42 Mercado Financiero de Derivados Futuros y Opcioneswebs.ucm.es/info/jmas/mon/42.pdf · Mercado de...

MercadoMercado de Derivados de Derivados Financieros: Futuros y Financieros: Futuros y

OpcionesOpciones

© Juan Mascareñas Universidad Complutense de Madrid

Primera versión: enero 1991 - Última versión: diciembre 2012

- Introducción a los futuros financieros, 1

- La cámara de compensación (clearing house), 5

- Algunas características de los contratos de futuros, 9

- La paridad entre el precio del futuro y el de contado, 16

- Introducción a las opciones financieras, 17

- Descripción de las opciones, 19

- Opciones de compra (call options), 23

- Opciones de venta (put options), 29

- Factores que determinan el valor de una opción, 34

- El método binomial de valoración de opciones, 39

- El modelo de Black & Scholes, 46

- La cobertura de riesgo con opciones, 48

- Bibliografía, 51

- Enlaces de interés, 51

- Anexos, 52

Monografías de Juan Mascareñas sobre Finanzas Corporativas ISSN: 1988-1878

42. Mercado de Derivados Financieros: Futuros y Opciones

1

Un activo derivado es un activo financiero o real cuyo precio depende – se deriva- del precio de otro activo al que se denomina subyacente. !

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!.2!K232@572!E./0123F!B$!%*!*-.7*!K+*!%"-L<!,(!7+($!/(3&/-!$!1$!'(7(,&/$/!/(!%$+$'.&4$+!2)(!1$,!*(+,-'$,!7-'!*-,&57&-'(,!,-3+(!E).)+-,!7)#*1$'!7-'!,),!-31&%$7&-'(,!11(%$/-!(1!#-#('.-9!!:1!7@17)1-!/(!1$,!*O+/&/$,!G!%$'$'7&$,!,(!+($1&4$!+).&'$+&$#('.(!*-+! 1-,!,!&?#!$!$!.+$CO,!/(!1$,!)3#-.*$'(#'+#!)*-)7*$!K)&++&(".D'*))&3-.$<!2)(!('!1-,!7-'.+$.-,!/(!E).)+-,!E&'$'7&(+-,!,(!/('-#&'$'!)3#-.*$'(#'#2#)."5&L!/(!,),!71&('.(,9!:1!,$1/-!'(.-!/(!(,.$!7)('.$!,(!-3.&('(!,)#$'/-!(1!/&'(+-!182)&/-!-!,&#&1$+!K3-'-,!/(1!X(,-+-<!*-+!(D(#*1-L<!#@,!1$,!%$'$'7&$,!/(!1$,!-*(+$7&-'(,!$3&(+.$,<!#('-,!1$,!*O+/&/$,!/(!1$,!#&,#$,9!!B$!%$+$'.8$!,(!(,.$31(7(!*$+$!$,(%)+$+!2)(!)'$!7)('.$!/(!(E(7.&C-!.&('(!)'!,$1/-!,)E&7&('.(!7-'!+(1$7&6'!$1!.$#$0-!/(!1$,!*-,&7&-'(,!$3&(+.$,<!/(!.$1!#$'(+$!2)(!1$!*+-3$3&1&/$/!/(!$157$'4$+!)'!,$1/-!'(%$.&C-!,($!#)G!*(2)(0$9!B$!%*!*-.7*'"-")"*0!K"-"."*0'+*!%"-L<!2)(!,(!$,&%'$!$!)'$!*-,&7&6'!+(7&O'!$3&(+.$<!,)(1(!-,7&1$+!('.+(!(1!l5W;a!/(1!C$1-+!/(1!7-'.+$.-\9!B$!%*!*-4

6 >-+!(D(#*1-<!('!(1!E).)+-!,-3+(!(1!`-'-!W;!/(!f:UU!1$!%$+$'.8$!&'&7&$1#('.(!*+(C&,.$!*-+!)'!7-'.+$.-!(,!/(!kol!*)'.-,!3@,&7-,!Kk<olaL



13

.7*'(#'+*-.#-"+"#-.&!K+*"-.#-*-)#'+*!%"-L<!*-+!/(3$D-!/(1!7)$1!'-!,(!*(+#&.(!2)(!7$&%$!(1!,$1/-!/(!1$!7)('.$!,&'!.-#$+!#(/&/$,!7-++(7.-+$,<!,)(1(!,(+!/(1!ol5=;a!/(!1$!%$+$'.8$!&'&7&$19!H)$'/-!/&7F-! ,$1/-! 7$(!*-+!/(3$D-!/(! 1$! %$+$'.8$!/(!#$'.('&#&('.-<! 1-,! 71&('.(,! +(7&3('!)'$!!#)0*+*)"8-'(#'%*!*-.7*' K+*!%"-')*00L<!/(!.$1!#$'(+$!2)(!,&!(1!71&('.(!'-!*-'(!/&'(+-!$/&7&-'$1!*$+$!7)3+&+!1-!2)(!E$1.$<!(1!,!&?#!!7-#('4$+@!$!7(++$+1(!*-,&7&-'(,!F$,.$!2)(!(1!,$15/-!$17$'7(!1-,!'&C(1(,!(,.&*)1$/-,!KC(+!E&%)+$!=L9!!

!U&%9=!T(*6,&.-!/(!%$+$'.8$<!%$+$'.8$!/(!#$'.('&#&('.-!G!+(71$#$7&6'!/(!%$+$'.8$!

!!O<;?=6@!2J9635<!E4563)23760687F!!:'!.-/-!#-#('.-<!(1!5&03+#-'*,"#!.&!(,!&%)$1!$!1$!7$'.&/$/!2)(!(,.@'!-31&%$/-,!$!('.+(%$+!$2)O11-,!&'C(+,-+(,!2)(!.&('('!*-,&7&-'(,!7-+.$,9!B$!7)$1!7-&'7&/&+@!7-'!1$!7$'.&/$/!2)(!(,5.@'!-31&%$/-,!$!$7(*.$+!G!*$%$+!1-,!&'C(+,-+(,!2)(!.&('('!*-,&7&-'(,!1$+%$,9!!:'!+($1&/$/<!*-7-,!7-'.+$.-,!/(!E).)+-,!E&'$'7&(+-,!K#('-,!/(1!W;a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

Wv9!Q&!'-!([&,.(!)'!*)."5&'#-.!#%*,0#! 1$!1&2)&/$7&6'!/(!1$!*-,&7&6'!,(!F$+@!#(/&$'.(!1$!/(C-1)7&6'!/(1! ,$1/-!/(1! /(*6,&.-!/(!%*!*-.7*' "-")"*0! '(.-!/(! 1$,! *O+/&/$,!-! $)5#('.$/-!('!1$,!%$'$'7&$,!/(!1$!_1.&#$!,(,&6'9!

Vv9!Q&!([&,.(!)'!*)."5&'#-.!#%*,0#<!O,.(!,(+@!C('/&/-!*-+!(1!C('/(/-+!$! 1$!CA+*!*'(#')&+/#-$*)"8-<!2)&O'!$!,)!C(4!1-!C('/(+@!$1!7-#*+$/-+!$1!*+(7&-!2)(!+(,)1.$!/(!1$!7-.&4$7&6'!/(1!#(+7$/-!/(!E).)+-,!$1!7&(++(!/(1!#&,#-9!B$!H@#$+$!,(!('7$+%$+@!/(!



14

*$%$+!$1!C('/(/-+!/(1!7-'.+$.-!(1!/&'(+-!('.+(%$/-!$!7$#3&-!/(1!$7.&C-!*-+!*$+.(!/(1!7-#*+$/-+9!

ev9!Q&!([&,.('!C$+&-,!*)."5&$'#-.!#%*,0#$<!(1!C('/(/-+!(1(%&+@!7)@1!/(!(11-,!('.+(%$+@!$!1$!H@#$+$!/(!H-#*(',$7&6'<!G!O,.$!,(!1-,!.+$,*$,$+@!$1!7-#*+$/-+<!*-+!1-!2)(!O,5.(! *-/+8$! ('7-'.+$+,(! 7-'!)'! $7.&C-! 2)(!'-! (,! (1!#&,#-!2)(!F$38$! 7-#*+$/-o9!H1$+-!2)(!1$!([&,.('7&$!/(!C$+&-,!$7.&C-,!('.+(%$31(,!F$7(!2)(!,($!'(7(,$+&-!/(E&5'&+!)'$!,(+&(!/(!+(%1$,!2)(!E&D('!(1!C$1-+!/(!7$/$!)'-!/(!(11-,!('!1$!1&2)&/$7&6'!*-+!7(,&6'!KC(+!$'([-!WL9!

!.2!J286!>$+$! )'! 7-3(+.)+&,.$! 2)(! .('%$! )'$! *-,&7&6'! ('! (1! &',.+)#('.-! E&'$'7&(+-! G! )'$! *-,&7&6'!-*)(,.$!/(!&%)$1!#$%'&.)/!('!)'!7-'.+$.-!/(!E).)+-,<!1$!/&E(+('7&$!('.+(!(1!*+(7&-!/(1!23.3!&!G!(1!*+(7&-!/(!)&-.*(&!/(1!$7.&C-!E&'$'7&(+-!7-++(,*-'/&('.(!(,!7+)7&$19!T&7F$!/&E(+('7&$!,(!/('-#&'$!,*$#'K,*$"$L!G!,(!*)(/(!/(E&'&+!/(!/-,!E-+#$,N!!

base = precio de contado - precio del futuro !-!.$#3&O'<!!!

base = precio del futuro - precio de contado !7)$'/-!,(!F$31$!/(!p*+(7&-!/(1! E).)+-p!'-,!+(E(+&+(#-,!$1!*+(7&-!$D),.$/-!*-+!(1!2*).&!'(#')&-5#!$"8-!7)$'/-!(,.(#-,!F$31$'/-!/(!)'!$7.&C-!,)3G$7('.(!('.+(%$31(!G!'-!/(!)'-!.(6+&57-!-!'-7&-'$19!!!H-#-!,(!-3,(+C$!('!1$!E&%)+$!n<!1$!3$,(!C$+8$!7-'!(1!.&(#*-!G!7-'C(+%(!F$7&$!7(+-!7-'E-+#(!,(!$*+-[&#$!1$!E(7F$!/(!C('7&#&('.-!/(1!7-'.+$.-!/(!E).)+-,!/(3&/-!$!2)(!('!(,(!#-#('.-!.$'.-!(1!*+(7&-!/(1!E).)+-!7-#-!(1!/(1!$7.&C-!,)3G$7('.(!/(3(+@'!7-&'7&/&+9!T(!F(7F-<!/&7F$!7-'C(+%('7&$!'-!(,!)'&E-+#(!K,&!$,8!E)(,(!7-&'7&/&+8$!7-'!1$!/&$%-'$1!/&3)D$/$!('!1$!E&%)+$!n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

7 H)$'/-!(,.-!-7)++(!(,!*-+2)(!(1!$7.&C-!,)3G$7('.(!(,!I'-7&-'$1J<!(,!/(7&+<!.(6+&7-!'-!+($19!>-+!(D(#*1-<!,&!7-#*+-!)'!g&1-!/(!#$'4$'$,! .&*-!%-1/('!$! E).)+-9!:1! $7.&C-! ,)3G$7('.(!'-7&-'$1!*)(/(! ,(+N! 7)$.+-!#$'4$'$,!/(!Vl;!%+$#-,!7$/$!)'$9!H)$'/-!11(%)(!(1!#-#('.-!/(!('.+(%@+#(1$,!(,!/&E87&1!2)(!([&,.$'!,)E&7&('.(,!#$'4$'$,!/(!Vl;!%+$#-,!([$7.-,<!*-+!1-!2)(!(1!C('/(/-+!(1(%&+@!$2)O11$,!2)(!#@,!,(!$*+-[&#('!$!(,(!*(,-9!M!1$!+(1$7&6'!('.+(!(1!$7.&C-!,)3G$7('.(!'-7&-'$1!G!(1!$7.&C-!+($1!$!('.+(%$+!,(!1(!/('-#&'$!IE$7.-+!/(!7-'C(+,&6'J9!



15

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1$+%-!/(!1-,!VW!/8$,!/(1!*(+8-/-L!

!w)&('! F$G$! 7)3&(+.-! 7-#*1(.$#('.(! ,)! *-,&7&6'! ,-3+(! (1! &',.+)#('.-! E&'$'7&(+-<! F$3+@!7$#3&$/-!(1!!"#$%&'(#0'/!#)"&!K-,7&1$7&6'!/(!O,.(L!*-+!(1!!"#$%&'(#'0*',*$#!KC$+&$3&1&/$/!/(!O,5.$<!(,!/(7&+<!(1!+&(,%-!/(!2)(!(1!*+(7&-!/(1!E).)+-!G!(1!*+(7&-!/(!7-'.$/-!'-!,(!#)(C$'!$1!)'85,-'-L9!:,.-!(,<!,61-!*(+#$'(7(+@!1$!&'7(+.&/)#3+(!,-3+(!1$!/&E(+('7&$!('.+(!1-,!/-,!*+(7&-,!,(0$1$/-,9!!

m!>$+$!1-,!3-'-,!/(*('/(+@!/(!1$!/&E(+('7&$!('!1-,!C('7&#&('.-,<!7$1&/$/!/(!1$!&'C(+5,&6'<!(.79!

m!>$+$!1$,!7$+.(+$,!/(!$77&-'(,!(!8'/&7(,!/(!$77&-'(,<!/(*('/(+@!/(!1$,!/&E(+('7&$,!('!1$,!,(',&3&1&/$/(,!7-'!+(,*(7.-!$!C$+&-,!E$7.-+(,!/(!+('/&#&('.-9!

!Q(!/('-#&'$!*(B3"!"!' 0*',*$#!$!1$!-*(+$7&6'!7-',&,.('.(!('!7-#*+$+!)'!$7.&C-!E&'$'7&(+-!G!C('/(+!,)!E).)+-!7-++(,*-'/&('.(!K*-+!(D(#*1-<!7-#*+$+!W;;!$77&-'(,!/(1!`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

!!

8 Q&!1$!,*$#!K*+(7&-!/(!7-'.$/-!#('-,!*+(7&-!/(1!E).)+-L!(,!*-,&.&C$!G!,)!C$1-+!$)#('.$!7-'!(1!.&(#*-<!,(!/&7(!2)(!(1!7-'5.+$.-!(,.@!('!,*)?E*!(*."&-!K(1!*+(7&-!/(1!E).)+-!+(.+-7(/(!7-'!+(,*(7.-!$1!/(!7-'.$/-!7$/$!C(4!#@,L9!:'!(1!7$,-!-*)(,.-<!3$,(!'(%$.&C$!G!7$/$!C(4!#@,!'(%$.&C$<!,(!/&7(!2)(!(,.@!('!)&-.*-%&!K(1!*+(7&-!/(1!E).)+-!,(!$/(1$'.$!$1!/(!7-'.$/-!7$/$!C(4!#@,L9!



