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时间:2019-08-04
《Betting with the Kelly Criterion》由会员上传分享,免费在线阅读,更多相关内容在学术论文-天天文库。
1、BettingwiththeKellyCriterionJaneHungJune2,2010Contents1Introduction22KellyCriterion23TheStockMarket34Simulations55Conclusion81Page2of9Hung1IntroductionGamblinginallforms,whetheritbeinblackjack,sports,orthestockmar-ket,mustbeginwithabet.Inthispaper,wesummarizeKelly'scriterionfordeterm
2、iningthefractionofcapitaltowagerinagamble.WealsotestKelly'scriterionbyrunningsimulations.Inhisoriginalpaper,Kellyproposedadierentcriterionforgamblers.Theclassicgamblerthoughttomaximizeexpectedvalueofwealth,whichmeantshewouldneedtoinvest100%ofhercapitalforeverybet.Ratherthanmaximizin
3、gexpectedvalueofcapital,Kellymaximizedtheexpectedvalueoftheutilityfunction.Utilityfunctionsareusedbyeconomiststovaluemoneyandareincreasingasafunctionofwealthundertheassumptionthatmoremoneycanneverbeworsethanless[1].Kellytookthebase2logarithmofcapitalashisutilityfunction[2],butwewillu
4、sethebaseelogarithm(thenaturallog)instead.2KellyCriterionThefollowingderivationismodiedfromThorp[1].Weassumethattheprob-abilityofeventsareknownandindependentandthattheprobabilityofawinisp(1>p>1=2)andtheprobabilityofalossisq=1 p.Supposeafractionf(05、ndLnrepresentthenumberofwinsandlossesafternbets,respectfully.Ratherthanevenpayo(i.e.,awinof1unitperunitbetperwin),weconsiderthemoregeneralscenariothatbunitsarewonperunitbetperwinandaunitsarelostperunitbetperloss.GiveninitialcapitalX0,thecapitalafternbetsisX=X(1 af)Ln(1+bf)Wn:n0Nowde6、ne1Xn1ng(f)=log=(Lnlog(1 af)+Wnlog(1+bf));X0ntheexponentialrateofincreasepertrial.Theexpectedvalueofg(f)isG(f)=E(g(f))=qlog(1 af)+plog(1+bf)becausetheratioofexpectedwinsorlossestotrialsisgivenbytheprobabilitiespandq,respectively.WewanttomaximizeG(f)because11G(f)=E(g(f))=Elo7、g(Xn) log(X0);nnsomaximizingG(f)wouldinturnmaximizeE(log(Xn)),theexpectedvalueofthelogarithmofwealth.AcriticalpointofG(f)canbefoundbysettingthederivativeto0:0aqbpG(f)= +=01 af1+bf2Page3of9HungFigure1:ExpectedValueofLogarithmofWealthvs.BetasaFractionofWealthbp aq abf==0;(1+bf)(1 af)s8、othecritical
5、ndLnrepresentthenumberofwinsandlossesafternbets,respectfully.Ratherthanevenpayo(i.e.,awinof1unitperunitbetperwin),weconsiderthemoregeneralscenariothatbunitsarewonperunitbetperwinandaunitsarelostperunitbetperloss.GiveninitialcapitalX0,thecapitalafternbetsisX=X(1 af)Ln(1+bf)Wn:n0Nowde
6、ne1Xn1ng(f)=log=(Lnlog(1 af)+Wnlog(1+bf));X0ntheexponentialrateofincreasepertrial.Theexpectedvalueofg(f)isG(f)=E(g(f))=qlog(1 af)+plog(1+bf)becausetheratioofexpectedwinsorlossestotrialsisgivenbytheprobabilitiespandq,respectively.WewanttomaximizeG(f)because11G(f)=E(g(f))=Elo
7、g(Xn) log(X0);nnsomaximizingG(f)wouldinturnmaximizeE(log(Xn)),theexpectedvalueofthelogarithmofwealth.AcriticalpointofG(f)canbefoundbysettingthederivativeto0:0aqbpG(f)= +=01 af1+bf2Page3of9HungFigure1:ExpectedValueofLogarithmofWealthvs.BetasaFractionofWealthbp aq abf==0;(1+bf)(1 af)s
8、othecritical
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