UNDERLYING ASSETS AND PATH-INDEPENDENTVERSUS PATH-DEPENDENT PAYOFFSStock prices Bond valueSUUSU RS0 SUD = SDU 1SD RSDDDate 0 Date 1 Date 2 Date 0 Date 1 Date 2Path-independent asset with payoffs at Date 2 Path-dependent asset with payoffs at Date 2PayoffUU0 0Value today PayoffUD = PayoffDU Value today0 0PayoffDDDate 0 Date 1 Date 2 Date 0 Date 1 Date 2R2R2R2PayoffUUPayoffUDPayoffDUPayoffDDUNDERLYING ASSETS AND PATH-INDEPENDENTVERSUS PATH-DEPENDENT PAYOFFSBond valueDate 0 Date 1 Date 2Path-dependent asset with payoffs at Date 2Date 0 Date 1 Date 2基础资产和路径独立收益与路径依赖收益对比股票价格 债券价值SUUSU RS0 SUD = SDU 1SD RSDD时期0 时期1 时期2 时期0 时期1 时期2时期2路径独立资产收益 时期2路径依赖资产收益收益UU0 0今天的价值 收益UD =收益DU 今天的价值0 0收益DD时期0 时期1 时期2 时期0 时期1 时期2R2R2R2收益UU收益UD收益DU收益DD基础资产和路径独立收益与路径依赖收益对比债券价值时期2路径依赖资产收益Up 1.4Down 0.9SUU = 98 = 50*1.42SU = 70S0 = 50 SUD = SDU = 63 = 50*1.4*0.9SD = 45 SDD = 40.5 = 50*0.9 2Date 0 Date 1 Date 2上涨 1.4下跌 0.9SUU = 98 = 50*1.42SU = 70S0 = 50 SUD = SDU = 63 = 50*1.4*0.9SD = 45 SDD = 40.5 = 50*0.9 2时期0 时期1 时期2SIMPLE SIMULATION:TWO PATHS IN A TWO-DATE MODELInitial stock price 50X 50Up 1.4Down 0.9R 1.05State pricesqu 0.2857 $B$15,1,0)Second period, up (1) or down (0)? 0 0 $B$15,1,0)Total ups 0 0 $B$15,1,0)第二时期, 上涨(1)或下跌(0)? 0 1 $B$15,1,0)上涨总计 0 2 - =SUM(C18:C19)最终股票价格 40.5 98 - =$B$2*upC21*down(2-C21)期权收益 0 48 - =MAX(C22-$B$3,0)平均收益折现 21.7687 - =AVERAGE(23:23)/R_2用状态价格计算真实的期权价格收益高 48 - =MAX(B2*up2-B3,0)中 13 - =MAX(B2*up*down-B3,0)低 0 - =MAX(B2*down2-B3,0)真实股票价格 8.8707 - =q_up2*B29+2*q_up*q_down*B30+q_down2*B31A B C D1234567891011121314151617181920212223242526272829303132SIMPLE SIMULATION: A TWO-DATE MODELas many price paths as there are columnspressing F9 runs the simulation and will change the value in cell B25This value should be compared to the actual option price in cell B32Initial stock price 50X 50Up 1.4Down 0.9R 1.05State pricesqu 0.2857 - =(B7-B6)/(B7*(B5-B6)qd 0.6667 - =(B7-B6)/(B7*(B5-B6)Risk-neutral probabilitiespu 0.3000 - =(B7-B6)/(B7*(B5-B6)pd 0.7000 - =(B7-B6)/(B7*(B5-B6)Random paths and the Monte Carlo priceFirst period, up (1) or down (0)? 1 1 1 1 0 0Second period, up (1) or down (0)? 