fm3_chapter23.1plainvanilla - UNDERLYING ASSETS AND...

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Unformatted text preview: UNDERLYING ASSETS AND PATH-INDEPENDENT VERSUS PATH-DEPENDENT PAYOFFS Stock prices SUU SU R Bond value S0 SUD = SDU 1 SD SDD Date 0 Date 1 Date 2 Date 0 R Date 1 Date 2 Path-independent asset with payoffs at Date 2 PayoffUU 0 Path-dependent asset with payoffs at Date 2 0 Value today PayoffUD = PayoffDU Value today 0 Date 0 Date 1 PayoffDD Date 2 Date 0 0 Date 1 Date 2 NT value R2 R2 R2 1 Date 2 t with payoffs at Date 2 PayoffUU PayoffUD PayoffDU 1 PayoffDD Date 2 Up Down 1.4 0.9 SUU = 98 = 50*1.42 SU = 70 S0 = 50 SUD = SDU = 63 = 50*1.4*0.9 SD = 45 Date 0 Date 1 SDD = 40.5 = 50*0.9 2 Date 2 A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Initial stock price X Up Down R State prices qu qd Risk-neutral probabilities πu πd B C SIMPLE SIMULATION: TWO PATHS IN A TWO-DATE MODEL 50 50 1.4 0.9 1.05 0.2857 0.6667 ### ### 0.3000 0.7000 ### ### Random paths and the Monte Carlo price First period, up (1) or down (0)? 1 Second period, up (1) or down (0)? 0 Total ups Terminal stock price Option payoff Average discounted payoff 1 63 13 ### 0 1 1 63 13 ### Computing the actual option price with state prices Payoffs top 48 middle 13 bottom 0 Actual option price 8.8707 ### ### ### ### D LE SIMULATION: 1 S IN A TWO-DATE MODEL 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? A B C D E F G 1 SIMPLE SIMULATION: A TWO-DATE MODEL as many price paths as there are columns pressing F9 runs the simulation and will change the value in cell B25 This value should be compared to the actual option price in cell B32 Initial stock price X Up Down R State prices qu qd Risk-neutral probabilities πu πd 50 50 1.4 0.9 1.05 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0.2857 0.6667 ### ### 0.3000 0.7000 ### ### 1 0 1 63 13 ### 0 1 1 63 13 0 0 0 40.5 0 0 0 0 40.5 0 1 0 1 63 13 Random paths and the Monte Carlo price First period, up (1) or down (0)? 1 Second period, up (1) or down (0)? 1 Total ups Terminal stock price Option payoff Average discounted payoff 2 98 48 10.0840 Computing the actual option price with state prices Payoffs top 48 middle 13 bottom 0 Actual option price 8.8707 ### ### ### ### H I J K L M N O P ODEL umns 1 e value in cell B25 n price in cell B32 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 0 1 63 13 1 0 1 63 13 0 1 1 63 13 0 1 1 63 13 1 1 2 98 48 1 0 1 63 13 0 1 1 63 13 1 0 1 63 13 0 0 0 40.5 0 Q R S T U V W X Y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 0 1 63 13 1 0 1 63 13 0 0 0 40.5 0 1 0 1 63 13 0 0 0 40.5 0 0 1 1 63 13 0 0 0 40.5 0 1 1 2 98 48 0 0 0 40.5 0 Z AA AB AC AD AE AF AG AH 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 1 2 98 48 0 0 0 40.5 0 0 0 0 40.5 0 1 1 2 98 48 1 1 2 98 48 1 0 1 63 13 1 0 1 63 13 0 1 1 63 13 1 1 2 98 48 AI AJ AK AL AM AN AO AP AQ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0 0 0 40.5 0 0 1 1 63 13 0 0 0 40.5 0 0 0 0 40.5 0 1 0 1 63 13 0 1 1 63 13 0 0 0 40.5 0 1 0 1 63 13 1 0 1 63 13 AR AS AT AU AV AW AX AY AZ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0 1 1 63 13 0 0 0 40.5 0 0 0 0 40.5 0 0 0 0 40.5 0 1 1 