Smooth Trading with Overconﬁdence and Market Powerapps.olin.wustl.edu/.../kyle_slides2.pdf ·...

Smooth Trading with Overconfidence

and Market Power

Albert S. Kyle Anna A. Obizhaeva Yajun Wang

University of Maryland New Economic School University of Maryland

College Park, MD Moscow College Park, MD

Finance Theory GroupSummer School

Washington University, St. LouisAugust 17, 2017

Kyle, Obizhaeva, and Wang Smooth Trading with Overconfidence and Market Power 1/49

Main Idea

We present a dynamic model of informed trading which derivesendogenously the speed with which traders trade and study theproperties of markets when all traders smooth out their trading.

We model a trading game of oligopolistic traders withoverconfidence and market power, where speed of trading isgoverned by trade-off:

• Reducing market impact motivates traders to trade slowly.

• Decaying private information motivates them to trade quickly.


Practical Implications

Order shredding significantly changes equilibrium properties.

• Model explains why dumping large quantities on the markettoo quickly has large temporary price impact. This is relevantfor understanding flash crashes.

• Model explains why even though prices immediately revealaverage signal of traders, traders continue to trade inequilibrium.

• Model explains why even though prices are fully revealing, aneconomist will be finding anomalies.


Key Feature

We develop an equilibrium model where speed matters. For eachtrader, price is linear function of traders inventory (permanentprice impact) and derivative of inventory (temporary impact).


Speed of Trading and Kyle (1985)

The speed of trading usually plays a limited role in theoreticalmodels. In Kyle (1985), the speed of trading is important neitherfor informed trader nor noise trader:

• The informed trader trades smoothly, but his profits do notdepend on the rate of trading. Price impact costs of changinginventory levels continuously by a given amount usually do notdepend on the derivative of the trader’s inventory levels. Inthis sense, there is no temporary price impact.

• The noise trader would be better off by smooth out theirtrading over time, as he would walk up or down the demandcurve as a price-discriminating monopolist.


Instantaneous Liquidity Disappears When AllTraders Smooth Trading

In unrestricted model, each trader tries to “walk the demandcurve” of all the other traders. This slows down trading.

Order flow becomes predictable. The market offers noinstantaneous liquidity, and the equilibrium collapses.

Developing intuition about dynamic properties of liquidity inmarkets with smooth trading is the main focus of our paper.


Related Dynamic Theoretical Papers:

• Vayanos (1999): Oligopolistic traders receive randomuncorrelated shocks to endowments and trade to share therisks. Adverse selection due to trader’s private informationabout hedging needs leads to smoothing of trading. Speed oftrading based on risk aversion (impatience).

• Du and Zhu (2013): Traders have private values, smooth outtrading. Adverse selection due to private information aboutprivate values leads to smoothing of trading.

Impatience is based on risk aversion, not decay of information dueto competition from other traders. Our paper differs in that privateinformation is correlated, implying private information decays asother traders acquire more information. Implies a trade-offbetween trading slowly due to price impact and trading fast due toinformation decays.


Other Related Theory Papers

• Kyle and Lin (2001) and Scheinkman and Xiong (2003):Same dynamic information structure. Competitive models.

• Ostrovsky (2012): Shows equilibrium must have a fullyrevealing steady state. Our paper assumes a fully revealingsteady state.

• Kyle (1989): Traders exercise monopoly by bid-shading. noregret pricing. Trade results because of noise trading, notdisagreement.

• Rostek and Weretka (2012): Bid-shading with private values.No regret pricing.

• Grossman and Miller (1988): Impatience based on riskaversion, not decay of information. Big difference!


“Partial Equilibrium” Theoretical Models ofDynamic Trading Costs

• Brunnermeier and Pedersen (2005) study price effects of a large traderunwinding his position in the presence of strategic traders with exogenouslyspecified maximum trading rate at which transaction costs are avoided.

• Longstaff (2001) analyzes portfolio choice problem of investors restricted totrade not faster than a specified rate per period.

• Carlin, Lobo, Vish (2007) study interaction between traders facing liquidityshocks given exogenously specified price dynamics as function of traders’ speedof trading.

• Grinold and Kahn (1999), Almgren and Chriss (2000), Obizhaeva and Wang(2012) and other literature on optimal execution studies optimal tradingstrategies given exogenous specification of price impact function.

We derive price impact functions and the speed of tradingendogenously. Some partial equilibrium effects disappear.


Trading Costs and Speed: Empirical Evidence

A conventional wisdom holds that speed of trading affectstransaction costs. There is a link between trading pressure andasset prices.

