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Dimensional Analysis,Leverage Neutrality, andMarket Microstructure InvarianceAlbert S. KyleAnna A. ObizhaevaUniversity of MarylandNew Economic SchoolSummer School, MoscowAugust 2019Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance1/60
Big QuestionsWe think of trading a stock as playing a trading game: Long-termtraders buy and sell shares to implement “bets,” and intermediarieswith short-term strategies–market makers, high frequency traders,and other arbitragers–clear markets. Can we derive quantitative predictions about microstructurevariables? Is there a simple empirical measure of liquidity andhow can theoretical liquidity parameters, like market impactcoefficient λ, be implemented empirically? Trading games look different across assets that different interms of their trading activity: dollar trading volume, volatilityetc. Are there any fundamental (universal) laws in financialmarkets as in physics?Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance2/60
Answers: Use General Principles Trading games look different across assets only at first sight!They are similar if one looks through invariance lenses. Invariance implies a simple measure of liquidity as function ofvolume V · P and returns volatility σ 2 :(L m2 · P · VC · σ2)1/3where constants m2 0.25 and C 2000. Liquidity L can be mapped to permanent price impact λ,temporary price impact κ, funding and trading liquidity etc. Theoretical dynamic models can link liquidity L to resiliency ofprices ρ and error variance of prices Σ.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance3/60
A General PictureThe scaling laws can be derived using different approaches: “Market Microstructure Invariance: Empirical Hypotheses”(Ecma, 2016): Empirical conjectures and tests. “Market Microstructure Invariance: A Dynamic EquilibriumModel”: Dynamic equilibrium model of speculative trading inwhich liquidity constrained investors seek to profit fromtrading on signals with invariant cost. “Adverse Selection and Liquidity: From Theory to Practice”:A meta-model. This paper: Physicists’ approach, apply dimensional analysis(consistency of units, Buckingham π-theorem)Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance4/60
APPROACH I: DIMENSIONAL ANALYSISAND LEVERAGE NEUTRALITYDerivation of Invariance for PhysicistsPete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance5/60
OverviewThis paper combines dimensional analysis, leverage neutrality, anda principle of invariance to derive scaling laws. Scaling laws relate transaction costs functions, bid-askspreads, bet sizes, number of bets, and other financialvariables in terms of dollar trading volume and volatility. These laws are tested using a data set of trades in the Russianand U.S. stock markets and find a strong support in the data. These scaling laws provide useful metrics for risk managersand traders; scientific benchmarks for evaluating issues relatedto high frequency trading, market crashes, and liquiditymeasurement; and guidelines for designing policies.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance6/60
Dimensional AnalysisPhysics researchers obtain powerful results by using dimensionalanalysis to reduce the dimensionality of problems (the size andnumber of molecules in a mole of gas, the size of the explosiveenergy, turbulence). Physics: fundamental units of mass, distance, and time &conservation laws based on laws of physics. Finance: fundamental units of time, currency, and shares &conservation laws based on no-arbitrage restrictions.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance7/60
Oscillation of a Pendulum?Suppose the time of oscillation T of a pendulum T f (M, L, g ).[M] kg[T ] sand[L] m[g ] m/s2 .Buckingham π theorem: Rescaled T is a function of N 3rescaled dimensionless variables LLL· f (dimensionless variables) · f (·) · const.T gggThe law of conservation of energy implies that const 2π. If T f (M, L, g , x1 , x2 ), then T gL · f (scaled x1, scaled x2).Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance8/60
