 5 & 6. Figure 6 is on the next page.
30 32 34 36 38 40 42 44 46 48 50-1
0
1
2
3
4
5
6
7
8
9
Stock Price in 5 days
Figure 5 (Problem 13.12)
Black Scholes
∆ Approx.
∆ Γ Approx.
∆ Γ θ Approx.
191
Part 3 Options
30 32 34 36 38 40 42 44 46 48 50-1
-0.5
0
0.5Figure 6 (Problem 13.12)
Stock Price in 5 days
∆ Error
∆ Γ Error
∆ Γ θ Error.Zero Error
Question 13.13.
Using the parameters and values from Table 13.1, the market maker profit from equation (13.9) is
−(
1
2.09 (40)2 (.06516) − .0173 (365) + .08 (.5842) 40 − .08 (2.7804)
)(10)
= − (4.6912 − 6.3325 + 1.8637 − .2224) = 0. (11)
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 perday. Equation (13.9) uses annualized rates (i.e. h is in the equation. Hence for equation (13.9) weshould use −.9139. For equation (13.9) we have a market-maker profit of
−(
.09
2402 (.045446) − .9139 + .08 ((−.6028) 40 − 5.3659)
)h (12)
= − (3.2721 − .9139 − 2.3582) h = 0. (13)
Question 13.15.
For our 45-strike call that we own: C1 = 2.1004, �1 = .3949, �1 = .0457. The 40-strike has C2 =4.1217,�2 = .6151,�2 = .0454. Our gamma hedge implies we must write�1/�2 = .0457/.0454 =
192
Chapter 13 Market-Making and Delta-Hedging
1.007 40-strike calls. Our option position have a total delta of .3949 − (1.007) .6151 = −.2247hence we have to buy .2247 shares. This will cost 2.1004 − 1.007 (4.1217) + .2247 (40) = 6. 9378.Using primes to denote next day prices, our one-day profit will be
C′1 − 1.007C′
2 + .2247S′ − 6.9378e.08/365 (14)
We use Black Scholes with T − t = 179/365 to arrive at our profit in Figure 7.
30 32 34 36 38 40 42 44 46 48 50-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4Figure 7 (Problem 13.15)
Stock Price($)
OvernightProfit($)
Question 13.16.
For our 45-strike put that we’ve written: P1 = 5.3596, �1 = −.6051, �1 = .0457. The 40-strikehas C2 = 4.1217, �2 = .6151, �2 = .0454. Since we are “short” gamma (we wrote an option), wemust buy �1/�2 = .0457/.0454 = 1.007 40-strike calls. Our option position will have a total deltaof .6051 + (1.007) .6151 = 1.2245 hence we have to short 1.2247 shares. Our total initial cash flowwill be 5.3596 − 1.007 (4.1217) + 1.2247 (40) = 50.20. Using primes to denote next day prices,our one-day profit will be
−P ′1 − 1.007C′
2 − 1.2247S′ + 50.20e.08/365 (15)
We use Black Scholes with T − t = 179/365 to arrive at our profit in Figure 8 on the next page.
193
Part 3 Options
30 32 34 36 38 40 42 44 46 48 50-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5Figure 8 (Problem 13.16)
Stock Price ($)
OvernightProfit($)
Question 13.17.
The relevant values of the spread are: f = 6.1315 − 2 (2.7804) + .9710 = 1.5417, � = .8642 −2 (.5824) + .2815 = −.0191, and � = .0364 − 2 (.0652) + .0563 = −.0377. Since we wrote thespread, to � hedge we need to write .0377/.04536 = .8311 options. The delta of the spread and thecall will become .0191 − (.8311) (.6151) = −.4921; therefore we need to buy .4921 shares. Thegraph of our profit is given in Figure 9.
35 36 37 38 39 40 41 42 43 44 45-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01Figure 9 (Problem 13.17)
Stock Price ($)
OvernightProfit($)
194
Chapter 13 Market-Making and Delta-Hedging
Question 13.18.