16

!!

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

!

rf = F0 + D - S0S0

!

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

F0 = S0 x (1 + rf) - D = S0 x (1 + rf - d) !M!(,.$!+(1$7&6'!,(!1$!7-'-7(!7-#-!1$!/*!"(*('#-.!#'0&$'/!#)"&$'(#')&-.*(&'D'(#'23.3!&<!2)(!*+-*-+7&-'$!1$!7-++(7.$!+(1$7&6'!.(6+&7$!('.+(!$#3-,!*+(7&-,9!Q&!1$!*$+&/$/!'-!E)(,(!+(,*(.$5/$!/$+8$!1)%$+!$!)'$,!-*-+.)'&/$/(,!/(!$+3&.+$D(!2)(!*+-/)7&+8$'!)'!3('(E&7&-!,&'!+&(,%-!2)(!'-!+(2)(+&+8$!'&'%)'$!&'C(+,&6'!&'&7&$19!B$!(,.+$.(%&$!/(!$+3&.+$D(!,(!*)(/(!+(*+(,('.$+!#@,!%('(+$1#('.(!/(!1$!,&%)&('.(!E-+#$N!!

Acción Flujo de caja inicial Flujo de caja en 1 año 1. Pedir prestado S0 S0 -S0 x (1 + rf) 2. Comprar activo por S0 - S0 S1 + D 3. Vender futuros por F0 0 F0 - S1 TOTAL 0 F0 - S0 x (1 + rf) + D

!B$!+(1$7&6'!/(!*$+&/$/!.$#3&O'!,(!7-'-7(!7-#-!1$!!#0*)"8-'(#0')&$.#'(#'+*-.#-"+"#-.&<!*-+52)(!$E&+#$!2)(!(1!*+(7&-!/(1!7-'.+$.-!/(!E).)+-,!C&('(!/(.(+#&'$/-!*-+!(1!7-,.(!+(1$.&C-!/(!$/2)&+&+!)'!$7.&C-!7-'!('.+(%$!/&E(+&/$!('!(1!#(+7$/-!/(!E).)+-,!pC(+,),p!1$!$/2)&,&7&6'!/(1!#&,#-!('!(1!#(+7$/-!/(!7-'.$/-!7-'!('.+(%$!&'#(/&$.$!G!,)!#$'.('&#&('.-!('!(1!&'C('.$5+&-!F$,.$!(1!E&'$1!/(1!*(+8-/-9!>-+!-.+-!1$/-<!1$!+(1$7&6'!/(!*$+&/$/!*)(/(!%('(+$1&4$+,(!*$+$!.!*(+8-/-,N!



17

!

F0 = S0 x (1 + rf - d)t

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18

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Cámara de compensación Intermediario Intermediario

Inversor Inversor



19

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20

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21

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22

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!.2!K232@572!<!W./0123W!:1!7-#*+$/-+!/(!)'$!-*7&6'!/(,($+@!$,(%)+$+,(!2)(!(1!C('/(/-+!*)(/(!('.+(%$+1(! 1$,!$757&-'(,!-!(1!/&'(+-!K,(%_'!2)(!$2)O11$!,($!/(!7-#*+$!-!/(!C('.$L!7)$'/-!$,8!,(!1-!+(2)&(+$9!B$!CA+*!*!/(!7-#*(',$7&6'!%$+$'.&4$!/&7F$!('.+(%$!G!*$+$!(11-!([&%(!$1!C('/(/-+!2)(!*+-5*-+7&-'(!$1%_'!.&*-!/(!%*!*-.7*!K+*!%"-L!7-'!-3D(.-!/(!$,(%)+$+!1$!+($1&4$7&6'!/(!,)!-31&%$57&6'9!>-+!(D(#*1-<!,(! 1(,!*-/+8$!([&%&+!)'!/(*6,&.-!/(1!la!/(1!C$1-+!/(!#(+7$/-!/(1! .8.)1-<!#@,!-!#('-,!1$!7$'.&/$/!*-+!1$!2)(!1$!-*7&6'!,(!('7)('.+(!"-'.>#'+&-#D!-!&3.'&2'.>#'+&-#D!K/&7F$!%$+$'.8$!,(!7$17)1$+8$!/&$+&$#('.(L9!!!

10 11 12 13 14 15 16 Precios de ejercicio

Precio de mercado del activo subyacente: 13,1!

OTM ITM ATM



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ZZ ## !!(D+$(%1/!)1!+(CD'-!E(D+$(%1/!)1!+(CD'-!E!$"")+9:&+'-!$"")+9:&+'- FF !!D?@5<!:6!V9852!:6;!4<=M32:<3!Q)*-'%$#-,! 2)(! )'! &'C(+,-+! /(,($! $/2)&+&+! )'$! $77&6'! /(! y+&E-1,! *-+2)(! *&(',$! 2)(! ,)!7-.&4$7&6'!C$!$!,)3&+<!*(+-!*-+!$1%_'!#-.&C-!'-!*)(/(<!-!'-!2)&(+(<!*$%$+!1-,!Vo!()+-,!2)(!(1!#(+7$/-!1(!/(#$'/$<!('!(,.(!7$,-!*-/+8$!$/2)&+&+!)'$!-*7&6'!/(!7-#*+$!K)*00L!,-3+(!1$!#&,5#$9!!M1!$/2)&+&+!)'$!-*7&6'!/(!7-#*+$!,(!*-/+@!3('(E&7&$+!/(!)'!$)#('.-!('!(1!*+(7&-!/(1!$7.&C-!,)3G$7('.(!,&'!F$3(+1-!7-#*+$/-9!M,8!2)(!(1!&'C(+,-+!$/2)&(+(!)'$!-*7&6'!/(!7-#*+$!,-3+(!)'$!$77&6'!/(!y+&E-1,!7-'!)'!*+(7&-!/(!(D(+7&7&-!/(<!*-+!(D(#*1-<!Vo!()+-,9!:1!*+(7&-!/(!#(+57$/-!/(!/&7F$!-*7&6'!K1$!/!"+*L!('!(,(!#-#('.-!(,!/(!W<e!()+-,9!!!:1!*-,((/-+!/(! 1$!-*7&6'!/(!7-#*+$!,-3+(!y+&E-1,!K,(!/&7(!2)(!.&('(!)'$!/&$")"8-' 0*!%*!('!-*7&-'(,!/(!7-#*+$!*-+2)(!1$,!*-,((!G!)'$!/&$")"8-!)&!.*!('!$77&-'(,!*-+2)(!'-!1$,!.&('(L!*-/+@!/(7&/&+!,&!(D(+7(!-!'-!1$!-*7&6'9!q3C&$#('.(<!1$!(D(+7(+@!7)$'/-!1$!7-.&4$7&6'!/(!1$!$757&6'!,)*(+(!$1!*+(7&-!/(!(D(+7&7&-9!>-+!(1!7-'.+$+&-<!,&! 11(%$/$!1$!E(7F$!/(!C('7&#&('.-!/(!1$!-*7&6'<!(1!*+(7&-!/(!(D(+7&7&-!,&%)(!,&('/-!,)*(+&-+!$!1$!7-.&4$7&6'!K,&.)$7&6'!&3.'&2'.>#'+&4-#DL!1$!-*7&6'!'-!,(+@!(D(+7&/$<!/(3&/-!$!2)(!,(!*)(/(!$/2)&+&+!(1!$7.&C-!/&+(7.$#('.(!('!(1!#(+7$/-!$!)'!*+(7&-!&'E(+&-+!$1!/(!1$!-*7&6'9!Q&!1$!-*7&6'!'-!,(!(D(+7(!1$!*O+/&/$!#@[&#$!,(+@!/(!W<e!()+-,9!!



24

B-,!7-#('.$+&-,!*-,.(+&-+(,!,(!3$,$'!,-3+(!(1!,&%)&('.(!(D(#*1-N!pQ)*-'%$#-,!2)(!(1!*+(57&-!/(!)'$!$77&6'!/(!y+&E-1,<!('!(1!#-#('.-!/(!(#&.&+!1$!-*7&6'<!(,!/(!Vo!()+-,!('!(1!#(+7$5/-!/(!C$1-+(,!#$/+&1(0-9!:1!*+(7&-!/(!(D(+7&7&-!K$.!"?#'/!")#L!/(!1$!-*7&6'!/(!7-#*+$!()+-*($!(1(%&/$!(,!.$#3&O'!/(!Vo!()+-,9!:1!7-#*+$/-+!/(!1$!-*7&6'!*$%$!)'$!*+&#$!/(!W<e!()+-,9!B$!.+$',$77&6'!.&('(!1)%$+!('!,(*.&(#3+(!G!(1!7-'.+$.-!([*&+$!('!/&7&(#3+(9p!!! b(,)#&('/-N! Precio de la acción (S): 27 ! Precio de ejercicio de la opción de compra [X]: 27 !. Prima [c]: 1,3 ! Vencimiento del contrato: Diciembre !:1!&'C(+,-+!2)(!$/2)&(+(!)'$!-*7&6'!/(!7-#*+$!,-3+(!/&7F-!$7.&C-!$/2)&(+(!(1!/(+(7F-!$!$/52)&+&+1-!$!)'!*+(7&-!/(!(D(+7&7&-!(,*(7&E&7$/-!KVo!iL<!*(+-!'-!.&('(!1$!-31&%$7&6'!/(!(D(+7(+1-<!('!1$!E(7F$!/(!C('7&#&('.-9!>-+!/&7F-!/(+(7F-<!O1!-!(11$!*$%$!)'$!*+&#$!KW<e!iL9!!:'! 1$! E(7F$! /(! C('7&#&('.-! /(1! 7-'.+$.-! (1! 7-#*+$/-+! ,(! *)(/(! ('7-'.+$+<! *-+! (D(#*1-<!$'.(! 1-,! ,&%)&('.(,! 7$,-,! K*-+! +$4-'(,!/(! ,('7&11(4!'-! ,(! .&('('!('! 7)('.$!('! 1-,! 7@17)1-,!*-,.(+&-+(,!1-,!7-,.(,!/(!.+$',$77&6'<!'&!1-,!&#*)(,.-,<!$,8!7-#-!.$#*-7-!(1!C$1-+!.(#*-+$1!/(1!/&'(+-LN!!-[!Q&!(1!*+(7&-!/(!1$!$77&6'!(,!Q!h!eV!i!:1!&'C(+,-+!(D(+7(!1$!-*7&6'!$/2)&+&('/-!1$!$77&6'!$1!*+(7&-!/(!(D(+7&7&-!/(!Vo!i!G!+(C('/&O'5/-1$!,(%)&/$#('.(!('!(1!#(+7$/-!$1!*+(7&-!/(!eV!i9!q3.('&('/-!1-,!,&%)&('.(,!+(,)1.$/-,W\N!! Precio de compra 27,0 ! Prima 1,3 ! Coste total … … … … … … … 28,3 ! Ingreso total … … … … … … … 32,0 ! Beneficio de la operación: 3,7 ! !\[!Q&!(1!*+(7&-!/(!1$!$77&6'!(,!Q!h!V=!i!:1! &'C(+,-+!(D(+7(! 1$!-*7&6'!$1!*+(7&-!/(!(D(+7&7&-!/(!Vo!i!G! +(C('/(!(1!$7.&C-!$1!*+(7&-!/(!#(+7$/-!/(!V=!i9!q3.('&('/-!1-,!,&%)&('.(,!+(,)1.$/-,N!! Precio de compra 27,0 ! Prima 1,3 ! Coste total … … … … … … … 28,3 ! Ingreso total … … … … … … … 28,0 ! Beneficio de la operación: -0,3 ! H1$+-!2)(!,&!'-!(D(+7&(,(!1$!-*7&6'!*(+/(+8$!(1!7-,.(!/(!1$!#&,#$<!(,!/(7&+<!W<e!i!1-!2)(!,(+8$<!,&'!/)/$<!*(-+9!

16 b(7)(+/(!2)(!('!(1!f:UU!7$/$!7-'.+$.-!&#*1&7$!W;;!-*7&-'(,<!$,8!2)(!+($1#('.(!F$3+@!%$'$/-!eo;!()+-,9



25

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26

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!Å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



27

!M,8!2)(!(1!(#&,-+!/(!)'$!-*7&6'!/(!7-#*+$!KE!".#!L!'-!*)(/(!/(.(+#&'$+!,&! 1$!#&,#$!,(+@!(D(+7&/$!-!'-9!M,)#(!)'!*$*(1!*$,&C-!('!(,*(+$!/(!1$!/(7&,&6'!/(1!7-#*+$/-+!/(!1$!#&,#$9!>-+!.-/-!1-!7)$1<!+(7&3(!)'$!*+&#$!K(1!*+(7&-!/(!1$!-*7&6'L<!2)(!#(D-+$!,)!+('/&#&('.-9!>-+!-.+$!*$+.(<! /(3(+@!(,.$+! *+(*$+$/-!*$+$! ('.+(%$+! 1$,! $77&-'(,!2)(! 1(! ,($'! ,-1&7&.$/$,!*-+!*$+.(!/(1!*-,((/-+!/(!1$,!-*7&-'(,!('!(1!7$,-!/(!2)(!O,.(!_1.&#-!/(,((!(D(+7(+!,)!/(+(7F-9!j($#-,!$!.+$CO,!/(1!#&,#-!(D(#*1-!/(1!(*8%+$E(!$'.(+&-+!(1!7$,-!/(!1$!(#&,&6'!/(!)'$!-*57&6'!/(!7-#*+$!$1!/(,7)3&(+.-<!('!1$!E(7F$!/(!C('7&#&('.-!/(!1$!#&,#$9!!-[!Q&!(1!*+(7&-!/(!1$!$77&6'!(,!Q!h!eV!i!:1!7-#*+$/-+!(D(+7(!1$!-*7&6'!$1!*+(7&-!/(!(D(+7&7&-!/(!Vo!i9!:1!C('/(/-+!/(!1$!-*7&6'!-3.('5/+@!1-,!,&%)&('.(,!+(,)1.$/-,N!! Precio de venta 27,0 ! Prima 1,3 ! Ingreso total … … … … … … … … 28,3 ! Precio de mercado de la acción 32,0 ! Resultado de la operación: -3,7 ! !M,8!2)(!,&!(1!C('/(/-+!'-!*-,((!1$!$77&6'!*(+/(+@!e<o!()+-,9!q3,O+C(,(<!2)(!,&!1$!*-,(G(,(!G!1$!F)3&(,(!7-#*+$/-!(1!#&,#-!/8$!2)(!(#&.&6!1$!-*7&6'!1(!F$3+8$!7-,.$/-!Vo!i!G!('!1$!E(7F$!/(!C('7&#&('.-!1(!F$3+8$'!*$%$/-!*-+!(11$!Vo!i!2)(!,)#$/-,!$!1$!*+&#$<!/$+8$'!)'$!%$'$'57&$!*$+$!(1!(#&,-+!/(!W<e!i9!M)'2)(<!71$+-!(,.@<!/(D$+8$!/(!%$'$+!e<o!i!#@,!K1-,!l!()+-,!2)(!F)3&(+$!%$'$/-!/(!'-!F$3(+!(#&.&/-!1$!-*7&6'!#('-,!1$!*+&#$!/(!O,.$L9!!\[!Q&!(1!*+(7&-!/(!1$!$77&6'!(,!Q!h!V=!i!:1!7-#*+$/-+!(D(+7(!1$!-*7&6'!$1!*+(7&-!/(!(D(+7&7&-!/(!Vo!i9!:1!C('/(/-+!/(!1$!-*7&6'!-3.('5/+@!1-,!,&%)&('.(,!+(,)1.$/-,N!! Precio de venta 27,0 ! Prima 1,3 ! Ingreso total … … … … … … … … 28,3 ! Precio de mercado de la acción 28,0 ! Resultado de la operación: 0,3 ! !Q&!(1!(#&,-+!'-!*-,((!1$!$77&6'!/(3(+@!7-#*+$+1$!$!V=!()+-,!G!C('/(+1$!$!Vo!i<!*(+-!7-#-!('!,)!/8$!+(7&3&6!)'$!*+&#$!/(!W<e!i!,)!%$'$'7&$!,(+@!/(!;<e!i!K2)(!(,!1-!2)(!*&(+/(!(1!7-#*+$5/-+!5CO$,(!(1!(*8%+$E(!$'.(+&-+L9!MF-+$!3&('<!,&!1$!*-,(G(,(<!G!,)!*+(7&-!/(!$/2)&,&7&6'!F)3&(5+$!,&/-!/(!Vo!()+-,<!F$3+8$!-3.('&/-!)'$!%$'$'7&$!.-.$1!/(!V<e!i!,&!1$!-*7&6'!E)(+$!(D(+7&/$9!!!:'!(,.(!7$,-<!(1!7-#*+$/-+!/(3(!(D(+7(+!(1!/(+(7F-!*$+$!+(7)*(+$+!*$+.(!/(1!*+(7&-!*$%$/-!*-+! 1$!-*7&6'<!7-#-!G$!C&#-,!('!(1!(*8%+$E(!$'.(+&-+9!>-+! +(%1$!%('(+$1<!O,.(!(,!(1! .&*-!/(!.+$',$77&6'!#@,!&'.(+(,$'.(!*$+$!(1!(#&,-+!/(!-*7&-'(,!/(!7-#*+$!K)*00'E!".#!L!G!,)(1(!-7)5++&+!7)$'/-!(1!#(+7$/-!*(+#$'(7(!(,.$31(9!!



28

+[!Q&!(1!*+(7&-!/(!1$!$77&6'!(,!Q!h!Vl!i!:1!7-#*+$/-+!'-!(D(+7(+@!1$!-*7&6'!G!1$!*+&#$!,(+@!.-.$1#('.(!/(1!(#&,-+<!2)(!F$3+8$!%$'$5/-!W<e!()+-,9!>-+!-.+$!*$+.(<!,&!O,.(!_1.&#-!F)3&(+$!7-#*+$/-!1$!$77&6'!$!Vo!()+-,!-3.('/+8$!1-,!,&%)&('.(,!+(,)1.$/-,N!!!!!!! Precio inicial de la acción 27,0 ! Precio de mercado de la acción 25,0 ! Pérdida … … … … … … … … … … … … 2,0 ! Ingreso por la venta de la opción. ... 1,3 ! Resultado… … … … … … … … … … … - 0,7 ! !M,8!2)(!,&!(1!C('/(/-+!/(!1$!-*7&6'!/(,($!C('/(+!,),!$77&-'(,!('!(1!#(+7$/-<!('!7-'D)'.-!-3.('/+@!)'$!*O+/&/$!E&'$1!/(!o;!7O'.&#-,!/(!()+-!('!C(4!/(!)'$!*O+/&/$!/(!V!()+-,!,&!'-!F)3&(,(!(#&.&/-!1$!-*7&6'!*(+.&'('.(9!M,8!(,!7-#-!,(!*+-.(%(!/(1!+&(,%-!/(!*O+/&/$,<!.('&('5/-!$7.&C-,!K/&$")"8-'0*!%*L!G!(#&.&('/-<!$1!#&,#-!.&(#*-<!-*7&-'(,!/(!7-#*+$!,-3+(!1-,!#&,5#-,!K/&$")"8-')&!.*LP!('!7)G-!7$,-!(1!*+(7&-!/(!O,.$,!_1.&#$,!+(/)7&+@'!,),!*O+/&/$,!('!(1!7$5,-!/(!)'$!7$8/$!/(1!C$1-+!/(!$2)O11-,9!!:'!1$!E&%)+$!Wo<!,(!#)(,.+$!1$!%+@E&7$!/(1!+(,)1.$/-!/(!)'$!-*7&6'!/(!7-#*+$!('!1$!E(7F$!/(!,)!C('7&#&('.-<!/(,/(!(1!*)'.-!/(!C&,.$!/(1!C('/(/-+9!H-#-!,(!*)(/(!$*+(7&$+!('!/&7F$!E&5%)+$!1$!#@[&#$!%$'$'7&$!/(1!(#&,-+!C('/+@!/$/$!*-+!1$!*+&#$!/(!1$!-*7&6'!K7L9!f&('.+$,!2)(!1$!*O+/&/$!/(*('/(+@!/(!1$!/&E(+('7&$!('.+(!(1!*+(7&-!/(!#(+7$/-!(1!/8$!/(1!C('7&#&('.-!G!(1!*+(7&-!/(!(D(+7&7&-!K7!5!f@[!YQ5zP;^L!,&(#*+(!2)(!/&7F$!/&E(+('7&$!'-!,($!'(%$.&C$!*)(,<!,&!$,8!E)(,(<!,(!.-#$+8$!)'!C$1-+!')1-!*$+$!1$!#&,#$!/$/-!2)(!(1!3('(E&7&-!#@[&#-!*$+$!(1!(#&,-+!/(!1$!-*7&6'!(,!(1!C$1-+!/(!1$!*+&#$9!>(+-!,&! 1$!#@[&#$!%$'$'7&$!(,.@! 1&#&.$/$!'-!-7)++(!1-!#&,#-!7-'!1$,!*O+/&/$,!2)(!*)(/('!,(+!&1&#&.$/$,<!$1!#('-,!('!.(-+8$9!!

!U&%9Wo!y+@E&7$!/(1!+(,)1.$/-!,-3+(!)'$!-*7&6'!/(!7-#*+$!K,(%_'!(1!(#&,-+L!('!,)!E(7F$!/(!C('7&#&('.-!

!



29

b(,)#&('/-<!('!(,.$!*-,&7&6'!1$!*+&#$!2)(!+(7&3(!(1!(#&,-+!$)#('.$!1$!+('.$3&1&/$/!/(!,)!&'C(+,&6'<!$/(#@,<!('!(1!7$,-!/(!2)(!1-,!*+(7&-,!/(!1$!$77&6'!,)3G$7('.(!,)3$'<!1$!*+&#$!+(5/)7(!1$!*O+/&/$!2)(!(1!C('/(/-+!/(!1$!-*7&6'!F)3&(,(!.('&/-9!!!!

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pQ)*-'%$#-,!2)(!(1!*+(57&-!/(!)'$!$77&6'!/(!y+&E-1,<!('!(1!#-#('.-!/(!(#&.&+!1$!-*7&6'<!(,!/(!Vo!()+-,!('!(1!#(+7$5/-!/(!C$1-+(,!#$/+&1(0-9!:1!*+(7&-!/(!(D(+7&7&-!K$.!"?#'/!")#L!/(!1$!-*7&6'!/(!C('.$!(1(%&/$!(,!/(!Vo!()+-,9!:1!7-#*+$/-+!/(!1$!-*7&6'!*$%$!)'$!*+&#$!/(!W<W!()+-,9!B$!.+$',$77&6'!.&('(!1)%$+!('!Q(*.&(#3+(!G!(1!7-'.+$.-!([*&+$!('!T&7&(#3+(9p!!! b(,)#&('/-N! Precio del activo [S]: 27 ! Precio de ejercicio de la opción de venta [X]: 27 ! Prima [p]: 1,1 ! Vencimiento del contrato: Diciembre !:1!7-#*+$/-+!/(!)'$!-*7&6'!/(!C('.$!.&('(!(1!/(+(7F-!$!C('/(+!1$!$77&6'!$1!*+(7&-!/(!(D(+7&57&-!KVo!iL!&'/&7$/-!('!(1!7-'.+$.-!-!/(D$+!2)(!1$!-*7&6'!([*&+(!,&'!(D(+7(+1$<!/(*('/&('/-!/(!1$!(C-1)7&6'!/(1!#(+7$/-9!Q)*-'/+(#-,!2)(!(1! &'C(+,-+!'-!*-,((!(1!$7.&C-!,)3G$7('.(<!$,8!2)(!/(!&'.(+(,$+1(!C('/(+1-<!*+(C&$#('.(!/(3(+@!$/2)&+&+1-!$1!*+(7&-!/(!#(+7$/-!G<!,(%)&/$5#('.(<!,(!/(,F$+@!/(!O1!$!7$#3&-!/(1!*+(7&-!/(!(D(+7&7&-9!! !-[!Q&!(1!*+(7&-!/(!1$!$77&6'!(,!Q!h!eV!i!:1!/)(0-!/(!1$!-*7&6'!1$!/(D$+@!([*&+$+!,&'!(D(+7(+1$<!,&('/-!,),!*O+/&/$,!/(!W<W!i!(,!/(7&+<!(1!7-,.(!/(!1$!#&,#$9!!\[!Q&!(1!*+(7&-!/(!1$!$77&6'!(,!Q!h!V\!i!:1!*-,((/-+!/(!1$!-*7&6'!/(!C('.$!1$!(D(+7(+@<!*)(,.-!2)(!,&'-!*(+/(+@!1$!.-.$1&/$/!/(1!7-,.(!/(!1$!#&,#$N!W<W!i!! Precio de venta de la acción (X) 27,0 ! Precio pagado por la opción - 1,1 ! Ingreso total … … … … … … … … 25,9 ! Precio de coste de la acción … … 26,0 !



30

Resultado de la operación: - 0,1 ! !+[!Q&!(1!*+(7&-!/(!1$!$77&6'!(,!Q!h!V;!i!:1!*-,((/-+!/(!1$!-*7&6'!/(!C('.$!1$!(D(+7(+@<!*)(,.-!2)(!,&'-!*(+/(+@!1$!.-.$1&/$/!/(1!7-,.(!/(!1$!#&,#$N!W<W!i!! Precio de venta de la acción (X) 27,0 ! Precio pagado por la opción - 1,1 ! Ingreso total … … … … … … … … 25,9 ! Precio de coste de la acción … … 20,0 ! Resultado de la operación: 5,9 ! !!

!U&%9!W=!y+@E&7$!/(1!+(,)1.$/-!,-3+(!)'$!-*7&6'!/(!C('.$!K,(%_'!(1!7-#*+$/-+L!('!1$!E(7F$!/(!C('7&#&('.-!!

!:'! 1$! E&%)+$! W=! ,(!#)(,.+$! 1$! %+@E&7$! +(*+(,('.$.&C$! /(1! 3('(E&7&-! 2)(! *)(/(! -3.('(+,(! $!.+$CO,!/(!1$!*-,(,&6'!K7-#*+$L!/(!)'$!-*7&6'!/(!C('.$!G!2)(!')#O+&7$#('.(!F(#-,!$'$1&54$/-!*+(C&$#('.(!KM!&'/&7$!(1!*+(7&-!/(!1$!-*7&6'!/(!C('.$L9!!:'!+(,)#('<!1$!#@[&#$!*O+/&/$!*$+$!(1!7-#*+$/-+!/(!1$!-*7&6'!/(!C('.$!C('/+@!/(.(+#&'$5/$!*-+!(1!7-,.(!/(!1$!#&,#$!K*L9!f&('.+$,!2)(!1-,!+(,)1.$/-,!/(!,)!*-,&7&6'!&+@'!#(D-+$'/-!7)$'.-!#@,!/(,7&('/$!(1!*+(7&-!/(!#(+7$/-!/(!1$!$77&6'!,)3G$7('.(!Kf@[!Yz5QP;^!5!*L<!F$,.$!11(%$+!$!1$!#@[&#$!%$'$'7&$!2)(!,(!-3.&('(!7)$'/-!1$!7-.&4$7&6'!,($!')1$!Kz5*L9!!D?@5<!:6!V9852!:6;!6=98<3!:1!(#&,-+!/(!)'$!-*7&6'!/(!C('.$!7+((!2)(!1$!.('/('7&$!/(1!*+(7&-!/(!1$!$77&6'!,)3G$7('.(!,(+@!'().+$!-!1&%(+$#('.(!$17&,.$!G!1$!(#&,&6'!/(!(,.(!.&*-!/(!-*7&6'!1$!-E+(7(!1$!-*-+.)'&5/$/!/(!-3.('(+!)'!&'%+(,-!('!E-+#$!/(!*+&#$9!!



