chapter23.3barrier-

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PRICING A KNOCKOUT BARRIER OPTIONInitial stock price 30Up 1.40Down 0.80Interest 1.08Option exercise price 30Barrier 50.00qU 0.4321 - =(B5-B4)/(B5*(B3-B4)qD 0.4938 - =(B3-B5)/(B5*(B3-B4)Risk-neutral probability, up 0.4667 - =B9*B5Risk-neutral probability, down 0.5333 - =B10*B5STOCK PRICEPaths Period 1 Period 2 Period 3 Period 4 Period 0 Period 1All up (1 path) up up up up 30.00 42.00One down (4 paths) down up up up 30.00 24.00up down up up 30.00 42.00up up down up 30.00 42.00up up up down 30.00 42.00Two down (6 paths) down down up up 30.00 24.00down up down up 30.00 24.00down up up down 30.00 24.00up down down up 30.00 42.00up up down down 30.00 42.00up down up down 30.00 42.00Three down (4 paths) up down down down 30.00 42.00down up down down 30.00 24.00down down up down 30.00 24.00down down down up 30.00 24.00Four down (1 path) down down down down 30.00 24.00A B C D E F G H12345678910111213141516171819202122232425262728293031323334353637383940414230A B C D E F G H4344454647Formula in cell M17: =MAX(G17:K17)$B$7Period 2 Period 3 Period 4 Max(St)Barrier?Pathrisk-neutralprobabilityKnockoutoptionpayoff58.80 82.32 115.25 FALSE 0.0474 0.0033.60 47.04 65.86 FALSE 0.0542 0.0033.60 47.04 65.86 FALSE 0.0542 0.0058.80 47.04 65.86 FALSE 0.0542 0.0058.80 82.32 65.86 FALSE 0.0542 0.0019.20 26.88 37.63 TRUE 0.0619 7.6333.60 26.88 37.63 TRUE 0.0619 7.6333.60 47.04 37.63 TRUE 0.0619 7.6333.60 26.88 37.63 TRUE 0.0619 7.6358.80 47.04 37.63 FALSE 0.0619 0.0033.60 47.04 37.63 TRUE 0.0619 7.6333.60 26.88 21.50 TRUE 0.0708 0.0033.60 26.88 21.50 TRUE 0.0708 0.0019.20 26.88 21.50 TRUE 0.0708 0.0019.20 15.36 21.50 TRUE 0.0708 0.0019.20 15.36 12.29 TRUE 0.0809 0.00Knockout value 1.7375Formula in cell N38:=SUMPRODUCT(O17:O36,N17:N36)/B54Formula in cell O36:=M36*MAX(K36-$B$6,0)PRICING A KNOCKOUT BARRIER OPTIONSTOCK PRICEI J K L M N O123456789101112131415161718192021222324252627282930313233343536373839404142敲出障碍期权定价初始股票价格 30上涨 1.40下跌 0.80利率 1.08期权执行价格 30障碍 50.00qU 0.4321 - =(B5-B4)/(B5*(B3-B4)qD 0.4938 - =(B3-B5)/(B5*(B3-B4)风险中性概率, 上涨0.4667 - =B9*B5风险中性概率, 下跌0.5333 - =B10*B5股票价格路径时期 1 时期 2 时期 3 时期 4 时期 0 时期 1 时期 2全部上涨 (1条路径)上涨上涨上涨上涨 30.00 42.00 58.80一次下跌 (4条路径)下跌上涨上涨上涨 30.00 24.00 33.60上涨下跌上涨上涨 30.00 42.00 33.60上涨上涨下跌上涨 30.00 42.00 58.80上涨上涨上涨下跌 30.00 42.00 58.80两次下跌 (6条路径)下跌下跌上涨上涨 30.00 24.00 19.20下跌上涨下跌上涨 30.00 24.00 33.60下跌上涨上涨下跌 30.00 24.00 33.60上涨下跌下跌上涨 30.00 42.00 33.60上涨上涨下跌下跌 30.00 42.00 58.80上涨下跌上涨下跌 30.00 42.00 33.60三次下跌 (4条路径)上涨下跌下跌下跌 30.00 42.00 33.60下跌上涨下跌下跌 30.00 24.00 33.60下跌下跌上涨下跌 30.00 24.00 19.20下跌下跌下跌上涨 30.00 24.00 19.20四次下跌 (1条路径)下跌下跌下跌下跌 30.00 24.00 19.20单元格M17公式=MAX(G17:K17)$B$7时期 3 时期 4 Max(St)障碍?