chapter21

UN-14EBlack-Scholes Option Pricing FormulaApplied to General Pills PutS 56 Stock priceX 50 Exercise priceT 1 Time remaining r 8.00% Risk-free rate of interestSigma 30% Stock volatilityd1 0.7944 - (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)d2 0.4944 - d1-Sigma*SQRT(T)N(d1) 0.7865 - Uses formula NormSDist(d1)N(d2) 0.6895 - Uses formula NormSDist(d2)Call price 12.22 - S*N(d1)-X*exp(-r*T)*N(d2)Put price 2.38 - call price - S + X*Exp(-r*T): by Put-Call parityA B C123456789101112131415UN-14E布莱克-斯科尔斯期权定价公式应用于General Pills公司股票的看跌期权S 56 股票价格X 50 执行价格T 1 剩余时间r 8.00% 无风险利率Sigma 30% 股票波动性d1 0.7944 - (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)d2 0.4944 - d1-Sigma*SQRT(T)N(d1) 0.7865 - 使用公式 NormSDist(d1)N(d2) 0.6895 - 使用公式 NormSDist(d2)看涨期权价格 12.22 - S*N(d1)-X*exp(-r*T)*N(d2)看跌期权价格 2.38 - 看涨期权价格 - S + X*Exp(-r*T): 用看跌-看涨期权平价原理A B C D123456789101112131415UN-14EBlack-Scholes Option Pricing FormulaApplied to General Pills PutS 56 Stock priceX 50 Exercise priceT 1 Time remaining r 8.00% Risk-free rate of interestSigma 30% Stock volatilityd1 0.7944 - (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)d2 0.4944 - d1-Sigma*SQRT(T)N(d1) 0.7865 - Uses formula NormSDist(d1)N(d2) 0.6895 - Uses formula NormSDist(d2)Call price 12.22 - S*N(d1)-X*exp(-r*T)*N(d2)Put price 2.38 - call price - S + X*Exp(-r*T): by Put-Call parityCalculating the portfolio insurance proportionsOmega 75.45% - =B2*B11/(B2+B15), proportion in shares1-omega 24.55% - =1-B18, proportion in bondsA B C12345678910111213141516171819UN-14E布莱克-斯科尔斯期权定价公式应用于General Pills公司股票的看跌期权S 56 股票价格X 50 执行价格T 1 剩余时间r 8.00% 无风险利率Sigma 30% 股票波动性d1 0.7944 - (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)d2 0.4944 - d1-Sigma*SQRT(T)N(d1) 0.7865 - 使用公式 NormSDist(d1)N(d2) 0.6895 - 使用公式 NormSDist(d2)看涨期权价格 12.22 - S*N(d1)-X*exp(-r*T)*N(d2)看跌期权价格 2.38 - 看涨期权价格 - S + X*Exp(-r*T): 用看跌-看涨期权平价原理计算投资组合中保险的比例Omega 75.45% - =B2*B11/(B2+B15), 股票的比例1-omega 24.55% - =1-B18, 债券的比例A B C12345678910111213141516171819UN-14EBlack-Scholes Option Pricing FormulaApplied to General Pills PutS 60 Stock priceX 50 Exercise priceT 0.9808 - =1-1/52 , time remainingr 8.00% Risk-free rate of interestSigma 30% Stock volatilityd1 1.0263 - (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)d2 0.7292 - d1-Sigma*SQRT(T)N(d1) 0.8476 - Uses formula NormSDist(d1)N(d2) 0.7671 - Uses formula NormSDist(d2)Call price 15.40 - S*N(d1)-X*exp(-r*T)*N(d2)Put price 1.63 - call price - S + X*Exp(-r*T): by Put-Call parityCalculating the portfolio insurance proportionsOmega 82.53% - =B2*B11/(B2+B15), proportion in shares1-omega 17.47% - =1-B18, proportion in bondsA B C12345678910111213141516171819UN-14E布莱克-斯科尔斯期权定价公式应用于General Pills公司股票的看跌期权S 60 股票价格X 50 执行价格T 0.9808 - =1-1/52 , 剩余时间r 8.00% 无风险利率Sigma 30% 股票波动性d1 1.0263 - (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)d2 0.7292 - d1-Sigma*SQRT(T)N(d1) 0.8476 - 使用公式 NormSDist(d1)N(d2) 0.7671 - 使用公式 NormSDist(d2)看涨期权价格 15.40 - S*N(d1)-X*exp(-r*T)*N(d2)看跌期权价格 1.63 - 看涨期权价格 - S + X*Exp(-r*T): 用看跌-看涨期权平价原理计算投资组合中保险的比例Omega 82.53% - =B2*B11/(B2+B15), 股票的比例1-omega 17.47% - =1-B18,债券的比例A B C12345678910111213141516171819Pages 581-583Page 7PORTFOLIO INSURANCE SIMULATIONX 50 Exercise price of puts in strategyMean, m 15% Mean of lognormal distributionSigma, s 30% Sigma of lognormal distributionr 8% Interest rateInitial 1000 Initial investmentPortfolio value, beginning of weekWeekStock priceat begin. ofweekStock priceat end ofweekd1 ,beginingof weekOmega,beginningofweekStocks Bonds Portfoliovalue0 56.00 60.00 0.7944 0.7545 754.50 245.50 1000.001 60.00 52.00 1.0263 0.8253 870.06 184.22 1054.272 52.00 ? 0.5419 0.6635 622.73 315.82 938.55Pages 581-583Page 8Portfolio value, end of weekStocks Bonds Portfoliovalue808.39 245.88 1054.27754.05 184.50 938.55? 316.30 ?PORTFOLIO INSURANCE SIMULATIONPages 581-583 中文Page 9投资组合保险的模拟X 50 策略中看跌期权的执行价格均值, m 15% 对数正态分布的均值Sigma, s 30% 对数正态分布的Sigmar 8% 利率初始 1000 初始投资投资组合价值, 星期开始星期星期开始时的股票价格星期结束时的股票价格 星期开始时的d1星期开始时的Omega, 股票 债券 投资组合价值0 56.00 60.00 0.7944 0.7545 754.50 245.50 1000.001 60.00 52.00 1.0263 0.8253 870.06 184.22 1054.272 52.00 ? 0.5419 0.6635 622.73 315.82 938.55Pages 581-583 中文Page 10投资组合价值, 星期结束股票 债券 投资组合价值808.39 245.88 1054.27754.05 184.50 938.55? 316.30 ?投资组合保险的模拟Page 585-86Page 11PORTFOLIO INSURANCE SIMULATIONX 50 Exercise price of puts in strategyMean, m 15% Mean of lognormal distributionSigma, s 30% Sigma of lognormal distributionr 8% Interest rateInitial 1000 Initial investmentPortfolio value, beginning of weekWeekStock priceat begin. ofweekStock priceat end ofweekd1,beginningof weekOmega,beginningof weekStocks Bonds Portfoliovalue0 56.00 53.61 0.7944 0.7545 754.50 245.50 1000.001 53.61 50.96 0.6470 0.7029 680.53 287.58 968.122 50.96 50.51 0.4734 0.6364 595.04 339.96 935.003 50.51 47.70 0.4392 0.6232 579.68 350.55 930.234 47.70 47.68 0.2366 0.5394 484.61 413.87 898.485 47.68 45.33 0.2295 0.5370 482.71 416.25 898.966 45.33 46.71 0.0440 0.4588 401.77 474.00 875.777 46.71 48.31 0.1434 0.5016 445.83 442.90 888.738 48.31 49.11 0.2587 0.5512 498.68 406.06 904.739 49.11 48.02 0.3134 0.5747 525.10 388.56 913.6510 48.02 45.45 0.2245 0.5380 485.59 416.95 902.5411 45.45 47.19 0.0117 0.4486 393.51 483.68 877.1912 47.19 47.85 0.1454 0.5059 451.73 441.26 892.9913 47.85 48.50 0.1920 0.5263 473.68 426.37 900.0514 48.50 44.49 0.2373 0.5460 495.30 411.78 907.0815 44.49 42.46 -0.1096 0.4012 347.79 519.03 866.8216 42.46 43.19 -0.3083 0.3234 275.45 576.27 851.7117 43.19 43.46 -0.2534 0.3453 296.04 561.28 857.3218 43.46 42.55 -0.2408 0.3510 301.91 558.17 860.0719 42.55 44.13 -0.3434 0.3126 267.14 587.44 854.5820 44.13 42.79 -0.2042 0.3671 317.70 547.70 865.4021 42.79 38.00 -0.3504 0.3116 266.96 589.67 856.6322 38.00 37.27 -0.8876 0.1467 121.40 706.26 827.6623 37.27 37.11 -1.0003 0.1220 100.82 725.59 826.4124 37.11 36.76 -1.0483 0.1128 93.28 733.82 827.1025 36.76 37.72 -1.1223 0.0993 82.17 745.19 827.3526 37.72 40.11 -1.0343 