simple montecarlo
TRANSCRIPT
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Risk256.comQuantitative Financial Risk Management
Introduction
To construct our Monte Carlo simulation, we need to use Excels !ataTa"le #eatureta"s %rovide a "rie# introduction to !ataTa"les. 'ou are alread' #amiliar wit$ !at$e Monte Carlo ta"s.
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. T$e next twoaTa"les ski% to
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Risk256.comQuantitative Financial Risk Management
Model !ataTa"le
( (2 ) )
*
2
+
)
5
6
-
*/
**
*2
*+
*)
*5
*6
(2
(2
0um"ers #romt$e 1rst columno# t$e !ataTa"leget su"stitutedinto t$is cell.
T$e result #romt$is cell ends u%in t$e secondcolumn o# t$e!ataTa"le.
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!ata Ta"le
DataTable Introduction
!ataTa"les are not ver' intuitive, "ut t$e' can "e extremel'automaticall' see $ow c$anging t$e in%uts to an e3uation orout%ut. &n t$is exam%le we $ave an extremel' sim%le e3uati
T$e 1rst column o# t$e !ataTa"le contains t$e in%uts. T$esesu"stituted into cell 4. T$ese can "e an't$ing t$at 'ou wanintegers **6.
e set t$e cell at t$e to% o# t$e second column e3ual to t$eadd additional columns wit$ additional out%uts, "ut, #or now,
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use#ul. !ataTa"le allows 'ou toseries o# e3uations will im%act t$en, cell C is sim%l' cell 4 s3uared.
are t$e values t$at will "e. &n t$is exam%le we $ave t$e
ut%ut #rom our e3uation. e couldwe limit ourselves to one out%ut.
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Risk256.comQuantitative Financial Risk Management
( (2 ) )
*
2
+
)
5
6
-
*/
**
*2
*+
*)
*5
*6
(2
(2
0um"ers #romt$e 1rst columno# t$e !ataTa"leget su"stitutedinto t$is cell.
T$e result #romt$is cell ends u%in t$e secondcolumn o# t$e!ataTa"le.
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!ata Ta"le
DataTable Introduction
0ext we insert our !ataTa"le. First select t$e !ataTa"le rangeasiest wa' to insert t$e !ataTa"le is to go to t$e !ata ta",
Tools "ox, and select !ataTa"le. n e3uation "ox wit$ 9Row is$ould a%%ear. 4ecause t$e in%uts and out%uts o# our !ataTaenter our in%ut cell :4: into t$e 9Column in%ut cell79 "ox ;'cell?, and 'ou are done. Excel s$ould t$en 1lls outwit$ t$e value we would $ave o"tained in cell C i# t$e value
T$is exam%le is a "it contrived. 8iven $ow sim%le our e3uatito enter t$e e3uation into t$e second column o# t$e ta"le dirco%' it down
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, cells F7 82+. &n Excel 2//, t$elick $at nal'sis in t$e !ata%ut cell79 and 9Column in%ut cell79
"le are arranged in columns, weu can t'%e 4, or sim%l' click on t$et$e second column o# t$e !ataTa"lein t$e 1rst column $ad "een in 4.
n is, it would $ave "een easier @ustctl' ;enter it into t$e 1rst cell andwe are dealing wit$ com%lex series
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Risk256.comQuantitative Financial Risk Management
AB/,* 0;/,*< 0;/,*</.2525* /.6665-) /.6665-
* /.6665-
2 /.6665-
!ataTa"le dumm' + /.6665-
) /.6665-
5 /.6665-
6 /.6665-
/.6665-
- /.6665-
/.6665-
*/ /.6665-
** /.6665-
*2 /.6665-
*+ /.6665-
*) /.6665-
*5 /.6665-
*6 /.6665-
mean /.6665-
standard deviation /./////
!ata
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Ta"le
Very Simple Monte Carlo
&n t$is exam%le, we created a standard normal random varia"le in cell C!ataTa"le to collect *6 draws #rom t$e random varia"le. e $ave calculstandard deviation o# t$is out%ut. Ditting t$e F ke' will recalculate all
Dow did we do t$is T$e 1rst row o# t$e !ataTa"le contains t$e integerst$ese num"ers would "e t$e in%uts #or our !ataTa"le, "ut, in t$is setu%,matter. e could @ust as easil' $ave le#t t$em "lank. T$e to%most cell o8, %oints to our standard normal random varia"le in cell C.
$en we insert t$e !ataTa"le, we select cells F782+, and set t$e 9Colu"lank cell ;in t$is case, cell !*/
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. e t$en use t$eated t$e mean and6 values.
* t$roug$ *6. 0ormall't$eir values dont reall't$e second column, cell
mn in%ut cell79 to an't$e values #rom t$e 1rstvaria"le is com%letel'
&ts as i# we $ad %us$ed
wit$out a !ataTa"le. T$e# t$is a%%roac$.