How Big was the Bomb?The first atomic blast, the Trinity Test in New Mexico in 1945, hadan explosive yield of about 20 kilotons, but this value was secret.Based on photographs of the Trinity Test released by the US Armyin 1947 and dimensional analysis, Taylor guessed the size E fromR ( E · t 2 )1/5,ρwhere R is radius, E is energy, t is time, ρ is density of air.[E ] kg · m2 /s2[R] mand[t] s[ρ] kg/m3 .Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance9/60
Dimensional Analysis and FinanceIn financial markets, institutional investors trade by implementingspeculative “bets” which move prices. A bet is a decision to buy orsell a quantity of institutional size.Trading is costly; bets tend to move market prices.Dimensional analysis can be used to find formulas for the numberof bets, their average size, market depth, transaction costs, etc.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance10/60
Market Microstructure VariablesEasy-to-observe quantities include price, volume, and volatility:Price Pjt 40.00 dollars/shareTrading Volume Vjt 1.00 million shares/dayReturns Volatility σjt 0.02/day1/2Hard-to-measure quantities that vary greatly across assets and timeinclude bet size, number of bets, and the price impact coefficient:Size of Bet Qjt 10 000 sharesNumber of Bets γjt 100/dayExecution Horizon Hjt 1 dayPrice Change per Bet Pjt 0.04 dollars/sharePrice Impact Coefficient λjt 5 10 5 dollars/share2( )Fjt1/2Price Error Σjt var1/2 {log} log(2)(dimensionless)PjtPrice Resiliency ρjt 0.0040/dayPete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance11/60
Transaction CostsTransaction costs are usually also hard to measure.Price Impact Pjt 0.04 dollars/sharePrice impact cost Gjt as fraction of value traded:Gjt Pjt Pjt · Qjt 10 basis pointsPjtPjt · QjtPrice impact cost in dollars:Dollar Price Impact Cost Pjt · Qjt 400 dollarsAvg Dollar Cost per Bet C :C EQ { Pjt · Qjt }.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance12/60
Dimensional Analysis and FinanceBasic idea: Use dimensional analysis like a physicist (units consistency,Buckingham π Theorem): “Guess” correct functional form, i.e, correct list of explanatoryvariables. Warning: Incorrect guess may lead to nonsense. Reduce dimensionality of problem by factoring out units, makingremaining parameters dimensionless. Add restriction of “leverage neutrality” (Modigliani–MillerTheorem) to reduce dimensionality further. The cost of exchangingcash is zero. Dollar market impact cost of exchanging a risky bundleof assets is the same for any positive or negative amount ofcash-equivalent assets included with the bundle. Make empirically motivated invariance assumption.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance13/60
Dimensional Analysis Approach: Example of γI. Assume that the variable of interest—the number of bets γjt —isdetermined by some unknown functions fγ , which take share volume Vjt ,share price Pjt , returns volatility σjt , and expected dollar costs C as theirarguments:γjt fγ (Vjt , Pjt , σjt2 , C ),II. Reduce the dimensionality by applying dimensional analysis:[σjt2 ] 1/day[γjt ] 1/dayand[Vjt ] shares/day[Pjt ] dollars/share[C ] dollars.(γjt σjt2 · gγ C ·Pete Kyle and Anna Obizhaeva( C · σ 2 )αγσjt2 )jt σjt2 ·.Vjt · PjtVjt · PjtDimensional Analysis and Market Microstructure Invariance14/60
Example Cont’d: Leverage NeutralityIII. Impose a leverage neutrality restriction:Exchanging cash-equivalent assets incurs zero cost. Exchanging riskysecurities is costly. The economic cost of trading bundles of riskysecurities and cash-equivalent assets is the same for any positive ornegative amount of cash-equivalent assets included into a bundle.γjt γjt ,Pjt Pjt · A,Vjt Vjt ,σjt σjt · A 1 ,C C,Pjt · σ Pjt · σjt .This restriction implies that αγ 2/3.γjt σjt2 ·( C · σ 2 )αγ( C · σ 2 ) 2/3jtjt γjt σjt2 ·.Vjt · PjtVjt · PjtPete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance15/60
Example Cont’d: InvarianceIV. Impose invariance restriction:Average dollar cost C is hard to observe. Suppose C is approximatelyconstant across assets and time, perhaps due to equilibrium in allocatingresources and skills across markets. Then,γjt σjt2 ·( C · σ 2 ) 2/3jtVjt · Pjt (In terms of liquidity measure Ljt ()2/3.γjt σjt · Vjt · Pjtm2 · Pjt · VjtC · σjt2)1/3, we getγjt σjt2 · L2jt .Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance16/60