2 98 48 0 0 0 40.5 0 1 1 2 98 48 1 0 1 63 13 0 0 0 40.5 0 BA BB BC BD BE BF BG BH BI 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 0 1 63 13 0 0 0 40.5 0 1 1 2 98 48 0 1 1 63 13 0 0 0 40.5 0 0 1 1 63 13 0 1 1 63 13 0 1 1 63 13 0 0 0 40.5 0 BJ BK BL BM BN BO BP BQ BR 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0 0 0 40.5 0 1 1 2 98 48 0 0 0 40.5 0 0 0 0 40.5 0 0 0 0 40.5 0 0 1 1 63 13 0 0 0 40.5 0 1 1 2 98 48 1 0 1 63 13 BS BT BU BV BW BX BY BZ CA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 0 1 63 13 0 1 1 63 13 0 1 1 63 13 0 0 0 40.5 0 0 0 0 40.5 0 0 0 0 40.5 0 0 0 0 40.5 0 0 0 0 40.5 0 1 0 1 63 13 CB CC CD CE CF CG CH CI CJ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0 0 0 40.5 0 0 0 0 40.5 0 0 1 1 63 13 0 1 1 63 13 0 0 0 40.5 0 1 0 1 63 13 1 0 1 63 13 0 0 0 40.5 0 0 1 1 63 13 CK CL CM CN CO CP CQ CR CS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0 0 0 40.5 0 0 1 1 63 13 0 0 0 40.5 0 0 0 0 40.5 0 0 0 0 40.5 0 1 1 2 98 48 0 0 0 40.5 0 1 0 1 63 13 0 0 0 40.5 0 CT CU CV CW CX CY CZ DA DB 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0 1 1 63 13 0 0 0 40.5 0 1 0 1 63 13 0 1 1 63 13 1 0 1 63 13 0 0 0 40.5 0 0 0 0 40.5 0 0 0 0 40.5 0 1 0 1 63 13 DC DD DE DF DG DH DI DJ DK 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0 0 0 40.5 0 0 0 0 40.5 0 1 1 2 98 48 0 0 0 40.5 0 1 0 1 63 13 0 0 0 40.5 0 1 1 2 98 48 0 0 0 40.5 0 1 0 1 63 13 DL DM DN DO DP DQ DR DS DT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0 0 0 40.5 0 0 0 0 40.5 0 1 0 1 63 13 1 0 1 63 13 0 0 0 40.5 0 1 0 1 63 13 1 0 1 63 13 1 1 2 98 48 0 1 1 63 13 DU DV DW DX DY DZ EA EB EC 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 0 1 63 13 0 1 1 63 13 1 0 1 63 13 0 1 1 63 13 0 1 1 63 13 1 0 1 63 13 0 0 0 40.5 0 0 0 0 40.5 0 1 0 1 63 13 ED EE EF EG EH EI EJ EK EL 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 1 2 98 48 0 0 0 40.5 0 1 0 1 63 13 0 1 1 63 13 0 0 0 40.5 0 0 0 0 40.5 0 1 0 1 63 13 0 0 0 40.5 0 0 0 0 40.5 0 EM EN EO EP EQ ER ES ET EU 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 0 1 63 13 1 0 1 63 13 0 1 1 63 13 0 1 1 63 13 0 0 0 40.5 0 0 0 0 40.5 0 1 0 1 63 13 1 1 2 98 48 0 1 1 63 13 EV EW EX EY EZ FA FB FC FD 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 1 2 98 48 0 0 0 40.5 0 1 0 1 63 13 0 0 0 40.5 0 0 0 0 40.5 0 0 1 1 63 13 0 0 0 40.5 0 0 0 0 40.5 0 0 1 1 63 13 FE FF FG FH FI FJ FK FL FM 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 0 1 63 13 0 0 0 40.5 0 0 1 1 63 13 0 1 1 63 13 0 0 0 40.5 0 1 1 2 98 48 0 0 0 40.5 0 0 1 1 63 13 0 0 0 40.5 0 FN FO FP FQ FR FS FT FU FV 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0 0 0 40.5 0 1 1 2 98 48 0 0 0 40.5 0 1 1 2 98 48 0 0 0 40.5 0 0 0 0 40.5 0 0 1 1 63 13 0 0 0 40.5 0 0 1 1 63 13 FW FX FY FZ GA GB GC GD GE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0 0 0 40.5 0 1 1 2 98 48 1 0 1 63 13 0 0 0 40.5 0 0 0 0 40.5 0 0 0 0 40.5 0 0 0 0 40.5 0 1 0 1 63 13 0 0 0 40.5 0 GF GG GH GI GJ GK GL GM GN 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0 0 0 40.5 0 0 0 0 40.5 0 0 0 0 40.5 0 1 0 1 63 13 0 0 0 40.5 0 1 0 1 63 13 0 0 0 40.5 0 0 1 1 63 13 0 0 0 40.5 0 GO GP GQ GR GS GT GU GV GW 