路径风险中性概率敲出期权现金流82.32 115.25 FALSE 0.0474 0.0047.04 65.86 FALSE 0.0542 0.0047.04 65.86 FALSE 0.0542 0.0047.04 65.86 FALSE 0.0542 0.0082.32 65.86 FALSE 0.0542 0.0026.88 37.63 TRUE 0.0619 7.6326.88 37.63 TRUE 0.0619 7.6347.04 37.63 TRUE 0.0619 7.6326.88 37.63 TRUE 0.0619 7.6347.04 37.63 FALSE 0.0619 0.0047.04 37.63 TRUE 0.0619 7.6326.88 21.50 TRUE 0.0708 0.0026.88 21.50 TRUE 0.0708 0.0026.88 21.50 TRUE 0.0708 0.0015.36 21.50 TRUE 0.0708 0.0015.36 12.29 TRUE 0.0809 0.00敲出期权价格 1.7375单元格N38公式=SUMPRODUCT(O17:O36,N17:N36)/B54单元格O36公式=M36*MAX(K36-$B$6,0)敲出障碍期权定价股票价格PRICING A KNOCKIN BARRIER OPTIONInitial stock price 30Up 1.40Down 0.80Interest 1.08Option exercise price 30Barrier 50.00qU 0.4321 - =(B5-B4)/(B5*(B3-B4)qD 0.4938 - =(B3-B5)/(B5*(B3-B4)Risk-neutral probability, up 0.4667 - =B9*B5Risk-neutral probability, down 0.5333 - =B10*B5STOCK PRICEPaths Period 1 Period 2 Period 3 Period 4 Period 0 Period 1All up (1 path) up up up up 30.00 42.00One down (4 paths) down up up up 30.00 24.00up down up up 30.00 42.00up up down up 30.00 42.00up up up down 30.00 42.00Two down (6 paths) down down up up 30.00 24.00down up down up 30.00 24.00down up up down 30.00 24.00up down down up 30.00 42.00up up down down 30.00 42.00up down up down 30.00 42.00Three down (4 paths) up down down down 30.00 42.00down up down down 30.00 24.00down down up down 30.00 24.00down down down up 30.00 24.00Four down (1 path) down down down down 30.00 24.00A B C D E F G H12345678910111213141516171819202122232425262728293031323334353637383940414230A B C D E F G H4344454647Formula in cell M17: =MAX(G17:K17)$B$7Period 2 Period 3 Period 4 Max(St)Barrier?Pathrisk-neutralprobabilityOptionpayoff58.80 82.32 115.25 FALSE 0.0474 85.2533.60 47.04 65.86 FALSE 0.0542 35.8633.60 47.04 65.86 FALSE 0.0542 35.8658.80 47.04 65.86 FALSE 0.0542 35.8658.80 82.32 65.86 FALSE 0.0542 35.8619.20 26.88 37.63 TRUE 0.0619 0.0033.60 26.88 37.63 TRUE 0.0619 0.0033.60 47.04 37.63 TRUE 0.0619 0.0033.60 26.88 37.63 TRUE 0.0619 0.0058.80 47.04 37.63 FALSE 0.0619 7.6333.60 47.04 37.63 TRUE 0.0619 0.0033.60 26.88 21.50 TRUE 0.0708 0.0033.60 26.88 21.50 TRUE 0.0708 0.0019.20 26.88 21.50 TRUE 0.0708 0.0019.20 