Russian Data One-minute data from the Moscow Exchange forJanuary–December 2015 provided by Interfax Ltd. 50 Russian stocks in the RTS index as of June 15, 2015. The Russian stock market is centralized with all tradingimplemented in a consolidated limit-order book. Small tick and lot sizes.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance17/60
U.S. Data One-minute data from the Trades and Quotes (TAQ) datasetfor January–December 2015. 500 U.S. stocks in the S&P 500 index as of June 15, 2015. The U.S. stock market is fragmented, and securities aretraded simultaneously at dozens of exchanges. Tick size of one cent, and lot sizes of 100 shares.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance18/60
Tests for Number of TradesLet Njt denote the number of trades. SupposeNjt γjtThen, from predictionγjt σjt2 · L2jt ,we getlog (Njt ) const 2 · log(σjt Ljt ).Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance19/60
Number of Trades: Results for Russian DataTuesday, April 19, 2016 01:23:43 AM 1141210ln N8642012345678ln L sdIn aggregate sample, the slope is close to 2! R-square is 0.882.log(Njt ) 3.085 2.239 · log(σjt Ljt )Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance20/60
Number of Trades: Results for U.S. DataIn aggregate sample, the slope is close to 2! R-square is 0.702.log(Njt ) 1.005 1.842 · log(σjt Ljt )Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance21/60
Dimensional Analysis: Transaction CostsLet Gjt denote the price impact cost as a fraction of the valuetraded Qjt · Pjt . The price impact Gjt is dimensionless, e.g. in basispoints, and it is a function ofGjt : Pjt (Qjt ) g (Qjt ; Pjt , Vjt , σjt2 , C ).Pjt bet size Qjt in units of shares, stock price Pjt in units of dollars per share, share volume Vjt in units of shares-per-day, volatility σjt2 in units of per-day, bet cost C in units of dollars.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance22/60
Dimensional AnalysisSince the value of Gjt : g (Qjt , Pjt , Vjt , σjt2 , C ) is dimensionless,consistency of units implies that it cannot depend on thedimensional quantities Pjt , Qjt , and σjt2 .Thus, dimensional analysis implies that the function g () can befurther simplified by writing it as function of two dimensionlessvariables.Gjt Pjt0 · Qjt0 · (σjt2 )0 · f (two dimensionless variables) f (two dimensionless variables).Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance23/60
Dimensional AnalysisThere are three sets of distinct units and five dimensionalquantities—Qjt , Pjt , Vjt , σjt2 , C .Form two independent dimensionless quantities:(Ljt : m2 · Pjt · Vjtσjt2 · C)1/3,Zjt : Pjt · Qjt,Ljt · Cwhere m2 is a dimensionless scaling constant.Thus, dimensional analysis implies that the function g can befurther simplified by writing it as g (Ljt , Zjt ).Gjt : g (Ljt , Zjt ).Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance24/60
Leverage NeutralityThe cost of exchanging cash is zero. Adding cash/riskless debt toa risky asset or changing margin requirements must not affecteconomic costs and trading.If (A 1)P dollars of cash or debt is added to Pjt , thenPjt Pjt · Aσjt2 σjt2Qjt QjtVjt VjtLjt Ljt · A 2·AZjt ZjtC CGjt Gjt · A 1The dollar costs Gjt · Qjt · Pjt are the same, but dollar bet sizeQjt · Pjt changes. 1/Ljt has the same leverage scaling as Gjt .Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance25/60
Leverage NeutralityPercentage cost Gjt of executing a bet of Qjt shares changes by afactor A 1 , since dollar cost did not change but dollar valuechanged. Leverage neutrality implies thatg (A · Ljt , Zjt ) A 1 · g (Ljt , Zjt ). 1If A L 1jt , then g (Ljt , Zjt ) Ljt · g (1, Zjt ).Define f (Zjt ) : g (1, Zjt ) and get a very important formula:Gjt g (Ljt , Zjt ) Pete Kyle and Anna Obizhaeva1· f (Zjt ).LjtDimensional Analysis and Market Microstructure Invariance26/60
Transaction Costs ModelA general specification for transaction costs functions consistentwith the scaling implied by dimensional analysis and leverageneutrality:(g (Qjt , Pjt , Vjt , σjt2 , C ) σjt2 ·C2m ·Pjt ·Vjt(()1/3·fσjt2 ·C2m ·Pjt ·Vjt))1/3·Pjt ·QjtC.It is consistent with different assumptions about the shape of thefunction f .Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance27/60
