(i) Using the information from the Friday, October 26, 1996 Wall Street Journal, calculate

the implied volatility from the S&P 100 index (put only). In particular, calculate for all

strike/exercise prices between 660 and 680 for options expiring in December and January

only. Note that by design index options expire on the 3rd Friday of each month so

calculate the time to maturity accordingly (assume 365 day year). In addition, use the

risk-free rate (T-Bills) closest to expiration date of the option. Remember that the RF rate

is quoted in annualized terms.

! 2

Graph the implied volatility against the strike price on a separate graph for December and

January.

Is the volatility constant?

To solve for the implied volatility you may use trial and error by equating the theoretical

price from B-S (which you must compute) to the actual market (closing price) by trial

and error, i.e. varying , but I recommend using either the Solver or Goal Seek function

in Excel.

(ii) Use the above data to estimate delta. Create graph in excel.

(iii) Use the above data to estimate gamma. Create graph in excel.

THE WALL STREET JOURNAL FRIDAY, OCTOBER 25, 1996 1:1 INDEX OPTIONS TRADING Volume Change *3*53}sss Clesing Price - Open Interest RANGES FOR UNDERLYING INDEXES Thursday, October 20, 1946 Net Frum High Low Close Chr. Deca Chr. SLP 100 ( OK) 885.23 678.48 685.01 + 101 + 99.09 + 16.9 S&P 500 A.M.Ex. 707.31 700.98 707.27 + 0.70 + 91.84 + 2.8 S&P Banks (DO 122.64 418.83 (72,6 + 0.75 + 92.10 + 27.9 CB-Tech (Th... 183.62 178.85 18.36 + 2.81 + 28.8 +. 17.1 CB-Mexico (ME).. 82.59 80,59 82.55' + 0.12 + 10,81 + 15.1 CB-Lps Mer (V). 8.26 8.06 8.26 + 0.04 + 209 + 15.2 Nasdaq IM INDO__ 750.59 7323749.70 + 13.06 + 173.47 + 30.1 Russell 2000 (Rt).. 342.93 341.31 342.93 + 0.11 + 26.96 + 8.5 Lps S&P 100 (OEM). 68.52 67.85 68.50 + 0.10 + 9.91 + 16.9 Lps S&P 500 (SF) 70.73 70.10 70.73 + 0.07 + 9.14 + 14.8 S&P Midcap (Mm).. 263.74 241.95 23.60 + 0.78 + 23.76 + 11.8 Major Mkt ... 26.32 619.46 225.49 -0.34 + 89.89 + 16.8 Leaps MMKT (LT). 62.63 61.95 62.55 0.03 + 8.99 + 16.8 Hong Kong (HKO).... 252.86 + 1.00 + 49.96 + 24.6 Leaps HK (HKL)... 25.29 0.10 + 5.00 + 20 internet (x).... 246.34 238.53 246.01 1.80 + 12.81 + AM-Mexico 007)... 92.01 89.53 91.82 + 0.74 + 3.21 + 9 Institut A.M.). 755.21 748.16 754.95 -0.26 +178.23 + 1 Japan JPN) ... 214.21 -0.47 + 12.27 + MS Cyclical (CYC)... 374.69 371.19 373.96 - 0.73 + 33.74 + MS Consum (CMR) 326.33 522.95 326.07 -0.26 + 41.12 + 12 MS Hi Tech (MSH)... 352.47 346.97 351.57 +244 + 35.80 + UK Pharm (DRG). 356,00 352.10 356.79 + 0.01 + 59.84 + Blotech (BTK) 133.38 131.41 133.31 + 0.30 - 0.46 0.33 Comp Tech ixch... 302.02 294.25 301.50 + 5.31 + 72.86 + 329 NYSE (NYA 376.72 373.24 375.93 - 0.31 + 46.10 + 14.1 Gold/Silver (XAU)... 