AStylizedHistoryof

QuantitativeFinance

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Emanuel DermanColumbia University

Modern Finance in One SentenceFeynman:Physics/Medicineinonesentence.

Finance:ForAnySecurity:

IFyoucanhedgeawayallcorrelatedrisk

ANDyoucanthendiversifyoveralluncorrelatedrisk

THENyoushouldexpectonlytoearntherisklessrate

_________________________________________________________Whatdoesthisassume?

• Stablefrequentiststatisticsofdiffusion• Replication (ofarisklessbond)• TheLawofOnePrice

Thehistory…

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Summary

• Derivativessans Diffusion• Diffusionsans Derivatives• 1952:Markowitz:RiskasVolatility• 1958:TheIdeaofReplication andTheLawofOnePrice• Ifyoucaneliminateallrisk,youmusthavereplicated arisklessbondwhosereturnyouknow:CAPM• WhyisCAPMbad?• 1960s:OptionsModels:Derivatives +Diffusion +Volatility + Hedging + Replication• WhyisBlack-Scholes-MertonbetterthanCAPM?• BusinessTimevsCalendarTime• 1976:Calibration:TheInventionofImpliedVolatility• VolatilityasanAssetClass• 1977:Vasicekonmodelingparametersratherthanassets• TheSmile• EvenMoreCalibration• TheFuture:Whatmakesamarket?• BehavioralFinance• MarketMicrostructure

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The Great Idea of Derivatives

▪ Thiscertainlygoesbackthousandsofyears,but…jumpingto17th Century:Spinoza

▪ Spinoza’streatsemotionslikeEuclidtreatsgeometry:emotionsarederivatives.

- PrimitivesareDesire,Pleasure,Pain (cf.Equity,FixedIncome,Credit)

▪ Good iseverythingthatbringspleasure,andEvil iseverythingthatbringspain.

▪ Love:Pleasure associatedwithanexternalobject.

▪ Hate: Painassociatedwithanexternalobject.

▪ Envy:Pain atanother’sPleasure.

▪ Schadenfreude

▪ Cruelty:Desire toinflictPain onasomeoneLoved.

▪ Threemoreprimitives:

- Vacillation,Wonder,Contempt.

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Derivatives sans Diffusion• Spinozaisverymodern:thepassionsgaintheirvaluefromtheirrelationshiptounderlyingsensations.

• Spinozabelievesthathumanbehaviorbehaveslaws,thatnothingisrandom.

• Buthisschemeanddefinitionsarestatic:thereisalmostnopossibilityofmotion,exceptfor

Vacillation:thecyclicalternationbetweentwodifferentpassions.

JealousyistheoscillationbetweenHatred andEnvy inrelationtoaLoveobjectandarival

HatredisPain associated EnvyisPain atanother’sPleasure.withanexternalPerson

Vacillationinvolvesvolatility– themorerapidlyandintenselyoneVacillates,thegreatertheJealousy.

• SpinozahasnoAnxietyinhissystem.Variousopinions:

Anxietyis avacillationbetweenHopeandFear;

AnxietyisnotaPassion;

TherewasnoAnxietyinthe17th Century.

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Diffusion sans Derivatives

•1831:ThomasGraham“…gases…whenbroughtintocontact,donotarrangethemselvesaccordingtotheirdensity,…buttheyspontaneouslydiffuse,mutuallyandequally,througheachother,andsoremainintheintimatestateofmixtureforanylengthoftime.”

•1858:JamesClerkMaxwellthefirsttheoryofatomsmovingingases

•LudwigBoltzmann,Boltzmannequation:kinetictheoryderivesthepropertiesofmatterfromthepropertiesofatoms

•Early20thCentury.AlbertEinstein,MarianSmoluchowski andJean-BaptistePerrin confirmatomictheoryofmatter.

•Physicistsunderstooddiffusionnotderivatives.Theydiscussedthebehaviorofunderliers,butnotfunctionsofunderliers.

•Except:Bachelier intheopeningyearofthe20th CenturydevelopedandapplieddiffusiontofinanceandderivedtheequationsforBrownianmotionappliedtooptionswithunderliers thatundergoarithmeticBrownianmotion.(Aheadofhistime,rediscoveredinthe1960s)

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1952:Risk = Volatility

• HarryMarkowitzmeasures“expectedreturn”against“risk”.

Risk=thestandarddeviationofreturns.

Suggestsfindingtheportfoliowiththemostreturnforagivenrisk

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1958: The Idea of Replicationand The Law of One Price

•ModiglianiandMiller: Youcanrecreate (andvalue) aleveredfirmfor

yourselfbyborrowingmoneytobuythestockofanunleveragedfirm.