The relevant values of the spread are: f = 5.0824 − 2(1.9905) = 1.1014, � = −.71845 −2(−.4176) = .11675, and � = .05633 − 2(.06516) = −.07399. Since we wrote the spread, to �
hedge we need to buy .07399/.04536 = 1.6312 options. The delta of the spread and the call willbecome .11675 + (1.6312)(.6151) = 1.120; therefore we need to short 1.120 shares. The graph ofour profit is given in Figure 10.
38 38.5 39 39.5 40 40.5 41 41.5 42-8
-6
-4
-2
0
2
4
6
8x 10
-3 Figure 10 (Problem 13.18)
Stock Price ($)
OvernightProfit($)
Question 13.19.
We purchased a 91-day 40-strike call, denoted option 1.
a) Using a 180 day 40-strike call (option 2) to delta-vega hedge we must write .7262 of theseoptions and short .1357 shares of stock. Our one day profit is given in Figure 11 on the next page.
b) Using option 2 as well as a one year (365 day) 45-strike put (option 3) to delta-gamma-vegahedge, we have the following solution: n2 = −2.1276, n3 = .9431, and nS = 1.1887. The one dayprofit is given in Figure 12 on the next page.
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Part 3 Options
35 36 37 38 39 40 41 42 43 44 45-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45Figure 11 (Problem 13.19a)
Stock Price ($)
OvernightProfit($)
37 38 39 40 41 42 43-7
-6
-5
-4
-3
-2
-1
0
1
2
3x 10
-3 Figure 12 (Problem 13.19b)
Stock Price ($)
OvernightProfit($)
Question 13.20.
We purchased a 91-day 40-strike call, denoted option 1.
196
Chapter 13 Market-Making and Delta-Hedging
a) Using a 180-day 40-strike call (option 2) to delta-rho hedge we must write 50.64 of theseoptions and short 27.09 shares of stock. Our one day profit is given in Figure 13.
35 36 37 38 39 40 41 42 43 44 45-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6Figure 13 (Problem 13.20a)
Stock Price ($)
OvernightProfit($)
b) Using option 2 as well as a one year (365 day) 45-strike put (option 3) to delta-gamma-vegahedge, we have the following solution: n2 = −1.2259, n3 = −.2874, and nS = .0307. The one dayprofit is given in Figure 14 on the next page. If we added another option, call it option 4, we can tryto hedge all of the greeks (note will be taken care of by the Black Scholes Equation). Let Vegabe noted by v
�2n2 + �3n3 + �4n4 + nS = −.5824 (16)
�2n2 + �3n3 + �4n4 = −.0652 (17)
v2n2 + v3n3 + v4n4 = −.0780 (18)
Rho2n2 + Rho3n3 + Rho4n4 = −.0511 (19)
These are four equations and four unknowns (the coefficients are from the Black Scholes model).Note we must try to solve the last three equations simultaneously, which give us the position of thethree options, and then use the underlying asset to delta hedge.
On a related note, occasionally you will find strange things may happen when we use options withthe same maturity. For a given time to maturity, vega and gamma are proportional (i.e. vi = ki�i).If two options have the same time to maturity, then k1 = k2. If we use option 2 to gamma hedge aposition of option 1, �2n2 = −n1�1; with the same maturity, we have
v2n2 = k2�2n2 = −k1n1�1 = −n1v1. (20)
197
Part 3 Options
37 38 39 40 41 42 43-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5x 10
-3 Figure 14 (Problem 13.20b)
Stock Price ($)
OvernightProfit($)
Hence gamma hedging takes care of vega hedging if the maturity matches. Similarly, if we use twooptions (call the 2 and 3) of the same maturity to hedge an option (call it 1) position with a differentmaturity we will have a problem for �2n2 + �3n3 = −n1�1 implies
v2n2 + v3n3 = k2 (�2n2 + �3n3) = −k2n1�1 = −(
k2
k1
)n1v1. (21)
If k1 �= k2 (i.e. the option being hedged is different from the two traded options’ identical time tomaturity), it will be impossible to both gamma and vega hedge. A simple algebraic way of lookingat this is by trying to solve
ax + by = c (22)
2 (ax + by) = kc (23)
Unless k = 2 (in which case we have an infinite number of solutions), there will be no solution.
198