119.81 117.49 119.25 + 0.65 1.17 1.0 OTC OXOC 533.97 572.36 583.11 + 8.17 + 108.41 + 25.5 Utility (um -... 251.05 248,92 251.02 -0.04 -25.58 - 9.6 Value Line (YLE).... 645.58 642.03 445.57 + 3.46 + 75.67 + 13.3 500.26 495.64 500.20 + 0.87 + 106.35 + 27.0 Sernicond (SOX).... 193.82 187.34 193.48 + 1.38 - 7.18 - 3.6 Top 100 (TPX) 646.92 640.82 646.86 -0.44 + 93.84 + 17.0 PSE Tech (PSET 221.88 218.07 721.74 +2.00 + 19.97 + 9.9 Me Red 165 4 P 100 INDEX(OEA) Nov + NOV 100 NOV $70p 1.312 Nov TAA 197 DRC 08 Me 4,097 Nov 04* 324 2 LOM NOV 25p 110 Vie 772 NOV 090 211 3019 Doc 600D 244 249 + We 5.955 Nov 450 - 2781 6100 We 2.718 Dec 6108 2 905 Nov 4152 W 4000 Dec 4150 4 1.849 Nov 6200 324 14 1207 Dec 4200 4 10 NOV 14 + 7.216 Dec 02 10 44 + 2,129 Jen 750 10 4 + 1 10 NOV 4304 S7 Nov 670p 1,100 1% + 6936 Doc 420D 316 M + 4,221 Jan 20p 316 64 + 1157 Feb 4200 9 NOV 625p Des 4750 AP 12 NOV Nov 0p 7,157 he Dec GOD MOD NOV 6-455 105 3644 $ 12,02 NOV up 205$ W + 1% 20.585 Dec MSP 4 SH - TA 4,977 NOY 1906 to 314 - 64 2,219 Nov 6500 2.501 4 + 174 171 Dee 05 M# 4 2.276 Jan 10 552 Feb 6500 2 46 Nov 12 148 NOV 855 14 te 8550 + 3,137 Nov 60c 2.757 NOV 72670 6000 10 2.441 Dec 660p w + 14 6.836 6600 208 11. Fct 6000 20% 21% MOV 102 19 Nov p1,145 6th + z 11.726 Dec 0650 z2 101 1% 1,247 Jan MASP 113% + 1% NOV 6704 4 ISA - SV 9.047 NON 6700 4,00? 744 +24 14,093 6700 67 14 + 1.200 Jan 670p 12 10% - 2 1.961 HOV 675 1.762 12V - $ 6.077 NOV 4750 10.409 94 + 3 12,024 675p 77 1575 + 14 594 Jan 6752 21 + 1 725 NOV 600C 5.716 6 - 44 11.5 NOV 1900 19.240 1214 + 10 240 od m 1 AOC 2500 $c De Dec Bank IBIDA... TREASURY BILLS Bank (BX)..... 500.26 495,64 500.20 + 0.87 + 106.35 + 27.0 Semicond (SOX)...... 193.82 187.34 293.48 + 1.38 - 7.18 - 3.6 Top 100 (TPX)........ 646.92 640.82 646.86 - 0.44 + $.84 + 17.0 PSE Tech (PSE......., 21.88 218.07 221.74 + 2.00 + 19.97 + 9.9 Des MSP 675p 594 TREASURY BILLS DAY 1993 Nov 0 2787 + 7670 Ote 6605 10 Hi De 660p 10 + 14 6,636 660p 208 17. 618 Act 600D 2041 Nov 10219 Nov Ap 1,15 to + 244 11.320 0950 22 1074 # in Jan 1 132 NOV 670e 4 ISA - $149,047 Noy 6760 4.60 744 + 24 14,092 De 6700 67 + 1.201 Jan 670D 12 106 - 1 1,441 Hov 675 1.742 1217 - SV 6.077 Nov 6750 10.400 + 312624 2 199 + 144 6750 24 16 + 775 Nov 600C 5,716 11.375 NOY 4900 19.20 1714 + 4V 10,240 DEC 480c 205 1776 - 14 247 Dec 6900 + 3 Jan 6806 .30 21% -1% 996 Jan SHOP 627 2014 + Fb ooop 0 14 - 09 NOV 4055 11.6.36 7 - * 14,658 NOY asp 6,026 144 + 7% 11414) De 6850 100 16 - 2903 DES 685p 26 204 + 5V 1,150 Nov 6700 10,351 S 2 7471 440p 1.243 1744 + 4 ki be 3 - 11 1.159 6700 17 21% + 374 76 Jan 4904 207 17 361 18 6900 * 547 Feb POC 100 22V - 212 Feb 4900 103 77 + 2v 355 Occ 0950 15 10 1 Nov 700C 6,443 24. 