• Holthausen, et al. (1990) measure temporary and permanent price effectsassociated with block trades and find most of the adjustment occurring in thevery first trade.

• Chan and Lakonishok (1995) find high demand for immediacy tends to beassociated with larger market impact.

• Keim and Madahvan (1997) find that more aggressive trades of index funds andtechnical traders have larger costs than trades of more patient value investors.

• Dufour and Engle (2000) find that the price impact of trades increases whenduration between transaction decreases.

• Almgren et al. (2005) calibrate a specific model of transaction costs as functionof shares traded and speed of trading.


Theoretical Model Assumptions

We consider dynamic oligopoly model of informed trading incontinuous time, to make transparent the idea that each tradertrades “smoothly.”

• Overconfidence: Trade based on “relative” agreement todisagree. Each trader believes his flow of information is moreprecise than other traders believe it to be.

• Market Power: Each trader learns from prices and restrictstrading to reduce impact.

• Symmetry: No noise traders; no market makers; no rationaluninformed traders.

Our model is a fully-fledged dynamic version of Kyle (1989), buttrading is based on agreement-to-disagree, not noise trading.


Analogous One-Period Model

We first develop intuition and consider a one-period model:

• There is a risky asset with liquidation value v ∼ N(0, 1/τv )and a safe numeraire asset with liquidation value of one.

• There are N traders with CARA utility and risk aversion A.

• Traders observe a public signal i0 = τ1/20 τ

1/2v v + ǫ0,

ǫ0 ∼ N(0, 1).

• Trader n observes a private signal

in = τ1/2n τ

1/2v v + ǫn, ǫn ∼ N(0, 1).

• Each trader believes his own signal has precision τH and othersignals have precisions τL. Traders agree to disagree. Tradersagree total precisions is τ = τ0 + τH + (N − 1)τL.


A symmetric linear equilibrium: Demand

Trader n’s optimal equilibrium demand x∗n is a partial adjustmentto his target inventory STI

n :

x∗n = δ · (STIn − Sn),

STIn =

1

A·(

1−1

N

)

· (τ1/2H

− τ1/2L

) ·

(

in −1

N − 1ΣNm=1,m 6=n im

)

.

Trading is subdued by market power.


A symmetric linear equilibrium: Prices

The equilibrium price is the weighted average of signals:

P∗ =τ1/20

τ· i0 +

τ1/2H

+ (N − 1)τ1/2L

τ·1

N

N∑

m=1

im.

Price fully reveals the average of private signals. Traders agree todisagree with price.

“Relative overconfidence” (τe =1N(τH + (N − 1)τL), i.e.

τE = τ): Prices follow a martingale. There is representative agent.

“Absolute overconfidence” case has prices overreact toinformation and a mean-reversion (τE < τ). “Absolute lack ofconfidence” case has prices underreact to information andmomentum (τE > τ).


One-Period Model: Existence of EquilibriumExistence of equilibrium requires “enough” disagreement:

τ1/2H

τ1/2L

> 2 +2

N − 2

.Intuition: Walking demand curve implies marginal price impact istwice average price impact. Trader trades to point where marginalimpact fully reflects his information. This creates losses to othertraders unless they believe trader inflates signals by a factor of two.

One period model is like Kyle (1989), but noise traders and marketmakers replaced by relatively overconfident traders providingliquidity to one another. Almost equivalent to “more than half” oftraders trade on noise like it is information, see Fischer Black(1986). One-period model has “bid shading” like Rostek andWeretka (2012), also Du and Zhu (2013).


Dynamic vs. Static Model

One-period model highlights bid-shading in auctions, learning fromprices, partial trading towards target inventory.

One-period model misses important dynamic considerations, suchas

Speed of price adjustment,

Speed of quantity adjustment,

Dynamics of permanent and transitory price impact,

Effect of dynamic Bayesian learning on steady state liquidityand accuracy of prices.

Our dynamic continuous-time infinite-horizon model with marketpower and overconfidence highlights subtle properties arising indynamic setting.


Continuous-Time Model: Assumptions

• There are N risk-averse oligopolistic traders, who trade a riskyasset with a zero net supply against a risk-free asset.

• Risk-free rate is r .

• A risky asset pays out dividends D(t) growing at a rate G ∗(t).

dD(t) = −αD · D(t) · dt + G ∗(t) · dt + σD · dBD ,

dG ∗(t) = −αG · G ∗(t) · dt + σG · dBG .

Dividends D are observable. Growth rate G ∗ is unobservable.