1 0 1 1 0 1Total ups 2 1 2 2 0 1Terminal stock price 98 63 98 98 40.5 63Option payoff 48 13 48 48 0 13Average discounted payoff 9.2766 - =AVERAGE(23:23)/R_2Computing the actual option price with state pricesPayoffstop 48 - =MAX(B2*up2-B3,0)middle 13 - =MAX(B2*up*down-B3,0)bottom 0 - =MAX(B2*down2-B3,0)Actual option price 8.8707 - =q_up2*B29+2*q_up*q_down*B30+q_down2*B31A B C D E F G12345678910111213141516171819202122232425262728293031320 0 0 1 1 0 0 1 00 0 0 0 1 1 0 1 00 0 0 1 2 1 0 2 040.5 40.5 40.5 63 98 63 40.5 98 40.50 0 0 13 48 13 0 48 0SIMPLE SIMULATION: A TWO-DATE MODELas many price paths as there are columnspressing F9 runs the simulation and will change the value in cell B25This value should be compared to the actual option price in cell B32H I J K L M N O P12345678910111213141516171819202122232425262728293031320 0 0 1 0 1 1 0 00 0 0 0 1 0 0 0 00 0 0 1 1 1 1 0 040.5 40.5 40.5 63 63 63 63 40.5 40.50 0 0 13 13 13 13 0 0Q R S T U V W X Y12345678910111213141516171819202122232425262728293031320 1 0 0 0 0 1 0 10 0 1 0 1 0 1 1 00 1 1 0 1 0 2 1 140.5 63 63 40.5 63 40.5 98 63 630 13 13 0 13 0 48 13 13Z AA AB AC AD AE AF AG AH12345678910111213141516171819202122232425262728293031320 0 0 0 0 1 0 1 11 0 0 1 0 1 0 0 01 0 0 1 0 2 0 1 163 40.5 40.5 63 40.5 98 40.5 63 6313 0 0 13 0 48 0 13 13AI AJ AK AL AM AN AO AP AQ12345678910111213141516171819202122232425262728293031321 0 0 1 0 1 1 0 00 0 1 0 0 1 0 1 11 0 1 1 0 2 1 1 163 40.5 63 63 40.5 98 63 63 6313 0 13 13 0 48 13 13 13AR AS AT AU AV AW AX AY AZ12345678910111213141516171819202122232425262728293031320 0 1 0 1 0 0 0 11 0 0 1 0 1 0 0 01 0 1 1 1 1 0 0 163 40.5 63 63 63 63 40.5 40.5 6313 0 13 13 13 13 0 0 13BA BB BC BD BE BF BG BH BI12345678910111213141516171819202122232425262728293031320 0 0 0 0 0 0 0 00 1 1 0 0 1 0 0 00 1 1 0 0 1 0 0 040.5 63 63 40.5 40.5 63 40.5 40.5 40.50 13 13 0 0 13 0 0 0BJ BK BL BM BN BO BP BQ BR12345678910111213141516171819202122232425262728293031321 0 0 1 0 1 1 0 01 0 1 0 1 0 1 0 02 0 1 1 1 1 2 0 098 40.5 63 63 63 63 98 40.5 40.548 0 13 13 13 13 48 0 0BS BT BU BV BW BX BY BZ CA12345678910111213141516171819202122232425262728293031320 0 1 0 0 0 1 1 10 0 0 1 1 1 0 1 10 0 1 1 1 1 1 2 240.5 40.5 63 63 63 63 63 98 980 0 13 13 13 13 13 48 48CB CC CD CE CF CG CH CI CJ12345678910111213141516171819202122232425262728293031320 0 0 0 1 1 1 0 10 0 0 0 0 0 0 1 00 0 0 0 1 1 1 1 140.5 40.5 40.5 40.5 63 63 63 63 630 0 0 0 13 13 13 13 13CK CL CM CN CO CP CQ CR CS12345678910111213141516171819202122232425262728293031321 0 0 0 0 0 0 1 10 0 1 0 0 0 1 1 01 0 1 0 0 0 1 2 163 40.5 63 40.5 40.5 40.5 63 98 6313 0 13 0 0 0 13 48 13CT CU CV CW CX CY CZ DA DB12345678910111213141516171819202122232425262728293031320 