Black-Scholes GreeksDelta graphsS 100 Current stock priceX 90 Exercise priceT 0.5 Time to maturity of option (in years)r 6.00% Risk-free rate of interestk 2.00% Dividend yieldSigma 55% Stock volatilityCall price 20.9272 - =bsmertoncall(B2,B3,B4,B5,B6,B7)Put price 9.2623 - =bsmertonput(B2,B3,B4,B5,B6,B7)GreeksCall PutDelta 0.6904 - =deltacall(B2,B3,B4,B5,B6,B7) -0.2996 - =deltaput(B2,B3,B4,B5,B6,B7)Gamma 0.0116 - =gamma(B2,B3,B4,B5,B6,B7) 0.0116 - =gamma(B2,B3,B4,B5,B6,B7)Vega 24.4375 - =vega(B2,B3,B4,B5,B6,B7) 24.4375 - =vega(B2,B3,B4,B5,B6,B7)Theta -14.9467 - =Thetacall(B2,B3,B4,B5,B6,B7) -11.6864-=Thetaput(B2,B3,B4,B5,B6,B7)Rho 24.0570 - =rhocall(B2,B3,B4,B5,B6,B7) -19.6131 - =rhoput(B2,B3,B4,B5,B6,B7)Put delta wrt time to expiration: Below is a two-dimensional data table.The table header is in cell A25Exercise price-0.2996 120 100 800.10 -0.8243 -0.4553 -0.0816545530.125 -0.7904 -0.4500 -0.1017028070.15 -0.7623 -0.4451 -0.1182026340.175 -0.7386 -0.4406 -0.1318962660.20 -0.7181 -0.4365 -0.1433807310.25 -0.6842 -0.4288 -0.1614304860.30 -0.6569 -0.4219 -0.1748255570.35 -0.6341 -0.4156 -0.1850300250.40 -0.6146 -0.4097 -0.1929550260.45 -0.5975 -0.4041 -0.1991955250.50 -0.5823 -0.3989 -0.2041563830.55 -0.5687 -0.3939 -0.2081228030.60 -0.5563 -0.3891 -0.2113016320.65 -0.5449 -0.3846 -0.2138466410.70 -0.5344 -0.3802 -0.2158745890.75 -0.5246 -0.3760 -0.2174757620.80 -0.5155 -0.3719 -0.2187210950.85 -0.5069 -0.3680 -0.2196671080.90 -0.4987 -0.3641 -0.2203593880.95 -0.4911 -0.3604 -0.2208351151.00 -0.4838 -0.3568 -0.22112491P u t D e lta , T im e to M a tu rity , a n d M o n e y n e s s1 .00 .90 .80 .70 .60 .50 .40 .30 .20 .1-0 .9-0 .8-0 .7-0 .6-0 .5-0 .4-0 .3-0 .2-0 .10 .0T im e to o p tio n e x p ira tio n , TCalldeltaO u t o f th e m o n e yA t th e m o n e yIn th e m o n e yA B C D E F G H I J K L M12345678910111213141516171819202122232425262728293031323334353637383940414243444546布莱克-斯科尔斯希腊字母Delta 图S 100 当前股票价格X 90 执行价格T 0.5 期权到期时间(年)r 6.00% 无风险利率k 2.00% 股息收益率Sigma 55% 股票波动性看涨期权价格 20.9272 - =bsmertoncall(B2,B3,B4,B5,B6,B7)看跌期权价格 9.2623 - =bsmertonput(B2,B3,B4,B5,B6,B7)希腊字母看涨期权 看跌期权Delta 0.6904 - =deltacall(B2,B3,B4,B5,B6,B7) -0.2996 - =deltaput(B2,B3,B4,B5,B6,B7)Gamma 0.0116 - =gamma(B2,B3,B4,B5,B6,B7) 0.0116 - =gamma(B2,B3,B4,B5,B6,B7)Vega 24.4375 - =vega(B2,B3,B4,B5,B6,B7) 24.4375 - =vega(B2,B3,B4,B5,B6,B7)Theta -14.9467 - =Thetacall(B2,B3,B4,B5,B6,B7) -11.6864-=Thetaput(B2,B3,B4,B5,B6,B7)Rho 24.0570 - =rhocall(B2,B3,B4,B5,B6,B7) -19.6131 - =rhoput(B2,B3,B4,B5,B6,B7)看跌期权delta对于到期时间: 以下是两维度模拟运算表。