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0 1 1 63 13 0 1 1 63 13 0 1 1 63 13 0 0 0 40.5 0 0 0 0 40.5 0 1 0 1 63 13 0 1 1 63 13 0 0 0 40.5 0 0 1 1 63 13 GX GY GZ HA HB HC HD HE HF 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 1 2 98 48 0 1 1 63 13 0 1 1 63 13 0 0 0 40.5 0 0 1 1 63 13 0 1 1 63 13 0 1 1 63 13 1 0 1 63 13 1 1 2 98 48 HG HH HI HJ HK HL HM HN HO 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 0 1 63 13 0 0 0 40.5 0 0 0 0 40.5 0 0 1 1 63 13 0 0 0 40.5 0 0 0 0 40.5 0 1 0 1 63 13 1 1 2 98 48 0 0 0 40.5 0 HP HQ HR HS HT HU HV HW HX 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0 0 0 40.5 0 0 0 0 40.5 0 0 0 0 40.5 0 0 1 1 63 13 0 1 1 63 13 1 0 1 63 13 0 0 0 40.5 0 0 1 1 63 13 0 1 1 63 13 HY HZ IA IB IC ID IE IF IG 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0 0 0 40.5 0 0 0 0 40.5 0 0 0 0 40.5 0 0 1 1 63 13 0 0 0 40.5 0 0 0 0 40.5 0 0 0 0 40.5 0 1 0 1 63 13 0 0 0 40.5 0 IH II IJ IK IL IM IN IO IP 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 0 1 63 13 1 0 1 63 13 0 0 0 40.5 0 0 0 0 40.5 0 0 0 0 40.5 0 0 1 1 63 13 0 0 0 40.5 0 1 1 2 98 48 0 0 0 40.5 0 IQ IR IS IT IU IV 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0 0 0 40.5 0 1 1 2 98 48 1 0 1 63 13 1 0 1 63 13 1 1 2 98 48 0 0 0 40.5 0 A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 S0 , current stock price X, exercise price r, interest rate T, time µ, mean stock return σ, sigma--standard deviation of stock return n, divisions of unit time Runs VanillaCall BS call B 50 50 10% 0.8 33% 30% 200 3,000 ### ### C MONTE CARLO PRICING OF PLAIN VANILLA CALLS #MACRO? #MACRO? A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 S0 , current stock price X, exercise price r, interest rate T, time µ, mean stock return σ, sigma--standard deviation of stock return n, divisions Runs VanillaCall BS call #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #DIV/0! B 50 50 10% 0.8 33% 30% 100 3000 #MACRO? #MACRO? C D RUNNING THE MONTE CARLO FUNCTION MANY TIMES #MACRO? #MACRO? Multiple runs of the function #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? #MACRO? E 1 FUNCTION 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 MANY TIMES #MACRO? #MACRO? #MACRO? A B 1 MONTE CARLO PRICING OF PLAIN VANILLA CALLS BetterVanillaCall: A somewhat more efficient function: If, after j random numbers which produce k Up moves, S0*Up(k+n-j)*Down(j-k)<X, then we abort the random price path and let the call value = 0 S0 , current stock price X, exercise price r, interest rate T, time µ, mean stock return σ, sigma--standard deviation of stock return n, divisions Runs VanillaCall BS call 50 45 6% 0.8 12% 30% 100 2,000 ### ### 2 3 4 5 6 7 8 9 10 11 12 13 14 C O PRICING OF PLAIN VANILLA CALLS ewhat more efficient function: If, after j random 1 k Up moves, S0*Up(k+n-j)*Down(j-k)<X, then we abort price path and let the call value = 0 2 3 4 5 6 7 8 9 10 11 12 13 14 #MACRO? #MACRO? ...
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This note was uploaded on 01/23/2011 for the course FGB 780 taught by Professor Edwardchang during the Spring '09 term at Missouri State University-Springfield.

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