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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 10 "IMAFA-ESSI" }}{PARA 256 
"" 0 "" {TEXT -1 52 "Processus Stochastiques Finis de la Finance - TD \+
n\2603" }}{PARA 258 "" 0 "" {TEXT -1 28 "Prix d'une Option Europ\351en
ne" }}{PARA 257 "" 0 "" {TEXT 256 30 "(c)1997 M. Diener - UNSA, CNRS" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "
Bonjour!" }}{PARA 0 "" 0 "" {TEXT -1 37 "Nous reprenons les donn\351es
 du TD n\2602:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "A pr\351sent, calculons et dessino
ns :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(plottools):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "d:=40;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "delta_t:=evalf(1.0/d);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "sigma:=0.4;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "rho
:=.1247;R:=ln(1+rho); r:=d*(exp(R/d)-1); " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "mu:=R*1.3;  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"d
G\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(delta_tG$\"+++++D!#6" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaG$\"\"%!\"\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%$rhoG$\"%Z7!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"RG$\"+ML;v6!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG$\"*?\"*o
<\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#muG$\"+MBrF:!#5" }}}
{EXCHG {PARA 259 "" 0 "" {TEXT -1 67 "Valeur (du portefeuille de couve
rture) d'une option Call europ\351enne" }}{PARA 0 "" 0 "" {TEXT -1 
358 "La th\351orie de Cox-Ross-Rubinstein \351tabli qu'\340 chaque ins
tent la valeur (du portefeuille de couverture) de l'option est egale \+
\340 la valeur actualis\351e de l'esp\351rence de la valeur du sous-ja
cent \340 la p\351riode suivante, pour la probabilit\351 dite \"risque
-neutre\". Nous notons  actu  le facteur d'actualisation,  p_up et p_d
own  les valeurs de la proba risque-neutre:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 22 "actu:=1/(1+r*delta_t);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "up:=1+sigma*sqrt(delta_t)+mu*delta_t;" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 40 "down:=1-sigma*sqrt(delta_t)+mu*delta_t; " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "p_down:=0.5 +(mu-r)*sqrt(delta_t)/(
2*sigma);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "p_up:=0.5 -(mu-r)*sqrt
(delta_t)/(2*sigma);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%actuG$\"+LS
mq**!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#upG$\"+M[1n5!\"*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%downG$\"+wst0%*!#5" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%'p_downG$\"+:rLp]!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%p_upG$\"+&)GmI\\!#5" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 174 "Red\351finissons la procedure donnant la valeur du sous-
jacent en fonction du nombre  n  d'intervalles de dur\351e delta_t (ou
 \"tics\") \351coul\351s et le nombre  d  de  up  intervenus, " }
{TEXT 257 64 "en introduisant la valeur initiale  S0 comme troisi\350m
e param\350tre" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 61 "Nous en
 profitons pour d\351r\351cursiver la d\351finition de S(ouja)." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "SouJa:= proc(n,j,S0)options \+
remember;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evalf(S0*up^j*down^(n-
j))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&SouJaG:6%%\"nG%\"jG%#S0G6\"6#%)rememberGF*-%&evalfG6
#*(9&\"\"\")%#upG9%F2)%%downG,&9$F2F5!\"\"F2F*F*" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 88 "Introduisons la valeur  CE  du portefeuille de cou
verture du Call Europ\351en  en fonction " }}{PARA 0 "" 0 "" {TEXT -1 
37 "   - du nombre  n  de  tics  \351coul\351s " }}{PARA 0 "" 0 "" 
{TEXT -1 34 "   - nombre  d  de  up  intervenus" }}{PARA 0 "" 0 "" 
{TEXT -1 46 "   - de la valeur initiale  S0  du sous-jacent" }}{PARA 
0 "" 0 "" {TEXT -1 53 "   - de la valeur d'exercice (strike)  K  de l'
option" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "CE:=proc (n,j,S0,
K) option remember;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "if n=d then \+
