Market-making, inventories and martingale pricing

(1)

HAL Id: halshs-00178162

https://halshs.archives-ouvertes.fr/halshs-00178162

Submitted on 10 Oct 2007

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Market-making, inventories and martingale pricing

Christian At, Laurent Flochel, Patrick Roger

To cite this version:

Christian At, Laurent Flochel, Patrick Roger. Market-making, inventories and martingale pricing. 2002. �halshs-00178162�

(2)

DOCUMENTS DE TRAVAIL - WORKING PAPERS

W.P. 02-03

Market-making, inventories and martingale pricing

Patrick ROGER

Christian AT

Laurent FLOCHEL

Juin 2002

GATE Groupe d’Analyse et de Théorie Économique

UMR 5824 du CNRS

93 chemin des Mouilles – 69130 Écully – France B.P. 167 – 69131 Écully Cedex

Tél. +33 (0)4 72 86 60 60 – Fax +33 (0)4 72 86 60 90 Messagerie électronique gate@gate.cnrs.fr

Serveur Web : www.gate.cnrs.fr

GATE

Groupe d’Analyse et de Théorie Économique

(3)

Market-making, inventories and martingale pricing

Patrick Roger, LARGE, University Louis Pasteur, Strasbourg I

Christian At, GATE, UMR 5824 CNRS, University Lumière Lyon II

Laurent Flochel, GATE, UMR 5824 CNRS, University Lumière Lyon II

Abstract

We discuss Shen and Starr(2002) results and show that the bid-ask spread of a monopolistic market-marker doesn't depend on his inventory when he posts ''martingale prices '' in an inventory model with random volumes and an unknown direction of trade.

Key words : martingale pricing, market maker, bid ask spread.

Résumé

Nous discutons les résultats de Shen et Starr (2002) et montrons que la fourchette de prix de réservation d’un teneur de marché ne dépend pas de son inventaire lorsque celui-ci fixe un prix suivant une martingale dans un modèle d’inventaire avec volumes aléatoires et direction de l’échange inconnnue.

JEL Classification : G12,D42

(4)

!

"#$%&'()$*&#

!" # $%&%"' (#(%$) *+%" #", *'#$$ -.//.0 ($%1%"' #" 2"3%"'4$5 64,%7 48 6#$9%':6#92"; <#1%, 4" '=4 &$>&2#7 #11>6('24"1? '+% @$1' 4"% 21 '+#' '+% 6#$9%':6#9%$A1 =%#7'+ 21 # 6#$'2";#7% #", '+% 1%&4", 4"% 21 $%7#'%, '+% %B21: '%"&% 48 #" %B4;%"4>1 ($2&% ($4&%11) '+% <2, #", #19 ($2&%1 <%2"; 1566%'$2& =2'+ $%1(%&' '4 '+21 %B4;%"4>1 ($2&%C