0.1168 96.99 733.64 830.6327 40.11 37.71 -0.7711 0.1801 150.89 687.02 837.9128 37.71 39.18 -1.1007 0.1050 87.14 742.82 829.9629 39.18 37.16 -0.9446 0.1381 115.25 719.25 834.5030 37.16 35.92 -1.2504 0.0806 66.87 762.78 829.6531 35.92 33.18 -1.4697 0.0523 43.37 785.23 828.6032 33.18 33.54 -1.9461 0.0176 14.59 811.91 826.4933 33.54 35.22 -1.9497 0.0177 14.63 813.28 827.9034 35.22 34.58 -1.7405 0.0296 24.52 805.36 829.8935 34.58 33.14 -1.9121 0.0198 16.45 814.23 830.6836 33.14 33.15 -2.2399 0.0085 7.08 824.17 831.25A B C D E F G H I12345678910111213141516171819202122232425262728293031323334353637383940414243444546Page 585-86Page 1237 33.15 33.47 -2.3266 0.0068 5.64 826.88 832.5238 33.47 34.20 -2.3629 0.0062 5.17 828.68 833.8539 34.20 35.10 -2.3234 0.0070 5.87 829.37 835.2440 35.10 35.18 -2.2544 0.0086 7.23 829.44 836.6741 35.18 37.68 -2.3553 0.0066 5.55 832.42 837.9742 37.68 36.82 -1.9677 0.0188 15.76 823.88 839.6443 36.82 37.28 -2.2774 0.0085 7.14 833.41 840.5544 37.28 36.70 -2.3313 0.0074 6.27 835.65 841.9245 36.70 37.70 -2.6567 0.0029 2.47 840.64 843.1146 37.70 35.60 -2.6295 0.0033 2.75 841.73 844.4747 35.60 37.72 -3.5220 0.0002 0.13 845.49 845.6248 37.72 38.96 -3.2727 0.0004 0.34 846.58 846.9349 38.96 38.99 -3.3629 0.0003 0.26 847.98 848.2450 38.99 41.97 -4.1436 0.0000 0.01 849.53 849.5551 41.97 42.84 -4.1505 0.0000 0.01 850.84 850.8552 42.84 852.17A B C D E F G H I47484950515253545556575859606162Page 585-86Page 13Portfolio value, end of weekStocks Bonds PortfoliovalueStandardnormaldeviates722.24 245.88 968.12 -1.119738646.97 288.02 935.00 -1.2849841589.75 340.48 930.23 -0.2841057547.40 351.09 898.48 -1.4467969484.45 414.51 898.96 -0.0774548458.87 416.89 875.77 -1.2863848414.00 474.73 888.73 0.65187578461.15 443.59 904.73 0.7427127506.97 406.68 913.65 0.32723278513.39 389.15 902.54 -0.6115295459.59 417.60 877.19 -1.3919225408.57 484.42 892.99 0.83319264458.11 441.94 900.05 0.26805196480.05 427.03 907.08 0.25198915454.40 412.42 866.82 -2.1408232331.88 519.83 851.71 -1.194594280.17 577.15 857.32 0.3392006297.92 562.15 860.07 0.08344159295.56 559.02 854.58 -0.5803736277.05 588.35 865.40 0.80632162308.08 548.54 856.63 -0.8078064237.09 590.58 827.66 -2.9218791119.07 707.35 826.41 -0.5369145100.39 726.71 827.10 -0.171777492.40 734.95 827.35 -0.29639384.30 746.34 830.63 0.54612656103.13 734.77 837.91 1.40720203141.88 688.07 829.96 -1.548660390.54 743.96 834.50 0.85074817109.28 720.36 829.65 -1.346043164.65 763.95 828.60 -0.881809740.05 786.44 826.49 -1.979751714.75 813.16 827.90 0.1930118315.36 814.53 829.89 1.1028578224.08 806.60 830.68 -0.511113315.77 815.48 831.25 -1.08739197.09 825.44 832.52 -0.0628847P o r t f o l i o I n s u r a n c e S i m u l a t i o n0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0 4 2 4 4 4 6 4 8 5 0 5 202 0 04 0 06 0 08 0 01 0 0 01 2 0 001 02 03 04 05 06 0W e e kInsuredportfoliovalueStockpriceI n s u r e d p o r t f o l ioS t o c k p r i c ePortfolio value, beginning of weekJ K