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Risk256.comQuantitative Financial Risk Management
&n%uts >ut%ut ;one trial<
!iusion Mean /./5G total log return=td !ev *.//G total return
Hum% IBHum% /.25G
Mean 2.//G
=td !ev +.//G
>ne Trial
t AB/,* AB/,* AB/,* !iusion Hum% Hum%
/
* /.+5) /.*6-6 /.)+* /.2G ).--G /
2 /.+6* /.*5/ /.6/ /.+/G ).-/G /
+ /.6))6 /.++* /./-55 /.)2G +.+/G /
) /.-2++ /.2/ /.)26/ /.-G /.2)G /
5 /.*62 /.))+ /.+ /.+G 2.)/G /
6 /.*-* /.5+6- /./5 /.6+G *.2G /
/./*+5 /.52+ /.6*- 2.*6G +.//G /
- /.*5+2 /.5-) /./-+ /.G *.+)G /
/.2)6 /.2-6* /.-/- /.6+G +.6G /
*/ /.-22 /./225 /.52) /.G -./*G /
** /.5/-) /.2*) /.2/2 /./G +.-+G /
*2 /.+52 /.*/ /.252* /.++G ).62G /
*+ /.**2 /./56 /.5+ *.)/G 6.+G /
*) /./5+ /.*-/ /.+62/ *.26G /.2G /
*5 /.* /.*+ /.-*)/ /.6+G +./G /
*6 /./-) /.)*5 /.5)6/ *.+-G 2./6G /
* /.*)5/ /.*26 /./*2 *./*G 5.)+G /
*- /.65 /.55+5 /.2+/ *.-G *.6/G /
* /.)/+) /./+56 /.+-6 /.*G .)*G /
2/ /.2// /.-+2 /.6*5 /.6G *.)2G /
2* /.-- /.-)*/ /.-// *.22G *.//G /
22 /.2// /.2)2 /.)+5 *.)5G 2.+/G /
2+ /.-5 /.2/2 /.+2-+ /.-2G ).)G /2) /.2662 /.2- /./52 /.5G +.6G /
25 /.-6*/ /.2/65 /.+/ *.*+G ).)6G /
26 /.252 /.-)2 /.5-26 /.)G *.))G /
2 /.+/) /.*+- /.-/* /.)6G 2./G /
2- /./6* /.+6- /.56 /.5G +.//G /
2 /.)2+ /.+-2 /.++) *.62G 2.62G /
+/ /.+26 /.++5) /.6+ /.)/G +.2G /
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+* /.6) /.)2- /./)- /.52G 2.5)G /
+2 /.5/+ /.-5 /.+/ /./G ).2G /
++ /.-52 /.-6-5 /.-5*5 *.*/G *.+6G /
+) /.+655 /.-/5 /.)*) /.2G *.6G /
+5 /.5- /.+222 /.5262 /.25G +.+-G /
+6 /.++- /.*6)5 /.--2* /.+G ).+G /
+ /.55+ /.)2 /.555) /.*G 2.*6G /+- /.2)22 /.)5*) /.+62 /.65G 2.+G /
+ /.5*/ /.+6/5 /.56/+ /./-G +./G /
)/ /.++6 /.+// /.2/5 /.6G 2.-)G /
)* /./5-* /.52- /.-6 *.52G *.-G /
)2 /.)//* /.2)* /.)62 /.2/G /.2*G /
)+ /.))25 /.-65 /.2)++ /./G ).6)G /
)) /.+++) /.+)6 /.*- /.+-G +.*6G /
)5 /.5-6/ /.522 /.6-/) /.2G *.)5G /
)6 /.2)) /.6+/ /.6/5 /.6+G +.+6G /
) /.52/5 /.-6) /.+6 /.*/G *.+G /)- /.2+- /.*-* /.*/)/ /.6)G /.2G /
) /.+5-/ /./5) /.522 /.+*G 6.6-G /
5/ /.-5* /.+--2 /.2-6 *./G 2.-5G /
5* /.+2/ /./5) /.56)* /.)/G 6.-/G /
52 /.66+ /.*+6 /./+5 /.5*G 5.2G /
5+ /.5*+ /./6/) /.5+ /.2-G 6.65G /
5) /.)+) /.522 /.**- /./+G *.-+G /
55 /.// /./5)5 /.2*26 /.5-G 6.-*G /
56 /.2/ /.+/66 /.)22 /.6+G +.52G /
5 /.2++2 /.-+2+ /./-*) /.6-G /.-G /
5- /.** /.+6/ /.+*52 /./G +.//G /
5 /.*)* /./5/- /.+5+2 /.62G 6.*G /
6/ /.)55 /./2+6 /.))25 /./6G .5G /
6* /.*/- /.++6 /.*26+ *.*G 2.-*G /
62 /.*6/2 /.55* /./6+* /.)G *.5G /
6+ /.)2/) /.)/*5 /.)/26 /.*5G 2.5G /
6) /.665) /.565 /.2/ /.)-G /./G /
65 /.6)+* /./+/ /.*-+- /.)2G 5.G /
66 /.)/ /.-* /.5/ /./2G /.)G /
6 /.+62/ /.656 /.-+)5 /.+/G /.G /
6- /.-*2 /.*/)+ /.)+ *.*-G 5.G /
6 /.*/ /.2)55 /.5 /.6+G )./G /
/ /.**55 /.)* /.-/5 *.*5G 2.*G /
* /.//2) /.2+*5 /.5** 2.G ).2/G /
2 /.65+ /.5*5) /.+-2 *.-G *.--G /
+ /.**-5 /.-/25 /.6+- *.*+G /.55G /
) /.*//+ /.))/5 /.56* *.2+G 2.)5G /
5 /.-))* /.566* /.6*+ *./6G *.5/G /
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6 /.++26 /.-)- /.*)*/ /.+-G *./-G /
/.626 /.)/26 /.5+ /.6G 2.)G /
- /.+*6/ /.+-* /.+-66 /.)+G 2.+G /
/.)56 /.2+ /.+6) *.65G +.5-G /
-/ /.6262 /.**** /./)6- /.+G 5.66G /
-* /./56* /.6--* /.6**- *.5)G /.5+G /
-2 /.-/)* /.6/6 /.+*/ /.*G +.2G /-+ /.)/22 /.*/)6 /./25+ /.2/G 5.G /
-) /.))* /.)/ /.22 /.*/G /.)6G /