Market Microstructure InvarianceExtra assumptions are necessary to make our predictionsoperational. Three of the quantities—asset price Pjt , trading volume Vjt ,and return volatility σjt —can be observed directly or readilyestimated from public data feeds. Qjt is a characteristic of a bet privately known to a trader. Invariance: the dollar value of C and the dimensionlessscaling parameter m2 are the same!These assumptions are related to bet size and transaction costsinvariance hypotheses. Preliminary calibration gives C 2, 000and m2 0.25.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance28/60
Economic IntuitionScale m and define C so thatE { Zjt } 1andC E {Gjt · Pjt Qjt }.The variables Ljt and Zjt have an intuitive interpretation: CE {Pjt · Q̃jt }is “illiquidity index” measuring average cost. Zjt size.Pjt ·Q̃jtE {Pjt · Q̃jt }is “scaled bet size” relative to the average1Ljt m E { Qjt }1/2(E {Qjt2 })is moment ratio.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance29/60
Liquidity IndexThe liquidity index Ljt is consistent in terms of units. It is thecorrect way to construct empirical measure of Kyle’s lambda.(Ljt : m2 · Pjt · Vjtσjt2 · CPete Kyle and Anna Obizhaeva)1/3( Pjt · Vjtσjt2)1/3.Dimensional Analysis and Market Microstructure Invariance30/60
Liquidity Index and Other VariablesLiquidity index Ljt can be linked to many variables, includingcomposition of order flow. More liquid markets are associated withmore bets of larger sizes (2-to-1 ratio): Bet size E {Pjt · Q̃jt } C · Ljt . Number of bets per day γjt 1m2· σjt2 · L2jt .Liquidity index Ljt appears in market impact formula:Gjt g (Ljt , Zjt ) Pete Kyle and Anna Obizhaeva1· f (Zjt ).LjtDimensional Analysis and Market Microstructure Invariance31/60
Transaction Costs ModelsSuppose f (·) is a power function of the form f (Zjt ) const · Zjt ω . A percentage bid-ask spread cost (ω 0) impliesGjt Sjt1 const ·.PjtLjt A linear market impact cost (ω 1) impliesGjt const ·Pjt · Qjt .C · L2jt A square-root market impact cost (ω 1/2) implies(Gjt const · σjt ·Pete Kyle and Anna Obizhaeva Qjt Vjt)1/2.Dimensional Analysis and Market Microstructure Invariance32/60
Tests for Bid-Ask SpreadLet Sjt denote the percentage bid-ask spread (ω 0). SinceSjt1 const ·,PjtLjtwe get(logSjtPjtPete Kyle and Anna Obizhaeva)( const 1 · log1Ljt).Dimensional Analysis and Market Microstructure Invariance33/60
Spread: Results for Russian DataTuesday, April 19, 2016 01:22:20 AM 1-2-3-4ln S/P-5-6-7-8-9-10-12-11-10-9-8-7-6-5-4ln 1/LIn aggregate sample, the slope is close to 1! R-square is 0.876.log(Sjt /Pjt ) 2.093 0.998 · log(1/Ljt )Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance34/60
Spread: Results for U.S. DataIn aggregate sample, the slope is close to 1! R-square is 0.450.log(Sjt /Pjt ) 1.011 0.961 · log(1/Ljt )Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance35/60
ExtensionsThe empirical implications of dimensional analysis, leverageinvariance, and market microstructure invariance can begeneralized to incorporate various trading frictions.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance36/60
Generalized Transaction Costs FormulaAdd the execution horizon Tjt (in units of time), the tick sizeKjtMIN (in dollars per share), and the lot size QjtMIN (in shares).Re-scale variables to make them dimensionless and leverage neutralusing the four variables Pjt , Vjt , σjt2 , and C : Qjt Tjt Qjt Vjt ·Tjt , KjtMIN KjtMIN ·LjtPjt , QjtMIN QjtMIN ·σjt2 ·L2jtVjt .Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance37/60
Generalized Transaction Costs Formula1Gjt ·fLjt(2MINMIN2Pjt · Qjt Qjt Kjt · Ljt Qjt · σjt · Ljt;,,C · Ljt Vjt · TjtPjtVjtPete Kyle and Anna Obizhaeva)Dimensional Analysis and Market Microstructure Invariance.38/60
Optimal Execution HorizonSuppose the optimal execution horizon Tjt for an order of Qjtshares depends on Pjt , Vjt , σjt2 , C , KjtMIN , and QjtMIN .Since Qjt /(Vjt · Tjt ) is dimensionless and leverage neutral, thesame logic implies: Qjt h Vjt · Tjt (MINMIN22Pjt · Qjt Kjt · Ljt Qjt · σjt · Ljt,,C · LjtPjtVjt).If tick size and lot size do not affect execution horizon, Qjt /(Vjt · Tjt ) depends only on Zjt : Pjt · Qjt /(C · Ljt ).Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance39/60
Optimal Tick Size and Lot SizeSetting optimal tick size and minimum lot size is of interest forexchange officials and regulators.Let KjtMIN and QjtMIN denote optimal tick size and optimalminimum lot size, respectively.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance40/60