D+% 6#$9%':6#9%$ -$%8%$$%, '4 #1 '+% EE 2" '+% 84774=2";0 '+%" (41'1 <2, #", #19 $%1%$3#'24" ($2&%1 #' '+% <%;2""2"; 48 %#&+ (%$24,) =+%$% F$%1%$3#'24" ($2&%1F 6%#"1 ($2&%1 =+2&+ %G>#72H% '+% $%#72H%, =%#7'+ #' ,#'% ! #", '+% %B(%&'%, =%#7'+ #' ,#'% ! I J" &4",2'24"%, 4" &>$$%"' 2"84$6#'24"C !' 21 '+%" #" 2"3%"'4$5 64,%7 2" '+% 1(2$2' 48 K62+>, #", E%",%714" -JLM/0) 4$ N4 #", *'477 -JLMJ:JLMO0 2" '+% 1%"1% '+#' '+% 2"3%"'4$5 7%3%7 21 # 9%5 3#$2#<7% '4 ,%'%$62"% '+% "%B' (%$24, <2, #", #19 ($2&%1 #", '+% 1($%#, <%'=%%" '+% '=4C D+% 6#2" &4"&7>124"1 #$% '+#' '+% 1($%#, 21 (412'23%75 $%7#'%, '4 '+% #<147>'% 3#7>% 48 '+% "%' (412'24" 48 '+% 6#$9%':6#9%$) '+% 347#'272'5 48 347>6%1 #", '+% 347#'272'5 48 ($2&%1C !" '+21 (#(%$) =% (>$1>% '=4 4<P%&'23%1 ? @$1') =% ,%1&$2<% '+% '$#,2"; ($4&%11 #", '+% <%+#324$ 48 # 64"4(4721'2& $219 #3%$1% EE '4 277>1'$#'% '+#') 2" 1>&+ # '$#,2"; 6%&+#"216) '+% ($2&% ($4&%11 21 "4' %B4;%"4>1 =+%" 2' 21 ,%@"%, #1 '+% 62,(42"' <%'=%%" '+% <2, #", '+% #19 ($2&%) #1 &4"12,%$%, <5 *+%" #", *'#$$C *%&4",) 28 $219 #3%$124" 21 2"'$4,>&%, <5 6%#"1 48 # &4"&#3% >'272'5 8>"&'24" #", 28 '+% %B(%&'%, >'272'5 -=2'+ $%1(%&' '4 '+% '%$62"#7 (#56%"' 48 '+% $2195 #11%'0 48 '+% EE 21 # 6#$'2";#7%) '+%" '+% <2,:#19 1($%#, 21 2",%(%",%"' 48 '+% 2"3%"'4$5 7%3%7 =+27%) #' '+% 1#6% '26%) <2, #", #19 ($2&%1 #$% ,%'%$62"%, <5 '+21 7%3%7 48 2"3%"'4$5C !" 4'+%$ =4$,1) =% #$;>% '+#' '+% $%7#'24"1+2( <%'=%%" 1($%#, #", 2"3%"'4$5) %1'#<721+%, <5 *+%" #", *'#$$) &46%1 8$46 '+% $#'+%$ 1'$4"; #11>6('24" 48 '+% 4<1%$3#<272'5 48 #" %B4;%"4>1 ($2&% ($4&%11 =+%") 2" 8#&') 4"75 <2, #", #19 ($2&%1 #$% 4<1%$3#<7%) '+% '$#"1#&'24" ($2&%1 <%2"; $%#72H%, 4" 4"% 48 '+%1% '=4 <4>",1C !' 21 =4$'+ "4'2&2"; '+#') %3%" 28 '$#,%1 #$% %B%&>'%, 2"12,% '+% <2,:#19 2"'%$3#7) '+% '$#"1#&'24" ($2&% 21 "4' %B4;%"4>1 <>' ,%(%",1 4" '+% "%' (412'24" 48 '+% EE) +21 ($%8%$%"&%1 #", +21 %B(%&'%, '%$62"#7 3#7>% 48 '+% $2195 #11%'C

D+% $%7#'24"1+2( <%'=%%" 1($%#, #", 2"3%"'4$5 +#1 <%%" 1'>,2%,) 2" # 1'#'2& 8$#6%=4$9) <5 Q%&94>,':R4;%$ -JLLL0 2" '+% &4"'%B' 48 %B(%&'%, >'27: 2'5 64,%71 #", <5 R4;%$ -.///0 2" '+% RSQT 64,%7C D+%5 1+4= '+#' '+%

(5)

%B(%&'%, 3#7>% 48 '+% $2195 #11%' -=+2&+ 6#5 <% 2"'%$($%'%, #1 '+% %B4;%: "4>1 ($2&% 48 *+%" #", *'#$$0 21 "4' 2" '+% <2,:#19 2"'%$3#7 #1 144" #1 '+% EE +47,1 #" 2"3%"'4$5C D+%1% #>'+4$1 #714 2777>1'$#'% # "4":64"4'4"2& $%: 7#'24"1+2( <%'=%%" 1($%#, #", 2"3%"'4$5 <5 &4"12,%$2"; # $2195 #11%' =+2&+ (#56%"' 21 <4>",%,C N4=%3%$) '+%2$ 64,%7 21 <#1%, 4" "4" 72"%#$ ($2&2"; $>7%1) '+#' 21 '4 1#5) '+% ($2&2"; 1&+%,>7% (41'%, <5 '+% EE 21 4"%:'4:4"% =2'+ $%1(%&' '4 G>#"'2'2%1C !" '+21 (#(%$) =% ,%#7 =2'+ 72"%#$ ($2&%1 2" '+% 1%"1% '+#' '+% EE (41'1 >"2' ($2&%1 #", #&&%('1 #77 4$,%$1 >( '4 # ;23%" 6#B26>6 G>#"'2'5C !" 1%&'24" .) =% @$1' ,%1&$2<% '+% @"#"&2#7 6#$9%' #", '+%" '+% '$#,2"; 6%&+#"216) 2"&7>,2"; '+% 6#$9%':6#9%$ <%+#324$C !" 1%&'24" O) =% ($43% '+% 2"3#$2#"&% 48 '+% <2,:#19 1($%#, =2'+ $%1(%&' '4 '+% 2"3%"'4$5? '+% #11>6('24" 48 # 6%#":3#$2#"&% >'272'5 8>"&'24") 48'%" >1%, 2" 62&$41'$>&'>$% 64,%71 -1%% 84$ %B#6(7% U2#21) JLLO0 #774=1 >1 '4 4<'#2" 126(7% #"#75'2&#7 84$6>7#'24"1 84$ ($2&%1 #", 1($%#,1C K <$2%8 &46(#$214" =2'+ '+% 1($%#, 4<'#2"%, 2" '+% N4 #", *'477 64,%7 21 ($432,%,C *%&'24" V &4"&7>,%1 '+% (#(%$ <5 1>;;%1'2"; ,2$%&'24"1 84$ 8>'>$% $%1%#$&+C

+

,-. $%/'*#0 1%&).22

+3!