•Replicationasastrategyforvaluation

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Early 1960s: CAPM/APT

• Finance’s Key Question: What return should you expect from taking on risk?

• Use Markowitz definition of risk as standard deviation

• Replication and the Law of One Price leads to CAPM and APT as well as BSM.

• The only currently known return is r, that of a riskless bond. The valuation strategy is to link the unknownreturn on any security to r by reducing the security’s risk to zero by including it in a portfolio.

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How Can You Reduce Risk?

• Dilution: Combine security with a riskless bond

• Diversification: Combine security with many uncorrelated securities

• Hedging: Combine security with a correlated security

• Apply this in three successively more realistic worlds

• Dilution: Combine weight w of a risky stock S (µ,s) with a weight (1 - w) of a riskless bond B (r,0) to create a new security with lower risk & return

• Law of One Price: All uncorrelated stocks with risk ws earn excess return w(µ – r)

• One parameter fixes everything

• Same Sharpe ratio for all stocks!

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Simple World 1 A few uncorrelated stocks and a riskless bond:

More risk, more returnAll stocks have same Sharpe Ratio

Less Simple World 2:Many Uncorrelated Stocks: Diversify!

Every stock is expected to earn the riskless rate rThe Sharpe Ratio is zero

• Suppose there are countless uncorrelated stocks

• Put them all in a portfolio with weights:

• Then the portfolio risk diversifies to zero.

• Thus the portfolio is riskless:

• But the portfolio return is the sum of individual returns:

therefore

• Thus every stock is expected to earn the riskless rate!

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More Realistic World 3: CAPMAll Stocks are Correlated with the Market:

Hedge Market Risk, Then Diversify!Sharpe Ratio of Every Market-Neutral Stock is Zero.

Every market-neutral stock must earn the riskless rate.

• Suppose there are countless stocks Si correlated with the market M

• Then the market-neutral stock is uncorrelated with M

• After diversification each market-neutral stock earns riskless rate.

• Which means

• CAPM just says that if you hedge every stock with the market, and then diversify over all remaining risk, you should earn only the riskless rate.

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Why is CAPM Bad?

• BecauseRiskisNotReallytheStandardDeviationofReturns

• AndthemarketMandthestockSarenotreallystablycorrelated.

• Marketsarenotexactlylikeflippingcoins.Thereisn’tawell-definedfrequentistprobabilityofamarketcrash.

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1960s: Early Options Models:

Diffusion and Volatility but No Replication

• Samuelson,Sprenkel,Ayres,Boness…valuestockoptionsactuarially,astheexpecteddiscountedpayoff oftheoptionunderalognormaldistributionwithagrowthrateandavolatility.

• Butatwhatratedoesthestockgrow?

• Andwhatdiscountratetouse?

• Ifyoudemandconsistencywithput-callparity,youcanguessthattherightrateistherisklessrate.Whydidnooneguessthat?

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1973: Putting everything togetherBlack-Scholes-Merton

Diffusion+Volatility+Hedging+Replication

• Usediffusionforthemoveinthe underlyingstockprice dS• Usestochasticcalculustofindthemoveinthederivative dC(S)• Hedgetoeliminatestockriskfromoption:dC - D dS• Requirethathedgedportfolio,whichisriskless,earnstheknownrisklessrate r:

dC-DdS =r(C-DdS)

• Thenwegetthesameformulaastheactuarialone,butwhereallgrowthanddiscountratesarereplacedbytherisklessrater.

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Black-Scholes-Merton

• Youcanreplicate/hedgeanoptionwithstock

• OptionCandstockSmusthavesameSharperatio

• Ito’sLemmaappliedtoaCleadstoBlack-Scholes

• AunifiedtreatmentofBSMandCAPMfromoneprinciple

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Why is BSM better than CAPM?

• Becauseyou really canhedgeanoptionwithastock,becausethecorrelationisreally1.

• Soevenifyoudon’tbelievetheriskisthestandarddeviationofreturns,thetwosecuritiesreallyaretrulyconnected,unlikethestatisticalconnectionbetweentwodifferentstocks.

• Wehaveassumedthatvolatilityisunchanging!Ifvolatilityisrandom,thenthederivativeisnotreallyaderivativeexceptatexpiration.

1970s: Using the BS Equation

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• Now,insteadofforecastingthereturnofthestock,tradersmustforecastthevolatilityofthestock.