15.36 21.50 TRUE 0.0708 0.0019.20 15.36 12.29 TRUE 0.0809 0.00Knockin value 9.0334Formula in cell N38:=SUMPRODUCT(O17:O36,N17:N36)/B54Formula in cell O36:=(1-M36)*MAX(K36-$B$6,0)PRICING A KNOCKIN BARRIER OPTIONSTOCK PRICEI J K L M N O123456789101112131415161718192021222324252627282930313233343536373839404142敲入障碍期权定价初始股票价格 30上涨 1.40下跌 0.80利率 1.08期权执行价格 30障碍 50.00qU 0.4321 - =(B5-B4)/(B5*(B3-B4)qD 0.4938 - =(B3-B5)/(B5*(B3-B4)风险中性概率, 上涨0.4667 - =B9*B5风险中性概率, 下跌0.5333 - =B10*B5股票价格路径时期 1 时期 2 时期 3 时期 4 时期 0 时期 1 时期 2全部上涨 (1条路径)上涨上涨上涨上涨 30.00 42.00 58.80一次下跌 (4条路径)下跌上涨上涨上涨 30.00 24.00 33.60上涨下跌上涨上涨 30.00 42.00 33.60上涨上涨下跌上涨 30.00 42.00 58.80上涨上涨上涨下跌 30.00 42.00 58.80两次下跌 (6条路径)下跌下跌上涨上涨 30.00 24.00 19.20下跌上涨下跌上涨 30.00 24.00 33.60下跌上涨上涨下跌 30.00 24.00 33.60上涨下跌下跌上涨 30.00 42.00 33.60上涨上涨下跌下跌 30.00 42.00 58.80上涨下跌上涨下跌 30.00 42.00 33.60三次下跌 (4条路径)上涨下跌下跌下跌 30.00 42.00 33.60下跌上涨下跌下跌 30.00 24.00 33.60下跌下跌上涨下跌 30.00 24.00 19.20下跌下跌下跌上涨 30.00 24.00 19.20四次下跌 (1条路径)下跌下跌下跌下跌 30.00 24.00 19.20单元格M17公式=MAX(G17:K17)$B$7时期 3 时期 4 Max(St)障碍?路径风险中性概率期权现金流82.32 115.25 FALSE 0.0474 85.2547.04 65.86 FALSE 0.0542 35.8647.04 65.86 FALSE 0.0542 35.8647.04 65.86 FALSE 0.0542 35.8682.32 65.86 FALSE 0.0542 35.8626.88 37.63 TRUE 0.0619 0.0026.88 37.63 TRUE 0.0619 0.0047.04 37.63 TRUE 0.0619 0.0026.88 37.63 TRUE 0.0619 0.0047.04 37.63 FALSE 0.0619 7.6347.04 37.63 TRUE 0.0619 0.0026.88 21.50 TRUE 0.0708 0.0026.88 21.50 TRUE 0.0708 0.0026.88 21.50 TRUE 0.0708 0.0015.36 21.50 TRUE 0.0708 0.0015.36 12.29 TRUE 0.0809 0.00敲入价值 9.0334单元格N38公式=SUMPRODUCT(O17:O36,N17:N36)/B54单元格O36公式=(1-M36)*MAX(K36-$B$6,0)敲入障碍期权定价股票价格KNOCKIN + KNOCKOUT = PLAIN VANILLAInitial stock price 30Up 1.40Down 0.80Interest 1.08Option exercise price 30Barrier 50.00qU 0.4321 - =(B5-B4)/(B5*(B3-B4)qD 0.4938 - =(B3-B5)/(B5*(B3-B4)Risk-neutral probability, up 0.4667 - =B9*B5Risk-neutral probability, down 0.5333 - =B10*B5STOCK PRICEPaths Period 1 Period 2 Period 3 Period 4 Period 0 Period 1All up (1 path) up up up up 30.00 42.00One down (4 paths) down up up up 30.00 24.00up down up up 30.00 42.00up up down up 30.00 42.00up up up down 30.00 42.00Two down (6 paths) down down up up 30.00 24.00down up down up 30.00 24.00down up up down 30.00 24.00up down down up 30.00 42.00up up down down 30.00 42.00up down up down 30.00 42.00Three down (4 paths) up down down down 30.00 42.00down up down down 30.00 24.00down down up down 30.00 24.00down down down up 30.00 24.00Four down (1 path) down down down down 30.00 24.00A B C D E F G H123456789101112131415161718192021222324252627282930313233343536373839404142434430A B C D E F G H45464748Formula in cell M17: =MAX(G17:K17)$B$7Period 2 Period 3 Period 4 Max(St)Barrier?Pathrisk-neutralprobabilityKnockoutpayoffKnockinpayoff58.80 82.32 115.25 FALSE 0.0474 0 85.248033.60 47.04 65.86 FALSE 0.0542 0 35.856033.60 47.04 65.86 FALSE 0.0542 0 35.856058.80 47.04 65.86 FALSE 0.0542 0 35.856058.80 82.32 65.86 FALSE 0.0542 0 35.856019.20 26.88 37.63 TRUE 0.0619 7.632 0.000033.60 26.88 37.63 TRUE 0.0619 7.632 0.000033.60 47.04 37.63 TRUE 0.0619 7.632 0.000033.60 26.88 37.63 TRUE 0.0619 7.632 0.000058.80 47.04 37.63 FALSE 0.0619 0 7.632033.60 47.04 37.63 TRUE 0.0619 7.632 0.000033.60 26.88 21.50 TRUE 0.0708 0 0.000033.60 26.88 21.50 TRUE 0.0708 0 0.000019.20 26.88 21.50 TRUE 0.0708 0 0.000019.20 15.36 21.50 TRUE 0.0708 0 0.000019.20 15.36 12.29 TRUE 0.0809 0 0.0000Knockin 9.0334 - =SUMPRODUCT(N17:N36,P17:P36)/B54Knockout 1.7375 - =SUMPRODUCT(N17:N36,O17:O36)/B54Sum 10.7708 - =L38+L39Plain vanilla 10.7708 - =SUMPRODUCT(N17:N36,Q17:Q36)/B54Formula in cell P36:=(1-M36)*MAX(K36-$B$6,0)KNOCKIN + KNOCKOUT = PLAIN VANILLASTOCK PRICEI J K L M N O P1234567891011121314151617181920212223242526272829303132333435363738394041424344Formula in cell O36:=M36*MAX(K36-$B$6,0)I J K L M N O P45464748Plainvanilla85.248035.856035.856035.856035.85607.63207.63207.63207.63207.63207.63200.00000.00000.00000.00000.0000KNOCKIN + KNOCKOUT = PLAIN VANILLAQ1234567891011121314151617181920212223242526272829303132333435363738394041424344敲入 + 敲出= 基本期权初始股票价格 30上涨 1.40下跌 0.80利率 1.08期权执行价格 30障碍 50.00qU 0.4321 - =(B5-B4)/(B5*(B3-B4)qD 0.4938 - =(B3-B5)/(B5*(B3-B4)风险中性概率, 上涨 0.4667 - =B9*B5风险中性概率, 下跌 0.5333 - =B10*B5股票价格路径s 时期 1 时期 2 时期 3 时期 4 时期 0 时期 1全部上涨 (1条路径) 上涨上涨上涨上涨 30.00 42.00一次下跌 (4条路径) 下跌上涨上涨上涨 30.00 24.00上涨下跌上涨上涨 30.00 42.00上涨上涨下跌上涨 30.00 42.00上涨上涨上涨下跌 30.00 42.00两次下跌 (6条路径) 下跌下跌上涨上涨 30.00 24.00下跌上涨下跌上涨 30.00 24.00下跌上涨上涨下跌 30.00 24.00上涨下跌下跌上涨 30.00 42.00上涨上涨下跌下跌 30.00 42.00上涨下跌上涨下跌 30.00 42.00三次下跌 (4条路径) 上涨下跌下跌下跌 30.00 42.00下跌上涨下跌下跌 30.00 24.00下跌下跌上涨下跌 30.00 24.00下跌下跌下跌上涨 30.00 24.00四次下跌 (1条路径) 下跌下跌下跌下跌 30.00 24.00A B C D E F G H123456789101112131415161718192021222324252627282930313233343536373839404142434445单元格M17公式=MAX(G17:K17)$B$7时期 2 时期 3 时期 4 Max(St)障碍?