149.362 Nov OD 28 2499 + ms 700 14 2068 Dec 700D 700C 1774 06 Feb TOP 74 - 19 Nov 705C 3,416 17 18 1 3,002 NOV 7100 2702 7 B. 112 710 100 6 Dec 7100 13H + JA NOY The - 1 4,178 Dec 7156 3% 849 NOV 14 Dec 720 v 2058 720p 10 1% NOV 7250 1,346 21 3.330 NOV 7305 VA 3,670 Dec 700 3.589 Nov $30 Des 130 Call Val, 55,14 Opon in 166,288 Put Vol. $5,010 Openint, 304.450 496 JAN $11 Feb to - Auk Maturity Mat. Old Asked CH, Y. od 3 6 4.75 4.71 +0.04.13 NOV 07 V 11 40 4.76 Nov 14 06 2J 4 +0.01 4.15 Nov 21 8 77 LA 477 +0.01 4.25 Nov 2 # +0.64 Doc OS 941 4.90 4.1 +0.01 Dec 127 & 1.90 446 +0.011.96 Det ty 96 315 4.8 41 +0.47 2.91 Dec 2.0 47 +0.014 Jan 17 I 1.4 +0.00 Jan 097 2.4 1.95 Jan 14 1 20. 5.04 4.1 +0.025.10 van 290 1,0 Jan 2017 3.02 5.0 0.0 114 Jan ** n 5 TN #10 5.12 Feb 2 10 5.02 5.00 0.00 5.14 13 9 15.00 +0... 5.13 Feb 7 7 2 SM +0.01 $.10 27 11:5.01 4.9 +0.025,14 Mar Os 17 IR 5.00 5.04 0.02 5.31 Mar 13 7 13 5.0 5.04 +0.01 si Mar 27 145.0 5.0 5427 Mar 77 97 15 4.05 $.13 5.05 -0.02 Aor 10-97 167 5.11 5. Apr 1797 174 5.12 5.10 17 Apr 24 7 197 5.12 5.11 +0.01 May 01 77 ita 5.15 1.13 +0.015 May di 77 10 5.12 + 0.01 52 May 29 7 216 5.15 5.13 +0.01 31 Jun 26' 241 $.17 5.15 +0.01 5.37 Jul 22 07 2725.22 5.20 +0.025.4 Aug 21 y 300$.245,72 +0.01 5.0 Seo 18 97 33 5.2 5.21 5.48 Oct 16'97 356 5.24 5.25 +0.01 534 Feb 1.578 ; Apr 097 497 1605.08 20 V . 739 7400 + Black-Scholes Formulas for a Call and a Put and th Option Call Inputs X.r,1,0,So Output Put X,1,1,0, S. Po Call 4 X,1,1,Sg, market call price o = "implied volatility" 1 Put X,1,T, S., market call price o = "implied volatility" where 4, "{$x}+{r*0%A) OJT dy =d, -ovt and N(d) = = cumulative normal distribution da Put and the Implied Volatilities Formula Cu = SN(d) - Xe"N(22) P = SN(-d) + XeN(-d)) ility" o that makes: market call price = SoN(D1) - Xe"IN(d2) ility" o that makes: market put price =-SoN(-di) + Xe "N(-da) THE WALL STREET JOURNAL FRIDAY, OCTOBER 25, 1996 1:1 INDEX OPTIONS TRADING Volume Change *3*53}sss Clesing Price - Open Interest RANGES FOR UNDERLYING INDEXES Thursday, October 20, 1946 Net Frum High Low Close Chr. Deca Chr. SLP 100 ( OK) 885.23 678.48 685.01 + 101 + 99.09 + 16.9 S&P 500 A.M.Ex. 707.31 700.98 707.27 + 0.70 + 91.84 + 2.8 S&P Banks (DO 122.64 418.83 (72,6 + 0.75 + 92.10 + 27.9 CB-Tech (Th... 183.62 178.85 18.36 + 2.81 + 28.8 +. 17.1 CB-Mexico (ME).. 82.59 80,59 82.55' + 0.12 + 10,81 + 15.1 CB-Lps Mer (V). 8.26 8.06 8.26 + 0.04 + 209 + 15.2 Nasdaq IM INDO__ 750.59 7323749.70 + 13.06 + 173.47 + 30.1 Russell 2000 (Rt).. 342.93 341.31 342.93 + 0.11 + 26.96 + 8.5 Lps S&P 100 (OEM). 68.52 67.85 68.50 + 0.10 + 9.91 + 16.9 Lps S&P 500 (SF) 70.73 70.10 70.73 + 0.07 + 9.14 + 14.8 S&P Midcap (Mm).. 