Optimal Tick Size and Lot SizeSince the scaled optimal quantities KjtMIN · Ljt /Pjt andQjtMIN · L2jt · σjt2 /Vjt are dimensionless and leverage neutral, thescaling laws for these market frictions areKjtMIN const ·Pete Kyle and Anna ObizhaevaPjt,LjtQjtMIN const ·Vjt.· σjt2L2jtDimensional Analysis and Market Microstructure Invariance41/60
General Formula for Bid-Ask SpreadHere is a formula for bid-ask spread for the market with frictions:Sjt1 ·sPjtLjt(KjtMIN · Ljt QjtMIN · σjt2 · L2jt,PjtVjt).If tick size and minimum lot size have no influence on quotedbid-ask spreads, then the the relationship simplifies toSjt /Pjt 1/Ljt .Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance42/60
General Formula for Trading PatternsHere are general formulas for trade sizes X̃jt and number of tradesNjt :{ProbPjt · X̃jt zC · Ljt}( FjtQ(Njt σjt2 · L2jt · fPete Kyle and Anna ObizhaevaKjtMIN · Ljt QjtMIN · σjt2 · L2jtz,,PjtVjtKjtMIN · Ljt QjtMIN · σjt2 · L2jt,PjtVjt).).Dimensional Analysis and Market Microstructure Invariance43/60
Dimensional Analysis Looks Simple Ex PostSuppose the variables of interest are functions of share volume Vjt ,share price Pjt , returns volatility σjt (NOT dollar costs C )γjt fγ (Vjt , Pjt , σjt2 ),Then, we get empirically implausible prediction, also beinginconsistent with leverage neutrality principle:γjt σjt2 .Similar analysis for G results in the Barra sqrt model, G σ ·QV.Theory has to provide guidance on which arguments have to beused.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance44/60
ConclusionsThere is a growing empirical evidence that the scaling lawsdiscussed above match patterns in financial data, at leastapproximately.Future research: Checking the validity of invariance predictions in othersamples, Improving the accuracy of estimates and the triangulation ofproportionality constants.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance45/60
APPROACH II: META-MODELDerivation of Invariance for Econo-physicistsPete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance46/60
ReferencesThese slides are based on the following paper: Kyle and Obizhaeva, “Adverse Selection and Liquidity: FromTheory to Practice”. Kyle and Obizhaeva, “The Market Impact Puzzle”.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance47/60
Meta ModelBasic idea: Write down some simple generic equations that arelikely to be valid in most theoretical models: Orders add up to trading volume; Order flow creates returns volatility; Each bet moves prices.Can we derive invariance formulas from these equations? Yes.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance48/60
Meta-ModelSuppose power function price impact for a bet Q: P λ · Q β .Define γ number of bets per day.Now assume three-equation “meta-model”:V γ · E [ Q ][()2 ] P2σ γ·EP[]E {( P)2 } λ2 · E Q 2β(Definition of volume)(Bets generate all volatility),(Volatility from one bet).Three easy-to-measure quantities: V , σ, P.We have five unknown hard-to-measure quantities:[][]γ, E [ Q ] , E Q 2β , E ( P)2 ,λ.and only three log-linear constraints, so we need more equations.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance49/60
Empirical Motivation for Invariance Emprical Problem: Parameters like bet arrival rate γ and betsize E [ Q ] are hard to measure or estimate. Can they be replaced with a parameter that is either easier toestimate or does not vary much across assets? Empirical strategy: Introduce a parameter C (dollars), whichdoes not vary (much) across assets and time. Use “invariant” parameter C to replace parameter which arehard-to-measure and varying across assets, such as γ orE [ Q ]. Assume “transactions cost invariance”: Ex ante expecteddollar cost of a bet is constant (almost?)[]C E [ Q · P ] λ · E Q 1 β .Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance50/60
Augmented Four Equation Meta-ModelAdd transactions cost invariance to obtain four equations:V γ · E [ Q ][()2 ] P2σ γ·EP[][ 2β ]22E ( P) λ · E Q []C λE Q 1 β(Definition of volume)(Bets generate all volatility),(Price Impact of one bet),(Dollar impact cost of a bet).Need two invariant moment ratios: E [ Q ] · E [ Q 2β ]m : ,E [ Q β 1 ]β 1mβ : (E [ Q ]).E [ Q β 1 ]Assume six parameters are easy to measure or almost constant:P,V,σ,C,m,mβ .Solve six equations for six hard-to-measure parameters:[][][]γ, α, E P 2 , E [ Q ] , E Q 1 β , E Q 2β .Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance51/60