L M N O P Q R S12345678910111213141516171819202122232425262728293031323334353637383940414243444546Page 585-86Page 145.70 828.15 833.85 0.15812935.28 829.96 835.24 0.452344016.03 830.64 836.67 0.556099177.24 830.72 837.97 -0.01357885.94 833.70 839.64 1.5777322915.40 825.15 840.55 -0.62142827.23 834.69 841.92 0.226639266.17 836.94 843.11 -0.44658922.53 841.94 844.47 0.576485492.59 843.02 845.62 -1.44592830.14 846.79 846.93 1.318539940.35 847.89 848.24 0.708962490.26 849.29 849.55 -0.04664230.01 850.84 850.85 1.697821970.01 852.15 852.17 0.42271154J K L M N O P Q R S47484950515253545556575859606162Page 585-86Page 15P o r t f o l i o I n s u r a n c e S i m u l a t i o n0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0 4 2 4 4 4 6 4 8 5 0 5 202 0 04 0 06 0 08 0 01 0 0 01 2 0 001 02 03 04 05 06 0W e e kInsuredportfoliovalueStockpriceI n s u r e d p o r t f o l ioS t o c k p r i c eT U V W X Y Z12345678910111213141516171819202122232425262728293031323334353637383940414243444546Page 585-86 中文Page 16投资组合保险的模拟X 50 策略中看跌期权的执行价格均值, m 15% 对数正态分布的均值Sigma, s 30% 对数正态分布的Sigmar 8% 利率初始 1000 初始投资投资组合价值, 星期开始星期星期开始时的股票价格星期开始时的股票价格 星期开始时的d1 星期开始时的Omega 股票 债券 投资组合价值0 56.00 53.61 0.7944 0.7545 754.50 245.50 1000.001 53.61 50.96 0.6470 0.7029 680.53 287.58 968.122 50.96 50.51 0.4734 0.6364 595.04 339.96 935.003 50.51 47.70 0.4392 0.6232 579.68 350.55 930.234 47.70 47.68 0.2366 0.5394 484.61 413.87 898.485 47.68 45.33 0.2295 0.5370 482.71 416.25 898.966 45.33 46.71 0.0440 0.4588 401.77 474.00 875.777 46.71 48.31 0.1434 0.5016 445.83 442.90 888.738 48.31 49.11 0.2587 0.5512 498.68 406.06 904.739 49.11 48.02 0.3134 0.5747 525.10 388.56 913.6510 48.02 45.45 0.2245 0.5380 485.59 416.95 902.5411 45.45 47.19 0.0117 0.4486 393.51 483.68 877.1912 47.19 47.85 0.1454 0.5059 451.73 441.26 892.9913 47.85 48.50 0.1920 0.5263 473.68 426.37 900.0514 48.50 44.49 0.2373 0.5460 495.30 411.78 907.0815 44.49 42.46 -0.1096 0.4012 347.79 519.03 866.8216 42.46 43.19 -0.3083 0.3234 275.45 576.27 851.7117 43.19 43.46 -0.2534 0.3453 296.04 561.28 857.3218 43.46 42.55 -0.2408 0.3510 301.91 558.17 860.0719 42.55 44.13 -0.3434 0.3126 267.14 587.44 854.5820 44.13 42.79 -0.2042 0.3671 317.70 547.70 865.4021 42.79 38.00 -0.3504 0.3116 266.96 589.67 856.6322 38.00 37.27 -0.8876 0.1467 121.40 706.26 827.6623 37.27 37.11 -1.0003 0.1220 100.82 725.59 826.4124 37.11 36.76 -1.0483 0.1128 93.28 733.82 827.1025 36.76 37.72 -1.1223 0.0993 82.17 745.19 827.3526 37.72 40.11 -1.0343 0.1168 96.99 733.64 830.6327 40.11 37.71 -0.7711 0.1801 150.89 687.02 837.9128 37.71 39.18 -1.1007 0.1050 87.14 742.82 829.9629 39.18 37.16 -0.9446 0.1381 115.25 719.25 834.5030 37.16 35.92 -1.2504 0.0806 66.87 762.78 829.6531 35.92 33.18 -1.4697 0.0523 43.37 785.23 828.6032 33.18 33.54 -1.9461 0.0176 14.59 811.91 826.4933 33.54 35.22 -1.9497 0.0177 14.63 813.28 827.9034 35.22 34.58 -1.7405 0.0296 24.52 805.36 829.8935 34.58 33.14 -1.9121 0.0198 16.45 814.23 830.6836 33.14 33.15 -2.2399 