4&$/$&#2 /#' '.5#$*&#2

W% &4"12,%$ # ,21&$%'%:'26% 64,%7 =2'+ # $%7%3#"' 1%' 48 ,#'%1_{! X "/" ####" $ # ?} '+% @"#"&2#7 6#$9%' 21 $%,>&%, '4 # 12";7% $2195 1%&>$2'5 =+2&+ ;%"%$#'%1 # 1G>#$%:2"'%;$#<7% $#",46 '%$62"#7 (#56%"' % =2'+ &-%0 X '! #",

(!-%0 X (!!C D+% >"&%$'#2"'5 21 ,%1&$2<%, <5 # @7'%$%, ($4<#<272'5 1(#&%

-Y"_{$" %" ) 0 =+%$% Y 21 '+% 1%' 48 1'#'%1 48 "#'>$% #", % 21 # @7'$#'24" 3%$2:} 852";_%" X"&" Y# #", %" X$# E4$%43%$) 84$ $%#14"1 =+2&+ =277 <%&46% &7%#$

7#'%$) =% #11>6% '+#' _%# 'X %##$ 6%#"2"; '+#' '+% 1%G>%"&% 48 (:#7;%<$#1%

21 1'$2&'75 2"&$%#12"; 84$ 2"&7>124"C

K1 =% =#"' '4 84&>1 4" '+% 2"3%"'4$5 %Z%&') =% 1>((41% '+#' % 21 _%" :

6%1>$#<7% <>' 2",%(%",%"' 48 _%#" ! * $ ? 2" 4'+%$ =4$,1) "4'+2"; 21 $%3%#7%,

'4 '+% 6#$9%' <%84$% ,#'% $" &4"&%$"2"; '+% '%$62"#7 (#56%"' %# D+% '$#,2"; ($4&%11 21 ,$23%" <5 # $219:#3%$1% EE =+41% P4< 21 4"75 '4 ($432,% 72G>2,2'5 '4 4'+%$ '$#,%$1C N% ,4%1"A' <%#$ #"5 4(%$#'2"; &41' #", (41'1) #8'%$ ,#'%:! '$#"1#&'24"1) <2, #", #19 ($2&%1 #' =+2&+ +% #&&%('1 #77 4$,%$1 &462"; 8$46

(6)

'+% 6#$9%' >( '4 # 6#B26>6 347>6% +$ -4" %#&+ 12,%0? '+%1% 4$,%$1 #$% %B%&>'%, #' ,#'% ! I JC D+% 4$,%$ [4= 21 84$6#72H%, #1 84774=1? 7%' ,# #", -# '+% 84774=2"; %3%"'1 \ ,# X "<>52"; 4$,%$ 8$46 '+% 6#$9%' #' ,#'% !# -# X "1%772"; 4$,%$ 8$46 '+% 6#$9%' #' ,#'% !# W% #11>6% %G>#7 ($4<#<272'2%1 84$ '+% '=4 %3%"'1) ) -,#0 X ) --#0 X . <>' ,2Z%$%"' ($4<#<272'2%1 =4>7, "4' &+#";% '+% "#'>$% 48 4>$ $%1>7'1C W+%" # <>52"; -$%1(C 1%772";0 4$,%$ #$$23%1 #' ,#'% !" 2'1 $#",46 347>6% 21 ,%"4'%, #1 /#%-/#&0# D4 9%%( $%1>7'1 126(7%) =% #11>6% >"284$6 ,21'$2<>'24"1 4" ]/? +$^ 84$ '+% 3#$2#<7%1 /%