31

:1!C('/(/-+!-!(#&,-+!/(!)'$!-*7&6'!/(!C('.$!/(3(+@!$/2)&+&+!1$!$77&6'!,)3G$7('.(!$1!*+(7&-!/(!(D(+7&7&-!(,.&*)1$/-!KVo!()+-,L<!,&!(1!7-#*+$/-+!/(!1$!-*7&6'!1$!(D(+7(!/('.+-!/(1!*1$4-!$1!2)(!.&('(!/(+(7F-9!>-+!&'7)++&+!('!(,.(!+&(,%-!+(7&3&+@!)'$!*+&#$!K(1!*+(7&-!/(!1$!-*7&6'!/(!C('.$N!W<W!()+-,L9!!-[!Q&!(1!*+(7&-!/(!1$!$77&6'!(,!Q!h!eV!i!B$!-*7&6'!'-!,(+@!(D(+7&/$9!B$!$77&6'!'-!1(!,(+@!('.+(%$/$!*-+!(1!7-#*+$/-+!/(!1$!-*7&6'!G!(1!(#&,-+!/(!O,.$!F$3+@!%$'$/-!1$!*+&#$!/(!W<W!i9!!\[!Q&!(1!*+(7&-!/(!1$!$77&6'!(,!Q!h!V\!i!:1!*+-*&(.$+&-!/(!1$!-*7&6'!/(!C('.$!1$!(D(+7(+@<!*-+!1-!2)(!('.+(%$+@!$1!C('/(/-+!/(!1$!#&,5#$!,)!$77&6'!$1!*+(7&-!/(!(D(+7&7&-!/(!Vo!i! 1-!2)(!.('/+@! 1-,!,&%)&('.(,!+(,)1.$/-,!*$+$!,)!(#&,-+N! Precio de compra de la acción (X) 27,0 ! Precio cobrado por la opción - 1,1 ! Gasto total … … … … … … … … … 25,9 ! Precio de mercado de la acción … 26,0 ! Resultado de la operación: 0,1 ! !+[!Q&!(1!*+(7&-!/(!1$!$77&6'!(,!Q!h!V;!i!:1!7-#*+$/-+!/(!1$!-*7&6'!/(!C('.$!1$!(D(+7(+@<!*-+!1-!2)(!('.+(%$+@!$1!C('/(/-+!/(!1$!#&,5#$!,)!$77&6'!$1!*+(7&-!/(!(D(+7&7&-!/(!Vo!i! 1-!2)(!.('/+@! 1-,!,&%)&('.(,!+(,)1.$/-,!*$+$!(1!(#&,-+N!! Precio de compra de la acción (X) 27 ! Precio cobrado por la opción - 1,1 ! Gasto total … … … … … … … … … 25,9 ! Precio de mercado de la acción … 20,0 ! Resultado de la operación: -5,9 ! !

!U&%9Wn!!y+@E&7$!/(1!+(,)1.$/-!,-3+(!)'$!-*7&6'!/(!C('.$!K,(%_'!(1!(#&,-+L!('!1$!E(7F$!/(!C('7&#&('.-!

!



32

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!U&%9V;!b(,)#('!/(!1$,!*-,&7&-'(,!,&#*1(,!7-'!-*7&-'(,!,(%_'!1$,!([*(7.$.&C$,!2)(!.('%$!(1!&'C(+,-+!,-3+(!(1!

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E).)+-,!

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33

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Los inversores europeos se forran con los warrants Cotizalia 7/08/2012 Están encantados con el desplome de Facebook, no desean bajo ningún concepto la salida del fundador Marck Zuckerberg del puesto de primer ejecutivo y ni les suena la expresión de nuevo cuño ‘zucked’, con la que se denomina a la legión de damnifi-cados por el desastre bursátil de la red social. Son los osos17 europeos de Facebook, que gracias a sus apuestas bajistas contra la acción la red a través de ‘put wa-rrants’ han conseguido rentabilidades de hasta el 500%. Casi tres meses después del debut, la foto fija de Facebook refleja unos nú-meros rojos brutales para los miles de inversores que compraron acciones en la ofer-ta pública de venta. Paradójicamente, el desplome del 42% de la cotización se ha convertido en una oportunidad de ganar dinero para quienes remaron contra el mer-cado y compraron productos estructurados para beneficiarse de un desplome de la cotización. Muchos de ellos son inversores europeos. Se da la circunstancia de que la OPV fue vendida casi íntegramente en Estados Unidos y que en países como España fue imposible comprar títulos. Un gran favor, visto lo visto. En algunos casos, los inversores han obtenido retornos de hasta el 500% con los ‘put warrants’ emitidos en el Viejo Continente por BNP Paribas, Commerzbank y UBS. Estos bancos ofrecieron warrants que permitían apostar por las caídas de Facebook hasta los 12 dólares y con subidas hasta los 150 dólares. Nueve de los diez títulos más negociados hoy son ‘puts’. Los ‘warrants’ son productos cotizados en bolsa, que a cambio del pago de una prima muy inferior al precio bursátil otorgan a su propietario el derecho teórico, pero no la obligación, de comprar a un precio determinado un activo a través de un ‘warrant call’ o de venderlo a mediante un ‘warrant put’ en un fecha determinada. Son productos de alto riesgo porque su característica más destacada es el apalancamiento. Permiten posicionarse en un activo con una inversión mucho más pequeña ampliando las variaciones del activo subyacente en su precio. Los inversores que no hicieron oídos sordos a los analistas que denunciaban que el precio de Facebook está inflado y apostaron porque el precio alcanzaría los 22 dólares, pagaron 0,07 dólares por este producto una semana después de que la red social empezara a cotizar a 38 dólares por acción. Hoy, con Facebook cotizando a 21,92 dólares, el warrant cuesta 0,45 dólares. Facebook, que nunca ha cotizado por encima por encima del precio de salida a bolsa, ha permitido a los osos hacer un negocio redondo. Máxime después de la pre-sentación de los primeros resultados de Facebook como empresa cotizada el pasado 26 de julio, que resultaron un fiasco que provocó un desplome del 21% sin solución de continuidad.

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X CALLS PUTS Enero Febrero Marzo Enero Febrero Marzo

9.800 210 423 557 97 260 353 9.850 182 397 532 118 283 376 9.900 159 372 507 148 307 400 9.950 134 348 484 179 333 425 10.000 113 326 461 202 359 451 10.050 93 304 438 230 387 478 10.100 77 283 417 266 416 506 10.150 64 264 396 303 446 534 10.200 50 245 376 340 477 563 10.250 42 228 357 379 510 594

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S0 = c + X1 + rf

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38

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c = S0 - X1 + rf

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!

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> 0 ; "c"X

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> 0 ; "c"#

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!

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39

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> 0 ; "c"t

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"S"S ## !! 1.!Cd&()(!\$%(C$-.!)1.!Cd&()(!\$%(C$-.!)1!O-.('-+$,%!)1!(D+$1!O-.('-+$,%!)1!(D+$(%1/(%1/ !! !!H-[<! b-,,! G! b)3&',.(&'V;! /(,$++-11$+-'!(,.(!#O.-/-!/(! C$1-+$7&6'!/(!-*7&-'(,<! 2)(! .&('(! 1$!C('.$D$!/(!2)(<!$/(#@,!/(!,(+!#)G!&'.)&.&C-<!).&1&4$!)'$!#$.(#@.&7$!#)G!,('7&11$9!>$+$!#-,.+$+!,)!E)'7&-'$#&('.-!C$#-,!$!$*1&7$+1-!$! 1$!C$1-+$7&6'!/(!$77&-'(,!-+/&'$+&$,!G!*$+$!F$7(+!#@,!,&#*1(!1$!([*-,&7&6'!7-#('4$+(#-,!,)*-'&('/-!2)(!1$!$77&6'!'-!+(*$+.(!/&C&/('/-,9!!1;!=e5<:<!J9@<=92;!M232!?@!M637<:<!Q)*-'%$!2)(!(1!C$1-+!/(! 1$!$77&6'!/(1!`$'7-!Q$'.$'/(+!(,!/(!=!i!G! 1-,! &'C(+,-+(,!*&(',$'!2)(!/('.+-!/(!)'!$0-!*)(/(!$17$'4$+!)'!C$1-+!/(!WV<;l!i!-!)'-!/(!l<eW!i9!>-+!,)*)(,.-<!),5.(/!*(',$+@<!$1!#('-,<!/-,!7-,$,N!?>-+!2)O!'-!*)(/(!.-#$+!#@,!C$1-+(,!/('.+-!/(!)'!$0-A!G<!/(!.-#$+!,61-!/-,<!?*-+!2)O!*+(7&,$#('.(!(,-,!/-,A9!B$!+(,*)(,.$!$!1$!*+&#(+$!*+(%)'.$!C&('(!/$/$!*-+!(1!'-#3+(!/(1!#O.-/-!/(!C$1-+$7&6'N!,"-&+"*0<!1$!,81$3$!I3&J!&#*1&7$!2)(!('!(1!*(+&-/-!/(!.&(#*-!,&%)&('.(!K)'!$0-!('!')(,.+-!7$,-L!,61-!,(!*)(/('!.-#$+!/-,!C$1-+(,P!(,.-!'-!(,!)'!*+-31(#$!7-#-!C(+(#-,!#@,!$/(1$'.(!$,8!2)(!/(!#-#('.-!I,8%$#(!(1! D)(5%-J9!B$!+(,*)(,.$!$!1$!,(%)'/$!*+(%)'.$!C&('(!/$/$!*-+!1$!C-1$.&1&/$/!/(!1$!$77&6'!/(1!Q$'5.$'/(+!('!(1!#(+7$/-!/(!C$1-+(,!K)'!kWaL9!q.+$!7-,$!&'.(+(,$'.(!(,!2)(!1$!*+-3$3&1&/$/!/(!2)(!-7)++$!)'!+(,)1.$/-!KWV<;l!iL!-!(1!-.+-!Kl<eW!iL!'-!&#*-+.$<!,61-!&'.(+(,$!(1!+$'%-!/(!+(5,)1.$/-,!*-,&31(,9!!!

V;!Hqz<c9<!bqQQ<Q9<!G!bS`"]QX:"]<f9N!pq*.&-',!*+&7&'%N!$!,&#*1&E&(/!$**+-$7Fp9!N&3!-*0'&2';"-*-)"*0'<)&-&+")$9!'t!o9!Wnon9!

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40

`&('!*$,(#-,!$F-+$!$!7$17)1$+!(1!C$1-+!2)(!.('/+8$!F-G!#&,#-!)'$!-*7&6'!/(!7-#*+$!/(!.&*-!()+-*(-!K,61-!,(!*)(/(!(D(+7(+!('!1$!E(7F$!/(!,)!C('7&#&('.-<!(,!/(7&+<!/('.+-!/(!)'!$0-L!,-3+(!1$!$77&6'!/(1!Q$'.$'/(+!/(,7+&.$!('!(1!*@++$E-!$'.(+&-+!G!2)(!*-,((!)'!*+(7&-!/(!(D(+7&57&-!/(!=!i9!Q)!C$1-+!$7.)$1<!2)(!(,!(1!2)(!*+(.('/(#-,!F$11$+<!(,!/(!4!i<!#&('.+$,!2)(!,)!C$51-+!&'.+8',(7-!('!1$!E(7F$!/(!C('7&#&('.-!/('.+-!/(!)'!$0-!,(+@!KE&%)+$!VoLN!!!

$L!7)!h!k<;l!i<!,&!1$!$77&6'!,(!,&._$!('!WV<;l!i!K(,!(1!#@[&#-!C$1-+!('.+(!WV<;l!5!=!G!;L!3L!7/!h!;!i!,&!1$!7-.&4$7&6'!/(!1$!$77&6'!/(,7&('/(!$!l<eW!i!K#@[&#-!C$1-+!('.+(!l<eW!5!=!

G!;L9!!!

!U&%9!Vo!>+(7&-,!/(!1$!$77&6'!-+/&'$+&$!G!C$1-+(,!/(!,)!-*7&6'!/(!7-#*+$!

!S'$!E-+#$!/(!C$1-+$+!)'!$7.&C-!E&'$'7&(+-!K)'$!-*7&6'!/(!7-#*+$!('!')(,.+-!7$,-L!7-',&,.(!('!,$3(+!7)@'.-!C$1(!-.+-!$7.&C-!E&'$'7&(+-!-!)'$!7-#3&'$7&6'!/(!$7.&C-,!E&'$'7&(+-,<!$1!2)(!,(!/('-#&'$!$7.&C-!I%(#(1-J<!2)(!%('(+(!([$7.$#('.(!1-,!#&,#-,!E1)D-,!/(!7$D$!2)(!(1!$7.&5C-!$!C$1-+$+!G!2)(!.('%$!(1!#&,#-!+&(,%-9!:,.(!#O.-/-!1-!C$#-,!$!).&1&4$+!*$+$!C$1-+$+!1$!-*57&6'!/(!7-#*+$!,-3+(!1$!$77&6'!/(1!Q$'.$'/(+9!B$!7$+.(+$!2)(!C$#-,!$!).&1&4$+!7-#-!7-#*$5+$7&6'!K.$#3&O'!7-'-7&/$!7-#-!)*!.#!*'!6/0")*<!*-+2)(!+(*1&7$!1-,!E1)D-,!/(!7$D$!/(1!$7.&C-!$!C$1-+$+L!,(!C$!$!7-#*-'(+!/(!Å!$77&-'(,!/(1!Q$'.$'/(+!G!/(!)'!*+O,.$#-!2)(!F(#-,!7-'5.+$8/-!*-+!`!i!$!)'!.&*-!/(!&'.(+O,!,&'!+&(,%-!K+EL!s!'-!.&('(!+&(,%-!*-+2)(!('!.-/-!#-#('.-!F$3+@!/&'(+-!*$+$!/(C-1C(+!(1!*+O,.$#-!7-#-!C(+(#-,!(',(%)&/$59!>-+!.$'.-<!/('.+-!/(!)'!*(+8-/-!$')$1<!1-,!E1)D-,!/(!7$D$!/(!/&7F$!7$+.(+$!*)(/('!.-#$+!1-,!/-,!C$1-+(,!,&%)&('.(,N!! Si Su = 12,05 ! Ü 12,05 H - (1 + rf) B = 4,05 ! Si Sd = 5,31 ! Ü 5,31 H - (1 + rf) B = 0 ! !(,!/(7&+<!('!(1!7$,-!/(!2)(!1$!$77&6'!$17$'7(!1-,!WV<;l!i!(1!C$1-+!/(!1$!7$+.(+$!,(+@!/(!Å!$757&-'(,!$!WV<;l!i!7$/$!)'$!KWV<;l!ÅL!#('-,!1$!/(C-1)7&6'!/(1!*+O,.$#-!7-'!,),!&'.(+(,(,!K`!Ç!`!+EL!.-/-!1-!7)$1!(,!&%)$1!$!k<;l!i9!T(!E-+#$!,(#(D$'.(!,(!([*1&7$!1$!,(%)'/$!(7)$7&6'9!b(,5.$'/-!$#3$,!(7)$7&-'(,!-3.('/+(#-,!(1!'_#(+-!KÅL!/(!$77&-'(,!/(1!Q$'.$'/(+!2)(!,(!/(53('!7-#*+$+!*$+$!7-',.&.)&+!1$!7$+.(+$N!! 6,74 H = 4,05 Ü H = 4,05/6,74 = 0,6009 acciones !Q&!(1!.&*-!/(!&'.(+O,!,&'!+&(,%-!+E!(,!&%)$1!$1!ea!$')$1!*-/(#-,!([.+$(+!(1!C$1-+!/(!`!('!7)$152)&(+$!/(! 1$,!/-,!(7)$7&-'(,!$'.(+&-+(,N!e<;n=!i9!q3,O+C(,(!2)(!,&(#*+(!F$G!/&'(+-!*$+$!*$%$+!(1!*+O,.$#-!#@,!,),!&'.(+(,(,!K,&!(1!C$1-+!/(!1$!$77&6'!E)(,(!&'E(+&-+!$!l<eW!i!(1!C$1-+!