-5 /.))2) /.+2 /.-56 /./G /./-G /
-6 /.*5++ /.*/+ /.2+*/ /.G ).-6G /
- /.+--- /.*++ /.+)) /.2+G 5.+2G /
-- /./)+ /.-+- /.2 *.62G /.+6G /
- /.-/2- /.-)-) /.622 /./G *./G /
/ /.6-56 /.)56 /./*22 /.5+G 2.++G /
* /.)/++ /.)*6 /./22 /.*G /./5G /
2 /.26 /.)2 /./+* /.65G 2.*G /+ /.)6* /.522 /.-/6- *.66G *.-+G /
) /.6*)2 /.**2/ /.//2- /.+)G 5.65G /
5 /.6)** /./+ /.6/* /.)*G /.+5G /
6 /./2 /.66+ /.)+5+ *.2G /.)6G /
/.-56 /.+65 /.6*+ *.+*G +./2G /
- /.)/ /./+- /.6)26 /./5G .++G /
/.)+/6 /.25** /.25)+ /.*2G )./*G /
*// /.)*6 /.)--) /.55) /./+G 2./G /
*/* /.+5/) /./2++ /.256- /.++G .G /
*/2 /./+ /.2/ /.*/- *.2G +.65G /
*/+ /.226 /.6+* /./**/ /./G /.*5G /
*/) /.+-2+ /.-*)/ /.)+/ /.25G /.6-G /
*/5 /.56 /.)-)/ /.522 *.G 2.*2G /
*/6 /.*+*+ /.6)/+ /.2) *./G /.2G /
*/ /.-) /.//) /.-6-* /.-G 6.)2G /
*/- /.6+-+ /.5*/ /.*++ /.)/G 2.6G /
*/ /.5*6* /.6+6* /.6+) /./G /.6G /
**/ /.252 /.2-* /.*) /.55G +.6G /
*** /.6)+ /.5/ /.2)5+ *.-5G /./G /
**2 /.556 /.*/ /.+ /.2)G /.2+G /
**+ /.2 /.6* /./)2 *.5*G /.2*G /
**) /.+6+ /.//5 /.65 /.+/G ./)G /
**5 /.56 /.*5/ /.-2/ /.5G /.+/G /
**6 /.)6- /.526) /.5-6 /./*G *.-/G /
** /.62-* /./- /.22- /.+-G /.22G /
**- /.5* /.-6) /.-)+5 /.25G *.-G /
** /.6*/ /.656 /.-525 /.++G +.)6G /
*2/ /.*)-) /.**)2 /.5- /.G 5.6*G /
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*2* /.66 /.6556 /.5/* /.)G /.-/G /
*22 /.-22- /.)*2 /.)52 /.-G 2./G /
*2+ /.2+/ /.5/- /.)+-* /.6G *.+G /
*2) /.)+62 /.)22 /.266) /.**G 2.5-G /
*25 /.-/-* /.+2 /.-/+ /.2G 2.-G /
*26 /.*)/ /.62+/ /.555) *.)2G *./6G /
*2 /.66)6 /.-/+ /.-2*+ /.)G /.5G /*2- /.6-/ /.-) /.*/5* /.5)G /.+/G /
*2 /.25+ /./-* /.2+/- *.)G 6.*-G /
*+/ /.--6* /.26+ /.6--) *.26G +.-6G /
*+* /.2)-- /.5*)+ /./-+ /.6+G *.-G /
*+2 /./** /.+5/ /.-* *.)2G 2.-/G /
*++ /.++2 /.)52- /.**5 /.+6G 2.+6G /
*+) /.+5) /.+// /.6)22 *.5G +.5G /
*+5 /.62 /.2* /.266 /.55G +.)G /
*+6 /.26- /.*/ /.+-22 /.5G /.++G /
*+ /.-/-) /.22 /.+*2* /.2G /.))G /*+- /.55 /.*+- /.2+- /.+/G 2./G /
*+ /.*/6 /./6+ /.*/2/ *.+G 6.5G /
*)/ /./* /.5*6 /.2-6 *.+G *.)6G /
*)* /.2)2 /.*665 /.)*) /.65G )./G /
*)2 /.52// /.22+) /.*5/5 /.*/G ).2-G /
*)+ /.6/ /.5-5) /.+*/ *.2G *.+5G /
*)) /.6)5) /./52 /.52* /.)2G 6.-6G /
*)5 /.)/2 /.)6*6 /.6*) /.*-G 2.2G /
*)6 /.+/6) /.+*-+ /.*+ /.)6G +.)2G /
*) /.+-*5 /.25+ /.-/ /.25G +.G /
*)- /.)/ /.-*2+ /./) /./2G /.66G /
*) /.6+2 /.-+5 /.2+ *.-)G *.)G /
*5/ /.6-- /.-* /./55/ /.5*G /.5G /
*5* /./66+ /.))6* /.5*2 *.)5G 2.)*G /
*52 /.)2/ /.5-- /.65 /./+G *.25G /
*5+ /.)6/) /.++ /.+-+ /./5G 2.65G /
*5) /.)56* /.)+5* /.))/ /./6G 2.)G /
*55 /.- /.-55 /.--6 /.-G ).55G /
*56 /.)25 /.**2+ /.62/+ /./2G 5.6)G /
*5 /.5/ /.)5+* /./+ /.+G 2.+5G /
*5- /./5- /./5* /./22 *.52G 6.)G /
*5 /.-/+ /.25+ /.-55 *.*-G +.G /
*6/ /.65 /.)/6* /.--+ /.5*G 2.*G /
*6* /.-6+ /.26/) /.226+ *.*G +.+G /
*62 /.)66 /.)5 /.-26 /./+G 2.*G /
*6+ /.22) /./* /.)/* /.6)G /.)2G /
*6) /.255* /./+// /.6/* /.6*G .6)G /
*65 /.)) /.++- /.)626 /.-G +.25G /
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*66 /.5-26 /.*6 /.*/2 /.26G ).5G /
*6 /.66/ /./+* /.*+ /.-G .+6G /
*6- /.**-5 /.)6 /.)5-2 *.*+G 2.2)G /
*6 /.2)+ /.55/+ /.2*+ /.65G *.62G /
*/ /.*+ /.*) /./5 *./)G 2.2/G /
** /.)+/2 /.5+-* /.// /.*+G *.*G /
*2 /.//)* /.- /.**/) 2.5G /.+*G /*+ /./-)* /./- /.6)-- *.++G /.+5G /
*) /.)/ /./522 /.+*- /./+G 6.-G /
*5 /.26*2 /.*+- /.**6 /.5G ).5G /