# #", /#&# E4$%43%$) %" !%!" !&!" /

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

!' 21 =4$'+ "4'2&2"; '+#' 126>7'#"%4>1 '$#,%1 #$% #774=%,) 12"&% ) -,#* -#0 X .! 2 /# S#'%:! '$#"1#&'24"1 #$% $%#72H%, #' ($2&%1 ! 3%#" 3&# " " (41'%, <5 '+% 6#$9%':6#9%$ #' ,#'% !+ J? 2" 8#&') =+%" '+% EE (41'1 +21 ($2&%1) +% ,4%1"A' 9"4= =+2&+ 347>6%1 #$% '4 <% '$#,%, #' '+%1% ($2&%1C D+% 1'4&+#1'2& ($4&%11 48 ($2&%1 !3%" 3&" 21 '+%" ($%,2&'#<7% =2'+ $%1(%&' '4 _%#

+3+

,-. 6/%7.$86/7.%

D+% ,5"#62&1 48 '+% EEA1 =%#7'+ #", 2"3%"'4$5 #$% ,%1&$2<%, <5 '+% 847: 74=2"; %G>#'24"1 \ 4# X 4#!$I !%!/#%3 % # + !&!/ & # 3 & # -J0 5# X 5#!$+ !%!/ % # I !&!/ & # O

(7)

D+21 84$6>7#'24" &7#$2@%1 '+% #11>6('24" 48 2",%(%",%"&% <%'=%%" 347>6%1 #", 2",2&#'4$ 8>"&'24"1 <%&#>1% 347>6%1 &+#";% 2"3%"'4$5 #", =%#7'+ 4"75 28 2",2&#'4$ 8>"&'24"1 #$% %G>#7 '4 JC D+% $%7#'24"1+2(1 -J0 #714 6%#" '+#' '+% &4",2'24"#7 %B(%&'%, =%#7'+ 48 '+% EE #' ,#'% ! I J &#" <% =$2''%" #1 \ &#-4##$I 5##$%0 X 4#I 5#'!I &# ! !%!!"/ % ##$ ! 3%_##$_{+ %}"_{+ !}&!!"/ & ##$ ! 3&_##$_{+ %}"" D+% EE 21 2"2'2#775 %",4=%, =2'+ # 1>$% =%#7'+ 4" #", "4 $2195 #11%'C N21 %B(%&'%, >'272'5 #' ,#'% ! 84$ # $#",46 =%#7'+ 7%3%7 #4 21 #11>6%, '4 <% 48 '+% 84774=2"; 84$6 \ &# $ 6%4#&'X &#-#4 0+ 7(!#-#4 0 =+%$% 7 21 # (#$#6%'%$ 48 $219 #3%$124") &#-#0 -(!# -#00 21 # 1+4$'+#", 84$ & -#_)%#0 - (!-#)%#00C K8'%$ '$#,%1 >( '4 ,#'% !" '+% $#",46 =%#7'+ 48 '+% EE 21 4#I 5#% #", =% &#" =$2'% \ &#-6 -4#I 5#%00 X 4#I 5#'!+ 75#!( ! ! <%&#>1% % 21 2",%(%",%"' 48 %#" ! * $ #", 4# #", 5# #$% %#:6%1>$#<7%C K1 '+% 6#$9%':6#9%$ 21 #11>6%, '4 1'#$' =2'+ # =%#7'+ 4" #", "4 $2195 #11%') +21 >'272'5 #' ,#'% / 21 %G>#7 '4 4"# N4=%3%$) &#-6 -4#I 5#%00 ,%(%",1

4" 5#" =+2&+ 21 '+% $%1>7' 48 '+% 4$,%$ [4=) "4' &4"'$477%, <5 '+% EE) <>'

#714 ,%(%",1 4" 4#" =+2&+ 21 ,%'%$62"%, <5 '+% EEA1 ($2&2"; (472&5C !" '+%

84774=2"; 1%&'24") =% #"#75H% '+% &#1% 48 F6#$'2";#7% ($2&2";F? 2' &4"121'1) 84$ '+% 6#$9%' 6#9%$) '4 (41' ($2&%1 '+#' %G>#72H% '+% %B(%&'%, >'272'5 %Z%&'23%75 4<'#2"%, #' ,#'% ! '4 '+% &4",2'24"#7 %B(%&'#'24" 48 ,#'% !IJ >'272'5C !" 4'+%$ =4$,1) &4",2'24"#7 4" '+% +21'4$5 48 '+% '$#,2"; ($4&%11 >( '4 ,#'% !" '+% EE (41'1 +21 $%1%$3#'24" ($2&%1 '4 <% 2",2Z%$%"' <%'=%%" '$#,2"; #", "4' '$#,2"; ,>$2"; '+% "%B' (%$24,C

9 :/%$#0/;. 1%)*#0

93!

