paul wilmott on quantitative finance bonus time

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1、CHAPTER75bonustimeInthisChapter...•modelingthebonuscompensationoftraders•incorporatingaskillfactorintovaluationmodels75.1INTRODUCTIONTradersarecompensatedbytheirbonus,inadditiontotheirbasicsalary.Atypicalbonusstructuremightbetopayoutfiveortenpercentofth

2、eprofittheyhavemadeforthebankinthepreviousyear.Thistypeofcompensationhasamajordrawback.Assumingthatthetraderwantstomaximizehisexpectedbonus,thisstructureencourageshimtotakesillyrisks.Forexample,atraderwithanegativetradingaccountwouldtakelargerisksintheh

3、opeofgettinghistradingaccountintotheblack.Ofcourse,thestoryisnotquiteassimpleasthis.Tradershavelimitsimposedontheiravail-ablecapitalandonthegearingandtypeofinstrumentstheyareallowedtotrade.Neverthelesslittleisknownabouthowtorewardagoodtraderoptimally.M

4、ostoftheacademicworkhastakenagame-theoreticapproachbuthasn’tseenmuchuseinpractice.Inthischapterwearegoingtobuildupaframeworkforthestudyofthisproblemandexploreavarietyofpossiblecompensationstructures.Themathematicsofthischapterissimilartothatrequiredint

5、heanalysisofthepassportoption,Chapter27.75.2ONEBONUSPERIODWe’veseentheanalysisoftheproblemofthepassportoptioninChapters27and63.Well,that’sprettymuchthesameasthetraderbonusproblemwhenthebonusisafractionofthe(positivepartofthe)profit.Thingsgetmoreinterest

6、ing,andsensibleperhaps,ifthebonusdependsalsoontherealizedSharperatio.WhyshouldabankwanttocompensateatraderdependingonhisSharperatio?Becausethatwaytheyexertsomecontrolovertheriskthatthetradertakes.75.2.1BonusDependingontheSharpeRatioAsalwaysSistheassetp

7、ricefollowingdS=µSdt+σSdX.1176PartFiveadvancedtopicsLetπbethevalueofthetradingaccount.Itfollowsthatdπ=rπdt+q(dS−rSdt),whereqisthepositionofthetrader.Hewillhavesomerestrictionsuchas

8、q

9、≤C,Cisthepositionlimit.LetIbethevarianceofthetradingaccount.Itfollows

10、that
tI=q2σ2S2dt0anddI=q2σ2S2dt.Attimet=0,π=I=0andthetraderbeginstotradetheunderlyingasset.Attheendoftheyear,thebankgivestothetraderabonusdependingontheprofitmade,π(T)andtheSharperatioπ√.II’mignoringtherisk-freeinterestratethatshouldbein