*6 /.*)-) /.-5+5 /.++5 /.G *.*6G /
* /.-5-- /.*-- /.*++ *.*+G ).6)G /
*- /.+)6+ /.2)/+ /.26/ /.+5G ).*2G /
* /.)- /.-5 /.-6 /.2G *.)6G /
*-/ /.5)6+ /.5- /.-65 /.*G *.++G /
*-* /./*2 /.*)) /.)/- 2.*G 5.*-G /
*-2 /.-** /.2+) /.)+2 /.+G ).*G /*-+ /.-)/ /.2+-- /./52 *.+/G ).*+G /
*-) /.565 /.65/* /.26 /.2)G /.-)G /
*-5 /.+- /./5-/ /./26 /.2+G 6.*G /
*-6 /.)+ /.)2* /.*+ /.**G 2.2G /
*- /./-55 /.2/*5 /.5/2 *.+2G ).5*G /
*-- /.*26+ /.-+*6 /.-+*6 *./G /.--G /
*- /.6/5+ /./2 /.-++- /.+2G /.+5G /
*/ /.*-+) /.+)* /.)55 /.-5G 2.6G /
** /.)- /.*6-/ /.*22 /.//G ).-G /
*2 /.225* /.+2)2 /.//22 /.*G +.+G *
*+ /.**- /.--/- /./655 /./G *.5)G /
*) /.**)5 /.5))) /.)5- *.*5G *.6G /
*5 /./6+ /.))* /.*)-- *.+-G 2.+-G /
*6 /.*)*6 /.--)- /.6- *./2G *.6/G /
* /./*- /.5)6/ /.5)+6 2./+G *.65G /
*- /.5-)2 /.--+/ /.*// /.26G *.5G /
* /.*2+2 /.-)*) /.+5 *.**G *.//G /
2// /.62*5 /./55 /./22 /.+6G 6.6-G /
2/* /./) /.5**5 /.+*/ *.62G *.*G /
2/2 /.)+*- /.2252 /./6)/ /.*2G ).26G /
2/+ /.*)// /.)25 /.-- *./+G 2.5+G /
2/) /.*+5+ /.+2 /.++)/ *./5G 2.G /
2/5 /.26*- /.*-26 /.+* /.5G ).2G /
2/6 /.--+5 /.*22 /.)*- *.2)G 5.)2G /
2/ /.)+* /.)6 /.+-- /./+G 5.6)G /
2/- /.56/* /.5+ /.22 /.2/G /./6G /
2/ /.)*+ /.)*5/ /.556 /./2G 2.6)G /
2*/ /.-55 /.+*+5 /.-+)6 *.2/G +.)6G /
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2** /.-+2- /.5- /.2)5/ *./2G *.)/G /
2*2 /.6)2 /.++/6 /.)+ /.G +.+*G /
2*+ /.+2/ /.+6*- /.+2 /.22G +./6G /
2*) /.6+52 /.2-+ /.))-2 /.)/G +.6G /
2*5 /./+56 /.5* /.6-+ *.5G *.)6G /
2*6 /.66 /.)2 /.65/ /.56G 2.+G /
2* /.2-)+ /.2) /.-/ /.52G )./2G /2*- /.*+5 /.+-+) /.+/6) *./)G 2.-G /
2* /.2/-* /.6)) /.55)2 /.6G +.)*G /
22/ /.))* /.-- /.5*-- /.*G *.5*G /
22* /.+6*6 /./-6 /.5--) /.+/G 2.//G /
222 /.55+6 /.)5* /.-) /.*-G 2.+6G /
22+ /.+)+ /.+- /.)6 /.22G 2.-)G /
22) /./-5 /.-*-6 /.5-)6 *.+*G /.+G /
225 /.5/-) /.+-*) /.52) /./G 2.*G /
226 /.-5/6 /./-2 /.6-- *./G 6.25G /
22 /.226/ /.2/- /.5-*+ /./G ).)2G /22- /.62* /.)/- /.5)6+ /.6G 2./G /
22 /.-52 /.5/2 /.*52 *.*+G *.-G /
2+/ /.+/// /.-/62 /.)) /.)G /.5G /
2+* /./*) /.22 /./++ /.5-G /.2+G /
2+2 /.2*5 /.)/) /./-5 /.+G 2.6G /
2++ /.6* /.++6+ /.- /.-G +.2G /
2+) /.**2 /.- /.5* /.6*G ).*5G /
2+5 /.62) /.-/2 /.56*5 /.5/G ).*G /
2+6 /.56+ /.*5/ /.*5* /.2*G ).5-G /
2+ /./5/ /.566 /.225 *.5+G *.)2G /
2+- /.)+ /.6*/ /.655 *.6G *.*6G /
2+ /.+)/- /.*2)/ /.2+2 /.+6G 5.)G /
2)/ /. /.*6 /.22- 2.*G 2.*5G /
2)* /.622 /.-6/- /.)55- /.+6G *.25G /
2)2 /.*-6 /.*-55 /.** /.-)G ).6-G /
2)+ /.)5** /.)2) /.+62 /./G 2.2G /
2)) /.+5- /.*55 /.++- /.6-G 5./)G /
2)5 /.26*5 /.2+) /.56+6 /.5G ).*2G /
2)6 /.)/ /.5+56 /.5)6/ /./)G *.+G /
2) /.5-- /.-252 /.*/-+ /.2-G /.-*G /
2)- /.++2 /.*+)5 /.)*)* /.+G 5.+2G /
2) /.555 /.-*+* /.6+/* /.2G /.6G /
25/ /.+/- /.+*+/ /.65) /.)5G +.)6G /
25* /./-* /.)--5 /.-62- *.+G 2./G /
252 /.**5 /.*+2* /.-//- *.*+G 5.+5G /
25+ /.5+2 /.)2 /./26+ /.+G 2.2*G /
25) /.2*) /.*62 /.)*-6 /.56G )./G /
255 /.+2 /.26 /.2226 /.2*G +.6/G /
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256 /.6+25 /./+/ /./6/- /.+G 6.+6G /
25 /.6-5 /.5)5 /.6-5/ /.5G /./G /
25- /./52) /.5+) /.+2+ *.5G *.)G /
25 /.-6 /.+)/+ /.2 *.2*G +.2+G /
26/ /.*5)5 /.)6 /.+2 /.G 2.//G /
26* /.+5-6 /./+-/ /.6/)) /.+*G .+2G /
262 /.25 /.-/2+ /.-)-- /.)-G /.55G /26+ /.*+26 /./++ /./+6 *./6G 6.+5G /
26) /.6*/2 /.+)5* /.265* /.++G +.2/G /