263.74 241.95 23.60 + 0.78 + 23.76 + 11.8 Major Mkt ... 26.32 619.46 225.49 -0.34 + 89.89 + 16.8 Leaps MMKT (LT). 62.63 61.95 62.55 0.03 + 8.99 + 16.8 Hong Kong (HKO).... 252.86 + 1.00 + 49.96 + 24.6 Leaps HK (HKL)... 25.29 0.10 + 5.00 + 20 internet (x).... 246.34 238.53 246.01 1.80 + 12.81 + AM-Mexico 007)... 92.01 89.53 91.82 + 0.74 + 3.21 + 9 Institut A.M.). 755.21 748.16 754.95 -0.26 +178.23 + 1 Japan JPN) ... 214.21 -0.47 + 12.27 + MS Cyclical (CYC)... 374.69 371.19 373.96 - 0.73 + 33.74 + MS Consum (CMR) 326.33 522.95 326.07 -0.26 + 41.12 + 12 MS Hi Tech (MSH)... 352.47 346.97 351.57 +244 + 35.80 + UK Pharm (DRG). 356,00 352.10 356.79 + 0.01 + 59.84 + Blotech (BTK) 133.38 131.41 133.31 + 0.30 - 0.46 0.33 Comp Tech ixch... 302.02 294.25 301.50 + 5.31 + 72.86 + 329 NYSE (NYA 376.72 373.24 375.93 - 0.31 + 46.10 + 14.1 Gold/Silver (XAU)... 119.81 117.49 119.25 + 0.65 1.17 1.0 OTC OXOC 533.97 572.36 583.11 + 8.17 + 108.41 + 25.5 Utility (um -... 251.05 248,92 251.02 -0.04 -25.58 - 9.6 Value Line (YLE).... 645.58 642.03 445.57 + 3.46 + 75.67 + 13.3 500.26 495.64 500.20 + 0.87 + 106.35 + 27.0 Sernicond (SOX).... 193.82 187.34 193.48 + 1.38 - 7.18 - 3.6 Top 100 (TPX) 646.92 640.82 646.86 -0.44 + 93.84 + 17.0 PSE Tech (PSET 221.88 218.07 721.74 +2.00 + 19.97 + 9.9 Me Red 165 4 P 100 INDEX(OEA) Nov + NOV 100 NOV $70p 1.312 Nov TAA 197 DRC 08 Me 4,097 Nov 04* 324 2 LOM NOV 25p 110 Vie 772 NOV 090 211 3019 Doc 600D 244 249 + We 5.955 Nov 450 - 2781 6100 We 2.718 Dec 6108 2 905 Nov 4152 W 4000 Dec 4150 4 1.849 Nov 6200 324 14 1207 Dec 4200 4 10 NOV 14 + 7.216 Dec 02 10 44 + 2,129 Jen 750 10 4 + 1 10 NOV 4304 S7 Nov 670p 1,100 1% + 6936 Doc 420D 316 M + 4,221 Jan 20p 316 64 + 1157 Feb 4200 9 NOV 625p Des 4750 AP 12 NOV Nov 0p 7,157 he Dec GOD MOD NOV 6-455 105 3644 $ 12,02 NOV up 205$ W + 1% 20.585 Dec MSP 4 SH - TA 4,977 NOY 1906 to 314 - 64 2,219 Nov 6500 2.501 4 + 174 171 Dee 05 M# 4 2.276 Jan 10 552 Feb 6500 2 46 Nov 12 148 NOV 855 14 te 8550 + 3,137 Nov 60c 2.757 NOV 72670 6000 10 2.441 Dec 660p w + 14 6.836 6600 208 11. Fct 6000 20% 21% MOV 102 19 Nov p1,145 6th + z 11.726 Dec 0650 z2 101 1% 1,247 Jan MASP 113% + 1% NOV 6704 4 ISA - SV 9.047 NON 6700 4,00? 744 +24 14,093 6700 67 14 + 1.200 Jan 670p 12 10% - 2 1.961 HOV 675 1.762 12V - $ 6.077 NOV 4750 10.409 94 + 3 12,024 675p 77 1575 + 14 594 Jan 6752 21 + 1 725 NOV 600C 5.716 6 - 44 11.5 NOV 1900 19.240 1214 + 10 240 od m 1 AOC 2500 $c De Dec Bank IBIDA... TREASURY BILLS Bank (BX)..... 500.26 495,64 500.20 + 0.87 + 106.35 + 27.0 Semicond (SOX)...... 193.82 187.34 293.48 + 1.38 - 7.18 - 3.6 Top 100 (TPX)........ 