<1%./' /#' *#=.#$&%>

D+% =#5 '+% EE (41'1 <2, #", #19 $%1%$3#'24" ($2&%1 21 ,%1&$2<%, 2" '+% 84774=2"; ,%@"2'24"C V

(8)

?.5#*$*&# ! !" #$"% !" "&% '%(%')$"*+, -'*.%( !3& ##$" 3%##$ " -+("%# /0 "&% 11 )%'*20 3 &#-6 -4#I 5#%00 X &#-6 -4##$I 5##$%00 4&%'% 4##$ $,# 5##$ $'% 5*)%, /0 '%6$"*+, -J0 `>$ %11%"'2#7 $%1>7' 21 '+%" '+% 84774=2";C @%&1&2*$*&# + !" %)%'0 #$"% !" "&% (-'%$# *( %78$6 "+ 3 3%_# _{+ 3}&_# X 7(!_!+$ ( V O + . ) $,# *( "&%, *,#%-%,#%," 2'+9 5## @%&&A B *>((41% '+#') #' ,#'% !" '+% EE +#1 # "%' 2"3%"'4$5 5# X + 2 / #", # 1>$% =%#7'+ 4## D+% '=4 $%1%$3#'24" ($2&%1 84$ '$#,2"; #' ,#'% ! I J &#" <% 84$6>7#'%, <5 6%#"1 48 # 8>"&'24" 8##$( -9" :0 ,%@"%, <5 \ 8_##$( -9" :0 X 9!'_!_{+ 3}&_##$"_{+ :}!'_! _{+ 3}%_##$"_{+ 7(}!_!-9 I +_{+ :0}! -.0 4#I +'!I 8##$( -9" :0 21 '+% %B(%&'%, >'272'5 #8'%$ # '$#,% 347>6% 9 4" '+% <2, 12,% #", : 4" '+% #19 12,%) =+%" '+% 2"3%"'4$5 21 +# R%&#772"; '+% ,%@"2'24" 48 ." '+% %G>#72'2%1 ) -,_{* -})_{0 X ) -,}) * -0 X . -J_{+ .0 #", ) -, * -0 X .}!) '+% #11>6('24" #<4>' '+% ($4<#<272'5 ,21'$2: <>'24" 48 '$#,2"; 347>6%1 7%#,1 '4 '+% 84774=2"; ,%@"2'24"C

?.5#*$*&# 9 :&%, "&% 6%)%6 +2 *,)%,"+'0 *( %78$6 "+ +" "&% /*# $,# $(; -'*.%( 3& ##$ $,# 3%##$ $'% $ (+68"*+, +2 "&% 2+66+4*,5 %78$"*+,3 J +! $ * (" " * (" " .-J_{+ .0}+8_##$( -9" /0 I 8_##$( -/" :0,;9;: -O0 I J +! $ * (" " * (" " .!8##$( -9" :0 I -J+ .0 ! 8##$( -/" /0;9;: X &#]6 -4#I +%0^+ 4#+ +'! X+7+ !₍! ! a

(9)

W% &#" $%6#$9 '+#' 8( ##$-/" /0 21 2" 8#&' %G>#7 '4 &#]6 -4#I +%0+ 4#+ +%^ # b4"1%G>%"'75) ,%@"2'24" O &#" <% $%84$6>7#'%, 2" '+% 84774=2"; =#5 \ * (" " * (" " ! .-J_{+ .0}+8_##$( -9" /0 I 8_##$( -/" :0,I .!8_##$( -9" :0";9;: X . -._{+ .0 &}#]6 -4#I +%0+ 4#+ +%^ X +7+!(!!. -.+ .0 W% &#" "4= ($43% '+% ($4(412'24") <5 ,%@"2"; '+% '+$%% 84774=2"; &46(4: "%"'1 \ ,* X J +$ * (" " .-J_{+ .08}##$( -9" /0 ;9 ,+ X J +$ * (" " .-J_{+ .08}##$( -/" :0 ;: ,*+ X J +! $ * (" " * (" " .!8##$( -9" :0 ;9;: D+% &46(>'#'24" 48 c X ,* I ,+ I ,*+ 21 $%(4$'%, 2" '+% #((%",2B #", 7%#,1 '4 '+% ,%12$%, $%1>7'C Q3%" 28 '+% &#7&>7#'24"1 #$% '%,24>1) '+% 2"'>2'24" <%+2", '+21 $%1>7' 21 G>2'% 126(7%C T12"; 6#$'2";#7% ($2&2";) '+% EE ,4%1"A' &#$% '4 ($%324>1 ;#2"1 #", 7411%1 =+%" +% (41'1 ($2&%1? 64$% ($%&21%75) +% '#9%1 &#]6 -4#I +%0^