0 1 1 0 0 1 0 00 0 1 1 0 0 0 0 00 0 2 2 0 0 1 0 040.5 40.5 98 98 40.5 40.5 63 40.5 40.50 0 48 48 0 0 13 0 0DC DD DE DF DG DH DI DJ DK12345678910111213141516171819202122232425262728293031321 0 1 0 0 0 0 0 10 0 0 1 0 1 1 0 01 0 1 1 0 1 1 0 163 40.5 63 63 40.5 63 63 40.5 6313 0 13 13 0 13 13 0 13DL DM DN DO DP DQ DR DS DT12345678910111213141516171819202122232425262728293031320 0 0 0 0 0 0 1 00 0 1 1 0 0 0 0 10 0 1 1 0 0 0 1 140.5 40.5 63 63 40.5 40.5 40.5 63 630 0 13 13 0 0 0 13 13DU DV DW DX DY DZ EA EB EC12345678910111213141516171819202122232425262728293031321 1 0 0 0 0 0 0 10 0 0 1 0 0 0 1 01 1 0 1 0 0 0 1 163 63 40.5 63 40.5 40.5 40.5 63 6313 13 0 13 0 0 0 13 13ED EE EF EG EH EI EJ EK EL12345678910111213141516171819202122232425262728293031320 1 0 0 0 0 0 0 00 0 0 0 1 0 0 1 00 1 0 0 1 0 0 1 040.5 63 40.5 40.5 63 40.5 40.5 63 40.50 13 0 0 13 0 0 13 0EM EN EO EP EQ ER ES ET EU12345678910111213141516171819202122232425262728293031320 0 0 0 1 0 0 0 10 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 140.5 40.5 40.5 40.5 63 40.5 40.5 40.5 630 0 0 0 13 0 0 0 13EV EW EX EY EZ FA FB FC FD12345678910111213141516171819202122232425262728293031320 0 0 0 0 0 0 0 00 1 1 0 1 0 1 0 00 1 1 0 1 0 1 0 040.5 63 63 40.5 63 40.5 63 40.5 40.50 13 13 0 13 0 13 0 0FE FF FG FH FI FJ FK FL FM12345678910111213141516171819202122232425262728293031320 0 0 0 1 0 1 0 00 0 0 0 1 0 1 1 00 0 0 0 2 0 2 1 040.5 40.5 40.5 40.5 98 40.5 98 63 40.50 0 0 0 48 0 48 13 0FN FO FP FQ FR FS FT FU FV12345678910111213141516171819202122232425262728293031320 0 0 1 0 0 0 0 10 0 0 0 0 0 1 0 10 0 0 1 0 0 1 0 240.5 40.5 40.5 63 40.5 40.5 63 40.5 980 0 0 13 0 0 13 0 48FW FX FY FZ GA GB GC GD GE12345678910111213141516171819202122232425262728293031320 0 0 0 0 0 0 1 00 1 0 1 0 0 0 0 00 1 0 1 0 0 0 1 040.5 63 40.5 63 40.5 40.5 40.5 63 40.50 13 0 13 0 0 0 13 0GF GG GH GI GJ GK GL GM GN12345678910111213141516171819202122232425262728293031321 1 0 0 1 1 0 0 01 0 0 1 0 0 0 0 12 1 0 1 1 1 0 0 198 63 40.5 63 63 63 40.5 40.5 6348 13 0 13 13 13 0 0 13GO GP GQ GR GS GT GU GV GW12345678910111213141516171819202122232425262728293031321 1 0 0 0 0 0 1 10 0 0 0 1 1 1 0 01 1 0 0 1 1 1 1 163 63 40.5 40.5 63 63 63 63 6313 13 0 0 13 13 13 13 13GX GY GZ HA HB HC HD HE HF12345678910111213141516171819202122232425262728293031320 0 0 1 0 0 0 0 01 0 1 0 0 0 0 1 01 0 1 1 0 0 0 1 063 40.5 63 63 40.5 40.5 40.5 63 40.513 0 13 13 0 0 0 13 0HG HH HI HJ HK HL HM HN HO12345678910111213141516171819202122232425262728293031321 0 1 1 0 1 0 0 10 0 0 0 0 0 0 0 01 0 1 1 0 1 0 0 163 40.5 63 63 40.5 63 40.5 40.5 6313 0 13 13 0 13 0 0 13HP HQ HR HS HT HU HV HW HX12345678910111213141516171819202122232425262728293031321 1 0 1 0 1 0 0 10 1 0 0 0 1 0 0 01 2 0 1 0 2 0 0 163 98 40.5 63 40.5 98 40.5 40.5 6313 48 0 13 0 48 0 0 13HY HZ IA IB IC ID IE IF IG12345678910111213141516171819202122232425262728293031320 1 0 0 0 1 0 1 10 1 0 1 1 0 0 0 10 2 0 1 1 1 0 1 240.5 98 40.5 63 63 63 40.5 63 980 48 0 13 13 13 0 13 48IH II IJ IK IL IM IN IO IP12345678910111213141516171819202122232425262728293031321 0 0 0 0 11 0 0 1 1 02 0 0 1 1 198 40.5 40.5 63 63 6348 0 0 13 13 13IQ IR IS IT IU IV1234567891011121314151617181920212223242526272829303132简单模拟:一个两时期模型价格路径与列数相同按F9运行模拟将会改变单元格B25中的值该值应与单元格B32中的真实期权价格相比较初始股票价格 50X 50上涨 1.4下跌 0.9R 1.05状态价格qu 0.2857 - =(B7-B6)/(B7*(B5-B6)qd 0.6667 - =(B7-B6)/(B7*(B5-B6)风险中性概率pu 0.3000 - =(B7-B6)/(B7*(B5-B6)pd 0.7000 - =(B7-B6)/(B7*(B5-B6)随机路径和Monte Carlo价格第一时期, 上涨(1)或下跌(0)? 0 0 0 0 0 0第二时期, 上涨(1)或下跌(0)? 0 1 1 0 0 0上涨总数 0 1 1 0 0 0期末股票价格 40.5 63 63 40.5 40.5 40.5期权收益 0 13 13 0 0 0平均收益折现 7.9427 - =AVERAGE(23:23)/R_2用状态价格计算真实的期权价格收益高 48 - =MAX(B2*up2-B3,0)中 13 - =MAX(B2*up*down-B3,0)低 0 - =MAX(B2*down2-B3,0)真实的期权价格 8.8707 - =q_up2*B29+2*q_up*q_down*B30+q_down2*B31A B C D E F G12345678910111213141516171819202122232425262728293031320 0 0 0 0 0 0 0 01 0 0 0 0 1 0 0 01 0 0 0 0 1 0 0 063 40.5 40.5 40.5 40.5 63 40.5 40.5 40.513 0 0 0 0 13 0 0 0简单模拟:一个两时期模型价格路径与列数相同按F9运行模拟将会改变单元格B25中的值该值应与单元格B32中的真实期权价格相比较H I J K L M N O P12345678910111213141516171819202122232425262728293031320 1 0 1 0 0 1 0 10 1 0 0 0 1 1 1 10 2 0 1 0 1 2 1 240.5 98 40.5 63 40.5 63 98 63 980 48 0 13 0 13 48 13 48Q R S T U V W X Y12345678910111213141516171819202122232425262728293031321 0 0 0 0 0 1 0 10 0 1 0 0 0 0 0 11 0 1 0 0 0 1 0 263 40.5 63 40.5 40.5 40.5 63 40.5 9813 0 13 0 0 0 13 0 48Z AA AB AC AD AE AF AG AH12345678910111213141516171819202122232425262728293031320 0 1 0 0 1 1 1 00 1 1 0 0 1 0 0 00 1 2 0 0 2 1 1 040.5 63 98 40.5 40.5 98 63 63 40.50 13 48 0 0 48 13 13 0AI AJ AK AL AM AN AO AP AQ12345678910111213141516171819202122232425262728293031320 0 0 0 0 0 0 1 00 0 1 0 0 0 1 0 00 0 1 0 0 0 1 1 040.5 40.5 63 40.5 40.5 40.5 63 63 40.50 0 13 0 0 0 13 13 0AR AS AT AU AV AW AX AY AZ12345678910111213141516171819202122232425262728293031320 0 1 0 0 1 0 0 11 