模拟运算表头单元格A25执行价格-0.2996 120 100 800.10 -0.8243 -0.4553 -0.0816545530.125 -0.7904 -0.4500 -0.1017028070.15 -0.7623 -0.4451 -0.1182026340.175 -0.7386 -0.4406 -0.1318962660.20 -0.7181 -0.4365 -0.1433807310.25 -0.6842 -0.4288 -0.1614304860.30 -0.6569 -0.4219 -0.1748255570.35 -0.6341 -0.4156 -0.1850300250.40 -0.6146 -0.4097 -0.1929550260.45 -0.5975 -0.4041 -0.1991955250.50 -0.5823 -0.3989 -0.2041563830.55 -0.5687 -0.3939 -0.2081228030.60 -0.5563 -0.3891 -0.2113016320.65 -0.5449 -0.3846 -0.2138466410.70 -0.5344 -0.3802 -0.2158745890.75 -0.5246 -0.3760 -0.2174757620.80 -0.5155 -0.3719 -0.2187210950.85 -0.5069 -0.3680 -0.2196671080.90 -0.4987 -0.3641 -0.2203593880.95 -0.4911 -0.3604 -0.2208351151.00 -0.4838 -0.3568 -0.22112491看 跌 期 权 Delta, 到 期 时 间 和 价 值 状 况1 .00 .90 .80 .70 .60 .50 .40 .30 .20 .1-0 .9-0 .8-0 .7-0 .6-0 .5-0 .4-0 .3-0 .2-0 .10 .0期 权 执 行 时 间 , T看涨期权delta虚 值两 平实 值A B C D E F G H I J K L M12345678910111213141516171819202122232425262728293031323334353637383940414243444546Black-Scholes GreeksTheta graphsS 100 Current stock priceX 100 Exercise priceT 0.6 Time to maturity of option (in years)r 6.00% Risk-free rate of interestk 2.00% Dividend yieldSigma 35% Stock volatilityCall price 11.7402 - =bsmertoncall(B2,B3,B4,B5,B6,B7)Put price 9.3971 - =bsmertonput(B2,B3,B4,B5,B6,B7)GreeksCall PutDelta 0.5816 - =deltacall(B2,B3,B4,B5,B6,B7) -0.4064 - =deltaput(B2,B3,B4,B5,B6,B7)Gamma 0.0149 - =gamma(B2,B3,B4,B5,B6,B7) 0.0149 - =gamma(B2,B3,B4,B5,B6,B7)Vega 29.7763 - =vega(B2,B3,B4,B5,B6,B7) 29.7763 - =vega(B2,B3,B4,B5,B6,B7)Theta -10.3069 - =Thetacall(B2,B3,B4,B5,B6,B7) -6.4952-=Thetaput(B2,B3,B4,B5,B6,B7)Rho 27.8537 - =rhocall(B2,B3,B4,B5,B6,B7) -30.0247 - =rhoput(B2,B3,B4,B5,B6,B7)Theta wrt time to expiration: Below is a two-sided data table.Exercise price-6.4952 144 100 800.10 6.4593 -19.8388 -2.3163050590.125 6.2273 -17.4822 -3.0841082930.15 5.9022 -15.7388 -3.6608068770.175 5.5196 -14.3810 -4.0793876760.20 5.1120 -13.2842 -4.3766613440.25 4.3066 -11.6024 -4.7219276970.30 3.5834 -10.3563 -4.8610140690.35 2.9683 -9.3844 -4.8841850780.40 2.4574 -8.5984 -4.841206930.45 2.0375 -7.9454 -4.7603215930.50 1.6939 -7.3912 -4.65804430.55 1.4129 -6.9132 -4.5442803770.60 1.1831 -6.4952 -4.4250845740.65 0.9951 -6.1254 -4.3042020980.70 0.8414 -5.7953 -4.1839581310.75 0.7156 -5.4981 -4.0657867430.80 0.6130 -5.2287 -3.9505536620.85 0.5295 -4.9830 -3.8387577750.90 0.4619 -4.7576 -3.7306593710.95 0.4075 -4.5500 -3.6263631271.00 0.3642 -4.3578 -3.525872543What does a positive theta mean? In general, as time to maturity gets shorter, optionslose value if they are out of the money and gain value if they are in the money.S 60 Current stock priceX 100 Exercise priceT 0.3 Time to maturity of option (in years)r 6.00% Risk-free rate of interestk 2.00% Dividend yieldSigma 35% Stock volatilityCall price 0.0213 - =bsmertoncall(B53,B54,B55,B56,B57,B58)Put price 38.5963 - =bsmertonput(B53,B54,B55,B56,B57,B58)Put theta 4.358016P u t T h e ta , T im e to M a tu rity , a n d M o n e y n e s s1 .00 .90 .80 .70 .60 .50 .40 .30 .20 .1-2 5-2 0-1 5-1 0-5051 0T im e to o p tio n m a tu rityThetaO u t o f th e m o n e yA t th e m o n e yIn th e m o n e yA B C D E F G H I J K L M N O123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263布莱克-斯科尔斯希腊字母Theta图S 100 当前股票价格X 100 执行价格T 0.6 期权到期时间 (年)r 6.00% 无风险利率k 2.00% 股息收益率Sigma 35% 股票波动性看涨期权价格 11.7402 - =bsmertoncall(B2,B3,B4,B5,B6,B7)看跌期权价格 9.3971 - =bsmertonput(B2,B3,B4,B5,B6,B7)希腊字母看涨期权 看跌期权Delta 0.5816 - =deltacall(B2,B3,B4,B5,B6,B7) -0.4064 - =deltaput(B2,B3,B4,B5,B6,B7)Gamma 0.0149 - =gamma(B2,B3,B4,B5,B6,B7) 0.0149 - =gamma(B2,B3,B4,B5,B6,B7)Vega 29.7763 - =vega(B2,B3,B4,B5,B6,B7) 29.7763 - =vega(B2,B3,B4,B5,B6,B7)Theta -10.3069 - =Thetacall(B2,B3,B4,B5,B6,B7) -6.4952-=Thetaput(B2,B3,B4,B5,B6,B7)Rho 27.8537 - =rhocall(B2,B3,B4,B5,B6,B7) -30.0247 - =rhoput(B2,B3,B4,B5,B6,B7)Theta 对于到期时间: 以下是两维度模拟运算表.执行价格-6.4952 144 100 800.10 6.4593 -19.8388 -2.3163050590.125 6.2273 -17.4822 -3.0841082930.15 5.9022 -15.7388 -3.6608068770.175 5.5196 -14.3810 -4.0793876760.20 5.1120 -13.2842 -4.3766613440.25 4.3066 -11.6024 -4.7219276970.30 3.5834 -10.3563 -4.8610140690.35 2.9683 -9.3844 -4.8841850780.40 2.4574 -8.5984 -4.841206930.45 2.0375 -7.9454 -4.7603215930.50 1.6939 -7.3912 -4.65804430.55 1.4129 -6.9132 -4.5442803770.60 1.1831 -6.4952 -4.4250845740.65 0.9951 -6.1254 -4.3042020980.70 0.8414 -5.7953 -4.1839581310.75 0.7156 -5.4981 -4.0657867430.80 0.6130 -5.2287 -3.9505536620.85 0.5295 -4.9830 -3.8387577750.90 0.4619 -4.7576 -3.7306593710.95 0.4075 -4.5500 -3.6263631271.00 0.3642 -4.3578 -3.525872543正的theta意味着什么? 