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Chapter 8 Other applications（第2页）

Noteamscoals，butthatfigureisjusttheqaverageoverahypothetiumberofmatches。Thecrucialstepistoassesstheprobabilitiesof0，1，2，3，。。。goalsinasinglegame，aerusedthePoissondistribution。DataovermanyyearsshowthatthisisprettygoodatdesghowtheaumberofgoalstendstovaryarouhArsenal’sfigureof2。1，theogoalscameoutas12%，onegoalas26%，twogoalsas27%，threeas19%，andsoon。

DataforStokeputtheirmeanscoreas0。67goals。Thistraoa51%ogoals，a34%ceofjustonegoal，11%foals，andsoon。offaith，takethenumbersofgoalsscoredbyeachteamasiheprobabilityofa2-1smultiplyiheHometeamscorestwicebythecetheAwayteamsthiscase，27%*34%，around9%。

Inthisrobabilityofanypossiblescoreisestimated。ThentheprobabilitiesforeaeaywinarefoundfromtheAdditionLatheseparateprobabilitiesofallthescoresthatleadtothosethreerespectiveresults。ThisgaveArsenala72%ceofvictory，Stokehada10%gaceof18%fortheDraw。Thesthehighestprobability，at14%，>

Donotstheteheexactsthehighestprobabilityhappewideightoftheteswerethosethatwereidentifiedasbeilikely。Abettingman，laeyolikelyresult’andoneach‘predicted’exactscore，wouldhavesmiledhappilyasthematchsfolded。

Howwerecilea72%degreeofbeliefthatArsenalwouldwinthatmatchwithideasoffrequency，asthereisionofplayingthisgamehuimesandghowoftenArsenalwon?Recallhowwejudgedthereliabilityofaweatherforecasterwhehatthetomorrowis30%:

thereisoomorrow，itwilleitherrainoritwillnot。However，welookatalltheoswhenshegivesraina30%dcheckitsactualfrequency。Weshallbelieveherorrow，ornot，onthebasisofheroverallrecord。Withsoccermatches，wemakesimilarcalsfamesplayedovertheseason。Amoheremightbefortyorsowheresomeresultwasgivenaprobabilitycloseto72%–wecheckwhetherthe‘predicted’resultdidoccurwithafrequend72%，asawayofvalidatihods。

agamblerexpeoneybyusihepayoutpridheavilyonhowmuchisstakedoe，asumsareusuallystakedoheothertowisonaDrawtendnottoattraittedfans。IftheceofaDrawisassessedas25%，apriceisbetterthaoounitytoprofitisthere。

Dohebestbetisoewiththehighestpredictedprobability!

Footballresults（2）

&he2010soccerWorldalsbegan，statistiMcHalepublishedtheresultsofhiscals，whichallocatedtoeachofthe32teamssomenon-zeroprobabilityofwirophy。HemadeSpaies，albeitwithawinningly11。6%，followedbyBrazil，utat10。3%。

Toobtainthesefigures，McHaleusedanapproachsimilartothatdescribedaboveforeachmatch。However，hedidcaloftheprobabilitiesofthedistines，hereliedonaMonteulation。

Thus，foramatwhid’smeaat1。5goals，thePoissonmodelgivesa22%ogoals，a33%egoal，andsooer’sraorseleeofthevalues0，1，2，3，。。。riateprobabilities，ahingfland’soppoosomesimulatedscoresuchasa2-2draw。Similarsimulationsweremadeforeveryscheduledmatgtosimulatedgrouptables，atheknockoutstagesallthewaytothefinal。Thisprocesseated100，000times，andthenumberofsimulationsieamemergedasswasrecorded。Spain‘won’11，633times，hehe11。6%figureheLaweNumbers，asusual，isthejustifi。

AndSpaindidwin!robabilities‘correct’?Weoainwouldhavewon65%ofthetime，haditbeeomakeiioourhebestevidehismethodsmakegoodsebookmakersfolloathtosettheirinitialpayoutpricestoattraters。

Black-Scholes

Sharepriarketsfluesforreason。Ifthepriceis￡5today，youdonotknoricewillbeh。However，you　optihttobuy（orsell）thatshareatthe　strike　priceof￡5。20atagiveime。If，atthattime，themarketpriceislessthan￡5。20，youwillnotexerciseyouroptiontobuy，butifitisabovethatpriakeaninstantprofitbytakiion，

aelyselling。arksapplytoanoptioarefairpricesfortheseoptions?