265 /./6-) /.))-) /.+5/) *.))G 2.+G /
266 /.-6+* /.65 /.5/ *.*)G /.6+G /
26 /.+/66 /.- /.2/) /.)6G .*G /
26- /.6*+/ /./2/ /.6*5 /.+)G -.**G /
26 /.*+-+ /.255* /.-+*6 *./)G +.-G /
2/ /./5-+ /.)622 /.)-* *.52G 2.2-G /
2* /.*))2 /.6-*/ /.-*5- *./*G /.5G /
22 /.) /./-2 /.-25 /.2G 5.--G /2+ /.-6- /.-*6 /.-/-5 /.-5G /.2G /
2) /.-2 /./+22 /.)6*- 2.*6G .55G /
25 /.6-/ /.252 /.)/65 /.5G +.6G /
26 /.)+) /.))6 /.6* /.**G 2.)/G /
2 /.-625 /.6-) /.6)- *.*)G /.))G /
2- /./*/ /.2/- /.)25* *.)2G ).)+G /
2 /.65) /.6*+ /.-/) /.)6G *.*+G /
2-/ /.52/ /./-+/ /./2)2 *.*G 6.*6G /
2-* /.)-- /.2*6 /.5** /./2G ).+5G /
2-2 /./)+/ /.-/ /.*)5/ *.6G ).-G /
2-+ /.6*55 /.5) /.6/-/ /.+)G *.6+G /
2-) /.)+)- /.65-* /.56 /.**G /.-G /
2-5 /./2 /.*/5/ /.*-)/ *.+G 5.6G /
2-6 /.6-2) /.2/6 /.562 /.52G 2.2+G /
2- /.-)+* /.*/* /.6)/ *./6G ).6+G /
2-- /.*6+ /.+-5 /.-/ /.+G 2.-G /
2- /.*)) /.-* /.--* *./*G /.+/G /
2/ /.-22 /.25- /.5)/ /.-G +.6*G /
2* /.)/- /.//* /.+5+ /.*-G .+5G /
22 /.5- /.+) /.-2 /.+/G /.*2G /
2+ /.-*2 /.+5)* /.62*- *.*-G +.*2G /
2) /.)5/ /.6+2 /.22/2 /./5G /.)-G /
25 /.2-5/ /.*- /.)-** /.52G ).5+G /
26 /./5 /.6*6) /.25// /.-6G *.**G /
2 /.5*6 /.))+) /.+/2 *.*G 2.)+G /
2- /.++ /./6+ /.*/) /.2*G 6.2G /
2 /.)*2 /.**-2 /.-+) /.*G 5.55G /
+// /.)/2/ /.2- /.6)- /.2/G 2.)/G /
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8/19/2019 Simple Montecarlo
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+/* /.*2/ /.--) /./+/ *./-G ).-*G /
+/2 /.22) /.6--2 /.*- /.5/G /.5+G /
+/+ /.--/ /.++ /.)2 *.2+G 2.-*G /
+/) /.)55 /.25-* /.5/ /.*G +.5G /
+/5 /.*/*) /.-+5+ /.*/ *.22G /.+G /
+/6 /.--5 /.++6 /.5+5 *.26G /.*+G /
+/ /.526 /.6-/* /./6+- /.*2G /.6/G /+/- /.-+/5 /.2)26 /.*-56 *./*G )./G /
+/ /./)*+ /.555/ /.2/*2 *.6G *.5-G /
+*/ /.-) /./+ /.+-/- /.-2G 6.2+G /
+** /.22- /.-*2- /.)55 /.6G /.66G /
+*2 /./* /./))+ /.66- *.+G .**G /
+*+ /.//5- /./625 /.*-/6 2.)G 6.6/G /
+*) /.5)6) /./62 /.)5** /.*G /.+G /
+*5 /.-*5- /.+/ /./6/ /.5G 2.-+G /
+*6 /.6*)+ /.-/ /.*-6) /.+)G ).22G /
+* /./62- /.+-/ /.2-/ *.)-G 2.-6G /+*- /.-66 /.2)2 /.2+)- 2.26G )./+G /
+* /.*+* /.* /.5*- /.-G ).G /
+2/ /.662 /.5655 /.-*6 /.)-G *.5*G /
+2* /.5-5/ /.5) /.*6* /.26G +./-G /
+22 /.)-- /.-) /.-/66 /.//G *.-/G /
+2+ /.)6 /./*26 /./2* *.6G -.*G /
+2) /.-52- /.*/5 /.-55 *.*/G 2./+G /
+25 /.-*+ /.+6*5 /.2+** *.2-G +./6G /
+26 /.6-*5 /.++5* /.+)* /.52G +.2-G /
+2 /.*/2 /./+ /.// *.*G .++G /
+2- /.*/- /.*/ /.5/2) *.*-G 2.*6G /
+2 /.2)/- /.+6 /.*525 /.65G 5.)G /
++/ /.+26+ /.*56 /.6625 /.)/G 2.*+G /
++* /.)*66 /.6)) /.6+52 /.*6G /.-G /
++2 /.26/- /.6 /./-+ /.5G /.)+G /
+++ /.5// /.62+) /.65/) /./5G *./6G /
++) /.)-)6 /.25 /.5-5+ /./*G 5.2G /
++5 /.6-* /.2+* /.6+ /.5)G ).*+G /
++6 /.)2) /.-)/+ /.2*)- /.*+G /.G /
++ /.-6) /.5-/ /.*/ 2.26G *.+)G /
++- /.//6 /./-*6 /.)55 *.+)G 6.*-G /
++ /.//-) /.--6 /.*)* 2.+)G *.6+G /
+)/ /.*5) /.52)+ /.26/ /.--G *.-2G /
+)* /./// /./6+ /./+5 /.5G /.+G /
+)2 /.62/ /.))5 /.-)** /.+6G 2.)*G /
+)+ /.5/62 /.56)/ *.//// /./G *.52G /
+)) /.)*/) /.*55- /.*2 /.*-G 5./)G /
+)5 /.)5 /.+-+ /.*5* /.*G 2.-)G /
-
8/19/2019 Simple Montecarlo
21/60
+)6 /.2) /.2)/5 /.+6)+ /.)G ).**G /
+) /.6) /.-+25 /.)2+6 2.)G /.-G /
+)- /.5+/- /.+6 /.**+ /.*+G 2.6G /
+) /.-6// /.6* /.*6* *.*+G /.)+G /
+5/ /.2- /.//// /.)65 /.-/G *6.-6G /
+5* /./2)- /./5 /.))* *.*G 6.2+G /
