Conditional Probability and Conditional Expectation

Transcription

1 Hadou #8 for B prig 2002 lecure dae: 3/06/2002 Codiioal Probabiliy ad Codiioal Epecaio uppose X ad Y are wo radom variables The codiioal probabiliy of Y y give X is } { }, { } { X P X y Y P X y Y P The codiioal epecaio of Y give X is Ω X y Y yp X Y E } { } { where Ω cosiss of hose samples whose X value is

2 Eample N, T 2, k 4 Price process is: sae 0 2 rue probabiliy ω /3 ω /6 ω /6 ω /3 a Wha is he codiioal disribuio of 2 give 8? The codiioal probabiliies ad epecaio? b Wha is he codiioal disribuio of 2 give 4? The codiioal probabiliies ad epecaio? 2

3 Eed he Idea E[Y F ] is he radom variable ha akes he value E[Y A i ] o he se A i i he pariio correspodig o he algebra where E[Y A i ] yp { Y y; Ai }/ P Ai Y ω P ω / P Ai y ω E[Y F ]ω is hus equal o E[Y A i ] if ω is i A i uesio: Is E[Y F ] measurable? Properies of he CE E[ E[Y F] ] E[ Y ] 2 E[ E[Y F 2 ] F ] E[Y F ] if F _ F 2 i paricular, E[ E[Y F2]] E[Y] 3 If X is measurable wih respec o F he E[X Y F ] X E[Y F ] Thus, E[X F ] X 4 Lieariy: Le a ad b be cosas E[aX + by F ] a E[X F ] + b E[Y F ] 3

4 Eample sae 0 2 rue probabiliy ω /4 ω /4 ω /4 ω /4 4

5 Defiiio: A process {Z 0, Z,, Z T } adaped o he filraio {F 0, F,, F T} is said o be marigale if Also called a fair game E[ Z + F ] Z Eample N Number of heads a ed of -h oss Probabiliy of geig a head 03 how ha N - 03 is a marigale 5

6 APT i Muliperiod Markes Game of Pricig ad Tradig Idea: The properies of a muliperiod marke are deermied by all of is uderlyig sigle period markes Arbirage Opporuiies AO eis if oe ca fid a TH such ha a V 0 0 b V T 0 for all ϖ Ω c V T > 0 for a leas oe ϖ Ω equivalely, 0, EG > 0 G T T Is here arbirage? Risk Neural Probabiliy Measure RNPM eiss if oe ca fid a ω, ω 2,, ω k such ha a is a probabiliy measure b ω > 0 for all ϖ Ω c Each is a marigale uder, or equivalely Ε [ + ], for all ad,, T is called a marigale MTG measure Ierpreaios: Uder RNPM, each risky asse is epeced o grow a he risk free ieres raes bewee 0 ad T 2 Uder RNPM, ay porfolio T is epeced o ear he risk free ieres raes bewee 0 ad T 6

7 Eample N, T 2, k 4 Price process is: sae 0 ω ω ω ω a uppose r / 9 ad r 2 / 9 Are here AOs? Does here eis a RNPM? b uppose r / 9 ad r 2 / 8 Are here AOs? Does here eis a RNPM? Resul If here is a MTG measure he here is o AO i he muliperiod marke Resul 2 If here is o AO i he muliperiod marke he here eiss a MTG measure 7

8 Icludig Divideds Defie he reur process as R R / 0 else / 0 else if if > 0 > 0 The MTG measure ca be rephrased as Thus E [ F - ] 0 E [ R F - ] 0 which should be epeced Defiiio: Le he divideds paid a ime for he period -, ] be D Thus, he cumulaive divided up o ime Value process: Where are divideds i his? V H D D u u N 0 + B + H + 8

9 Gai Process G u H N N 0 u Bu + H u u + H u D u u u elf-fiacig good eercise V V 0 + G Le / B, D D / B oice i makes o sese o discou he cumulaive divideds Le V V / B Discoued gais: G + For a self-fiacig porfolio: good pracice eercise Arbirage: A T provides arbirage if 0, EG > 0 G T T uesio: How o modify he MTG-measure? N Clue: G H u + D u u aoher clue: epeced discoued reur 0 Eesio of MTG Measure + D 0 E F 9

10 0 This is eacly he same as if: D + is a MTG uder ad is sricly posiive for each ω Thus, Claim : If is a MTG-measure he 0 G E Claim 2: If here is o AO <> here is a MTG-measure ome people say whe a sock pays divideds he i is uder-priced Wha i may mea is ha D + is a -marigale: D E D D E F F Pricig Formula: Eesio o ime +s Ierpreaio

11 Homework due 03/20/02 The GMAT eam cosiss of muliple choice quesios, each of which has five opios Eacly oe of he five opios is he correc aswer For each quesio, a perso ca choose o pick a opio, i which case oe poi is awarded if he opio is correc ad /4 poi is deduced if he opio is icorrec, or leave i blak, i which case zero poi is give uppose a perso s choices pick a opio or leave i blak for differe quesios are idepede of oe aoher Le X be he cumulaive pois received for he firs quesios, for,2,3, how ha X is a marigale for someoe who guesses he aswers radomly picks a opio radomly for each quesio hus does o leave ayhig blak Wha is he moral of his resul? 2 I he eample below r Creae a radig sraegy ha eplois he arbirage opporuiy Assume ha you have $0 moey i he bak o begi wih For he radig sraegy you foud, compue he value of your porfolio, he gai, discoued value, he discoued gai, for every ode i he ree