Solution with InvarianceDefine “illiquidity” 1/L as volume-weighted expected cost:()1/31Cσ2 · C: .LE [ P · Q ]m2 · P · VThen, expected bet size and number of bets are given byE [ P · Q ] C · L,γ 1· σ 2 · L2 .m2Price impact is P1 · mβ · Z β ,PLwhereZ : QP ·Q ,E [ Q ]C ·LIf C , m, and mβ are invariant across assets, then we have auniversal market impact formula and universal formula for size andnumber of bets. They require estimation of only these threeparameters and β! All formulas turned out to be the same as indimensional analysis!Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance52/60
Calibration of Constants(Preliminary) Calibration of constants: C 2, 000; if β 1, thenm 0.25 and mβ m2 . If β 1/2, then mβ m 0.40.Future research: Checking the validity of invariance predictions in othersamples, Improving the accuracy of estimates and the triangulation ofproportionality constants.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance53/60
ConclusionMeta-model derives an empirical formula for liquidity L without anunderlying model of adverse selection. This is consistent withmechanical aspects of trading experienced in markets, whereinformation is invisible.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance54/60
ConclusionWe need theories based in economics to link meta-model toadverse selection. This enables further link to pricing accuracy,probability of informed trading, and precision of signals.The meta-model and dimensional approach are consistent withtheoretical models of both block trading and smooth trading.Theoretical dynamic models link L to resiliency of prices ρ anderror variance of prices Σ.See Kyle and Obizhaeva “Market Microstructure Invariance: ADynamic Equilibrium Model”.Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance55/60
05.3.10 15 10 505 15 10 50505 15 10 505 15 10 505.2.1.2.30.1.2.20.3.2.10 15 10 5.3 15 10 5.1.20.3.2.10Pete Kyle and Anna Obizhaeva.2.3.10550500.30.1.2.10.3.2.10.2.3 15 10 5 15 10 5 15 10 5.15550000.3 15 10 5 15 10 5 15 10 5.255.1000 15 10 5 15 10 5.35.30.3 15 10 50.1.2.3.2.100.1.2.3Invariant Log-Normality of Portfolio TransitionOrder SizeDimensional Analysis and Market Microstructure Invariance56/60
Linear versus Square Root Modelf*(.)/L x 10 410 4 x f*(.)/L*80806060404020200-200-8-40-6-4-2-40ln( f I)SQRT modelPete Kyle and Anna Obizhaeva-20LIN modelDimensional Analysis and Market Microstructure Invariance57/60
Switching Points: Korean Data14The fitted line for the regression of the number of switching pointson trading activity is ln(Sit ) 11.156 0.675 · ln(Wit /W ). Theinvariance-implied slope is 2/3.0246ln(Sit)81012y 11.156 0.675x 10 8 6 4 202ln(Wit W*)Pete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance58/60
InterceptNews Articles3210200320042005200620072008slope 00720082009200420052006200720082009864202003All Firms, ArticlesPete Kyle and Anna ObizhaevaAll Firms, TagsTR Firms, ArticlesTR Firms, TagsDimensional Analysis and Market Microstructure Invariance59/60
NYSE TAQ Prints, 1993epv group 1volume group 10.50.40.40.45M 1600.5N 180.40.0 70.40.50.45M 370.0 70.5N 1060.40.50.40.20.105M 260.0 70.5N 1780.40.105M 200.0 70.5N 0 70.0 70.0 70.0 70.505M 1815M 360.5N 140.400.5N 820.40.405M 100.5N 1610.45M 120.0 70.5N 2370.40.30.30.30.30.20.20.20.20.20.50.105M 1550.105M 90.5N 120.40.0 70.5N 710.40.0 70.40.105M 50.0 70.5N 1000.45M 40.0 70.5N 2340.40.30.30.30.30.20.20.20.20.20.10.10.10.10.0 70.0 70.0 70.0 7505055M 14N 55305M 7N 1,0280.100.3000.100.30.0 7M 25N 1,1390.30.20.10volume group 10M 15N 6670.30.20.10volume group 9M 14N 3010.30.20.10.1epv group 40.50.30.20.5volume group 7M 38N 126effective price volatilityepv group 20.50.30.0 7epv group 3volume group 4M 111N 2705M 1N 2850.1050.0 705dollar volumePete Kyle and Anna ObizhaevaDimensional Analysis and Market Microstructure Invariance60/60
The U.S. stock market is fragmented, and securities are traded simultaneously at dozens of exchanges. Tick size of one cent, and lot sizes of 100 shares. Pete Kyle and Anna Obizhaeva Dimensional Analysis an