41

/(!`!.$#3&O'!,(+8$VW!&'E(+&-+!$!e<;n=!i!/(!.$1!#$'(+$!2)(!,&(#*+(!F$3+8$!/&'(+-!*$+$!*$%$+!(1!,(+C&7&-!/(!1$!/()/$L9!B)(%-!(1!C$1-+!/(!1$!-*7&6'!/(!7-#*+$<!4<!('!1$!$7.)$1&/$/<!,(+@!&%)$1!$1!C$1-+!$7.)$1!/(!1$!7$+.(+$!E-+#$/$!*-+!Å!$77&-'(,!#@,!)'$!/()/$!/(!`!()+-,<!(,!/(7&+N!! c = S H – B = 8 x 0,6009 – 3,098 = 1,7092 ! !M'.(,!/(!7-'.&')$+<!-3,(+C(!2)(!1$!7-#3&'$7&6'!E-+#$/$!*-+!Å!$77&-'(,!/(1!Q$'.$'/(+!G!1$!C('.$!/(!)'$!_'&7$!-*7&6'!/(!7-#*+$!*+-*-+7&-'$!(1!#&,#-!+(,)1.$/-!/('.+-!/(!)'!$0-!,($!7)$1!,($!(1!C$1-+!E).)+-!/(!1$!$77&6'!/(1!3$'7-N!! Si Su = 12,05 ! Ü 12,05 x 0,6009 – 4,05 = 3,191 ! Si Sd = 5,31 ! Ü 5,31 x 0,6009 – 0 = 3,191 ! !(,!/(7&+<!'-!F$G!+&(,%-<!-.+-!#-.&C-!#@,!*$+$!).&1&4$+!(1!.&*-!/(!&'.(+O,!,&'!+&(,%-!K+EL9!U8D(,(!2)(!,&!/(,7)('.$!$1!ea!/(!&'.(+O,!e<WnW!-3.&('(!)'!C$1-+!/(!e<;n=!i!2)(!(,!(1!C$1-+!/(1!*+&'57&*$1!/(!1$!/()/$9!!1;!=<:6;<!K6@632;!Q&!$F-+$!2)(+(#-,!-3.('(+!(1!C$1-+!/(!1$!-*7&6'!/(!7-#*+$!#(/&$'.(!)'$!([*+(,&6'!#$.(5#@.&7$!%('(+$1<!1-!*+&#(+-!2)(!F$+(#-,!,(+@!+(*+-/)7&+!(1!C$1-+!&'.+8',(7-!/(!1$!-*7&6'!/(!7-#*+$!/('.+-!/(!)'!*(+8-/-!(!&%)$1$+1-!$!1-,!E1)D-,!/(!7$D$!/(!1$!)*!.#!*'!6/0")*N!! cu = Su H - (1 + rf) B cd = Sd H - (1 + rf) B !/-'/(!Q!(,!(1!*+(7&-!$7.)$1!/(!1$!$77&6'!,)3G$7('.(!K1$!/(1!Q$'.$'/(+!('!')(,.+-!(D(#*1-L<!Q)!,(+@!(1!*+(7&-!/(!1$!$77&6'!/('.+-!/(!)'!*(+8-/-!,&!(,!$17&,.$<!*)(,!,&!E)(,(!3$D&,.$!,(! 1(!/('-#&'$+8$!Q/!K/-'/(!?!G!:!,-'!1-,!7-(E&7&('.(,!*-+!1-,!2)(!F$G!2)(!#)1.&*1&7$+!Q!*$+$!-35.('(+!(1!*+(7&-!/(!1$!$77&6'!$1!E&'$1!/(1!*(+8-/-P!?!a!6!

!G!:!a!6"!!a!"f?!s('!')(,.+-!(D(#*1-VV!)!h!(;<kW!h!W<l;\e!G!/!h!WR)!h!;<\\k5L9!>-+!-.+$!*$+.(<!(1!*+(7&-!/(!1$!-*7&6'!/(!7-#*+$!('!1$!$75.)$1&/$/!,(+8$!4<!,&('/-!7)!G!7/<!+(,*(7.&C$#('.(<!*$+$!1-,!7$,-,!('!2)(!(1!*+(7&-!/(!1$!$77&6'!F$G$!$,7('/&/-!-!F$G$!3$D$/-9!Q&!$F-+$!+(,.$#-,!)'$!(7)$7&6'!/(!1$!-.+$!G!/(,*(D$#-,!(1!C$1-+!/(!Å<!-3.('/+(#-,!(1!C$1-+!/(1!/('-#&'$/-!!*."&'(#')&,#!.3!*N!!

!

H = cu - cd

S (u - d) !!:1!,&%)&('.(!*$,-<!,(+@!/(,*(D$+!`!('!)'$!/(!1$,!(7)$7&-'(,!$'.(+&-+(,N!! cu = Su H – B (1+rf) ! B = (Su H – cu ) ÷ (1 + rf)

21 q3,O+C(,(!2)(!,&!7/!h!Q/!Å!5!KWÇ+EL!`!!!`!h!YQ/!Å!s!7/^!r!KWÇ+EL9!b(7)(+/(<!7/!á!;!1)(%-!`!~!;!*-+2)(!Q/!Å!á!KWÇ+EL!`! 22 Q&!1$!C-1$.&1&/$/<!!<!(,.@!+(E(+&/$!$1!#&,#-!*(+&-/-!$')$19!Q&!(,.)C&(+$!+(E(+&/$!$!)'!*(+&-/-!/&,.&'.-!1$!([*+(5

,&6'!,(+8$!

!

u = e" t/n !('!1$!2)(!'!&'/&7$+8$!(1!'_#(+-!/(!,)3*(+&-/-,!('!1-,!2)(!,(!,)3/&C&/(!.9!>-+!(D(#*1-<!,&!1$!C-1$.&1&/$/!E)(,(!$')$1!*(+-!1-,!*(+&-/-,!/(1!@+3-1!E)(,('!#(',)$1(,!.!h!W!G!'!h!WV9



42

!G!,),.&.)&+!,)!C$1-+!('!1$!(7)$7&6'!4!h!Q!Å!s!`!! c = S H - (Su H - cu ) ÷ (1 + rf) c (1 + rf) = S H + S H rf - Su H - cu c (1 + rf) = S H (1 + rf - u) + cu !,),.&.)G('/-!$F-+$!Å!*-+!,)!C$1-+N!!

c (1 + rf) = S

!

cu - cdS (u - d)

(1 + rf - u) + cu

!! MF-+$<!F$7&('/-!)'!$1.-!('!')(,.+$!/(#-,.+$7&6'<!C$#-,!$!/('-#&'$+N!!

! a) p =

!

1 + rf - du - d

b) 1- p =!!

!

u - (1 + rf )u - d !

!:,.-,!C$1-+(,!+(*+(,('.$'!1$!*+-3$3&1&/$/!&#*187&.$!/(!$,7(',-!K*L!G!/(!/(,7(',-!KW5*L!/(1!C$51-+!/(!1$!$77&6'!,)3G$7('.(9!M,8<!*-+!(D(#*1-<!,&!,),.&.)&#-,!('!1$!(7)$7&6'!/(!*!1$,!C$+&$31(,!*-+!1-,!/$.-,!/(1!(D(#*1-!7-'!(1!2)(!C('&#-,!.+$3$D$'/-!-3.('/+(#-,!/&7F$,!*+-3$3&1&/$5/(,N!! p = (1 + 0,03 - 0,664) ÷ (1,5063 - 0,664) = 43,46% de que ascienda 1-p = 56,54% de que descienda !>-+! .$'.-<! ,&! $F-+$! +(.-#$#-,! ')(,.+$! /(#-,.+$7&6'! G! ,),.&.)&#-,! *$+.(! /(! 1$! (7)$7&6'!$'.(+&-+!*-+!(1!C$1-+!/(!W5*<!-3.('/+(#-,N!!

c (1 + rf) = S

!

cu - cdS (u - d)

(1 + rf - u) + cu =

!

cu - cdu - d

(1 + rf - u) + cu = (cu - cd) (p-1) + cu

!$F-+$!/(,*(D$'/-!4<!-3.('/+(#-,!1$!([*+(,&6'!2)(!7$17)1$!(1!C$1-+!/(!1$!-*7&6'!/(!7-#*+$!,(%_'!(1!#O.-/-!3&'-#&$1!2)(<!7-#-!,(!*)(/(!$*+(7&$+<!7-',&,.(!('!7$17)1$+!1$!#(/&$!*-'5/(+$/$!/(!1-,!E1)D-,!/(!7$D$!*+-*-+7&-'$/-,!*-+! 1$!-*7&6'!/(!7-#*+$!.$'.-!,&!(1!*+(7&-!/(1!$7.&C-!,)3G$7('.(!$,7&('/(!7-#-!,&!/(,7&('/(<!G!).&1&4$'/-!7-#-!*-'/(+$7&-'(,!1$,!*+-3$53&1&/$/(,!&#*187&.$,!/(!2)(!/&7F-!*+(7&-!/(1!$7.&C-!,)3$!-!7$&%$9!{!.-/-!(11-!$7.)$1&4$/-!$1!.&5*-!1&3+(!/(!+&(,%-N!!

c = cu p - cu - cd p + cd + cu = cu p + cd (1-p) !

!

c = cu p + cd (1-p)1 + rf

!

!!



43

H-'7+(.$'/-<!(1!*+(7&-!.(6+&7-!/(! 1$!-*7&6'!/(!7-#*+$!(,! &%)$1!$1!C$1-+!$7.)$1!/(! 1$!#(/&$!*-'/(+$/$!/(! 1-,! E1)D-,!/(! 7$D$!2)(!*+-*-+7&-'$9!>$+$!/(#-,.+$+!2)(!O,.$!(,! 1$!(7)$7&6'!2)(!3),7$#-,!,),.&.)&+(#-,!1$,!C$+&$31(,!*-+!,),!C$1-+(,VeN!! c = (4,05 x 0,4346 + 0 x 0,5654) ÷ (1,03) = 1,709 ! !M!M!G!"bM!,(!1$,!7-'-7(!7-#-!I*+-3$3&1&/$/(,!'().+$1(,!$1!+&(,%-J!K/(!$,7(',-!G!/(!/(,7('5,-L!*-+2)(!*$+(7('!*+-3$3&1&/$/(,!*(+-!à!'-!1-!,-'9!b($1#('.(!,-'!1-,!*+(7&-,!.&(#*-5(,5.$/-!/(!1-,!/-,!*-,&31(,!(,.$/-,!K$,7(',-5/(,7(',-L!#)1.&*1&7$/-,!*-+!WÇ+E9!:1!'-#3+(!*-+!(1!2)(!,-'!7-'-7&/$,N!/!&,*,"0"(*('-#3.!*0'*0' !"#$%&! C&('(!/$/-!*-+N!$L!$#3$,!,)#$'! 1$!)'&5/$/<! 7-#-! 1$,! *+-3$3&1&/$/(,! ,)3D(.&C$,P! 3L! $#3$,! ,-'!*-,&.&C$,<! 7-#-! 1$,! *+-3$3&1&/$/(,!,)3D(.&C$,P!G<!7L!7)$'/-!,(!).&1&4$'!*$+$!(,.&#$+!(1! +('/&#&('.-!(,*(+$/-!/(!)'!$7.&C-!7-'!+&(,%-!F$7('!2)(!1$!*+&#$!/(!+&(,%-!/(,$*$+(47$!K,-'!)'$!(,*(7&(!/(!#B3"5*0#-.#$'(#')#!.#41*!/(!1-,!E1)D-,!/(!7$D$!&'7&(+.-,L9!!1;!=e5<:<!J9@<=92;!M232!:<8!M637<:<8!H-'!-3D(.-!/(!-3.('(+!(1!C$1-+!/(!1$!-*7&6'!/(!7-#*+$!()+-*($!*$+$!C$+&-,!*(+8-/-,<!*+&#(5+$#('.(!C$#-,!$!$*1&7$+!(1!#O.-/-!3&'-#&$1!*$+$!)'!*$+!/(!(11-,9!{!C$#-,!$!F$7(+1-!).&1&54$'/-! 1-,!/$.-,!/(1! (D(#*1-! $'.(+&-+!/(! .$1!#$'(+$!2)(! C$#-,!$! C-1C(+! $! 7$17)1$+1-!*(+-!,)3/&C&/&O'/-1-!('!/-,!*(+&-/-,!,(#(,.+$1(,9!>$+$!(11-!/(3(#-,!+($/$*.$+!1-,!/$.-,!/(!1-,!2)(!/&,*-'(#-,!*$+$!*-/(+!#$'.('(+!1$!C-1$.&1&/$/!$')$1!/(1!kWa!$!.+$CO,!/(!/-,!*(+&-/-,!,(#(,.+$1(,!/(!.$1!#$'(+$!2)(!1$!C-1$.&1&/$/!,(#(,.+$1!(2)&C$1('.(Vk!,(+@!/(1!Vna!1-!2)(!&#5*1&7$!)'!7-(E&7&('.(!;!,(#(,.+$1!&%)$1!$!W<ee\e!G!)'$!<!,(#(,.+$1!/(!;<ok=e9!:'!7)$'.-!$1!.&5*-!/(!&'.(+O,!,&'!+&(,%-!,(#(,.+$1!*$,$!$!,(+!$F-+$!&%)$1!$!KW<;eLWRV!s!W!h!W<kna9!!!b(,)#&('/-!1-,!/$.-,!3@,&7-,!/(1!(D(#*1-!C$'!$!,(+N!! - Precio actual de la acción Santander = 8 ! - Precio de ejercicio de la opción de compra = 8 ! - Tiempo: 2 semestres - Tipo de interés sin riesgo: 1,49% semestral - Coeficiente de ascenso u = 1,3363 - Coeficiente de descenso d = 0,7483 !:1!*+&#(+!*$,-!(,!/&3)D$+!(1!@+3-1!3&'-#&$1!/(!1$!(C-1)7&6'!/(1!*+(7&-!/(!1$!$77&6'!/(1!Q$'.$'5/(+!$! 1-! 1$+%-!/(! 1-,!/-,!*+6[&#-,!,(#(,.+(,<!@+3-1!2)(!$*$+(7(!('! 1$! E&%)+$!V=9!>$+$!(11-!#)1.&*1&7$#-,!*-+!?! G!*-+!:!(1!C$1-+!$7.)$1!/(! 1-,!=!i! 1-!2)(!'-,!*(+#&.(!-3.('(+! 1-,!/-,!*+(7&-,!2)(!1$!$77&6'!*)(/(!.-#$+!$1!E&'$1!/(1!*+&#(+!,(#(,.+(N!W;<\n!i!G!l<n=o!i9!MF-+$!+(5*&.&('/-!1$!-*(+$7&6'!*$+$!7$/$!)'-!/(!(,.-,!/-,!*+(7&-,!/(1!*+&#(+!,(#(,.+(!-3.('/+(#-,!1-,!.+(,!*+(7&-,!K+($1#('.(!,-'!7)$.+-!*(+-!/-,!7-&'7&/('L!/(1!E&'$1!/(1!,(%)'/-!,(#(,.+(N!Wk<V=\!i<!=!i!G!k<k=!i9!