• BlackandScholessetaboutusingtheequationbyusinghistoricalvolatilitiestoestimatefuturevolatilities.Butwhoknowswhatfuturevolatilitywillbe?

1973: Use Business Time for Measuring Risk and Return

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• P.C.Clark:Theideaofexaminingmarketmovementswithaclockthatdoesn’ttickseconds,butticksnumberoftradesorvolumeoftrades. Abehavioralapproachtotimeperception.

• Returnsseemtobenormallydistributedwhenexaminedasreturnpertickratherthanreturnpersecond. (Geman,Ané)

• IfallstockshavethesameSharpeRatioinbusinesstime,then,incalendartime:

Expectedreturnishighwhenvolatilityortradingfrequencyislarge.

• KyleObizhaeva

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• LataneandRendelmansuggestfittingoptionmarketpricestotheBlack-ScholesformulaandextractingtheWISD(weightedimpliedstandarddeviationofthestock).TheythensuggestcalculatinghedgeratiosfromthemodelusingtheWISD.

• “…theWISDisgenerallyabetterpredictoroffuturevariabilitythanstandarddeviationpredictorsbasedonhistoricaldata.”

• Butimpliedvolatilitiesareunstable.Thisprocessmustberepeatedastheimpliedvolatilitykeepschanging,sothereissomethingnotquiteright.

• Impliedvolatilitiestellyouthatifyoubelievethemodel,givenanoptionprice,thisiswhatthefuturemustbelike.Butitisn’t.

• Nevertheless,fromnowoneveryonecalibrates (andrecalibrates) models.

• Mostpeopledon’tevenrealizeitwasaninvention.

Trading Volatility is Now a Possibility

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• BSMsaysthat

• Impliedvolatilityisanestimateoffuturevolatility.

• Butitkeepschanging.

• Soyoucan’treally replicate anoption

• Instead,youcanspeculateonvolatility,usingoptions.

• Ifyouusethemodel,optionsnowbecomeawayoftradingvolatilityratherthanspeculatingonthestock price.

• Volatilityasanassetclass.

1977: Vašíček

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• Modelingtheyieldcurve– extensionofBlack-Scholestoparametersratherthanassets.

• Youcanvalueoptionsontwostockindependently,butyoucannotvalueoptionsontwoTreasurybondsindependently.

• Thereareno-arbitrageconstraintsonbondprices.

• Andsoon…tootherextensionsofthehedgingparadigmfor40years…

1987: The Smile

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• Whenyoufitthemodeltodifferentoptionstrikes,eachoneimpliesadifferentfuturevolatilityfortheunderlyingstock.Butinthemodelastockcanonlyhaveonevolatility.Nowsomethingisreallywrong– theBSmodelcanNOTaccommodatedifferentvolatilitiesforthesamestock.

• Nevertheless,peoplekeepusingthemodelinconsistentlytoestimatehedgeratiosastheycalibratethemodeltoagivenoption price.

Before 1987

1994 - present: The Smile

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• Extensions:localvolatility,stochasticvolatility,jumps…

• Morecomplexitywithoutaccurate knowledgeoftheparameters.

• Usemarket prices toimplytheparameters- e.g.thevolatilityofvolatilityinacalibratedstochasticvolatilitymodel.

• Butmarketschangeandtheseimpliedparametersarethemselvesunstableandrandom,buttherearenowmoreofthem.So,forexample,volofvolisnowstochastic.

• Sonowusingthemodelyouhaveamarketfortradingvolatilityofvolatility

The Future

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• Aninfiniteregressofmodels?

• Eachmodelisinadequate,introducesnewparameters,which,asthemodelisembraced,becomequantitiesthemarketcanspeculateandtradeon.

• “Derivativesarenot(really)derivative,exceptatexpiration.”

• Ittakesallofthesesecurities—

• stocks,options,optionsonoptions,volatility,volatilityofvolatility…

• todefinethepossibilitiesofthemarket.

• Theseinstrumentsarenottrulyderivative.Theprobabilisticapproachtodistributionsisafallacy.The“probabilities”onlycomeintoexistenceafteranevent,whichis the offeringaprice.(Ayache,TheMediumofContingency)

• “Allwehaveinthemarketarepricesofcontingentclaimsofvaryingcomplexity.Probabilisticmodelsandpricingkernelsandderivativevaluationtoolsareonlyinternalepisodesthatwerequireinorderlocallyandalwaysimperfectlytohedgesomething.Wehavetokeepinmindthatthoseepisodesarepresentandusefulonlyinsofarastheywillberecalibrated.”

Human Affairs

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