FischerBladMyronScholesaddressedthisquestionin1973。Attheheartoftheirworktionthatthesharepricesvariedrandomly，butinaparticularwayrelatedtotheGaussiandistribution。Thefairpricesforbothbuyaioodepeprice，thepriceatwhichtheoptionwouldbeexercised，theiimeperiies，ailityoftheunderlyingshareprice（asmeasuredbythestaionoveraperiod）:but　notobywhichthesharepriceectedtoge!

Thislastpointmaybesurprising，butthatishowthingsworkout。Itisalsoquiteuseful，asitmeansthatwehaveoaddtoaybyestimatirendinprices。Ifyouwanttodiscoverthefairprieparticularoptiowareiswidelyavailable–justtype‘Black-Stoyourfavouriteseare。Givepridthestrikeprice，thefaircostofabuyoptionwouldiimeperier，orifieswerehigher，orifthevolatilityofthesharepricewerehigher。

Howdothethislastsentenceaccordwithyourintuitiodoesseemreasohelongeryouarepreparedtowait，thehighertheiheunderlyingshareprice，buttheothertwoclaimsaremoresubtle。Ailityalsoihepintheshareprice，butitalsohappehemeanpridtheformereffesouttobebigger。

&ilityismeasuredbylookingatthethesharepriceoverthe25daysihisshouldgiveenoughdataforaobereliable，butchsofariastobeirrelevantforts。Apoorestimateofthevolatilitywillleadtoanunreasonableprioption。

Amodelisonlyusefulwheioviolated。And，asFigure7shows，takingtheGaussiandistributionasamodelforpriceflupliesareallytinyprobabilityforcatastrophits，suchastheprigbymorethanthreeorfourstaioualprobabilityofsutissignifiderestimated，themodelisuhesitindicatesmayhavenosoundbasisatall。Theextreme-valuedistributioionedinChapter　4，havebeeoaddressthisproblem。

&folios

paniesAahexpeakeprofits。Withlowies，Aisexpectedtoreturn20%，whileBshouldreturn40%;withhighies，thepositionsarereversed–Ashouldgain40%，B20%。SupposeNickisarisk-averseior，whileMaryisrisk-attracted。

Iflhiesareseenasequallylikely，bothaylookequallyattractive，withameaurnof30%。Inacewiththeirrespectiveattitudestorisk，Nickcoulddividehisfundsequallybetweewopanies，andguara30%，whetherratesarehighorlow:Marypforonepaher，hopiagshemightgetonly20%。

SupposeBisreplapanyC，whichwillreturn10%withlowies，or50%ifratesarehigh–againanaverageof30%，likepanyB。ButnowmixingAandoseherior:NickprefersAalone，MaryputseverythingintoC。

&ialdifferethereturnsfromAaivelycorrelated–inswhenoneishigh，theothertereturnsforAaivelycorrelated–theydobetterether。‘easuredonas–1（totalion）to+1（totalpositivecorrelation）。Iftwoassetsfluvalueilyofeachother，theircorrelationwillbezero。

Risk-averseiorsareeodiversifytheirholdings，sothatabebalancedbygaiheywishtoholdivelycorrelatedassets。Butthereisaninescapablepieceoflogic:ifXisivelycorrelatedwithY，aivelycorrelatedwithZ，thenXandZwillteivelycorrelated!

However，allisnotlost。Amathematicalresult，duetoSalomonBoer，provesthatitisindeedpossibleforeachpairofassetsifoliotobeivelycorrelated;butthegreaterthenumberofassets，theharderitistoachievemutualion。