Ve!B$!#8'&#$!/&,7+(*$'7&$!('!1$!7)$+.$!7&E+$!/(7&#$1!,(!/(3(!$!1-,!(++-+(,!/(!+(/-'/(-9!24 !!,(#(,.+$1!h!!!$')$1!*-+!1$!+$84!7)$/+$/$!/(!WRV9!Vna!h!kWa![!KWrVLWRV9



44

!U&%9!V=!:C-1)7&6'!,(#(,.+$1!/(1!C$1-+!/(!1$!$77&6'!/(1!Q$'.$'/(+!

!:1!*$,-!,&%)&('.(!(,!7$17)1$+!(1!C$1-+! &'.+8',(7-!/(! 1$!-*7&6'!/(!7-#*+$!$1! E&'$1!/(1!$0-!.(5'&('/-!('!7)('.$!1-,!.+(,!*+(7&-,!*-,&31(,N!! Cuu = Máx [14,286 - 8 ; 0] = 6,286 ! Cud = Máx [8 - 8 ; 0 ] = 0 ! Cdd = Máx [4,48 - 8 ; 0] = 0 ! !

!U&%9!Vn!j$1-+(,!&'.+8',(7-,!/(!1$!-*7&6'!/(!7-#*+$!$1!E&'$1!/(1!$0-!

!:'!1$!E&%)+$!Vn!,(!-3,(+C$'!1-,!.+(,!C$1-+(,!&'.+8',(7-,!$'.(+&-+(,9!M,8!2)(!$F-+$!,(!.+$.$!/(!&+! 7$17)1$'/-!F$7&$! 1$! &42)&(+/$!/(1!@+3-1!3&'-#&$1! 1-,!C$1-+(,! &'.(+#(/&-,!/(! 1$!-*7&6'!/(!7-#*+$9!>$+$!(11-!).&1&4$+(#-,!1$,!*+-3$3&1&/$/(,!'().+$1(,!$1!+&(,%-9!!B$!*+-3$3&1&/$/!/(!$,7(',-!M!(,!&%)$1!$N!!

! p =

!

1 + rf - du - d =

1,0149 " 0,74831,3363 " 0,7483= 45,33%

!MF-+$!*-/(#-,!7$17)1$+!7)!G!7/!KCO$,(!1$!E&%)+$!e;LN!!

!!!!!!!cu =

!

cuu p + cud (1-p)1 + rf

= 6,286 x 0,4533 + 0 x (1- 0,5467)1, 0149 = 2,808 !

cd =

!

cud p + cdd (1-p)1 + rf

= 0 x 0,4533 + 0 x (1- 0,5467)1, 0149 = 0 !

!



45

!U&%9!e;!j$1-+(,!&'.+8',(7-,!/(!1$!-*7&6'!/(!7-#*+$!$1!E&'$1!/(1!$0-!G!C$1-+(,!/(!1$!-*7&6'!$1!E&'$1!/(1!*+&#(+!

,(#(,.+(9!

!>-+!_1.&#-!7$17)1$#-,!(1!C$1-+!/(!4N!!

! c =

!

cu p + cd (1-p)1 + rf

= 2,808 x 0,4533 + 0 x (1- 0,5467)1, 0149 = 1,254 !

!

!U&%9!eW!:C-1)7&6'!/(!1-,!C$1-+(,!.(6+&7-,!/(!1$!-*7&6'!/(!7-#*+$!

!:,.(!C$1-+!/(!1$!-*7&6'!/(!7-#*+$!,-3+(!1$!$77&6'!/(1!Q$'.$'/(+!(,!/&E(+('.(!/(1!7$17)1$/-!$'.(+&-+#('.(9!T(!F(7F-!,&!/&C&/&#-,!(1!$0-!('!.+(,!7)$.+&#(,.+(,!(1!C$1-+!/(!1$!-*7&6'!,(+@!/(!W<lW!i<!,&!1-!,)3/&C&/&#-,!('!7)$.+-!.+&#(,.+(,!(1!C$1-+!/(!1$!-*7&6'!,(+@!/(!W<eV!i<!,&!1-!,)3/&C&/&#-,!('!7&'7-!*(+&-/-,!(1!C$1-+!(,!/(!W<k\!i!G!$,8!,)7(,&C$#('.(!(1!C$1-+!,(!&+@!$*+-5[&#$'/-!$!W<k;!i!2)(!(,!(1!C$1-+!+($1!/(!1$!-*7&6'!/(!7-#*+$!KCO$,(!1$!E&%)+$!eVL9!!

U&%9!eV!:C-1)7&6'!/(1!C$1-+!/(!1$!-*7&6'!/(!7-#*+$!,(%_'!1$,!&.(+$7&-'(,!/(1!#-/(1-!3&'-#&$1!

!



46

H-'E-+#(!C$G$#-,!$)#('.$'/-!(1!'_#(+-!/(!,)3*(+8-/-,!G<!*-+!7-',&%)&('.(<!+(/)7&('/-!(1! .&(#*-!/(! 1-,!#&,#-,!*$,$+(#-,!/(!7-',&/(+$+!(1! .&(#*-!7-#-!)'$!C$+&$31(!/&,7+(.$!$!7-',&/(+$+1-!)'$!C$+&$31(!7-'.&')$9!:'!+($1&/$/<!*$+$!)'-,!+(,)1.$/-,!C@1&/-,!(1!.&(#*-!F$,5.$! (1! C('7&#&('.-! K)'! $0-!('!')(,.+-! (D(#*1-L! /(3(+8$! ,(+! /&C&/&/-! $1!#('-,! ('!)'-,! l;!,)3*(+8-/-,9!!!!!

"""" ## !! 1.!C()1.(!)1!\.-+g!R1.!C()1.(!)1!\.-+g!R !/+h(.1/!/+h(.1/ !!:1!#-/(1-!/(,$++-11$/-!*-+!1-,!*+-E(,-+(,!U&,7F(+!`1$7g!G!fG+-'!Q7F-1(,Vl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

W9!:1!*+(7&-!/(1!$7.&C-!,&%)(!)'$!/&,.+&3)7&6'!'-+#$1!1-%$+8.#&7$<!*-+!1-!2)(!1-,!+('/&5#&('.-,!,(!/&,.+&3)G('!'-+#$1#('.(9!

V9!:1!C$1-+!/(!1-,!+('/&#&('.-,!(,!7-'-7&/-!G!(,!/&+(7.$#('.(!*+-*-+7&-'$1!$1!*$,-!/(1!.&(#*-9!

e9!]-!F$G!7-,.(,!/(!.+$',$77&6'<!$,8!2)(!,(!*)(/(!(,.$31(7(+!)'$!7-3(+.)+$!,&'!+&(,%-,!('.+(!(1!$7.&C-!G!1$!-*7&6'!,&'!'&'%_'!7-,.(9!

k9!B-,!.&*-,!/(!&'.(+O,!,-'!7-'-7&/-,!G!7-',.$'.(,9!l9!T)+$'.(!(1!*(+8-/-!/(!(D(+7&7&-<!1$!$77&6'!,)3G$7('.(!'-!*$%$+@!/&C&/('/-,9!\9!B$,!-*7&-'(,!,-'!/(!.&*-!()+-*(-!

!:1!#-/(1-!/(,$++-11$/-!*-+!`1$7g!G!Q7F-1(,<!7)G$!E6+#)1$!/(!C$1-+$7&6'!/(!-*7&-'(,!()+-5*($,!#-,.+$#-,!,(%)&/$#('.(<! +(,)(1C(!(1!*+-31(#$! E)'/$#('.$1!/(! 1$!C$1-+$7&6'!/(! 1$,!#&,#$,!2)(!7-',&,.(!('!2)(!/$/-,!(1!.&(#*-!2)(!E$1.$!F$,.$!,)!C('7&#&('.-!K5L<!(1!.&*-!1&3+(!/(!+&(,%-!K3

>L<!(1!*+(7&-!/(!(D(+7&7&-!/(!1$!-*7&6'!KcL!G!1$!C$+&$'4$!/(!1$!.$,$!/(!+('.$3&1&/$/!

&',.$'.@'($!K!VL<!F$3+@!2)(!/(.(+#&'$+!1$!+(1$7&6'!([&,.('.(!('.+(!(1!7-,.(!/(!1$!-*7&6'!/(!7-#*+$!()+-*($!K4L!G!(1!*+(7&-!/(!1$!$77&6'!,-3+(!1$!2)(!+(7$(!KQ

-L9!T&,*-'&('/-!/(!)'!#-5

/(1-!2)(!-E+(7&(,(!.$1!+(1$7&6'<!7$/$!/8$!,(!*-/+8$!/(.(+#&'$+!2)O!-*7&-'(,!,(!('7)('.+$'!&'E+$C$1-+$/$,!G!7)@1(,!,-3+(C$1-+$/$,!#(/&$'.(! 1$!,&#*1(! &'.+-/)77&6'<!('! 1$! E6+#)1$<!/(1!*+(7&-!/(!1$!$77&6'!(,(!/8$9!!!

c = So N(d1) - X e-rf t N(d2)

! Vl!`BMHd<!U&,F(+!G!QHÅqB:Q<!fG+-'N!pXF(!>+&7&'%!-E!q*.&-',!$'/!H-+*-+$.(!B&$3&1&.&(,p9!N&3!-*0'&2'K&0".")*0'<)&-&+D!'t!=W9!f$G-5c)'&-!Wnoe9!:,!/(!D),.&7&$!/(,.$7$+!1$!$*-+.$7&6'!2)(!.)C-!b-3(+.!f(+.-'!('!(1!#-/(1-!/(!`1$7g!G!Q7F-1(,9!!!



47

T-'/(!]K/&L! (,! 1$! E)'7&6'!/(!/&,.+&3)7&6'!/(! 1$! C$+&$31(! $1($.-+&$!'-+#$1!/(!#(/&$!')1$! G!

/(,C&$7&6'!.8*&7$!)'&.$+&$!K*+-3$3&1&/$/!/(!2)(!/&7F$!C$+&$31(!,($!#('-+!-!&%)$1!$!/&L9!

!

!

d1 = Ln(S0/X) + rft + 1

2"2t

" t ; d2 = d1 - " t !!>$+$!`1$7g!G!Q7F-1(,!)'!&'C(+,-+!+$7&-'$1!')'7$!(D(+7(+8$!)'$!-*7&6'!/(!7-#*+$!$'.(,!/(!,)!7$/)7&/$/V\!G<!*-+!1-!.$'.-<!(1!C$1-+!/(!1$!-*7&6'!/(!7-#*+$!$#(+&7$'$!7-&'7&/&+@!7-'!1$!()5+-*($9!>-+!-.+$!*$+.(<!/$/-!2)(!1$!-*7&6'!/(!C('.$!$#(+&7$'$!&'7-+*-+$!,-3+(!1$!()+-*($!1$!C('.$D$!/(!*-/(+!,(+!(D(+7&/$!('!7)$12)&(+!#-#('.-!/(1!*(+8-/-<!,)!C$1-+!,)*(+$+@!$!1$!7-+5+(,*-'/&('.(!()+-*($<!*+-*-+7&-'$'/-!1$!C$1-+$7&6'!/(!O,.$!_1.&#$!)'!18#&.(!#8'&#-!*$+$!$2)O11$9!:,.(!18#&.(!#8'&#-!,(!7$17)1$+8$!$!.+$CO,!/(!1$!+(1$7&6'!/(!*$+&/$/<!-3.('&O'/-,(!(1!

,&%)&('.(! C$1-+! *$+$! )'$! -*7&6'! /(! C('.$! ()+-*($! K/-'/(! (5+E.! (,! (1! E$7.-+! /(! /(,7)('.-!7-'.&')-LN!!

*!h!7!5!Q-!Ç!jMKzL!

!

*!h!YQ-!]K/

WL!5!z!!(5+E!.!]K/

VL^!5!Q

-!Ç!z!!(5+E!.!

!

*!h!Q-!Y]K/WL!5W^!5!z!(5+E!.!Y]K/VL!5W^!

!H-#-!,(!*)(/(!$*+(7&$+!1$!([*+(,&6'!/(!`1$7g!G!Q7F-1(,!$*1&7$!)'$!*-'/(+$7&6'!/(!]K/

WL!$!Q

-!

G!-.+$!/(!]K/VL!$!z!(5+E.9!Q&!1$!(7)$7&6'!,(!+(11('$!/$'/-!C$1-+(,!*-+!('7&#$!/(1!18#&.(!&'E(5

+&-+<!('.-'7(,!]K/WL!/(3(+@!,(+!#$G-+!2)(!]K/

VL9!

!Ejemplo: El precio actual de una acción (S0) es de 10 euros El precio de ejercicio (X) es de 11 euros La tasa libre de riesgo (rf) es del 10% El tiempo hasta el vencimiento (t) es de 0,5 años La volatilidad expresada mediante la desviación típica (!) de los rendimientos del

activo subyacente es del 30% Calcular el valor de la opción de compra (c) Q),.&.)G('/-!(,.-,!C$1-+(,!('!1$,!(7)$7&-'(,!+(*+(,('.$.&C$,!/(!/W!G!/V<!-3.('/+(#-,N!

!d1 = -0,10753 ————> N(d1) = 0,457185 d2 = -0,31966 ————> N(d2) = 0,374613

26 b(7)(+/(<!1$!-*7&6'!/(!7-#*+$!C$1(!#@,!IC&C$J!2)(!I#)(+.$J!1)(%-!1$,!-*7&-'(,!/(!7-#*+$!$#(+&7$'$,!C$1('!1-!#&,#-!2)(!1$,!()+-*($,9



48

!! {!,),.&.)G('/-!('!1$!(7)$7&6'!/(!`1$7g!G!Q7F-1(,N!!

c = 10 x 0,457185 - 11 x e-0,5x0,10 x 0,374613 c = 0,65 euros

!Q&!(1!C$1-+!/(!1$!-*7&6'!/(!7-#*+$!('!(1!#(+7$/-!(,!&'E(+&-+!$!;<\l!()+-,<!1$!$/2)&+&+8$#-,!G!,&! E)(,(! ,)*(+&-+! 1$! C('/(+8$#-,9! >-+! -.+$! *$+.(<! ,&! 7$17)1@,(#-,! (1! C$1-+! /(! 1$! -*7&6'!/(!C('.$!()+-*($!-3.('/+8$#-,!)'!C$1-+!/(!*!h!W<WWl\!()+-,9!!:'!$2)(11-,!#(+7$/-,!7-'!)'!$1.-!%+$/-!/(!(E&7&('7&$!,(!).&1&4$!(,.(!#-/(1-!*$+$!F$11$+! 1$!C$+&$'4$!/(!)'$!$77&6'!7)$12)&(+$!K!V

&L!7-#-!#(/&/$!/(!,)!+&(,%-!.-.$1!K1$!5&0*."0"(*('"+/074)".*GM!>$+$!(11-!,(!,)*-'(!2)(!(1!C$1-+!&'.+8',(7-!/(!1$!-*7&6'!7-&'7&/(!7-'!(1!/(!#(+7$/-!G!,(!.+$.$!('.-'7(,!/(!$C(+&%)$+!2)(!C$1-+!/(!!V!F$7(!2)(!,(!7)#*1$! 1$!(7)$7&6'!/(!`1$7g!G!Q7F-1(,9!!!!

