UN-13BBlack-Scholes Option-Pricing FormulaS 50 Current stock priceX 45 Exercise pricer 4.00% Risk-free rate of interestT 0.75 Time to maturity of option (in years)Sigma 30% Stock volatility, sd1 0.6509 B4,B2*B5/B4,1000)Option formula 1100.00 B4,B2*B5/B4,1000)B4,B2*B5/B4,1000)期权公式 1100.00 B4,B2*B5/B4,1000)25% 1040.84 1045.48 1050.4430% 1047.16 1052.22 1057.6935% 1050.86 1056.29 1062.2640% 1052.66 1058.44 1064.8845% 1053.10 1059.19 1066.1050% 1052.55 1058.94 1066.29D a ta ta b le h e a d e r:= B 1 4A B C D E1234567891011121314151617181920212223242526272 1 0.00011000.59 1001.77 1000.201011.13 1013.78 1000.401028.28 1029.65 1000.591043.69 1044.54 1000.791055.14 1056.54 1000.991063.09 1065.58 1001.191068.39 1072.19 1001.391071.75 1076.95 1001.591073.70 1080.28 1001.791074.59 1082.53 1001.99ABN-AMRO AIRBAG SENSITIVITY TO TIME TO MATURITY AND SIGMA25% 1040.84 1045.48 1050.4430% 1047.16 1052.22 1057.6935% 1050.86 1056.29 1062.2640% 1052.66 1058.44 1064.8845% 1053.10 1059.19 1066.1050% 1052.55 1058.94 1066.29模 拟 运 算 表表 头 := B 1 4A B C D E1234567891011121314151617181920212223242526272 1 0.00011000.59 1001.77 1000.201011.13 1013.78 1000.401028.28 1029.65 1000.591043.69 1044.54 1000.791055.14 1056.54 1000.991063.09 1065.58 1001.191068.39 1072.19 1001.391071.75 1076.95 1001.591073.70 1080.28 1001.791074.59 1082.53 1001.99荷兰银行Airbag对到期时间和s的敏感性39,1000,B2*B3)In option terms 820.51 39,1000,B2*B3)期权条款的定义 820.51 B26,Yes,No)Technical note: For didactic clarity, the computations use 5.2% as the interest rate forvaluing both the bond component of the Goals (rows 10-14) and for the option valuation. Givena 2.6% semi-annual discrete interest rate, it would be technically more correct to use anequivalent continuously-compounded interest rate of LN(1.026)2) in the optioncomputations. The reader can confirm that the effect of this correction is negligible.A B C123456789101112131415161718192021222324252627282930定价UBS GOALS隐含看跌期权年无风险利率 5.20%票面利率 19.50%初始成本 1,000转换率: 股价较低时收到的Cisco股份数量 25.641 B26,Yes,No)技术注释: 为了简洁, 该计算使用5.2%作为定价Goals债券部分（行10-14）和期权价值的利率。考虑一个离散的2.6%的半年利率, 使用LN(1.026)2)的连续复合的等价利率在技术上将更加正确。读者可以确认这种正确的效果是可以忽略的。A B C123456789101112131415161718192021222324252627282930CREATING A RISKLESS SECURITY WITH THEUBS GOALS AND 25.641 PUTSInitial cash flowsBuy UBS security -1,000.00Buy 25.641 puts -300.21 - =-25.641*BSPut(B16,B17,B19,B18,B20)Cash flow of engineered security:GOALS + 25.641 bought putsDate Cash flow23-一月-01 (1,300.21) - =SUM(B3:B4)23-七月-01 97.5023-一月-02 97.5023-七月-02 1,097.50IRR of above -0.43% - =XIRR(B8:B11,A8:A11)Inputs for Black-Scholes formula in cell B4S 42.625 Current stock priceX 39 Exercise pricer 5.20% Risk-free rate of interestT 1.5 Time to maturity of option (in years)Sigma 80% Stock volatilityA B C1234567891011121314151617181920用UBS GOALS和25.641份看跌期权建立一个无风险证券初始现金购买UBS证券 -1,000.00购买25.641份看跌期权 -300.21 - =-25.641*BSPut(B16,B17,B19,B18,B20)工程化 证券的现金流:GOALS + 25.641份购买的证券日期 现金流23-一月-01 (1,300.21) - =SUM(B3:B4)23-七月-01 97.5023-一月-02 97.5023-七月-02 1,097.50以上的内部收益率 -0.43% - =XIRR(B8:B11,A8:A11)在单元格B4中输入布莱克-斯科尔斯公式S 42.625当前股票价格X 39执行价格r 5.20%无风险利率T 1.5期权到期时间（年)s 80%股票性A B C1234567891011121314151617181920BANG FOR THE BUCK WITH OPTIONSS 25 Current stock price Data table: Effect of S on bangX 25 Exercise prier 6.00% Risk-free rate of interestT 0.5 Time to maturity of option (in years) 15Sigma 30% Stock volatility 1617d1 0.2475 - (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T) 18d2 0.0354 - d1-sigma*SQRT(T) 1920N(d1) 0.5977 - Uses formula NormSDist(d1) 21N(d2) 0.5141 - Uses formula NormSDist(d2) 2223Call price 2.47 - S*N(d1)-X*exp(-r*T)*N(d2) 24Put price 1.73 - call price - S + X*Exp(-r*T): by put-call parity 2526Call bang 6.0483 - =B11*B2/B14 27Put bang 5.8070 - =NORMSDIST(-B8)*B2/B15 282930Data table: Effect of S and T on call bang6.048315161718192021222324252627282930D a ta ta b le h e a d e r:= B 1 7A B C D E123456789101112131415161718192021222324252627282930313233343536373839404142Calls Puts- Data table header2.3828 14.31002.6095 13.09252.8695 11.97703.1641 10.95343.4933 10.01343.8555 9.15034.2481 8.35814.6675 7.63215.1100 6.96765.5716 6.36056.0483 5.80706.5367 5.30347.0335 4.84637.5358 4.43218.0414 4.05788.5481 3.7202T-option time to exercise0.25 0.5 0.75 125.8566 14.1767 10.1698 8.111323.3203 12.9886 9.4124 7.562520.9931 11.9035 8.7218 7.062318.8591 10.9122 8.0913 6.605616.9055 10.0067 7.5154 6.188215.1222 9.1804 6.9891 5.806213.5006 8.4274 6.5082 5.456512.0334 7.7424 6.0691 5.136210.7137 7.1205 5.6682 4.84269.5347 6.5572 5.3025 4.57378.4892 6.0483 4.9691 4.32727.5694 5.5896 4.6655 4.10126.7664 5.1773 4.3892 3.89416.0706 4.8074 4.1379 3.70435.4720 4.4764 3.9094 3.53034.9598 4.1807 3.7019 3.3708Data table: Effect of S on bangData table: Effect of S and T on call bangF G H I123456789101112131415161718192021222324252627282930313233343536373839404142期权的最大化利润S 25当前的股票价格X 25执行价格r 6.00%无风险利率T 0.5期权到期时间（年)Sigma 30%股票波动性d1 0.2475 - (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)d2 0.0354 - d1-sigma*SQRT(T)N(d1) 0.5977 - 使用公式 NormSDist(d1)N(d2) 0.5141 - 使用公式 NormSDist(d2)看涨期权价格 2.47 - S*N(d1)-X*exp(-r*T)*N(d2)看跌期权价格 1.73 - 看跌期权价格 - S + X*Exp(-r*T): 用看跌-看涨平价定理看涨期权最大化利润 6.0483 - =B11*B2/B14看跌期权最大化利润 5.8070 - =NORMSDIST(-B8)*B2/B15模 拟 运 算 表表 头 := B 1 7A B C D123456789101112131415161718192021222324252627282930313233343536373839404142Data table: Effect of S on bangCalls Puts- Data table header15 2.3828 14.310016 2.6095 13.092517 2.8695 11.977018 3.1641 10.953419 3.4933 10.013420 3.8555 9.150321 4.2481 8.358122 4.6675 7.632123 5.1100 6.967624 5.5716 6.360525 6.0483 5.807026 6.5367 5.303427 7.0335 4.846328 7.5358 4.432129 8.0414 4.057830 8.5481 3.7202模拟运算表: S和T对看涨期权最大化利润的影响T-期权执行时间6.0483 0.25 0.5 0.75 115 25.8566 14.1767 10.1698 8.111316 23.3203 12.9886 9.4124 7.562517 20.9931 11.9035 8.7218 7.062318 18.8591 10.9122 8.0913 6.605619 16.9055 10.0067 7.5154 6.188220 15.1222 9.1804 6.9891 5.806221 13.5006 8.4274 6.5082 5.456522 12.0334 7.7424 6.0691 5.136223 10.7137 7.1205 5.6682 4.842624 9.5347 6.5572 5.3025 