总的来说, 随着到期时间越来越近, 如果期权为虚值期权,价值减少;如果期权为实值期权,价值增加.S 60 当前股票价格X 100 执行价格T 0.3 期权到期时间 (年)r 6.00% 无风险利率k 2.00% 股息收益率Sigma 35% 股票波动性看涨期权价格 0.0213 - =bsmertoncall(B53,B54,B55,B56,B57,B58)看跌期权价格 38.5963 - =bsmertonput(B53,B54,B55,B56,B57,B58)看跌期权 theta 4.358016看 跌 期 权 Theta, 到 期 时 间 和 价 值 状 态1 .00 .90 .80 .70 .60 .50 .40 .30 .20 .1-2 5-2 0-1 5-1 0-5051 0到 期 时 间ThetaO u t o f th e m o n e yA t th e m o n e yIn th e m o n e yA B C D E F G H I J K L M N O123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263Black-Scholes GreeksTheta graphsS 100 Current stock priceX 100 Exercise priceT 0.6 Time to maturity of option (in years)r 6.00% Risk-free rate of interestk 2.00% Dividend yieldSigma 35% Stock volatilityCall price 11.7402 - =bsmertoncall(B2,B3,B4,B5,B6,B7)Put price 9.3971 - =bsmertonput(B2,B3,B4,B5,B6,B7)GreeksCall PutDelta 0.5816 - =deltacall(B2,B3,B4,B5,B6,B7) -0.4064 - =deltaput(B2,B3,B4,B5,B6,B7)Gamma 0.0149 - =gamma(B2,B3,B4,B5,B6,B7) 0.0149 - =gamma(B2,B3,B4,B5,B6,B7)Vega 29.7763 - =vega(B2,B3,B4,B5,B6,B7) 29.7763 - =vega(B2,B3,B4,B5,B6,B7)Theta -10.3069 - =Thetacall(B2,B3,B4,B5,B6,B7) -6.4952-=Thetaput(B2,B3,B4,B5,B6,B7)Rho 27.8537 - =rhocall(B2,B3,B4,B5,B6,B7) -30.0247 - =rhoput(B2,B3,B4,B5,B6,B7)What does a positive theta mean? In general, as time to maturity gets shorter, optionslose value if they are out of the money and gain value if they are in the money.S 60 Current stock priceX 100 Exercise priceT 0.3 Time to maturity of option (in years)r 6.00% Risk-free rate of interestk 2.00% Dividend yieldSigma 35% Stock volatilityCall price 0.0213 - =bsmertoncall(B22,B23,B24,B25,B26,B27)Put price 38.5963 - =bsmertonput(B22,B23,B24,B25,B26,B27)Put theta 4.358016A B C D E1234567891011121314151617181920212223242526272829303132布莱克-斯科尔斯希腊字母Theta图S 100当前股票价格X 100执行价格T 0.6期权到期时间(年)r 6.00%无风险利率k 2.00%股息收益率Sigma 35%股票波动性看涨期权价格 11.7402 - =bsmertoncall(B2,B3,B4,B5,B6,B7)看跌期权价格 9.3971 - =bsmertonput(B2,B3,B4,B5,B6,B7)希腊字母看涨期权 看跌期权Delta 0.5816 - =deltacall(B2,B3,B4,B5,B6,B7) -0.4064 - =deltaput(B2,B3,B4,B5,B6,B7)Gamma 0.0149 - =gamma(B2,B3,B4,B5,B6,B7) 0.0149 - =gamma(B2,B3,B4,B5,B6,B7)Vega 29.7763 - =vega(B2,B3,B4,B5,B6,B7) 29.7763 - =vega(B2,B3,B4,B5,B6,B7)Theta -10.3069 - =Thetacall(B2,B3,B4,B5,B6,B7) -6.4952-=Thetaput(B2,B3,B4,B5,B6,B7)Rho 27.8537 - =rhocall(B2,B3,B4,B5,B6,B7) -30.0247 - =rhoput(B2,B3,B4,B5,B6,B7)正的theta意味着什么? 总的来说, 随着到期时间越来越近, 如果期权为虚值期权,价值减少;如果期权为实值期权,价值增加.S 60当前股票价格X 100执行价格T 0.3期权到期时间(年)r 6.00%无风险利率k 2.00%股息收益率Sigma 35%股票波动性看涨期权价格 0.0213 - =bsmertoncall(B22,B23,B24,B25,B26,B27)看跌期权价格 38.5963 - =bsmertonput(B22,B23,B24,B25,B26,B27)看跌期权theta 4.358016A B C D E1234567891011121314151617181920212223242526272829303132