"A"A ## !! .-.- !+(\1'&*'-!)1!'$1/H(!+(\1'&*'-!)1!'$1/H( !+(%!(D+$(%1/!+(%!(D+$(%1/ !!"A#"!+<J635?32!4<@!<M49<@68!:6!V6@52#!:,.(!.&*-!/(!7-3(+.)+$!E-+#$!*$+.(!/(!1-!2)(!,(!/('-#&'$!$#%3!&'(#')*!.#!*!K/&!.2&0"&'"-$34!*-)#L<!2)(!7-',&,.(!('!7)3+&+!)'$!*-,&7&6'!1$+%$!('!)'!$7.&C-!E&'$'7&(+-!#(/&$'.(!)'$!*-,&57&6'!7-+.$!('!-*7&-'(,!/(!7-#*+$<!-!1$+%$!('!-*7&-'(,!/(!C('.$<!,-3+(!/&7F-!$7.&C-9!B$,!-*57&-'(,!F$7('!(1!*$*(1!/(!)'!,(%)+-<!*-+2)(!$!.+$CO,!/(!(,.$!E-+#$!/(!7-3(+.)+$!,(!*)(/('!1&#&.$+!1$,!*O+/&/$,!$!)'!'&C(1!#@[&#-!*+(/(.(+#&'$/-!G!$*+-C(7F$+,(!/(!1$,!%$'$'7&$,!2)(!1$!7$+.(+$!*)(/$!-3.('(+9!"#$%8'(,(<!*-+!(D(#*1-<!(1!,(%)+-!7-'.+$!&'7('/&-,!/(!)'$!7$,$P!),.(/!*$%$!)'$!7$'.&/$/!$')$1!2)(!1(!*(+#&.(!+(/)7&+!,),!*O+/&/$,!('!7$,-!/(!&'7('/&-!*(5+-<!#&('.+$,!(,.-!'-!-7)++(<!),.(/!/&,E+).$!/(!,)!7$,$9!MF-+$!3&('<!7)$'.-!#('-+(,!*O+/&/$,!2)&(+$!),.(/!.('(+!#@,!7$+-!1(!+(,)1.$+@!(1!,(%)+-9!!B$!(,.+$.(%&$!7-',&,.('.(!('!$/2)&+&+!-*7&-'(,!/(!C('.$!7-'!-3D(.-!/(!7)3+&+!)'$!*-,&7&6'!1$+%$!('!)'!$7.&C-!,(!/('-#&'$!/3.'/!&.#).&!*9!Q(%)&/$#('.(<!C$#-,!$!$'$1&4$+1$!$!.+$CO,!/(!)'!(D(#*1-!2)(!).&1&4$!)'$!-*7&6'!/(!C('.$!,-3+(!8'/&7(,!3)+,@.&1(,9!!Q)*-'%$#-,! 2)(! (1! =! /(! ,(*.&(#3+(! .('(#-,! )'$! 7$+.(+$! E-+#$/$! *-+! $7.&C-,! /(! +('.$!C$+&$31(!C$1-+$/$!('!W;!#&11-'(,!/(!()+-,!7)G-!7-(E&7&('.(!/(!C-1$.&1&/$/!O#.*!(,!/(1!W<e!G!*+(.('/(#-,!7)3+&+1$!7-'!)'$!-*7&6'!/(!C('.$!,-3+(!(1! "3([5el!,&('/-!(1!C$1-+!/(!O,.(!$1!7&(++(!/(!=9oW;<W;!*)'.-,9! "'.('.$+(#-,!7)3+&+'-,! 1-!#@,!*-,&31(!*-+! 1-!2)(!$/2)&+&+(#-,!)'$!-*7&6'!/(!C('.$!7-'!*+(7&-!/(!(D(+7&7&-!=9o;;<!2)(!(,!(1!*+(7&-!/(!(D(+7&7&-!*.'.>#'+&4-#D9!:1!F-+&4-'.(!.(#*-+$1!1-!(,.$31(7(#-,!('!,(&,!#(,(,!K(1!.&*-!/(!&'.(+O,!,&'!+&(,%-!*$+$!1-,!,(&,!#(,(,!,(!,)*-'(!/(1!Va!,(#(,.+$1L9!!



49

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!

Nº teórico = 10.000.0008.700 x 10 = 114,94 opciones de venta!!Q(%)&/$#('.(!*+-7(/(+(#-,!$!$D),.$+!(,.(!'_#(+-!/(!-*7&-'(,!7-'!+(1$7&6'!$!1$!C-1$.&1&/$/!/(!1$!7$+.(+$<!*$+$!1-!7)$1!1-!#)1.&*1&7$+(#-,!*-+!(1!C$1-+!/(!1$!u(.$N!!

Nº = 114,94 x 1,3 = 149,43 " 150 opciones de venta !:1!*+(7&-!/(!1$!-*7&6'!/(!C('.$!/(!*+(7&-!/(!(D(+7&7&-!=9o;;!7-'!C('7&#&('.-!('!Q(*.&(#3+(!(+$! /(! eWk! *)'.-,<! (,! /(7&+<! e9Wk;! ()+-,! 1-! 2)(! &#*1&7$! *$%$+! *-+! 1$,! Wl;! -*7&-'(,! )'-,!koW9;;;!()+-,9!:1!7-,.(!/(1!,(%)+-!(,!&%)$1!$!eWk!r!=9o;;!h!e<\Wa!('!,(&,!#(,(,!K)'!o<VVa!'-#&'$1!$')$1L9!!Q&<!*-+!(D(#*1-<!(1!/8$!/(1!C('7&#&('.-!(1!8'/&7(!"3([5el!.-#$!)'!C$1-+!/(!=9;;;!*)'.-,<!(,!/(7&+<!F$!/(,7('/&/-!)'!=<Wla<!(1!/(,7(',-!/(!1$!7$+.(+$!F$3+@!,&/-!/(!=<Wla!*-+!W<e!1-!2)(!/$!)'!C$1-+!/(!W;<\a!KW<;\!#&11-'(,!/(!()+-,!/(!*O+/&/$L9!Q&!(,(!/8$!(D(+7(#-,!1$,!-*7&-'(,!/(!C('.$!-3.('/+(#-,!)'!C$1-+!&%)$1!$!K=9o;;!5!=9;;;L![!W;![!Wl;!h!W<;l!#&11-'(,!/(!()+-,9!:,.-!&#*1&7$!2)(!1$!*O+/&/$!/(!1$!7$+.(+$!7)3&(+.$!K&'71)G('/-!(1!7-,.(!/(1!,(%)+-!G!,)!7$*&.$51&4$7&6'!,(&,!#(,(,!#@,!.$+/(!$1!Va!/(!&'.(+O,!,(#(,.+$1L!,(+@!&%)$1!$N!!

-1.060.000 + 1.050.000 - 471.000 (1,02) = -490.420 euros !*O+/&/$! &'E(+&-+!$! 1$!2)(! ,(!F$3+8$!-3.('&/-! ,&!'-! ,(!F)3&(,(! 7)3&(+.-! 1$! 7$+.(+$9!q3,(+C(!2)(!7-'!)'!7-'.+$.-!/(!E).)+-,!,-3+(!(1!"3([5el!,(!*-/+8$!F$3(+!7)3&(+.-!7$,&!*(+E(7.$#('5.(!(1!+&(,%-!*(+-!'-!*-/+8$#-,!/&,E+).$+!/(!1$,!C('.$D$,!2)(!*-/+8$#-,!7-',(%)&+!,&!(1!C$1-+!/(1!"3([!F)3&(,(!$,7('/&/-!('!1)%$+!/(!7$(+9!!:,!&#*-+.$'.(!.('(+!('!7)('.$!2)(!,&!'-!*(',$#-,!#$'.('(+!1$,!-*7&-'(,!/(!C('.$!F$,.$!,)!C('7&#&('.-!'(7(,&.$+8$#-,!7-++(%&+!(1!C$1-+!$'.(+&-+!/&C&/&O'/-1-!*-+!1$!(#0.*!/(!1$!-*7&6'!KO,.$!&'/&7$!7)@'.-!C$+8$!(1!C$1-+!/(!1$!-*7&6'!$'.(!)'$!C$+&$7&6'!/(1!C$1-+!/(1!$7.&C-!,)3G$57('.(!sC$(+!$'([-!eL9!:,.-!(,!'(7(,$+&-!*-+2)(!7)$'.-!#@,!1(D-,!/(1!C('7&#&('.-!/(!1$!-*57&6'!'-,!('7-'.+(#-,!#$G-+!/&,*$+&/$/!([&,.&+@!('.+(!(1!#-C&#&('.-!/(1!*+(7&-!/(!1$!-*7&6'!G!(1!/(!,)!$7.&C-!,)3G$7('.(!K/(3&/-!$!1$!([&,.('7&$!/(1!C$1-+!.(#*-+$1<!C$1-+!2)(!(,!#)G!*(52)(0-!('!1$,!*+-[&#&/$/(,!/(1!C('7&#&('.-L<!*-+!1-!2)(!,)!7-3(+.)+$!,(+@!&#*(+E(7.$9!:,!/(57&+<!,&!(1!$7.&C-!,)3G$7('.(!C(!/(,7('/(+!,)!C$1-+!('!)'!W;a<!C(+(#-,!2)(!(1!*+(7&-!/(!1$!-*57&6'!/(!C('.$!'-!$)#('.$!('!1$!#&,#$!7$'.&/$/9!!:'!1$!E&%)+$!ee!,(!#)(,.+$!(1!(,2)(#$!/(!1-,!E1)D-,!/(!7$D$!/(!)'$!/3.'/!&.#).&!*!('!,)!E(7F$!/(!C('7&#&('.-9!:1!$7.&C-!$/2)&+&/-!K1$!$77&6'L!,(!+(*+(,('.$!#(/&$'.(!1$!18'($!/&$%-'$1!$4)1!2)(!7+)4$!(1!(D(!/(!$3,7&,$,!('!(1!C$1-+!l!&'/&7$.&C-!/(!,)!7-.&4$7&6'!$7.)$19!B$!-*7&6'!/(!C('5.$!$/2)&+&/$!-!I*).!7-#*+$/$J!s18'($!C(+/(5!,(!+(*+(,('.$!/(!.$1!#$'(+$!2)(!,)!*+(7&-!/(!



50

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52

-@6i<!"!Futuro sobre el Bono 10 en el MEFF

http://www.meff.com/aspx/Comun/Pagina.aspx?l1=Financiero&f=NuevoBono10

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53

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!!/-'/(!]<!&'/&7$!(1!'-#&'$1!/(1!3-'-!('.+(%$31(P!w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

"#*-+.(!$!*$%$+!h!>![!UH![!W9;;;!Ç!HH!!M!7-'.&')$7&6'!/(3(+(#-,!(1(%&+!7)@1!/(! 1-,!3-'-,!('.+(%$31(,!/(3(+@!,(+!7(/&/-!$1!7-#5*+$/-+9! >$+$! (11-! /(3(+(#-,! ('7-'.+$+! (1! /('-#&'$/-! 3-'-! #-.!#%*,0#' +A$' #)&-8+")&!K)>#*/#$.'.&'(#0"5#!L<!1-!2)(!,)*-'(!7-#*$+$+!(1!*+(7&-!/(!$/2)&,&7&6'!('!(1!#(+7$/-!$1!7-'5.$/-!/(!+('.$! E&D$! K>;L!7-'! 1$!7)$'.8$!2)(!*(+7&3&+@!*-+!,)!C('.$!('!(1!#(+7$/-!/(! E).)+-,!K(,.-!,(!F$+@!*$+$!1$!.-.$1&/$/!/(!1-,!.8.)1-,!('.+(%$31(,LN!!

Y>![!UH^!s!>;!!!:1!C$1-+!2)(!*+-*-+7&-'(!)'$!#$G-+!%$'$'7&$<!-!)'$!#('-+!*O+/&/$<!,(+@!(1!(1(%&/-9!:,!(,.(!.8.)1-!(1!2)(!&'.+-/)7(!)'$!7-'([&6'!('.+(!(1!#(+7$/-!/(!E).)+-,!G!(1!/(!+('.$!E&D$<!G!$/(#@,!,(+@!).&1&4$/-!(,*(7&$1#('.(!*-+!1-,!$+3&.+$D&,.$,!*-+!+$4-'(,!-3C&$,9!



54

-@6i<!A!Futuro sobre Ibex 35® en el MEFF

http://www.meff.com/aspx/Comun/Pagina.aspx?l1=Financiero&f=FuturosIBEX35!

!



55

-@6i<!G!La sensibilidad del valor de una opción: las “griegas”

Å$G!7&(+.$,!C$+&$31(,!([6%('$,!2)(!$E(7.$'!$1!*+(7&-!/(!1$,!-*7&-'(,!7)G-!(E(7.-!,(!(,.)/&$!$!.+$CO,!/(!)'-,!7-(E&7&('.(,<!2)(!,(+C&+@'!*$+$!(,.$31(7(+!7-3(+.)+$,!/(!+&(,%-!('!1$,!7$+.(5+$,!7-'!-*7&-'(,<!G!$!1-,!2)(!,(!7-'-7(!F$3&.)$1#('.(!7-#-!I1$,!%+&(%$,J!*-+!/('-#&'$+,(!#(/&$'.(!1(.+$,!%+&(%$,9!!1;!4<6>9496@56!)1.&-!:,.(!7-(E&7&('.(!,(!/(E&'(!7-#-!1$!C$+&$7&6'!*+-/)7&/$!('!(1!*+(7&-!/(!1$!-*7&6'!*-+!)'$!)'&/$/!/(!7$#3&-!('!(1!*+(7&-!/(!1$!$77&6'!,)3G$7('.(9!Q(!1(!7-'-7(!.$#3&O'!7-#-!+$.&-!/(!7-3(+.)+$!*-+2)(!&'/&7$!(1!'_#(+-!/(!$77&-'(,!,)3G$7('.(!'(7(,$+&-!*$+$!7)3+&+!)'$!*-,&7&6'!('!-*7&-5'(,9!:[*+(,$/-!('!E-+#$!/&,7+(.$!.('/+8$#-,N!!