路径风险中性概率敲出收益敲入收益一般期权58.80 82.32 115.25 FALSE 0.0474 0 85.2480 85.248033.60 47.04 65.86 FALSE 0.0542 0 35.8560 35.856033.60 47.04 65.86 FALSE 0.0542 0 35.8560 35.856058.80 47.04 65.86 FALSE 0.0542 0 35.8560 35.856058.80 82.32 65.86 FALSE 0.0542 0 35.8560 35.856019.20 26.88 37.63 TRUE 0.0619 7.632 0.0000 7.632033.60 26.88 37.63 TRUE 0.0619 7.632 0.0000 7.632033.60 47.04 37.63 TRUE 0.0619 7.632 0.0000 7.632033.60 26.88 37.63 TRUE 0.0619 7.632 0.0000 7.632058.80 47.04 37.63 FALSE 0.0619 0 7.6320 7.632033.60 47.04 37.63 TRUE 0.0619 7.632 0.0000 7.632033.60 26.88 21.50 TRUE 0.0708 0 0.0000 0.000033.60 26.88 21.50 TRUE 0.0708 0 0.0000 0.000019.20 26.88 21.50 TRUE 0.0708 0 0.0000 0.000019.20 15.36 21.50 TRUE 0.0708 0 0.0000 0.000019.20 15.36 12.29 TRUE 0.0809 0 0.0000 0.0000敲入 9.0334 - =SUMPRODUCT(N17:N36,P17:P36)/B54敲出 1.7375 - =SUMPRODUCT(N17:N36,O17:O36)/B54总计 10.7708 - =L38+L39一般期权 10.7708 - =SUMPRODUCT(N17:N36,Q17:Q36)/B54单元格P36公式=(1-M36)*MAX(K36-$B$6,0)单元格O36公式=M36*MAX(K36-$B$6,0)敲入 + 敲出= 基本期权股票价格I J K L M N O P Q123456789101112131415161718192021222324252627282930313233343536373839404142434445PRICING BARRIER OPTIONSBY MONTE CARLOUp 1.4Down 0.8Interest 1.08Initialprice 30Periods 4Exercise 30Barrier 50Runs 100Knockin option value 10.8490 - =mcbarrierin(B6,B8,B9,B2,B3,B4,B7,B11)Actual value 9.0334 - =Page 641!N38Knockout option value 1.3463 - =mcbarrierout(B6,B8,B9,B2,B3,B4,B7,B11)Actual value 1.7375 - Determined from fully-worked out example48 iterations of MCBarrierIn7.6430 7.9754 8.1617 10.2743 7.6949 8.80109.7260 13.3554 6.8090 8.2866 8.3649 7.837810.9305 11.2989 9.1037 10.4648 9.1248 8.46448.2569 11.8472 8.3384 11.3200 10.0499 9.483611.8905 10.4733 9.8943 11.1994 8.9396 9.107910.9750 6.5412 8.1828 11.3200 9.9250 10.94018.2823 11.8165 8.1140 10.9485 8.4210 8.48558.6369 9.8382 10.2224 8.1828 9.1471 9.7948Average of simulations 9.4769 - =AVERAGE(A20:F27)True value 9.0334 - =B146.5412 - =MIN(A20:F27)13.3554 - =MAX(A20:F27)1.4873 - =STDEV(A20:F27)A B C D E F12345678910111213141516171819202122232425262728293031323334用Monte Carlo定价障碍期权上涨 1.4下跌 0.8利率 1.08初始价格 30时期数 4执行价格 30障碍 50运行次数 100敲入期权价值 8.2008 - =mcbarrierin(B6,B8,B9,B2,B3,B4,B7,B11)真实的价值 9.0334- =C:UserslingjieDesktop财务金融建模（第三版）Chapter filesfm3_chapter23.3.xlsPage 641!N38敲出期权价值 1.7390 - =mcbarrierout(B6,B8,B9,B2,B3,B4,B7,B11)真实的价值 1.7375 - 取决于完全运行的例子MCBarrierIn的48次迭代8.2781 10.4733 8.8793 9.5186 8.4421 9.850910.7845 7.9150 7.6303 8.5723 11.8694 6.44178.3257 8.6369 8.4210 10.2097 9.8594 11.091411.0396 11.3677 12.0895 7.4356 8.9862 8.917411.2121 9.2677 8.1744 