0 0 0 1 0 0 0 01 0 1 0 1 1 0 0 163 40.5 63 40.5 63 63 40.5 40.5 6313 0 13 0 13 13 0 0 13BA BB BC BD BE BF BG BH BI12345678910111213141516171819202122232425262728293031321 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 11 0 0 0 0 0 0 1 163 40.5 40.5 40.5 40.5 40.5 40.5 63 6313 0 0 0 0 0 0 13 13BJ BK BL BM BN BO BP BQ BR12345678910111213141516171819202122232425262728293031320 1 0 0 0 0 0 1 00 0 1 0 0 0 0 0 00 1 1 0 0 0 0 1 040.5 63 63 40.5 40.5 40.5 40.5 63 40.50 13 13 0 0 0 0 13 0BS BT BU BV BW BX BY BZ CA12345678910111213141516171819202122232425262728293031321 0 0 1 1 1 0 1 00 0 0 1 0 0 0 0 11 0 0 2 1 1 0 1 163 40.5 40.5 98 63 63 40.5 63 6313 0 0 48 13 13 0 13 13CB CC CD CE CF CG CH CI CJ12345678910111213141516171819202122232425262728293031320 0 0 0 0 0 0 0 11 1 1 0 0 0 1 0 01 1 1 0 0 0 1 0 163 63 63 40.5 40.5 40.5 63 40.5 6313 13 13 0 0 0 13 0 13CK CL CM CN CO CP CQ CR CS12345678910111213141516171819202122232425262728293031320 0 0 0 1 0 0 1 00 0 0 0 0 0 0 0 10 0 0 0 1 0 0 1 140.5 40.5 40.5 40.5 63 40.5 40.5 63 630 0 0 0 13 0 0 13 13CT CU CV CW CX CY CZ DA DB12345678910111213141516171819202122232425262728293031321 0 0 0 1 0 0 0 00 1 0 0 0 1 0 0 11 1 0 0 1 1 0 0 163 63 40.5 40.5 63 63 40.5 40.5 6313 13 0 0 13 13 0 0 13DC DD DE DF DG DH DI DJ DK12345678910111213141516171819202122232425262728293031320 1 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 1 0 0 1 1 0 0 040.5 63 40.5 40.5 63 63 40.5 40.5 40.50 13 0 0 13 13 0 0 0DL DM DN DO DP DQ DR DS DT12345678910111213141516171819202122232425262728293031320 1 1 0 1 0 1 1 11 0 0 1 0 1 0 0 01 1 1 1 1 1 1 1 163 63 63 63 63 63 63 63 6313 13 13 13 13 13 13 13 13DU DV DW DX DY DZ EA EB EC12345678910111213141516171819202122232425262728293031321 0 0 1 0 0 0 0 10 0 0 1 0 1 0 0 01 0 0 2 0 1 0 0 163 40.5 40.5 98 40.5 63 40.5 40.5 6313 0 0 48 0 13 0 0 13ED EE EF EG EH EI EJ EK EL12345678910111213141516171819202122232425262728293031321 0 0 0 0 0 0 0 00 1 0 0 0 0 1 0 01 1 0 0 0 0 1 0 063 63 40.5 40.5 40.5 40.5 63 40.5 40.513 13 0 0 0 0 13 0 0EM EN EO EP EQ ER ES ET EU12345678910111213141516171819202122232425262728293031320 0 0 0 1 1 0 0 00 0 0 0 1 1 0 0 10 0 0 0 2 2 0 0 140.5 40.5 40.5 40.5 98 98 40.5 40.5 630 0 0 0 48 48 0 0 13EV EW EX EY EZ FA FB FC FD12345678910111213141516171819202122232425262728293031320 0 1 0 1 0 0 1 00 0 0 0 0 1 0 0 00 0 1 0 1 1 0 1 040.5 40.5 63 40.5 63 63 40.5 63 40.50 0 13 0 13 13 0 13 0FE FF FG FH