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Chapter 8 Other applications（第2页）

Buthowmightthepunterdeducehisdegreesofbeliefinthedifferentoutay2009，statistiDavidSpiegelhaltertookupthegeonBBoreorLessbyanalysiobeplayedinthePremierLeaguetwodayslater.Fame，heestimatedthenumberofgoalseachteammightsaverage，takingatitsthinattadtheiroppos’defensivecapabilities.Forexample，astroeam（Arseedtoals，aiy.

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.