4.573725 8.4892 6.0483 4.9691 4.327226 7.5694 5.5896 4.6655 4.101227 6.7664 5.1773 4.3892 3.894128 6.0706 4.8074 4.1379 3.704329 5.4720 4.4764 3.9094 3.530330 4.9598 4.1807 3.7019 3.3708E F G H I J123456789101112131415161718192021222324252627282930313233343536373839404142 B a n g fo r th e B u c k T h e P r ic e E la s tic ity o f C a lls a n d P u tsa s a F u n c tio n o f th e E x e r c is e P r ic e X3 53 02 52 01 5024681 01 21 41 6O p tio n e x e r c is e p r ic e , X ($ )Profitelasticity-bangC a lls P u ts 利 润 最 大 化 作 为 执 行 价 格 X 的 一 个 函 数 的看 涨 和 看 跌 期 权 的 价 格 弹 性3 53 02 52 01 5024681 01 21 41 6期 权 执 行 价 格 , X ( $ )利润弹性-最大化利润C a lls P u ts UN-13BUSING THE BLACK (1976) MODELTO PRICE A BOND OPTIONF 133.011 - Bond forward priceX 130.000 - Exercise pricer 4.00% -Risk-free rate of interestT 0.5Sigma 6% - Bond forward price volatility, sd1 0.5609 - =(LN(B2/B3)+B62*B5/2)/(B6*SQRT(B5)d2 0.5185 - =B8-SQRT(B5)*B6Call price 4.13 - =EXP(-B4*B5)*(B2*NORMSDIST(B8)-B3*NORMSDIST(B9)Put price 1.02 - =EXP(-B4*B5)*(B3*NORMSDIST(-B9)-B2*NORMSDIST(-B8)A B C123456789101112UN-13B用布莱克(1976)模型定价一个债券期权F 133.011 - 债券远期价格X 130.000 - 执行价格r 4.00% -无风险利率T 0.5Sigma 6% - 债券远期价格波动性, sd1 0.5609 - =(LN(B2/B3)+B62*B5/2)/(B6*SQRT(B5)d2 0.5185 - =B8-SQRT(B5)*B6看涨期权价格 4.13 - =EXP(-B4*B5)*(B2*NORMSDIST(B8)-B3*NORMSDIST(B9)看跌期权价格 1.02 - =EXP(-B4*B5)*(B3*NORMSDIST(-B9)-B2*NORMSDIST(-B8)A B C123456789101112THE FORWARD INTEREST RATEBond maturity, W 7Option maturity, T 4Year W pure discount rate 6%Year T pure discount rate 5%Discretely-compounded interest rates0 1 2 3 47-year deposit at 6.00% 100.004-year loan at 5.00% -100.00 121.55Sum of above: A 3-year deposit at year 4 0.00 121.55Discretely-compounded forward interest ratefrom year 4 to year 7 7.35% - =(-I11/F11)(1/(B2-B3)-1Continuously-compounded interest rates0 1 2 3 47-year deposit at 6.00% 100.004-year loan at 5.00% -100.00 122.14Sum of above: A 3-year deposit at year 4 0.00 122.14Continuously-compounded forward interestrate from year 4 to year 7 7.33% - =LN(-I19/F19)/(B2-B3)A B C D E F1234567891011121314151617181920215 6 7 8 9 10-150.36-150.365 6 7 8 9 10-152.20-152.20THE FORWARD INTEREST RATEG H I J K L123456789101112131415161718192021远期利率债券到期时间, W 7债权到期时间, T 4W年纯贴现率 6%T年纯贴现率 5%离散复合利率0 1 2 3 4在6.00%贴现率下的7年存款 100.00在5.00%贴现率下的4年存款 -100.00 121.55小计：在第4年的一个3年存款 0.00 121.55从第4年到第7年离散的复合远期利率 7.35% - =(-I11/F11)(1/(B2-B3)-1连续的复合利率0 1 2 3 4在6.00%贴现率下的7年存款 100.00在5.00%贴现率下的4年存款 -100.00 122.14小计：在第4年的一个3年存款 0.00 122.14从第4年到第7年连续的复合远期利率 7.33% - =LN(-I19/F19)/(B2-B3)A B C D E F1234567891011121314151617181920215 6 7 8 9 10-150.36-150.365 6 7 8 9 10-152.20-152.20远期利率G H I J K L123456789101112131415161718192021DETERMINING THE FORWARD PRICEOF THE BONDBonds maturity, N 2Option maturity, T 0.5Bond maturity value 147Interest rate to N 6%Interest rate to T 4%Bond forward price to T 133.011 - =B4*EXP(-B6*B2)*EXP(B7*B3)A B C123456789决定债券的远期价格债券到期时间, N 2期权到期时间, T 0.5债券到期价值 147对应N的利率 6%对应T的利率 4%对应T的债券远期价格 133.011 - =B4*EXP(-B6*B2)*EXP(B7*B3)A B C123456789