Continuous-Time Model: Information Flow

Let Ent . . . and varnt . . . use all information and n’s beliefs:

Gn(t) := Ent G

∗(t), Ω := varnt

G ∗(t)

σG

• Each trader observes the public information flow dI0(t) fromdividends (τ0 = σ2

G/σ2

D):

dI0(t) = τ1/20 ·

G ∗(t)

Ω1/2σG· dt + dB0.

• Each trader n has continuous flow of private informationdIn(t) about the unobserved growth rate G ∗(t):

dIn(t) = τ1/2n ·

G ∗(t)

Ω1/2σG· dt + dBn, n = 1, . . . ,N,

• Trader n infers from prices the average of other signals

I−n :=1

N − 1Σm 6=nIm(t).


Continuous-Time Model: Bayesian Updating

• Each trader n thinks that his own information has highprecision τn = τH and other traders have low precisionτm = τL, m 6= n, with τH > τL ≥ 0.

• Trader n constructs signals Hi , i = 0, ..N as weighted averageof information flow:

Hn(t) :=

∫

t

u=−∞e−(αG+τ)·(t−u) · dIn(u),

where τ = τ0 + τH + (N − 1) · τL, Ω−1 = 2 · αG + τ.

Gn(t) = σG ·Ω1/2·

τ1/20 · H0(t) + τ

1/2H

· Hn(t) +∑

m 6=n

τ1/2L

· Hm(t)

• The importance of each bit of information about G decaysexponentially at a rate αG + τ , the same for every trader.


Continuous-Time Model: Optimization

Each trader n chooses consumption path ct and trading rate xt tomaximize CARA utility function U(ct) = −e−A·ct ,

V (M,S ,D,H0,Hn,H−n) = maxct ,xt

Et

[∫ ∞

s=t

−e−ρs · U(cs) · ds

]

.

Inventory: dS(t) = xt · dt.

Cash: dM(t) = (r ·M(t) + S(t) · D(t)− ct − P(xt) · xt) · dt.

Dividends: dD(t) = −αD · D(t) · dt + G ∗(t) · dt + σD · dBD .

Information Flow Dynamics: dHn and dH−n =∑

m 6=ndHm

incorporate H0 into Hn.


Conjectured Linear Strategies

Trader n conjectures the other N − 1 traders, m = 1, . . .N,m 6= n,submit symmetric linear demand schedules of the form

Xm(t) = γD · D(t) + γH · Hm(t)− γS · Sm(t)− γP · P(t).

Let xn(t) := dSn(t)/dt. Assume continuous single price auctionlike Kyle (1989). Market clearing quantities are derivatives ofinventories.

P(xn(t)) =γDγP

·D(t)+γHγP

·H−n(t)+γSγP

1

N − 1·Sn(t)+

1

(N − 1)γP·xn(t).

All traders exercise monopoly power optimally, with “no regretpricing,” taking into account how their own trading affects othertraders’ beliefs about H−m inferred from “fully revealing” prices.


Conjecture: Quadratic Value Function

Value function depends on nine psi -parameters:

V (Mn, Sn,D, Hn, H−n) =

− exp [ψ0 + ψM ·Mn + ψSD · Sn · D

+1

2ψSS · S2

n + ψSn · Sn · Hn + ψSx · Sn · H−n

+1

2ψnn · H

2n +

1

2ψxx · H

2−n + ψnx · Hn · H−n

]

.

Note: Wealth does not enter value function; instead have separatecomponents cash M(t) and security holdings S(t).

Equilibrium problem is to solve for nine ψ-parameters and fourγ-parameters which sustain a symmetric equilibrium.


Theorem: Symmetric Linear Flow Equilibrium

Equilibrium defined by solution to six mostly quadraticpolynomials in six unknowns.

Numerical results consistent with “derived” existencecondition

τ1/2H

τ1/2L

> 2 +2

N − 2.


Theorem: Symmetric Linear Flow Equilibrium

• Market clearing quantities are derivatives of inventories,implying partial adjustment towards “target inventory”,depending on constant CL,

xn(t) = dSn(t)/dt = γS ·(

CL · (Hn(t)− H−n(t))− Sn(t))

.

• Equilibrium price is dampened average, i.e., 0 < CG < 1, ofbuy and hold valuations based on Gordon’s growth formula:

P∗(t) =D(t)

r + αD

+CG · 1

N

∑

N

n=1 Gn(t)

(r + αD)(r + αG ).


Implications

The model explains several realistic features of trading and pricesin speculative markets.


Implications: Prices Adjust Immediately

• Prices immediately reveal the average estimate of a growthrate and the average of signals

∑

n=1,..N Hn/N.

P∗(t) =D(t)

r + αD

+CG · 1

N

∑

N

n=1 Gn(t)

(r + αD)(r + αG ).