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "  if SouJa(d,j,S0)>K then SouJa
(d,j,S0)-K else 0 fi" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "else (p_dow
n*CE(n+1,j,S0,K)+p_up*CE(n+1,j+1,S0,K))*actu" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 2 "fi" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%#CEG:6&%\"nG%\"jG%#S0G%\"KG6\"6#%)remember
GF+@%/9$%\"dG@%29'-%&SouJaG6%F19%9&,&F5\"\"\"F4!\"\"\"\"!*&,&*&%'p_dow
nGF;-F$6&,&F0F;F;F;F8F9F4F;F;*&%%p_upGF;-F$6&FD,&F8F;F;F;F9F4F;F;F;%%a
ctuGF;F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Exemples pour S0:=1
40<160=:K" }}{PARA 0 "" 0 "" {TEXT -1 62 "Valeur finale  (n=d)  si le \+
titre n'a fait que descendre (j=0)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "S0:=140;K:=160;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#S0G\"$S\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG\"$g\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 122 "Repr\351sentons \340 pr\351sent les vale
ur CE du Call europ\351en en fonction du temps  mesur\351 par i=1..d e
t du prix S du sous jacent:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Va
leur  CE  du portefeuille de couverture du Call Europ\351en  en foncti
on " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 35 "branchedown:=proc (n,S0,K)local i; " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "seq(line([i,SouJa(i,n,S0),CE(i,n,S0
,K)],[i+1,SouJa(i+1,n,S0),CE(i+1,n,S0,K)]),i=n..d-1)" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,branch
edownG:6%%\"nG%#S0G%\"KG6#%\"iG6\"F,-%$seqG6$-%%lineG6$7%8$-%&SouJaG6%
F49$9%-%#CEG6&F4F8F99&7%,&F4\"\"\"F@F@-F66%F?F8F9-F;6&F?F8F9F=/F4;F8,&
%\"dGF@!\"\"F@F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "branc
heup:=proc (n,S0,K)local i; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "seq
(line([i,SouJa(i,i-n,S0),CE(i,i-n,S0,K)],[i+1,SouJa(i+1,i-n+1,S0),CE(i
+1,i-n+1,S0,K)]),i=n..d-1)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*brancheupG:6%%\"nG%#S0G%\"KG6#%\"
iG6\"F,-%$seqG6$-%%lineG6$7%8$-%&SouJaG6%F4,&F4\"\"\"9$!\"\"9%-%#CEG6&
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"" 1 "" {XPPMATH 20 "6#\"#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
76 "fillet:=[seq(branchedown(n,S0,K),n=0..d-1),seq(brancheup(n,S0,K),n
=0..d-1)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 12 "> " 1 "" {MPLTEXT 1 0 19 "theta:=-64:phi:=58:" }}{PARA 0 "> \+
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{TEXT -1 146 "Examinons le cas du Put am\351ricain: nous voyons qu'il \+
suffit de modifier la formule donnant le pay-off dans PE et nous obten
ons PA (Put Am\351ricain):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
36 "PA:=proc (n,j,S0,K) option remember;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "if n=d then " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "  \+
if SouJa(d,j,S0)<K then K-SouJa(d,j,S0) else 0 fi" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 83 "else max(max(0,K-SouJa(n,j,S0)),(p_down*PA(n+1,j,S0
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{XPPMATH 20 "6#>%#PAG:6&%\"nG%\"jG%#S0G%\"KG6\"6#%)rememberGF+@%/9$%\"
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{MPLTEXT 1 0 16 "PA(0,0,140,160);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"+^^kZF!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "PE(0,0,140,
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{PARA 0 "" 0 "" {TEXT -1 28 "Exemples pour S0:=140<160=:K" }}{PARA 0 "
" 0 "" {TEXT -1 62 "Valeur finale  (n=d)  si le titre n'a fait que des
cendre (j=0)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "S0:=140;K:=
160;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S0G\"$S\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"KG\"$g\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
122 "Repr\351sentons \340 pr\351sent les valeur PA du Put am\351riacin
 en fonction du temps  mesur\351 par i=1..d et du prix S du sous jacen
t:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Valeur  PA  du portefeuille
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1 0 87 "seq(line([i,SouJa(i,n,S0),PA(i,n,S0,K)],[i+1,SouJa(i+1,n,S0),P
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S0),PA(i,i-n,S0,K)],[i+1,SouJa(i+1,i-n+1,S0),PA(i+1,i-n+1,S0,K)]),i=n.
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