+52 /.2)/) /.-6*+ /.2) /.65G *.26G /+5+ /./+5) /./25 /.)2 *.6G .-)G /
+5) /.-* /.* /./) *.2G /.2G /
+55 /.-*-5 /.+5 /.+*/ /.6G 2.6/G /
+56 /.//56 /.+56- /./5 2.)G +.*/G /
+5 /.+/6 /.6//- /.**5* /.)5G *.2+G /
+5- /.2)+ /.-5* /.666 /.6+G /.+G /
+5 /.++65 /./)/ /.56** /.+G .22G /
+6/ /.++6/ /.)- /./6+ /.+G /.)G /
+6* /.+)5 /.5 /.)2- /.22G +.+G /
+62 /./-22 /.5655 /.62*5 *.+)G *.5*G /+6+ /.+2/* /.)/ /.2/*- /.)2G /.//G /
+6) /.2++- /.+ /.+22/ /.6-G 2.2G /
+65 /.+62 /.5/2 /.-*5 /.2*G *.5G /
+66 /.565* /.22/- /.*/-- /.2*G ).+*G /
+6 /.**)- /.-6 /.5/-2 *.*5G 6.G /
+6- /.-6* /.- /.226 *.2*G *.52G /
+6 /.2)- /.52 /./2- /.55G +./2G /
+/ /.//)2 /.-2) /.-2/ 2.5G *.)*G /
+* /.2 /.+)) /.-+/- 2./5G 2.5+G /
+2 /./*2 /./)2 /.2-5 *.2-G .*5G /
++ /.566 /.++ /.)56 /.2+G +.-/G /
+) /.5-*6 /.) /.-- /.26G 6.+G /
+5 /.2++ /.5655 /.222* *.)-G *.5/G /
+6 /.5)/ /.6/) /.)+* *.+G /.6-G /
+ /.*6- /.6/2/ /.)65/ /.*G *.22G /
+- /.*-55 /./*-5 /.6+)5 /.-)G -.26G /
+ /.-+-/ /.5-*2 /.-*+ *./)G *.+G /
+-/ /.**- /.6)* /.+/6 *.*)G +.)/G /
+-* /.-6 /.-- /.6-5) /.-2G )./G /
+-2 /./*52 /.665+ /.6*)2 2.**G /.2G /
+-+ /.-56 /.*65 /.5+5 *.2/G ).56G /
+-) /./-) /./5- /.+*+2 /.6/G 6.+/G /
+-5 /./2 /.-/ /.565 *.+5G /.+/G /
+-6 /.2-* /.*2) /.+++) /.)-G ).-+G /
+- /.*-+ /.2++2 /./*5 /.-/G ).*-G /
+-- /.)*/) /.+**5 /.-/6 /.*-G +.)G /
+- /.2) /./2/2 /.-** *.G -.*5G /
+/ /.-/ /./65 /.256- *.2-G 5.*G /
-
8/19/2019 Simple Montecarlo
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+* /.-/5* /.+//+ /.*+6- /.*G +.5G /
+2 /./-2+ /./6/ /.+--+ *.+)G 5.*G /
++ /.*62 /.2/5 /.)/25 /.+G ).)6G /
+) /.-+5 /.*+5 /.-2 /.-+G 5.+/G /
+5 /.*2- /./*) /.2-)6 /.-G -.++G /
+6 /.6- /.//2 /.+*- /.52G */.+)G /
+ /.)*6 /.+5- /.+26 /.*5G +./-G /+- /.6-** /.2/2 /.2-22 /.52G +.66G /
+ /.22* /.+)* /.+*- /.56G +.22G /
)// /.66*6 /.*2** /.2*6 /.)G 5.5*G /
)/* /.)* /.-66 /.+5 /./5G *.+)G /
)/2 /./5+- /.5* /.5+/ *.56G *.)2G /
)/+ /.)* /.6*6 /.--/ *.6G *.*/G /
)/) /./*++ /.*//5 /.-5)5 2.*G 5.-)G /
)/5 /.2-26 /.6* /.-/) /.5+G /.5/G /
)/6 /.)2/ /.6** /./-* /.*+G *.**G /
)/ /.*6 /.5*)/ /.-22/ /.G *.-G /)/- /.-* /.65- /.)) /.G /.G /
)/ /.*6+/ /./5- /.5/6- /.+G 6.6G /
)*/ /./-/ /.+** /.56/ *.+5G +.)+G /
)** /.)*6 /.6+-- /./-2/ /./2G /.+G /
)*2 /.56 /.)6 /.+*2/ /.5G /./*G /
)*+ /./2/+ /.-/ /./*/) 2.//G ).22G /
)*) /.*+/+ /.+6 /.6666 *./G /.*/G /
)*5 /.* /.552) /.525) /.-G *.6/G /
)*6 /.5*2 /.2)/6 /.6*- /.+G ).**G /
)* /.26) /.6--* /.2-/ /.)-G /.5+G /
)*- /./)+/ /.25- /.++ *.6G +.5G /
)* /.---+ /./* /.-// *.2G 2./*G /
)2/ /.-66* /.+//) /./2*+ *.*6G +.5G /
)2* /.6+-2 /.*6/- /.5/ /.)/G ).G /
)22 /./)-6 /.*+5+ /.*6+6 *.6*G 5.+*G /
)2+ /.)/+- /.552) /.*+2+ /.*G *.6/G /
)2) /.*)* /.**2- /.662* /.G 5.6+G /
)25 /.*//+ /.+2*2 /./2/5 *.2+G +.+G /
)26 /.*)/6 /.-)6 /.*--/ *./+G *.5G /
)2 /.+/- /.*5- /.--5) /.)5G 2.*+G /
)2- /.*5/ /.) /.-*5/ /.--G /.5/G /
)2 /.)5* /.2-** /.2)- /./G +.)G /
)+/ /.+/ /.**/ /.-652 /.2+G 5.5)G /
)+* /.66-+ /.+)) /.)5+2 /.)G 2.5+G /
)+2 /.+-2 /.)65 /.5 /.2*G 2.*-G /
)++ /.)*)- /.-/) /.255 /.*G /.6+G /
)+) /.-5* /.-** /.-66 2.22G /.65G /
)+5 /.6)) /.*)-6 /./22/ /.56G 5.*+G /
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)+6 /.6*/ /.//-* /.26 /.55G .2*G /
)+ /.+5)- /.+5) /.)26/ /.+2G +./-G /
)+- /.5+56 /./+5 /./) /.*)G 5.6G /
)+ /.6/*- /.*/2 /.22- /.+*G 2./+G /
))/ /.55 /.52* /.5 *.6G +.//G /