• Trading on information continues after signals revealed inprices. This is different from Milgrom and Stockey (1982).


Implications: Quantities Adjust Slowly

• Market power with private information makes quantitiesadjust slowly. Traders “shred orders.”

• Trading strategy is “partial adjustment” towards“steady-state” target inventory.

x∗n (t) = γS ·(

STI (t)− Sn(t))

.

STI (t) = CL · (Hn(t)− H−n(t)).

• “Half-life” of trading 1/γS depends on flow of informationinto the market. Size of target inventories CL dependsrisk-aversion and overconfidence.


Coefficients CL and γS

Coefficients CL related to target inventories and γS related tospeed of converging to target inventories as functions of riskaversion A, degree of competition N and overconfidence τH ,keeping total precision τ constant.

2 3 4 5 63.13.23.33.43.53.63.73.8

Ln@ND

CL

0 50 100 150 200 2500

10

20

30

40

ΤHΤL

ΓS


Short-term Trading on Long-Term Information

• Traders acquire private information about fundamentalsunfolding over long-term horizons.

• Traders build and reduce their positions over much shorterhorizons, when their information become known to othertraders.


Implications: Price Impact Functions

• From perspective of each oligopolistic trader, price is linearfunction of traders inventory (permanent price impact) andderivative of inventory (temporary impact).

P(S(t), x(t)) = λ0 + λS · S(t) + λx · x(t),

where constants λS = γs · λx and dS(t) = x(t) · dt. Priceimpact is linear in the first and second derivatives of inventory.

• Permanent and temporary impact is based on “adverseselection,” even though price reveals the average of privatesignals.

• Block traders are infinitely expensive.


Coefficients λS and λx

Permanent price impact λS and temporary price impact λx asfunctions of risk aversion A, degree of competition N andoverconfidence τH , keeping total precision τ constant.

0 50 100 150 200 2500

100

200

300

400

ΤHΤL

1Λ

0 50 100 150 200 2500

5000

10 000

15 000

ΤHΤL

1Κ

1 2 3 4 5 60

500

1000

1500

2000

2500

Ln@ND

1Λ

1 2 3 4 5 60

200 000

400 000

600 000

800 000

Ln@ND

1Κ


Implied Execution Cost

• The cost of buying B over time T at uniform rate:

EC =(

λS +λxT/2

)

·B2

2= λS ·

(

1 +1

γS·

1

T/2

)

·B2

2.

• The cost of buying B at a constant rate γ:

EC =(

λS + γ · λx

)

·B2

2= λS ·

(

1 +γ

γS

)

·B2

2.

Recall λS = γs · λx .


Implied Execution Cost - Summary

• Infinitely fast block trades are infinitely expensive due totemporary price impact costs.

• Infinitely slow trades incur only permanent price impact costs.

• In the equilibrium (γ = γS), the permanent cost λS · B2/2 isequal to the temporary cost λS · B2/2 · γ/γS ; both are equalto one half of the total cost EC = λS · B2.

Each trader expects to break even and pay out his potentialmonopoly profits to others in form of temporary price impact.


Literature of Optimal Execution

A conventional wisdom is that speed of trading affects incurredtransaction costs. Papers on optimal execution study optimalstrategies given exogenously specified price impact functions, witha lot of attention on costs due to temporary price impact.

Grinold and Kahn (1999) and Almgren and Chriss (2000) proposedto model price impact functions similar to ours and derived theoptimal execution strategy of buying S :

S(t) = S ·(

1−sinh(k · (T − t))

sinh(k · T )

)

,

where 1/k is the “half-life” of executing an order in the absence ofany external time constraint.


Fast Trading Can Lead to Flash Crashes

SP

Y v

olu

me

(co

ntr

act

s p

er

min

ute

)

SP

Y p

rice

• Joint CFTC-SEC report identified a large sale of 75,000 contracts asa trigger for Flash crash on May 6, 2010.

• Kyle and Obizhaeva (2013): invariance implies impact would be0.60%, which is smaller than the actual decline of 5%; costs wereinflated by too fast execution. The order was executed faster thanother large orders of similar magnitude.


Fast Trading Can Lead to Flash Crashes

Price path and inventories deviate from the equilibrium if the speedof trading γ deviates from the equilibrium rate γS :

0.00 0.05 0.10 0.15 t

200

400

600

800

0.05 0.10 0.15 t

-10

-8

-6

-4

-2

Panel A: Expected Prices, E P(t) Panel B: Expected Inventories, E S (t)0n

n 0n

Fast Trade, High Precision

Fast Trade, Low Precision

Slow Trade, High Precision

Slow Trade, Low Precision

E n

TP(T+t) = −γ − γS

(N − 1)γP·e−γt ·Sn(T ), E n

TSn(T+t) = e−γ·t·Sn(T ).