0.0085 7.08 824.17 831.25A B C D E F G H I12345678910111213141516171819202122232425262728293031323334353637383940414243444546Page 585-86 中文Page 1737 33.15 33.47 -2.3266 0.0068 5.64 826.88 832.5238 33.47 34.20 -2.3629 0.0062 5.17 828.68 833.8539 34.20 35.10 -2.3234 0.0070 5.87 829.37 835.2440 35.10 35.18 -2.2544 0.0086 7.23 829.44 836.6741 35.18 37.68 -2.3553 0.0066 5.55 832.42 837.9742 37.68 36.82 -1.9677 0.0188 15.76 823.88 839.6443 36.82 37.28 -2.2774 0.0085 7.14 833.41 840.5544 37.28 36.70 -2.3313 0.0074 6.27 835.65 841.9245 36.70 37.70 -2.6567 0.0029 2.47 840.64 843.1146 37.70 35.60 -2.6295 0.0033 2.75 841.73 844.4747 35.60 37.72 -3.5220 0.0002 0.13 845.49 845.6248 37.72 38.96 -3.2727 0.0004 0.34 846.58 846.9349 38.96 38.99 -3.3629 0.0003 0.26 847.98 848.2450 38.99 41.97 -4.1436 0.0000 0.01 849.53 849.5551 41.97 42.84 -4.1505 0.0000 0.01 850.84 850.8552 42.84 852.17A B C D E F G H I47484950515253545556575859606162Page 585-86 中文Page 18投资组合价值, 星期结束股票 债券 投资组合价值 标准正态偏差随机数722.24 245.88 968.12 -1.119738646.97 288.02 935.00 -1.2849841589.75 340.48 930.23 -0.2841057547.40 351.09 898.48 -1.4467969484.45 414.51 898.96 -0.0774548458.87 416.89 875.77 -1.2863848414.00 474.73 888.73 0.65187578461.15 443.59 904.73 0.7427127506.97 406.68 913.65 0.32723278513.39 389.15 902.54 -0.6115295459.59 417.60 877.19 -1.3919225408.57 484.42 892.99 0.83319264458.11 441.94 900.05 0.26805196480.05 427.03 907.08 0.25198915454.40 412.42 866.82 -2.1408232331.88 519.83 851.71 -1.194594280.17 577.15 857.32 0.3392006297.92 562.15 860.07 0.08344159295.56 559.02 854.58 -0.5803736277.05 588.35 865.40 0.80632162308.08 548.54 856.63 -0.8078064237.09 590.58 827.66 -2.9218791119.07 707.35 826.41 -0.5369145100.39 726.71 827.10 -0.171777492.40 734.95 827.35 -0.29639384.30 746.34 830.63 0.54612656103.13 734.77 837.91 1.40720203141.88 688.07 829.96 -1.548660390.54 743.96 834.50 0.85074817109.28 720.36 829.65 -1.346043164.65 763.95 828.60 -0.881809740.05 786.44 826.49 -1.979751714.75 813.16 827.90 0.1930118315.36 814.53 829.89 1.1028578224.08 806.60 830.68 -0.511113315.77 815.48 831.25 -1.08739197.09 825.44 832.52 -0.0628847投 资 组 合 保 险 的 模 拟0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0 4 2 4 4 4 6 4 8 5 0 5 202 0 04 0 06 0 08 0 01 0 0 01 2 0 001 02 03 04 05 06 0星 期保险投资组合的价值股票价格保 险 的 投 资 组 合股 票 价 格J K L M N O P Q R S12345678910111213141516171819202122232425262728293031323334353637383940414243444546Page 585-86 中文Page 195.70 828.15 833.85 0.15812935.28 829.96 835.24 0.452344016.03 830.64 836.67 0.556099177.24 830.72 837.97 -0.01357885.94 833.70 839.64 1.5777322915.40 825.15 840.55 -0.62142827.23 834.69 841.92 0.226639266.17 836.94 843.11 -0.44658922.53 841.94 844.47 0.576485492.59 843.02 845.62 -1.44592830.14 846.79 846.93 1.318539940.35 847.89 848.24 0.708962490.26 849.29 849.55 -0.04664230.01 850.84 850.85 1.697821970.01 852.15 852.17 0.42271154J K L M N O P Q R S47484950515253545556575859606162Page 585-86 中文Page 20投 资 组 合 保 险 的 模 拟0 2 4 6 8 1 0 1