Betting with the Kelly Criterion

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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.2KellyCriterionThefollowingderivationismodiedfromThorp[1].Weassumethattheprob-abilityofeventsareknownandindependentandthattheprobabilityofawinisp(1>p>1=2)andtheprobabilityofalossisq=1p.Supposeafractionf(0

5、ndLnrepresentthenumberofwinsandlossesafternbets,respectfully.Ratherthanevenpayo(i.e.,awinof1unitperunitbetperwin),weconsiderthemoregeneralscenariothatbunitsarewonperunitbetperwinandaunitsarelostperunitbetperloss.GiveninitialcapitalX0,thecapitalafternbetsisX=X(1af)Ln(1+bf)Wn:n0Nowde

6、ne1Xn1ng(f)=log=(Ln
log(1af)+Wn
log(1+bf));X0ntheexponentialrateofincreasepertrial.Theexpectedvalueofg(f)isG(f)=E(g(f))=q
log(1af)+p
log(1+bf)becausetheratioofexpectedwinsorlossestotrialsisgivenbytheprobabilitiespandq,respectively.WewanttomaximizeG(f)because11G(f)=E(g(f))=E
lo

7、g(Xn)
log(X0);nnsomaximizingG(f)wouldinturnmaximizeE(log(Xn)),theexpectedvalueofthelogarithmofwealth.AcriticalpointofG(f)canbefoundbysettingthederivativeto0:0aqbpG(f)=+=01af1+bf2Page3of9HungFigure1:ExpectedValueofLogarithmofWealthvs.BetasaFractionofWealthbpaqabf==0;(1+bf)(1af)s

8、othecritical