#1 '+% 4<P%&'23% 84$ '+% "%B' ,#'%C b4"1%G>%"'75) '+% 2"3%"'4$5 %Z%&' 21 "%>: '$#7 <%&#>1% 2' 21 5%' '#9%" 2"'4 #&&4>"' 2" '+% %B(%&'%, >'272'5 $%#72H%, <5 ,#'% !# N4=%3%$) 28 <2, #", #19 ($2&%1 =%$% &4"12,%$%, 1%(#$#'%75) =2'+4>' &4"12,%$2"; '+% >"&%$'#2"'5 #<4>' '+% ,2$%&'24" 48 '+% "%B' '$#,%) #1 2" '+% N4 #", *'477 -JLMO0 64,%7) %#&+ 48 '+% '=4 ($2&%1 =4>7, ,%(%", 4" 2"3%"'4$5C D+21 (42"' 21 <$2%[5 277>1'$#'%, +%$%#8'%$C

93+

C&61/%2&# D$- $-. E& /#' <$&;; 6&'.;

d4$ '+% '=4 64,%71 '4 <% &46(#$#<7%) =% &4"12,%$ # @B%, '$#,2"; 347>6% %G>#7 '4 (" ! " '+#' 21 '+% 6%#" 347>6% 48 '+% ($%&%,2"; 1%&'24"C D+% '=4 ($2&%1 #$% '+%" 147>'24"1 48 '+% 84774=2"; %G>#'24"1 \ +7+!(!_! X +$ . ! '!+ 3&##$ " + 7(!! %+$ . I + &! +7+!(!! X +$ . ! 3%##$+ '! " + 7(!! % +₊ +$ . &! e

(10)

8$46 =+2&+ =% ,%,>&% \ 3&_##$ X '_!_{+ 7(}!_!%.+ I +$ . & 3%_##$ X '_!I 7(!_!%+$ . + .+ & 3%_##$_{+ 3}&_##$ X 7(!_!+$ D+% 1($%#, 21 2",%(%",%"' 48 '+% 2"3%"'4$5 7%3%7? 2' 21 #714 ,2Z%$%"' 8$46 '+% 4"% 4<'#2"%, 2" '+% ($%&%,2"; 1%&'24" 84$ '=4 $%#14"1C d2$1') #1 =% &4"12,%$ $#",46 347>6%1) '+% (412'24" 48 '+% EE 21 $2192%$ #", 2' %B(7#2"1 '+% &4%f: &2%"' %_&# K' '+% 1#6% '26%) 126>7'#"%4>1 <>5 #", 1%77 4$,%$1 #$% (4112<7% =2'+ ($4<#<272'5 .!# D+%5 2",>&% # +%,;2"; %Z%&' =+2&+ ,%&$%#1%1 '+% 1($%#,) #1 '+21 %Z%&' #((%#$1 2" '+% 1($%#, <5 6%#"1 48 '+% "%;#'23% &4%f&2%"' _+.# !" 4'+%$ =4$,1) . 21 # 6%#1>$% 48 6#$9%' 72G>2,2'5? '+% 64$% 72G>2, 21 '+% 6#$9%') '+% ;$%#'%$ 21 '+% (4112<272'5 48 126>7'#"%4>1 '$#,%1) 7%#,2"; '4 #" 2"&$%#1% 2" '+% EEA1 =%#7'+C

N4=%3%$) %3%" 28 . X J" '+% EE 21 1'277 2" # $2195 12'>#'24" <%&#>1% '+% 347>6%1 4" %#&+ 12,% ,4 "4' &#"&%7 %#&+ 4'+%$) #1 2' 21 '+% &#1% 2" '+% N4 #", *'477 8$#6%=4$9 =2'+ 9"4=" #", %G>#7 347>6%1 4" %#&+ 12,%C

F

C&#);(2*&#

!" '+21 (#(%$) =% +#3% ,%64"1'$#'%,) 2" # 126(7% 8$#6%=4$9) '+#' '+% <2,:#19 1($%#, #", '+% 2"3%"'4$5 7%3%7 #$% 2",%(%",%"' =+%" '+% EE >1%1 F6#$'2": ;#7% ($2&2";F? 4>$ $%1>7' #714 277>1'$#'%1 '+#' '+% ($2&2"; ($4&%11) ,%@"%, #1 '+% 62,(42"' <%'=%%" '+% #19 #", '+% <2, ($2&%) 21 "4' %B4;%"4>1 #", 21 &7%#$75 72"9%, '4 '+% 2"3%"'4$5 7%3%7C D+% ,2Z%$%"' $%1>7'1 4<'#2"%, <5 *+%" #", *'#$$) 72"92"; '+% 1($%#, '4 '+% 2"3%"'4$5 7%3%7) &46% 8$46 '+% #11>6(: '24" '+#' <2, #", #19 ($2&%1 #$% 1566%'$2&#7 =2'+ $%1(%&' '4 #" %B4;%"4>1 "4" 4<1%$3#<7% ($2&% ($4&%11C `>$ 84$6>7#'24" #714 (42"'1 4>' '+% 2"&$%#1% 2" 1($%#, ,>% '4 $#",46 347>6%1 #", '+% ,%&$%#1% ,>% '4 6#$9%' 72G>2,2'5) $%($%1%"'%, <5 '+% ($4<#<272'5 48 126>7'#"%4>1 4$,%$1 4" '+% '=4 12,%1C

