chapter24
THE OPTION TO EXPANDYear 0 1 2 3CF of single machine -1000 220 300 400Discount rate for machine cash flows (risk-adjusted) 12%Riskless discount rate 6%Present value of machines future cash flows 932.52 - =NPV(B5,C3:G3)NPV of single machine -67.48 - =NPV(B5,C3:G3)+B3Number of machines bought next year 5Option value of single machine purchased in one more year 143.98 - =B24NPV of total project 652.39 - =B8+B10*B11Black-Scholes Option Pricing FormulaS 932.52 PV of machine CFsX 1000.00 Exercise price = Machine costr 6.00% Risk-free rate of interestT 1 Time to maturity of option (in years)Sigma 40% - Volatilityd1 0.1753 - (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)d2 -0.2247 - d1 - sigma*SQRT(T)N(d1) 0.5696 - Uses formula NormSDist(d1)N(d2) 0.4111 - Uses formula NormSDist(d2)Option value = BS call price 143.98 - S*N(d1)-X*exp(-r*T)*N(d2)Data Tables 652.39 - =B12 , data table header1% -63.4810% 97.1620% 283.0930% 468.4040% 652.3950% 834.5960% 1,014.5470% 1,191.81P r o je c t V a lu e a s F u n c tio n o f S ig m a7 0 %6 0 %5 0 %4 0 %3 0 %2 0 %1 0 %0 %- 2 0 002 0 04 0 06 0 08 0 01 , 0 0 01 , 2 0 01 , 4 0 0S ig m aA B C D E1234567891011121314151617181920212223242526272829303132333435363738394041424 5200 150THE OPTION TO EXPANDTime to maturity of option (in years)- (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)- Uses formula NormSDist(d1)- Uses formula NormSDist(d2)- =B12 , data table headerP r o je c t V a lu e a s F u n c tio n o f S ig m a7 0 %6 0 %5 0 %4 0 %3 0 %2 0 %1 0 %0 %- 2 0 002 0 04 0 06 0 08 0 01 , 0 0 01 , 2 0 01 , 4 0 0S ig m aF G H I J123456789101112131415161718192021222324252627282930313233343536373839404142扩张期权年 0 1 2 3每台机器的现金流量 -1000 220 300 400机器现金流量的折现率 (风险调整) 12%无风险折现率 6%机器未来现金流量的现值 932.52 - =NPV(B5,C3:G3)每台机器的净现金流量 -67.48 - =NPV(B5,C3:G3)+B3下一年购买的机器数量 5在一年或更多年中购买一台机器的期权价值 143.98 - =B24整个项目的净现金流量 652.39 - =B8+B10*B11布莱克-斯科尔斯期权定价公式S 932.52 机器现金流量的现值X 1000.00 执行价格 = 机器成本r 6.00% 无风险利率T 1 期权到期时间 (年)Sigma 40% - 波动性d1 0.1753 - (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)d2 -0.2247 - d1 - sigma*SQRT(T)N(d1) 0.5696 - 使用公式 NormSDist(d1)N(d2) 0.4111 - 使用公式 NormSDist(d2)期权价值 = 布莱克-斯科尔斯看涨期权价格 143.98 - S*N(d1)-X*exp(-r*T)*N(d2)模拟运算表s 652.39 - =B12 , 模拟运算表头1% -63.4810% 97.1620% 283.0930% 468.4040% 652.3950% 834.5960% 1,014.5470% 1,191.81作 为 标 准 差 的 函 数 的 项 目 价 值7 0 %6 0 %5 0 %4 0 %3 0 %2 0 %1 0 %0 %- 2 0 002 0 04 0 06 0 08 0 01 , 0 0 01 , 2 0 01 , 4 0 0标 准 差A B C D E1234567891011121314151617181920212223242526272829303132333435363738394041424 5200 150扩张期权- (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)- =B12 , 模拟运算表头作 为 标 准 差 的 函 数 的 项 目 价 值7 0 %6 0 %5 0 %4 0 %3 0 %2 0 %1 0 %0 %- 2 0 002 0 04 0 06 0 08 0 01 , 0 0 01 , 2 0 01 , 4 0 0标 准 差F G H I J123456789101112131415161718192021222324252627282930313233343536373839404142PRICING AN ABANDONMENT OPTIONMarket data State pricesExpected market return 12% qU 0.3087 - =(1+B5-B9)/(1+B5)*(B8-B9)Sigma of market return 30% qD 0.6347 - =(B8-1-B5)/(1+B5)*(B8-B9)Risk-free rate 6%One-period up and down of marketUp 1.521962 - =EXP(B3+B4) , note that a valid alternative is Up = EXP(B4)Down 0.83527 - =EXP(B3-B4) , note that a valid alternative is Down = EXP(-B4)Project cash flows State-dependent present value factors15010080-50 180-50-60State-by-state present value14.2981 - =E14*K1430.874015.6755 - =E16*K16-5015.6755 - =E18*K18-31.7328-24.1673 - =E20*K20Net present value -29.38 - =SUM(A23:E29)Cash flows with abandonment Present value with abandonment15010080-50 -50000Present value with abandonment= C 1 5 * I 1 5A B C D E F G H1234567891011121314151617181920212223242526272829303132333435363738394041424344450.0953 - =E320.30870.1959 - =E3*E40.1959 - =E3*E40.63470.4028 - =E4214.2981 - =E36*K1430.874015.6755 - =E38*K1600010.85 - =SUM(G36:K42)PRICING AN ABANDONMENT OPTION- =(1+B5-B9)/(1+B5)*(B8-B9)- =(B8-1-B5)/(1+B5)*(B8-B9)- =EXP(B3+B4) , note that a valid alternative is Up = EXP(B4)- =EXP(B3-B4) , note that a valid alternative is Down = EXP(-B4)State-dependent present value factorsPresent value with abandonmentPresent value with abandonmentI J K L123456789101112131415161718192021222324252627282930313233343536373839404142434445放弃期权定价市场信息 状态价格市场收益的期望值 12% qU 0.3087 - =(1+B5-B9)/(1+B5)*(B8-B9)市场收益的标准差 30% qD 0.6347 - =(B8-1-B5)/(1+B5)*(B8-B9)无风险利率 6%One-period up and down of market上涨 1.521962 - =EXP(B3+B4) , 注意正确的选择是上涨 = EXP(B4)下跌 0.83527 - =EXP(B3-B4) , 注意正确的选择是下跌 = EXP(-B4)项目现金流 取决于状态的现值因子15010080-50 180-50-60每个状态的现值14.2981 - =E14*K1430.874015.6755 - =E16*K16-5015.6755 - =E18*K18-31.7328-24.1673 - =E20*K20净现值 -29.38 - =SUM(A23:E29)允许放弃项目的现金流量 允许放弃项目的现值15010080-50 -50000允许放弃项目的现值= C 1 5 * I 1 5A B C D E F G H1234567891011121314151617181920212223242526272829303132333435363738394041424344450.0953 - =E320.30870.1959 - =E3*E40.1959 - =E3*E40.63470.4028 - =E4214.2981 - =E36*K1430.874015.6755 - =E38*K1600010.85 - =SUM(G36:K42)放弃期权定价- =(1+B5-B9)/(1+B5)*(B8-B9)- =(B8-1-B5)/(1+B5)*(B8-B9)取决于状态的现值因子I J K L123456789101112131415161718192021222324252627282930313233343536373839404142434445PRICING AN ABANDONMENT OPTIONMarket data State pricesExpected market return 12% qU 0.3087 - =(1+B5-B9)/(1+B5)*(B8-B9)Sigma of market return 30% qD 0.6347 - =(B8-1-B5)/(1+B5)*(B8-B9)Risk-free rate 6%One-period up and down of marketUp 1.521962 - =EXP(B3+B4) , note that a valid alternative is Up = EXP(B4)Down 0.83527 - =EXP(B3-B4) , note that a valid alternative is Down = EXP(-B4)Project cash flows State-dependent present value factors15010080-50 180-50-60State-by-state present value14.2981 - =E14*K1430.874015.6755 - =E16*K16-5015.6755 - =E18*K18-31.7328-24.1673 - =E20*K20Net present value -29.38 - =SUM(A23:E29)Cash flows with abandonment Present value with abandonment15010080-50 -500-100Present value with abandonment= C 1 5 * I 1 5A B C D E F G H1234567891011121314151617181920212223242526272829303132333435363738394041424344450.0953 - =E320.30870.1959 - =E3*E40.1959 - =E3*E40.63470.4028 - =E4214.2981 - =E36*K1430.874015.6755 - =E38*K160-6.346604.50 - =SUM(G36:K42)PRICING AN ABANDONMENT OPTION- =(1+B5-B9)/(1+B5)*(B8-B9)- =(B8-1-B5)/(1+B5)*(B8-B9)- =EXP(B3+B4) , note that a valid alternative is Up = EXP(B4)- =EXP(B3-B4) , note that a valid alternative is Down = EXP(-B4)State-dependent present value factorsPresent value with abandonmentPresent value with abandonmentI J K L123456789101112131415161718192021222324252627282930313233343536373839404142434445放弃期权定价市场信息 状态价格市场收益率的期望值 12% qU 0.3087 - =(1+B5-B9)/(1+B5)*(B8-B9)市场收益率的标准差 30% qD 0.6347 - =(B8-1-B5)/(1+B5)*(B8-B9)无风险利率 6%一期市场的上涨和下跌状态上涨 1.521962 - =EXP(B3+B4) ,注意正确的选择是 下跌 = EXP(B4)下跌 0.83527 - =EXP(B3-B4) , 注意正确的选择是 下跌 = EXP(-B4)项目现金流量 取决于状态的现值因子15010080-50 180-50-60每个状态的现值14.2981 - =E14*K1430.874015.6755 - =E16*K16-5015.6755 - =E18*K18-31.7328-24.1673 - =E20*K20净现值 -29.38 - =SUM(A23:E29)允许放弃项目的现金流量 允许放弃项目的现值15010080-50 -500-150允许放弃项目的现值= C 1 5 * I 1 5A B C D E F G H1234567891011121314151617181920212223242526272829303132333435363738394041424344450.0953 - =E320.30870.1959 - =E3*E40.1959 - =E3*E40.63470.4028 - =E4214.2981 - =E36*K1430.874015.6755 - =E38*K1609.5198020.37 - =SUM(G36:K42)放弃期权定价- =(1+B5-B9)/(1+B5)*(B8-B9)- =(B8-1-B5)/(1+B5)*(B8-B9)取决于状态的现值因子I J K L123456789101112131415161718192021222324252627282930313233343536373839404142434445MATCHING THE STATE PRICES TO THE COST OF CAPITALProject cost of capital 22% - This is the discount rate for the project if it has no optionsRisk-free rate 6%Project cash flows15010080-5080-50-60Project expected cash flows: Assumes equal state probabilitiesYear 0 1 2Expected CF -50 25 62.5Project NPV 12.48 - =NPV(B2,C17:D17)+A9State pricesqU 0.4241 - =1/(1+B3)-B23qD 0.5193 - Determined by SolverProject state-by-state discounting26.9797 - =E6*B22242.410517.6187 - =E8*B22*B23-5017.6187 - =E10*B22*B23-25.9646-16.1798 - =E12*B232State-by-state NPV 12.48 - =SUM(A27:E33)Target cell -0.00 - =B38-B18= A V E R A G E ( C 7 : C 1 )= A V E R A G E ( E 6 : E 1 2 )= C 7 * B 2 2= C 1 1 * B 2 3A B C D E F G H12345678910111213141516171819202122232425262728293031323334353637383940状态价格与资本成本的匹配项目的资本成本 22% - 这是如果该项目没有期权时的折现率无风险利率 6%项目现金流量15010080-5080-50-60项目期望现金流量:假设状态是等概率的年 0 1 2期望的现金流量 -50 25 62.5项目净现值 12.48 - =NPV(B2,C17:D17)+A9状态价格qU 0.4241 - =1/(1+B3)-B23qD 0.5193 - 由Solver决定项目的每期状态的折现26.9797 - =E6*B22242.410517.6187 - =E8*B22*B23-5017.6187 - =E10*B22*B23-25.9646-16.1798 - =E12*B232状态的 NPV 12.48 - =SUM(A27:E33)目标单元 -0.00 - =B38-B18= A V E R A G E ( C 7 : C 1 )= A V E R A G E ( E 6 : E 1 2 )= C 7 * B 2 2= C 1 1 * B 2 3A B C D E F G H12345678910111213141516171819202122232425262728293031323334353637383940STANDARD DCF PROJECT VALUATIONProject cash flowsYear 0 1 2 3 4Cash flow -750 100 200 300 400Risk-adjusted discount rate 12% The projects cost of capitalNPV without options -33.53 - =B4+NPV(B6,C4:F4)A B C D E F1234567项目的现金流量项目的现金流量年 0 1 2 3 4现金流量 -750 100 200 300 400风险调整折现率 12% 项目的资本成本无期权的NPV -33.53 - =B4+NPV(B6,C4:F4)A B C D E F1234567ABANDONMENT VALUE-DETAILS OF YEAR 1 CALCULATIONProject cash flowsYear 0 1 2 3 4Cash flow -750 100 200 300 400Risk-adjusted discount rate 12% The projects cost of capitalNPV without options -33.53 - =B4+NPV(B6,C4:F4)Valuing the year-1 abandonment putValue of project, end year 1 702.44 - =NPV(B6,D4:F4)Abandonment value 300 Like strike price in put formulaTime to option maturity (years) 3Risk-free rate 6%Sigma 50%Put value 19.53 - =bsput(B10,B11,B12,B13,B14)Black-Scholes Put Option Pricing FormulaS -100.00 PV of CFs, discounted at risk-adjusted discount rateX -100.00 Exercise price = Abandonment costr 6.00% Risk-free rate of interestT 5 Time to maturity of option (in years)A B C D E F1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950Sigma 80% - Volatilityd1 1.0621 - (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)d2 -0.7267 - d1 - sigma*SQRT(T)N(d1) 0.8559 - Uses formula NormSDist(d1)N(d2) 0.2337 - Uses formula NormSDist(d2)Option value = BS call price -68.28 - S*N(d1)-X*exp(-r*T)*N(d2)State pricesUp 122.6% qu 0.240829Down -55.1% qd 0.7025672Mean 200 300Cash flows1485.91445.11-750 300.0089.8760.57Year 1 2Project present value 143.1 - =SUMPRODUCT(A30:I37,K30:S37)Cash flows with abandonmentTerminal value on abandonment 1001485.91445.110.8 300.0089.8760.57A B C D E F51525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899A