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

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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

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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 =

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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.

195

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