K 64$% 8>",#6%"'#7 G>%1'24" $%72%1 4" '+% 6#$'2";#7% #11>6('24"? 2" 8#&' # EE 21 64$% 729%75 '4 (41' ($2&%1 2" # 6%#":$%3%$'2"; =#5) '+#' 21) #8'%$ # F<#, '$#,%F) +% =277 ($4<#<75 '$5 '4 $%&43%$ G>2&975 <5 74=%$2"; +21 <2, ($2&% #", 2"&$%#12"; +21 #19 ($2&%C D+21 12'>#'24" &#" <% '#9%" 2"'4 #&&4>"'

(11)

<5 &4"12,%$2"; '+#' '+% ;4#7 48 '+% EE 21 #7=#51 '4 &46% <#&9 '4 +21 2"2'2#7 12'>#'24") # 1+4$'+#", 84$ =+#' 21 >1>#775 &#77%, '+% ,%12$%, 2"3%"'4$5 7%3%7C !" 1>&+ # 12'>#'24") '+% <2,:#19 1($%#, 21 "4' 2",%(%",%"' 48 '+% 2"3%"'4$5 7%3%7? +4=%3%$) '+% $%7#'24"1+2( 21) 2" ;%"%$#7) "4' 64"4'4"2& #", ,%(%",1 %11%"'2#775 4" '+% +21'4$5 48 '$#,%1C !' 21 ,>% '4 '+% 8#&' '+#' ($2&%1 =+2&+ %G>#72H% 2"2'2#7 >'272'5 #", &4",2'24"#7 %B(%&'%, >'272'5 ;%"%$#'% <%"%@'1 84$ 74= '$#,% 347>6%1 #", 7411%1 84$ 7#$;% 347>6%1$_{C b4"1%G>%"'75) =+%" 347>6%1} #$% $#",46) '+% +21'4$5 48 '$#,%1 21 # 9%5 8#&'4$ '4 $%7#'% '+% <2,:#19 1($%#, #", '+% 2"3%"'4$5 7%3%7C

G.A.%.#).2

]J^ K62+>,) hC#", NC E%",%714") JLM/) S%#7%$1+2( 6#$9%' \ 6#$9%' 6#92"; =2'+ 2"3%"'4$5) <+8',$6 +2 =*,$,.*$6 >.+,+9*.( ?@ OJ:aOC

].^ K') bC) d74&+%7 _C #", iCR4;%$) .//.) R%1%$3#'24" ($2&%1) $219 #3%$124" #", 2"3%"'4$2%1 \ '+% 72"%#$ &#1%) 1*9%+@ T"23%$12'5 _4>21 i#1'%>$C ]O^ U2#21) UC) JLLO) i$2&% d4$6#'24" #", %G>272<$2>6 72G>2,2'5 2" 8$#;6%"'%,

#", &%"'$#72H%, 6#$9%'1C <+8',$6 +2 =*,$,.% A?@ Jag:JMaC

]V^ Q%&9+4>,') _C #", iC R4;%$) JLLL) R219 #3%$124" #", '+% <2,:#19 1($%#,C >8'+-%$, =*,$,.*$6 1$,$5%9%,"@ B@ O.O:OV/C

]a^ N4) DC #", NC *'477) JLMJ) `('26#7 ,%#7%$ ($2&2"; >",%$ '$#"1#&'24"1 #", $%'>$" >"&%$'#2"'5C <+8',$6 +2 =*,$,.*$6 >.+,+9*.( D@AEFEG