))* /.25/) /.-// /.-+/ /.62G /.5+G /
))2 /./+22 /.6)6 /.--- *.-/G /.-6G /))+ /.++2/ /.26) /.-)5 /.+-G +.-G /
))) /.)-+) /.2*-) /.+*6+ /./*G ).++G /
))5 /.266 /.-)22 /.65+ /.5G *./*G /
))6 /.5+/ /.5/5 /.6/* /.*+G *.)G /
)) /.+))5 /./)2 /.)+2+ /.+5G .*6G /
))- /.+6- /.2-+ /.5 /.2*G /.*-G /
)) /.2-66 /.+//) /.2*** /.5*G +.5G /
)5/ /.-/2) /.5*-) /.2*6 /./G *.-6G /
)5* /.*6 /.6*)) /.5555 /.*G *.*+G /
)52 /./555 /.*-26 /.6+-) *.5)G ).2G /)5+ /.+2-- /./5 /.+2)2 /.+G 5.2G /
)5) /.)+)5 /.5-2 /.+-2 /.**G *.+2G /
)55 /.*+52 /.2/*) /.+5-6 *./5G ).5*G /
)56 /.5// /.56*5 /.+55- /./5G *.5)G /
)5 /.+66* /.*6-/ /.655 /.2G ).-G /
)5- /.2/)/ /.-/-6 /.*6 /.-G /.62G /
)5 /.*56- /.+2+ /.6)5 /.6G +.++G /
)6/ /.)) /./5 /.-/+) /.*G 6.6G /
)6* /.*+/ /.--/- /.//*- /.-G *.5)G *
)62 /.)-2 /.*2* /.)) /./5G 5.5/G /
)6+ /.6*5 /.)/+- /.6*5 /.+5G 2.+G /
)6) /.)+/) /.6- /./+- /.*+G 6.*-G /
)65 /.*)-* /.2)-+ /.-/5 /.G )./)G /
)66 /.)62 /.2// /./++6 /./)G ).)+G /
)6 /.552 /.2*+ /.++2/ /.)G ).+-G /
)6- /.6/* /.*/+/ /.-5-+ /.6G 5.G /
)6 /.*66 /.+2 /.5*- /.2G +.+2G /
)/ /.+*56 /./+ /.+2*- /.)+G 2./*G /
)* /./ /.+**+ /.*+/6 /.6/G +.)-G /
)2 /.)/ /.5++ /.56- 2.56G *.)5G /
)+ /.5)+- /.2/+ /.-5/- /.*6G ).)+G /
)) /.2-6 /.6- /.6+) *.52G /.)G /
)5 /.6++ /.+)5- /.*5-2 /.+G +.*G /
)6 /.5-* /.-- /.6))+ /.25G ).-6G /
) /.6) /.++2- /.5) /.-*G +.+/G /
)- /./ /.)56 /.5222 /.G /./2G /
) /.-2 /.6*52 /.2 2.6G *.*2G /
)-/ /.*-- /.-)/5 /.+)5 /.-+G /.G /
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)-* /.266/ /.6555 /.5)2 /.5-G /.-/G /
)-2 /.*-/ /.)-) /./5- /.-)G 2.*2G /
)-+ /.26 /./62- /.6*-5 /.)-G 6.6/G /
)-) /.)+ /.5-66 /.+-5+ /.*/G *.+)G /
)-5 /.++26 /.)+2) /.6/ /.+-G 2.5*G /
)-6 /.)) /.)2) /./6 /./*G 2.5+G /
)- /.++5+ /.-26* /.*5/ /.+-G /.-2G /)-- /.65 /.*) /.5/- *.-G /.2G /
)- /.*)*6 /.-*/ /.52 *./2G *./G /
)/ /.5/2 /.//52 /.62+ /./6G .6G /
)* /.**)) /.5*/- /.)5+* *.*5G *.2G /
)2 /.22* /.++5+ /.-2/ /.2G +.2-G /
)+ /.)6+ /.5-/ /./* /.*G *.)*G /
)) /.5)// /.-5+ /.**5 /.*5G *.2+G /
)5 /.26) /.65/ /./-*2 /.5)G /.*G /
)6 /.5*5 /.-56- /.5/2 /.2-G *.2/G /
) /.5+6- /./+2/ /.2/66 /.*)G .56G /)- /.+/2) /.55/ /.+2+ /.)G *.)+G /
) /.5)- /.5* /.2++- /.*-G *.+/G /
5// /.5/- /./55 /.6+ /./G 6.G /
5/* /.)-6/ /./2+2 /.+*** /./*G .-G /
5/2 /./2++ /.*)* /.+--+ *.)G 2.*/G /
5/+ /.2/5 /./6*) /.5)- /.6+G 6.6+G /
5/) /.*/+) /.+2- /.))5 *.2*G 2.-2G /
5/5 /.6 /.-* /.///6 /.--G )./5G *
5/6 /.2)5 /.+) /.552 /.)G /.)6G /
5/ /.2--/ /.6)-/ /./6 /.5*G /.-6G /
5/- /.2*+* /.56 /.2/ /.5G 5.-5G /
5/ /.// /.)++ /./-* *.+6G 2.5G /
5*/ /.))+/ /.6+)5 /.-) /./G /.G /
5** /.-+) /.-2 /.5- *.2G *.)2G /
5*2 /./-+2 /.-26 /./5*6 *.++G /.-+G /
-
8/19/2019 Simple Montecarlo
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>ut%ut Monte Carlo
/.2/G mean /.2/G/.2/G std dev /.//G
min /.2/G
max /.2/G
!ataTa"le
r Trial Total Return
/.2/G
/.2G * /.2/G
/.+/G 2 /.2/G
/.)2G + /.2/G
/.-G ) /.2/G
/.+G 5 /.2/G
/.6+G 6 /.2/G
2.*6G /.2/G
/.G - /.2/G
/.6+G /.2/G
/.G */ /.2/G
/./G ** /.2/G
/.++G *2 /.2/G
*.)/G *+ /.2/G
*.26G *) /.2/G
/.6+G *5 /.2/G
*.+-G *6 /.2/G
*./*G * /.2/G
*.-G *- /.2/G
/.*G * /.2/G
/.6G 2/ /.2/G
*.22G 2* /.2/G
*.)5G 22 /.2/G
/.-2G 2+ /.2/G/.5G 2) /.2/G
*.*+G 25 /.2/G
/.)G 26 /.2/G
/.)6G 2 /.2/G
/.5G 2- /.2/G
*.62G 2 /.2/G
/.)/G +/ /.2/G
More Complicated Mont
&n t$is simulation we $averandom log returns ;colum%eriod.