646.92 640.82 646.86 - 0.44 + $.84 + 17.0 PSE Tech (PSE......., 21.88 218.07 221.74 + 2.00 + 19.97 + 9.9 Des MSP 675p 594 TREASURY BILLS DAY 1993 Nov 0 2787 + 7670 Ote 6605 10 Hi De 660p 10 + 14 6,636 660p 208 17. 618 Act 600D 2041 Nov 10219 Nov Ap 1,15 to + 244 11.320 0950 22 1074 # in Jan 1 132 NOV 670e 4 ISA - $149,047 Noy 6760 4.60 744 + 24 14,092 De 6700 67 + 1.201 Jan 670D 12 106 - 1 1,441 Hov 675 1.742 1217 - SV 6.077 Nov 6750 10.400 + 312624 2 199 + 144 6750 24 16 + 775 Nov 600C 5,716 11.375 NOY 4900 19.20 1714 + 4V 10,240 DEC 480c 205 1776 - 14 247 Dec 6900 + 3 Jan 6806 .30 21% -1% 996 Jan SHOP 627 2014 + Fb ooop 0 14 - 09 NOV 4055 11.6.36 7 - * 14,658 NOY asp 6,026 144 + 7% 11414) De 6850 100 16 - 2903 DES 685p 26 204 + 5V 1,150 Nov 6700 10,351 S 2 7471 440p 1.243 1744 + 4 ki be 3 - 11 1.159 6700 17 21% + 374 76 Jan 4904 207 17 361 18 6900 * 547 Feb POC 100 22V - 212 Feb 4900 103 77 + 2v 355 Occ 0950 15 10 1 Nov 700C 6,443 24. 149.362 Nov OD 28 2499 + ms 700 14 2068 Dec 700D 700C 1774 06 Feb TOP 74 - 19 Nov 705C 3,416 17 18 1 3,002 NOV 7100 2702 7 B. 112 710 100 6 Dec 7100 13H + JA NOY The - 1 4,178 Dec 7156 3% 849 NOV 14 Dec 720 v 2058 720p 10 1% NOV 7250 1,346 21 3.330 NOV 7305 VA 3,670 Dec 700 3.589 Nov $30 Des 130 Call Val, 55,14 Opon in 166,288 Put Vol. $5,010 Openint, 304.450 496 JAN $11 Feb to - Auk Maturity Mat. Old Asked CH, Y. od 3 6 4.75 4.71 +0.04.13 NOV 07 V 11 40 4.76 Nov 14 06 2J 4 +0.01 4.15 Nov 21 8 77 LA 477 +0.01 4.25 Nov 2 # +0.64 Doc OS 941 4.90 4.1 +0.01 Dec 127 & 1.90 446 +0.011.96 Det ty 96 315 4.8 41 +0.47 2.91 Dec 2.0 47 +0.014 Jan 17 I 1.4 +0.00 Jan 097 2.4 1.95 Jan 14 1 20. 5.04 4.1 +0.025.10 van 290 1,0 Jan 2017 3.02 5.0 0.0 114 Jan ** n 5 TN #10 5.12 Feb 2 10 5.02 5.00 0.00 5.14 13 9 15.00 +0... 5.13 Feb 7 7 2 SM +0.01 $.10 27 11:5.01 4.9 +0.025,14 Mar Os 17 IR 5.00 5.04 0.02 5.31 Mar 13 7 13 5.0 5.04 +0.01 si Mar 27 145.0 5.0 5427 Mar 77 97 15 4.05 $.13 5.05 -0.02 Aor 10-97 167 5.11 5. Apr 1797 174 5.12 5.10 17 Apr 24 7 197 5.12 5.11 +0.01 May 01 77 ita 5.15 1.13 +0.015 May di 77 10 5.12 + 0.01 52 May 29 7 216 5.15 5.13 +0.01 31 Jun 26' 241 $.17 5.15 +0.01 5.37 Jul 22 07 2725.22 5.20 +0.025.4 Aug 21 y 300$.245,72 +0.01 5.0 Seo 18 97 33 5.2 5.21 5.48 Oct 16'97 356 5.24 5.25 +0.01 534 Feb 1.578 ; Apr 097 497 1605.08 20 V . 739 7400 + Black-Scholes Formulas for a Call and a Put and th Option Call Inputs X.r,1,0,So Output Put X,1,1,0, S. Po Call 4 X,1,1,Sg, market call price o = "implied volatility" 1 Put X,1,T, S., market call price o = "implied volatility" where 4, "{$x}+{r*0%A) OJT dy =d, -ovt and N(d) = = cumulative normal distribution da Put and the Implied Volatilities Formula Cu = SN(d) - Xe"N(22) P = SN(-d) + XeN(-d)) ility" o that makes: market call price = SoN(D1) - Xe"IN(d2) ility" o that makes: market put price =-SoN(-di) + Xe "N(-da)