B C D E F100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150A B C D E F151152153154155156157158159160161162163164165166167168169170171172173174175176177178PV of CFs, discounted at risk-adjusted discount rateG H I J K L M N O1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950400 50012266.27 State dependent present-value factors4409.272476.52 0.06890.22 0.24500.00 1 0.34179.73 0.70100.95 0.4936.2920.383 412266.274409.272476.52890.22500.00179.73100.95136.29100.00- (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)G H I J K L M N O51525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899G H I J K L M N O100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150G H I J K L M N O1511521531541551561571581591601611621631641651661671681691701711721731741751761771780.000.010.040.120.170.360.330.350.24P Q R S51525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899放弃期权估价-第1年的计算那细节项目的现金流量年 0 1 2 3 4现金流量 -750 100 200 300 400风险调整的折现率 12% 项目的资本成本无期权的NPV -33.53 - =B4+NPV(B6,C4:F4)估计第1年的放弃看跌期权第1年底项目价值 702.44 - =NPV(B6,D4:F4)放弃期权 300 类似看跌期权公式的交易价格期权到期时间 (年) 3无风险利率 6%标准差 50%看跌期权价格 19.53 - =bsput(B10,B11,B12,B13,B14)Black-Scholes Put Option Pricing FormulaS -100.00 PV of CFs, discounted at risk-adjusted discount rateX -100.00 Exercise price = Abandonment costr 6.00% Risk-free rate of interestT 5 Time to maturity of option (in years)A B C D E F1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950Sigma 80% - Volatilityd1 1.0621 - (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)d2 -0.7267 - d1 - sigma*SQRT(T)N(d1) 0.8559 - Uses formula NormSDist(d1)N(d2) 0.2337 - Uses formula NormSDist(d2)Option value = BS call price -68.28 - S*N(d1)-X*exp(-r*T)*N(d2)State pricesUp 122.6% qu 0.240829Down -55.1% qd 0.7025672Mean 200 300Cash flows1485.91445.11-750 300.0089.8760.57Year 1 2Project present value 143.1 - =SUMPRODUCT(A30:I37,K30:S37)Cash flows with abandonmentTerminal value on abandonment 1001485.91445.110.8 300.0089.8760.57A B C D E F51525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899A B C D E F100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150A B C D E F151152153154155156157158159160161162163164165166167168169170171172173174175176177178PV of CFs, discounted at risk-adjusted discount rateG H I J K L M N O1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950400 50012266.27 State dependent present-value factors4409.272476.52 0.06890.22 0.24500.00 1 0.34179.73 0.70100.95 0.4936.2920.383 412266.274409.272476.52890.22500.00179.73100.95136.29100.00- (LN(S/X)+(r+0.5*sigma2)*T)/(sigma*SQRT(T)G H I J K L M N O51525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899G H I J K L M N O100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150G H I J K L M N O1511521531541551561571581591601611621631641651661671681691701711721731741751761771780.000.010.040.120.170.360.330.350.24P Q R S51525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899PRICING AN ABANDONMENT OPTION AS A SERIES OF PUTSProject cash flowsYear 0 1 2 3 4Cash flow -750 100 200 300 400Risk-adjusted discount rate 12% The projects cost of capitalNPV without options -33.53 - =NPV(B6,C4:F4)+B4Sigma 50%Risk-free rate 6%Abandonment value 300 Project can be abandoned at end of any year for this amountNPV of cash flows at RADR -33.53 - =B8Value of abandonment option 299.10 - =NPV(B11,C20:F20)Adjusted present value 265.57 - =B15+B14Function in cell D19