7.4313 6.2258 8.75338.8613 11.5190 13.6105 8.5459 6.9995 10.98358.4898 10.0457 7.7510 10.8660 10.1018 7.699110.9401 7.0291 7.0207 7.8240 10.8363 10.8279模拟的平均价值9.2927 - =AVERAGE(A20:F27)正确的价值 9.0334 - =B146.2258 - =MIN(A20:F27)13.6105 - =MAX(A20:F27)1.6692 - =STDEV(A20:F27)- =C:UserslingjieDesktop财务金融建模（第三版）Chapter filesfm3_chapter23.3.xlsPage 641!N38PRICING BARRIER OPTIONS-VBA FUNCTIONEach time interval is divided into n subintervals. In this simulation the initial stock price = 50.00, the exerciseprice = 45.00, the time to maturity = 0.40, and the unit time interval is divided into 80 subintervals. The stock priceprocess has mean return = 15.00% and standard deviation = 1.25%, and the interest rate = 8.00%.There are 100 runs in each Monte Carlo simulationS0, current stock price 50X, exercise price 45Barrier 50T, time to option exercise 0.4r, interest rate 8%m, mean stock return 15%s, standard deviation of stock return 22%n, number of sub-intervals of 1 period 80Delta t 0.0125 - =1/B10Up over 1 sub-interval 1.0268 - =EXP(B7*B11+B8*SQRT(B11)Down over 1 sub-interval 0.9775 - =EXP(B7*B11-B8*SQRT(B11)Interest over 1 sub-interval 1.0010 - =EXP(B6*B11)Runs 1007.3655 6.8915 7.7780 6.9108 7.8583 6.00526.6477 7.0292 6.6503 7.3228 5.8899 6.86437.3644 6.5328 7.5145 7.2878 6.6975 6.39747.5545 6.4182 7.3540 6.5154 7.1842 6.41876.9752 6.5637 6.5968 6.4313 6.9983 8.73665.8657 7.0768 7.0599 7.0812 7.4132 8.08596.1975 8.3787 6.7131 7.3604 7.6684 7.47496.6708 7.4352 6.4113 8.3344 5.2404 7.3852Average of above 7.0127 - =AVERAGE(A19:F26)Minimum 5.2404 - =MIN(A19:F26)Maxmimum 8.7366 - =MAX(A19:F26)Standard deviation 0.6850 - =STDEV(A19:F26)Data table: Sensitivity of results on number of runsRunsAverageof 48MCBarrierInMinimum Maximum Standarddeviation50 6.9072 4.6286 9.8825 0.9381100 6.8619 5.6949 8.5336 0.6004150 6.8746 5.5887 7.7955 0.4700200 6.9004 5.9512 7.7908 0.4203250 6.9241 6.0396 8.1116 0.4241300 6.9180 6.2277 7.7330 0.3624A B C D E F1234567891011121314151617181920212223242526272829303132333435363738394041350 7.0255 6.2962 7.6249 0.3097400 6.9689 6.1218 7.5116 0.3270450 6.9016 6.3760 7.4389 0.2491500 6.9313 6.1996 7.7104 0.3718550 6.9039 6.3118 7.5860 0.2470A B C D E F4243444546474849505152535455565758- =mcbarrierin($B$2,$B$3,$B$4,$B$13,$B$14,$B$15,INT($B$10*$B$5),$B$17)- =B31 , data table header (hidden)4 8 M C B a r r i e r I n S i m u l a t i o n s f o

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