!

DELTA = " precio de la opción" precio de la acción = " c

" S !

!f&('.+$,!2)(<!('!E-+#$!7-'.&')$<!1$!(#0.*!/(!1$,!-*7&-'(,!/(!7-#*+$!G!/(!C('.$!,(+8$!&%)$1!$!1$!/(+&C$/$!*$+7&$1!/(1!*+(7&-!/(!1$!-*7&6'!7-'!+(1$7&6'!$1!*+(7&-!/(1!$7.&C-!,)3G$7('.(N!!

!

"c = #c#S = N(d1 ) "p = #p

#S = N(d1 ) - 1 !!M,8<!,&!1$!(#0.*!/(!)'$!-*7&6'!/(!7-#*+$!(,!&%)$1!$!;<kloW=l!&'/&7$!2)(!,&!(1!*+(7&-!/(!1$!$757&6'!,)3G$7('.(!$)#('.$! K-!/(,7&('/(L!('!)'!()+-!(1!*+(7&-!/(! 1$!-*7&6'! 1-!F$+@!('!$1%-!#@,!/(!kl! 7O'.&#-,9! :'!(1! 7$,-!/(! 1$!-*7&6'!/(! C('.$! ,)!(#0.*! ,(+8$!/(N!;<kloW=l! 5! W!h! 5;<lkV=Wl!2)(!&'/&7$+8$!2)(!$'.(!)'!$,7(',-!K/(,7(',-L!/(1!*+(7&-!/(!1$!$77&6'!,)3G$7('.(!('!)'!()+-!(1!C$1-+!/(! 1$!-*7&6'!/(!C('.$!/(,7('/(+8$! K$)#('.$+8$L!('!)'-,!lk!7O'.&#-,9!X$#3&O'!,(!*)(/(!/(E&'&+!1$!(#0.*!/(!1$!-*7&6'!7-#-!1$!*+-3$3&1&/$/!/(!(D(+7(+!1$!#&,#$<!$,8!*-/+8$#-,!/(7&+!2)(!([&,.(!(1!kl<oa!/(!*+-3$3&1&/$/!/(!(D(+7(+! 1$!-*7&6'!/(!7-#*+$!$'.(5+&-+9!! !:'!1$!E&%)+$!e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



56

!U&%9e$!B$!(#0.*!G!1$!(1$,.&7&/$/!/(!)'$!-*7&6'!/(!7-#*+$!YzhW9W;;P!.h;<l;P!,he;aP!+EhW;a^9!!

!Q&!#(/&#-,!(1!*-+7('.$D(!/(!C$+&$7&6'!/(1!*+(7&-!/(! 1$!-*7&6'!7)$'/-!(1!*+(7&-!/(1!$7.&C-!,)3G$7('.(!C$+8$!)'!Wa<!-3.('/+(#-,!1$!#0*$.")"(*(!/(!1$!#&,#$9!:'!E-+#$!#$.(#@.&7$!*-5/+8$#-,!([*+(,$+1$!$,8N!!

!

Elasticidad = "c"S x Sc !

!B$!(1$,.&7&/$/!(,!)'$!#(/&/$!/(1!$*$1$'7$#&('.-!-3.('&/-!7-'!)'$!-*7&6'9!:'!1$!E&%)+$!e$!,(!-3,(+C$!1$!(1$,.&7&/$/!/(!1$!-*7&6'!/(1!(D(#*1-!$'.(+&-+<!/6'/(!,(!*)(/(!$*+(7&$+!7-#-<!('!(1! 18#&.(<! 1-,!C$1-+(,!#@[&#-!G!#8'&#-!/(! 1$!#&,#$!,-'<! +(,*(7.&C$#('.(<! &'E&'&.-! K-*7&6'!*+-E)'/$#('.(!&3.'&2'.>#'+&-#DL!G!7(+-!K-*7&6'!*+-E)'/$#('.(!"-'.>#'+&-#DL9!!1;!4<6>9496@56!H-CC-!f&/(! (1! (E(7.-! 2)(! 1$! &'(,.$3&1&/$/! /(1!#(+7$/-! *+-/)7(! ('! (1! C$1-+! /(!(#0.*9! M,8! 2)(! 1$!%*++*!/(!)'$!-*7&6'!#&/(!1$!.$,$!/(!7$#3&-!/(!1$!(#0.*!7)$'/-!(1!*+(7&-!/(!1$!$77&6'!C$+8$!)'$!)'&/$/9!f$.(#@.&7$#('.(!,(!*)(/(!/(E&'&+!7-#-!1$!,(%)'/$!/(+&C$/$!/(1!*+(7&-!/(!1$!-*7&6'!7-'!+(,*(7.-!$1!*+(7&-!/(1!$7.&C-!,)3G$7('.(N!!

!

Gamma = "Delta"S # $ = %

2c%S2 = N'(d1)

S& t = 12' e-0,5d1

2

S& t !

!:,!*)(,<!)'$!#(/&/$!/(!1$!,(',&3&1&/$/!/(!1$!(#0.*<!(,!/(7&+<!(,!1$!(#0.*!/(!1$!(#0.*9!T(,/(!)'!*)'.-!/(!C&,.$!7-'7(*.)$1<!,&!O,.$!_1.&#$!+(*+(,('.$!1$!C(1-7&/$/<!%*++*!+(*+(,('.$!1$!$7(51(+$7&6'9!j($#-,!)'!(D(#*1-N! ,)*-'%$#-,!)'$!C-1$.&1&/$/!/(1!e;a<!)'! .&*-!/(! &'.(+O,! ,&'!+&(,%-!/(1!W;a<!)'!.&(#*-!/(!C('7&#&('.-!/(!W=;!/8$,!G!)'!*+(7&-!/(!(D(+7&7&-!/(!WW!()+-,9!Q(%_'!1-!$'.(+&-+!(1!C$1-+!/(!1$!-*7&6'!/(!7-#*+$!,(+@!/(!;<\l!()+-,<!,&!(1!*+(7&-!/(!1$!$77&6'!(,!/(!W;!()+-,9!B$!(#0.*!.-#$+@!)'!C$1-+!/(!;<kloW=l!G!1$!%*++*Vn'/(!;<;;W=o9!B-!2)(!&#5

29 X$'.-! 1$!(#0.*! 7-#-! 1$!%*++*! ,(!*)(/('!([*+(,$+!('!*-+7('.$D(9!M,8<! .('/+8$#-,!)'$!(#0.*!/(!kl<oa!K-!kl<oL!G!)'$!%*++*!/(!;<W=a!K-!;<W=!(#0.*$!-<!,&#*1(#('.(<!;<W=L9



57

*1&7$! 2)(! ,&! (1! *+(7&-! /(! 1$! $77&6'! $,7('/&(,(! $! W;<;W! ()+-,! 1$!(#0.*! ,(! &'7+(#('.$+8$! ('!;<;;W=o!$17$'4$'/-!)'!C$1-+!/(!;<kln;l9!!

!U&%9e3!B$!%$##$!/(!)'$!-*7&6'!/(!7-#*+$!

!:1!7-(E&7&('.(!%*++*!/(!)'$!-*7&6'!(,!#$G-+!7)$'/-!1$!$77&6'!(,.@!('!1$!4-'$!*.'.>#'+&4-#D<!#&('.+$,!2)(! .('/(+@!$! 7(+-! ,(%_'!,(!$1(D(!/(!(11$!('!7)$12)&(+!/&+(77&6'! KC(+! E&%)+$!e3L9!B$!%*++*!+(E1(D$!(1!%+$/-!/(!1$!7)+C$.)+$!('!1$!18'($!+(*+(,('.$.&C$!/(1!*+(7&-!/(!1$!-*57&6'!/(!7-#*+$<!/(!.$1!E-+#$!2)(!7)$'/-!1$!7)+C$.)+$!(,!#@,!%+$'/(!K('!(1!*+(7&-!/(!(D(+7&57&-L! 1$!%*++*!$17$'4$+@!,)!C$1-+!#@[&#-!K('!(1!%+@E&7-!,(!#)(,.+$! 1$! 18'($!+(*+(,('.$.&C$!/(1!C$1-+!&'.+8',(7-!/(!1$!*+&#$!/(!1$!-*7&6'!7$17)1$/$!,(%_'!1$!([*+(,&6'!/(!`1$7g5Q7F-1(,<!*$+$!(1!(D(#*1-!2)(!C('&#-,!).&1&4$'/-L9!>-+!(1!7-'.+$+&-<!7)$'/-!'-!F$G!7)+C$.)+$!(1!C$1-+!/(1!7-(E&7&('.(!(,!')1-9!!1;!4<6>9496@56!&h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f$.(#@.&7$#('.(<!(,!1$!/(+&C$/$!*$+7&$1!/(1!*+(7&-!/(!1$!-*7&6'!7-'!+(,*(7.-!$1!.&(#5*-!F$,.$!(1!C('7&#&('.-N!!

!

Theta = "c"t # $ = %c

%t = S&2 t N'(d1) + X e-rft rf N(d2 ) !

!B$,!-*7&-'(,!2)(!,-'!/(1!.&*-!"-4.>#4+&-#D!C(+@'!,)!*+(7&-!/(,7('/(+!('!E-+#$!1&'($1!7-'5E-+#(!,(!$7-+.(!(1!.&(#*-!F$,.$!(1!C('7&#&('.-<!1-!2)(!(,!/(3&/-!$!2)(!.&('('!)'!*(2)(0-!C$1-+!.(#*-+$1!G!$!2)(!,(%)+$#('.(!1$!-*7&6'!*(+#$'(7(+@!('!/&7F$!4-'$9!>$+$!1$,!-*7&-5'(,!*.4.>#4+&-#D! 1$!.$,$!/(!+(/)77&6'!/(1!*+(7&-!$)#('.$!7-'E-+#(!/(,7&('/$!(1! .&(#*-P!(,.(!.&*-!/(!-*7&-'(,!,-'!#)G!,(',&31(,!*)(,.-!2)(!7-'E-+#(!,(!$*+-[&#('!$1!E&'$1!/(!,)!C&5/$!*)(/('!%&+$+!.$'.-!$!1$!4-'$!"-4.>#4+&-#D!7-#-!$!1$!&3.4&24.>#4+&-#D9!Q&!1$!-*7&6'!*(+5.('(7(!$!O,.(!_1.&#-!.&*-<!1$!.$,$!/(!+(/)77&6'!/(1!*+(7&-!,(!C$!+(/)7&('/-!7)$'.-!#('-,!1(!



58

2)(/(!/(!C&/$P!(11-!(,!$,8!/(3&/-!$!2)(!1$!*+-[&#&/$/!/(!1$!E(7F$!/(!C('7&#&('.-!+(/)7(!1$!*-,&3&1&/$/!/(!2)(!(1!*(2)(0-!C$1-+!.(#*-+$1!/(!1$!-*7&6'!$E(7.(!$!,)!*+(7&-!K2)(<!*-+!-.+$!*$+.(<!,(+@!7$,&!')1-L9!!1;!4<6>9496@56!'h(!:,.(!7-(E&7&('.(!&'/&7$!1$!,(',&3&1&/$/!/(1!*+(7&-!/(!1$!-*7&6'!/(3&/$!$!1-,!7$#3&-,!/(1!.&*-!/(!&'.(+O,!1&3+(!/(!+&(,%-9!:,!/(7&+<!#&/(!1$!7-3(+.)+$!/(!1$!-*7&6'!7-'!+(,*(7.-!$!/&7F-!.&*-!/(!&'.(+O,9!f&('.+$,!2)(!!>&!(,!*-,&.&C-!*$+$!1$,!-*7&-'(,!,-3+(!$77&-'(,<!+(,)1.$!,(+!'(%$5.&C-!*$+$!-.+-!.&*-!/(!$7.&C-,!7-#-!(,!(1!7$,-!/(!1$,!-*7&-'(,!,-3+(!E).)+-,<!-!/(!1$,!*+-*&$,!$77&-'(,<!*-+!(D(#*1-9!f$.(#@.&7$#('.(<! !>&! ,(!7$17)1$+8$!-3.('&('/-! 1$!/(+&C$/$!*$+7&$1!/(1!*+(7&-!/(!1$!-*7&6'!7-'!+(1$7&6'!$1!.&*-!/(!&'.(+O,N!!

!

Rho = "c"rf

# $ = %c%rf

= t X e-rft N(d2 ) !!V>&!(,! 1$!#('-,! &#*-+.$'.(!/(! 1$,!C$+&$31(,!2)(!&'7&/('!,-3+(!(1!C$1-+!/(! 1$!-*7&6'9!:'! 1$!E&%)+$!e7!,(!#)(,.+$!1$!+(1$7&6'!('.+(!(1!C$1-+!/(!1$!-*7&6'!G!(1!.&*-!/(!&'.(+O,<!G!('!(11$!,(!*)(/(!$*+(7&$+!7-#-!1$,!$1.(+$7&-'(,!/(1!.&*-!/(!&'.(+O,!,61-!$E(7.$'!1&%(+$#('.(!$1!*+(7&-!/(! 1$!-*7&6'9!M/(#@,<!F$G!2)(!.('(+!('!7)('.$!2)(!('!(1!7@17)1-!/(!!>&!,(!,)*-'(!2)(!$1!C$+&$+!(1!.&*-!/(!&'.(+O,!(1!*+(7&-!/(!1$!$77&6'!,)3G$7('.(!,(!#$'.&('(!7-',.$'.(<!1-!2)(!'-!(,!7&(+.-!,&'-!2)(!/(,7('/(+8$!/(!C$1-+!&#*)1,$'/-!$!1$!3$D$!$1!*+(7&-!/(!1$!-*7&6'9!!

!U&%9e7!B$!!>&!/(!)'$!-*7&6'!/(!7-#*+$!

!1;!4<6>9496@56!O1H-!:,.(!7-(E&7&('.(<!.$#3&O'!/('-#&'$/-!?*//*!)!&+#%*<!&'/&7$!(1!7$#3&-!('!(1!*+(7&-!/(!)'$!-*7&6'!7-'!+(,*(7.-!$!)'$!C$+&$7&6'!*+-/)7&/$!('!1$!C-1$.&1&/$/!/(!1$!$77&6'9!:[*+(,$/$!('!E-+#$!#$.(#@.&7$!5#%*!(,!1$!/(+&C$/$!*$+7&$1!/(1!*+(7&-!/(!1$!-*7&6'!7-'!+(1$7&6'!$!1$!C-1$.&51&/$/!/(1!$7.&C-!,)3G$7('.(9!!

!

Vega = "c"#

$ % = &c&#

= S t N'(d1) !!



59

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