FI FJ FK FL FM12345678910111213141516171819202122232425262728293031320 0 1 0 0 1 1 0 00 1 1 0 1 0 0 0 00 1 2 0 1 1 1 0 040.5 63 98 40.5 63 63 63 40.5 40.50 13 48 0 13 13 13 0 0FN FO FP FQ FR FS FT FU FV12345678910111213141516171819202122232425262728293031320 0 0 0 0 1 0 1 10 0 1 0 1 0 0 0 00 0 1 0 1 1 0 1 140.5 40.5 63 40.5 63 63 40.5 63 630 0 13 0 13 13 0 13 13FW FX FY FZ GA GB GC GD GE12345678910111213141516171819202122232425262728293031320 0 0 0 0 0 0 1 01 0 1 1 1 0 0 1 01 0 1 1 1 0 0 2 063 40.5 63 63 63 40.5 40.5 98 40.513 0 13 13 13 0 0 48 0GF GG GH GI GJ GK GL GM GN12345678910111213141516171819202122232425262728293031320 1 0 0 0 0 0 0 01 0 0 0 0 0 0 0 01 1 0 0 0 0 0 0 063 63 40.5 40.5 40.5 40.5 40.5 40.5 40.513 13 0 0 0 0 0 0 0GO GP GQ GR GS GT GU GV GW12345678910111213141516171819202122232425262728293031321 0 0 1 0 0 0 0 00 0 0 0 1 0 1 1 11 0 0 1 1 0 1 1 163 40.5 40.5 63 63 40.5 63 63 6313 0 0 13 13 0 13 13 13GX GY GZ HA HB HC HD HE HF12345678910111213141516171819202122232425262728293031320 0 0 0 0 1 1 0 01 1 0 0 1 0 0 0 01 1 0 0 1 1 1 0 063 63 40.5 40.5 63 63 63 40.5 40.513 13 0 0 13 13 13 0 0HG HH HI HJ HK HL HM HN HO12345678910111213141516171819202122232425262728293031320 0 0 1 1 0 1 0 11 1 0 1 1 0 0 1 01 1 0 2 2 0 1 1 163 63 40.5 98 98 40.5 63 63 6313 13 0 48 48 0 13 13 13HP HQ HR HS HT HU HV HW HX12345678910111213141516171819202122232425262728293031320 0 1 0 0 0 0 0 00 0 1 1 0 0 1 1 00 0 2 1 0 0 1 1 040.5 40.5 98 63 40.5 40.5 63 63 40.50 0 48 13 0 0 13 13 0HY HZ IA IB IC ID IE IF IG12345678910111213141516171819202122232425262728293031320 0 0 1 0 0 1 1 11 1 1 0 0 0 0 1 01 1 1 1 0 0 1 2 163 63 63 63 40.5 40.5 63 98 6313 13 13 13 0 0 13 48 13IH II IJ IK IL IM IN IO IP12345678910111213141516171819202122232425262728293031320 1 0 1 0 00 1 0 0 0 00 2 0 1 0 040.5 98 40.5 63 40.5 40.50 48 0 13 0 0IQ IR IS IT IU IV1234567891011121314151617181920212223242526272829303132MONTE CARLO PRICING OF PLAIN VANILLA CALLSS0 , current stock price 50X, exercise price 50r, interest rate 10%T, time 0.8m, mean stock return 33%s, sigma-standard deviation of stock return 30%n, divisions of unit time 200Runs 3,000VanillaCall 7.1131 - =vanillacall(B2,B3,B6,B7,B4,B5,B9,B10)BS call 7.2782 - =BSCall(B2,B3,B5,B4,B7)A B C1234567891011121314基本看涨期权的Monte Carlo定价S0 ,当前股票价格 50X,执行价格 50r,利率 10%T,时间 0.8m,平均股票收益 33%s,股票收益标准差 30%n, 单位时间的分割 200运行 3,000基本看涨期权价格 7.4931 - =vanillacall(B2,B3,B6,B7,B4,B