T$e !ataTa"le t$en runs t$ideviation, mean and max #
s "e#ore we could $ave dentire model and %asted itwould $ave "een ver' diJsee w$at was going on, an
note o# caution7 !ataTa"llarge ;lots o# trials
-
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/.52G +* /.2/G
/./G +2 /.2/G
*.*/G ++ /.2/G
/.2G +) /.2/G
/.25G +5 /.2/G
/.+G +6 /.2/G
/.*G + /.2/G/.65G +- /.2/G
/./-G + /.2/G
/.6G )/ /.2/G
*.52G )* /.2/G
/.2/G )2 /.2/G
/./G )+ /.2/G
/.+-G )) /.2/G
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/.6+G )6 /.2/G
/.*/G ) /.2/G/.6)G )- /.2/G
/.+*G ) /.2/G
*./G 5/ /.2/G
/.)/G 5* /.2/G
/.5*G 52 /.2/G
/.2-G 5+ /.2/G
/./+G 5) /.2/G
/.5-G 55 /.2/G
/.6+G 56 /.2/G
/.6-G 5 /.2/G
/./G 5- /.2/G
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/./6G 6/ /.2/G
*.*G 6* /.2/G
/.)G 62 /.2/G
/.*5G 6+ /.2/G
/.)-G 6) /.2/G
/.)2G 65 /.2/G
/./2G 66 /.2/G
/.+/G 6 /.2/G
*.*-G 6- /.2/G
/.6+G 6 /.2/G
*.*5G / /.2/G
2.G * /.2/G
*.-G 2 /.2/G
*.*+G + /.2/G
*.2+G ) /.2/G
*./6G 5 /.2/G
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/.+-G 6 /.2/G
/.6G /.2/G
/.)+G - /.2/G
*.65G /.2/G
/.+G -/ /.2/G
*.5)G -* /.2/G
/.*G -2 /.2/G/.2/G -+ /.2/G
/.*/G -) /.2/G
/./G -5 /.2/G
/.G -6 /.2/G
/.2+G - /.2/G
*.62G -- /.2/G
/./G - /.2/G
/.5+G / /.2/G
/.*G * /.2/G
/.65G 2 /.2/G*.66G + /.2/G
/.+)G ) /.2/G
/.)*G 5 /.2/G
*.2G 6 /.2/G
*.+*G /.2/G
/./5G - /.2/G
/.*2G /.2/G
/./+G *// /.2/G
/.++G
*.2G
/./G
/.25G
*.G
*./G
/.-G
/.)/G
/./G
/.55G
*.-5G
/.2)G
*.5*G
/.+/G
/.5G
/./*G
/.+-G
/.25G
/.++G
/.G
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/.)G
/.-G
/.6G
/.**G
/.2G
*.)2G
/.)G/.5)G
*.)G
*.26G
/.6+G
*.)2G
/.+6G
*.5G
/.55G
/.5G
/.2G/.+/G
*.+G
*.+G
/.65G
/.*/G
*.2G
/.)2G
/.*-G
/.)6G
/.25G
/./2G
*.-)G
/.5*G
*.)5G
/./+G
/./5G
/./6G
/.-G
/./2G
/.+G
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*.*-G
/.5*G
*.*G
/./+G
/.6)G
/.6*G
/.-G
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/.26G
/.-G
*.*+G
/.65G
*./)G
/.*+G
2.5G*.++G
/./+G
/.5G
/.G
*.*+G
/.+5G
/.2G
/.*G
2.*G
/.+G*.+/G
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*.+2G
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/.+2G
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/./G
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2./+G
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/.+6G
*.62G
/.*2G
*./+G
*./5G
/.5G
*.2)G
/./+G
/.2/G
/./2G
*.2/G
-
8/19/2019 Simple Montecarlo
30/60
*./2G
/.G
/.22G
/.)/G
*.5G
/.56G
/.52G*./)G
/.6G
/.*G
/.+/G
/.*-G
/.22G
*.+*G
/./G
*./G
/./G/.6G
*.*+G
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/.+6G
2.*G
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/.)5G
*.+G
*.*+G
/.+G
/.56G
/.2*G
-
8/19/2019 Simple Montecarlo
31/60
/.+G
/.5G
*.5G
*.2*G
/.G
/.+*G
/.)-G*./6G
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*.))G
*.*)G
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-
8/19/2019 Simple Montecarlo
32/60
*./-G
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/.*G
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/.*2G*./*G
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/.*G
-
8/19/2019 Simple Montecarlo
33/60
/.)G
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2.**G
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-
8/19/2019 Simple Montecarlo
34/60
/.*G
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/.-G
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-
8/19/2019 Simple Montecarlo
35/60
/.55G
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-
8/19/2019 Simple Montecarlo
36/60
/.5-G
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-
8/19/2019 Simple Montecarlo
37/60
!ataTa"le dumm'
Carlo
et u% a ver' sim%le @um%diusion model. T$e model generates 5*2&
-
8/19/2019 Simple Montecarlo
38/60
-
8/19/2019 Simple Montecarlo
39/60
-
8/19/2019 Simple Montecarlo
40/60
-
8/19/2019 Simple Montecarlo
41/60
-
8/19/2019 Simple Montecarlo
42/60
-
8/19/2019 Simple Montecarlo
43/60
-
8/19/2019 Simple Montecarlo
44/60
-
8/19/2019 Simple Montecarlo
45/60
-
8/19/2019 Simple Montecarlo
46/60
-
8/19/2019 Simple Montecarlo
47/60
-
8/19/2019 Simple Montecarlo
48/60
-
8/19/2019 Simple Montecarlo
49/60
-
8/19/2019 Simple Montecarlo
50/60
-
8/19/2019 Simple Montecarlo
51/60
-
8/19/2019 Simple Montecarlo
52/60
-
8/19/2019 Simple Montecarlo
53/60
-
8/19/2019 Simple Montecarlo
54/60
-
8/19/2019 Simple Montecarlo
55/60
-
8/19/2019 Simple Montecarlo
56/60
-
8/19/2019 Simple Montecarlo
57/60
-
8/19/2019 Simple Montecarlo
58/60
-
8/19/2019 Simple Montecarlo
59/60
-
8/19/2019 Simple Montecarlo
60/60