]e^ N4) DC) *'477) NCRC) JLMO) FD+% ,5"#62&1 48 ,%#7%$ 6#$9%'1 >",%$ &46: (%'2'24"F) <+8',$6 +2 =*,$,.%@ G?@ HIBGFHIEAC

]g^ R4;%$) iC) .///) Fi$4(%$'2%1 48 <2, #", #19 $%1%$3#'24" ($2&%1 2" '+% $#"9: ,%(%",%"' %B(%&'%, >'272'5 64,%7F) <+8',$6 +2 1$"&%9$"*.$6 >.+,+9*.(@ GA@ JKDFJ?BC ]M^ *+%") iC #", EC *'#$$) .//.) E#$9%':6#9%$1A 1>((75 48 @"#"&2#7 6#$9%' 72G>2,2'5) >.+,+9*.( L%""%'(@ EK@,"_H@BGFB?C !_{!"" #$% &'()*"'% +(,"- ./00/12} M

(12)

H11.#'*I

@%&&A &A 1%&1&2*$*&# + J0 Q3#7>#'24" 48 ,* ,* X .-J_{+ .0} +$ * (" " + 9 -'!+ 3&-+00+ 7(!!-9 I +0 !, ;9 !"'%;$#'24" =2'+ $%1(%&' '4 9 ;23%1 \ ,* X . -J+ .0 ( +$ . -'!+ 3&-+00+ 7(! ! O+$ ! -+$ I +0 & + +&" ) X . -J_{+ .0} ( +$ . -'!+ 3&-+00+ 7(! ! O ! +!_$ I O+!I O+$+ ") .0 Q3#7>#'24" 48 ,+ ,+ X .-J_{+ .0} +$ * (" " ! +:!'!+ 3%##$-+0 " + 7(!!-:+ +0! " ;: X . -J_{+ .0} ( ++$ . ! '_!_{+ 3}%_##$-+0"₊ 7( ! ! O ! +!_$ I O+!_{+ O++}$ ") O0Q3#7>#'24" 48 ,*,+ %#$'%# &# " -&" $ -&" $ ]*--'!.(!!"(())!+--'!.)!!"(()00!/0#'(*#(!+) # ^1*1+ '%# &" -&" $ ( &" # --'!.(!!"(())!+--'!.)!!"(()00!/0#'

(&" !&!$)%!&&!$'% %&" ) 1+ '2# ( &" # -. ) !!"(()!.(!!"(()0!/0#' (

!&(!(&" !&)(!(&_!&")(!&( "#&# " )) '2# ( &" # -. ) !!"(()!.(!!"(()0!/0#' ( &#_{" !)}&# ) )) W% '+%" ,%,>&% c X ,*I ,+I ,*+ \ c X . -J_{+ .0} ( +$ . ! '!+ 3&##$ " +7( ! ! O ! +!_$ I O+!I O+$+ ") I . -J_{+ .0} ( ++$ . ! '_!_{+ 3}%_##$"₊ 7( ! ! O ! +!_$ I O+!_{+ O++}$ ") I .! ( +$ . ! 3%##$+ 3&##$ " + 7(!! ( +! $ I e+! e )) L

(13)

c X .+$ . ! 3%##$+ 3&##$ " + . -J + .07( ! ! O ! .+!$ I e+! " + .!7(!! ( +!$ I e+! e ) X .+$ . ! 3%_##$_{+ 3}&_##$"₊7( ! ! O ( . -J_{+ .0}!.+!_$ I e+!"I .! ( +! $ I e+! . )) X .+$ . ! 3%##$+ 3&##$ " +7( ! ! O ( +!$ ( .. -J_{+ .0 I} . ! . ) I +!+e. -J_{+ .0 I O.}!, ) d2"#775 2' $%6#2"1 '4 1473% \ +7+!(!_!. -._{+ .0 X c} =+2&+ 21 %G>23#7%"' '4 \ 3%##$+ 3&##$ X 7(!!+$ ( V O + . ) J/

Market-making, inventories and martingale pricing

HAL Id: halshs-00178162

https://halshs.archives-ouvertes.fr/halshs-00178162

Market-making, inventories and martingale pricing

Christian At, Laurent Flochel, Patrick Roger

To cite this version:

DOCUMENTS DE TRAVAIL - WORKING PAPERS

W.P. 02-03

Market-making, inventories and martingale pricing

Patrick ROGER

Christian AT

Laurent FLOCHEL

Market-making, inventories and martingale pricing

!

"#$%&'()$*&#

+

,-. $%/'*#0 1%&).22

+3!

4&$/$*&#2 /#' '.5#*$*&#2

+3+

,-. 6/%7.$86/7.%

9

:/%$*#0/;. 1%*)*#0

93!

<1%./' /#' *#=.#$&%>

93+

C&61/%*2&# D*$- $-. E& /#' <$&;; 6&'.;

F

C&#);(2*&#

G.A.%.#).2

4&$/$&#2 /#' '.5#$*&#2

:/%$#0/;. 1%)*#0

C&61/%2&# D$- $-. E& /#' <$&;; 6&'.;