Speed of Trading and Price Patterns

• Speeding up selling leads to sharp price decline following byV-shaped recovery.

• Slowing up leads to price increase and then convergence toequilibrium level.

• Speeding up execution twice leads to twice bigger pricedecline relative to the equilibrium price change.

• Speed of recovery depends on price resilience, αG + τ .

• Selling may occur after prices crash, while the marketrecovers.


Implications: Dampened Price Reflects aKeynesian “Beauty Contest”

• Our model captures the intuition of the beauty contest:

For most of these persons are, in fact, largely concerned, not

with making superior long-term forecasts of the probable yield

on an investment over its whole life, but with foreseeing

changes in the conventional basis of valuation a short time

ahead of general public.

• In contrast to Keynes(1936), short-term trading dynamics inour paper dampens price volatility.


“Dampening Effect”

liquidate at Gn

liquidate at G-n

liquidate at P

20 40 60 80 100 120 140s

22

24

26

28

30

Present Value of Dividend and Liquidation Value

Even if all traders agree on the fundamental value today, each trader

believes that other traders will find out their information is incorrect in

the future. Despite fundamentals, each trader “agrees” on a dampened

equilibrium price. This effect does not exist in one-period model.


Keynes (1936) and “Dampening Effect”

Keynes:

• “It is not sensible to pay 25 for an investment of which you believe

the prospective yield to justify value of 30, if you also believe that

the market will value it at 20 three months hence.”

• When a trader purchases a stock, he is “attaching his hopes, not so

much to its prospective yield, as to a favorable change in the

conventional basis of valuation.”

• Speculation predominates over enterprise.

In our dynamic model, traders also seem to be preoccupied with“short-term price dynamics” rather than “hold-to-maturity”values. The market internalizes overconfidence of traders andcorrects equilibrium prices, so instead price deviations aredampened.


Dampened Price Fluctuations

Equilibrium price are similar to Gordon’s formula with growth ratebeing the average of all estimates Gn:

P∗(t) =D(t)

r + αD

+CG · G(t)

(r + αD)(r + αG ).

Since CG < 1, prices are not equal to the average estimates offundamentals. In the model with agreement to disagree, beautycontest dampens price fluctuations.


Coefficient CG

Coefficient CG does not change with risk aversion A. CoefficientCG decreases with degree of competition N and overconfidence τH ,keeping total precision τ constant.

50 100 150 200 2500.0

0.1

0.2

0.3

0.4

ΤHΤL

CG


Implications: Anomalies

• Price patterns have to be studied from the perspective of theeconomist, who may assign different precisions to informationsources and disagree with traders on how to construct signals.

• Regardless of beliefs, everybody finds anomalies such asmean-reversion and momentum in the model.

• Given reasonably calibrated of parameters, our model providesinsights on horizons and magnitudes of anomalies; see Kyle,Obizhaeva, Wang (2014) soon.


Implications: Valuation of Risky Asset

Value function parameters define how the trader values securityholdings as compromise between “buy-and-hold” value and“mark-to-market” value.

V (Mn, Sn,D, Hn, H−n) = − exp [ψ0 − rA · (Mn + Pn(t) · Sn)

+1

2ψnn · H

2n +

1

2ψxx · H

2−n + ψnx · Hn · H−n

]

.

where Pn(t) = P(t) + λS · STIn (t)− 1

2λS · Sn(t).

The “mark-to-market” value P(t) is adjusted for

• the difference between the market price and trader n’sappraisal value, λS · STI

n (t),

• the cost of liquidating his inventories, −12λS · Sn(t).

H’s terms capture the value of trading opportunities.


Implications: Market Liquidity

• Symmetry of strategies does not define liquidity parameter γPin Xn(t) = γD · D(t) + γH · Hn(t)− γS · Sn(t)− γP · P(t).

• Liquidity is “somewhat indeterminate.” It reflects a delicateand unstable balance between demand and supply of liquidity.

• Similar delicacy in determination of equilibrium arises in Kyle(1985), where market depth is not determined from theoptimization of informed trader either. The second ordercondition requires the market depth to be constant.

The level of market depth is determined from the martingalecondition.


Flow Equilibrium and Real Markets

The idea that securities markets offer a flow equilibrium ratherthan a stock equilibrium may seem far-fetched. Yet recent trendsare consistent with the way our model predicts liquidity to besupplied and demanded.

• Electronic processing of orders has reduced the fixed costs ofexecuting an order.

• “Order shredding” strategies are similar to strategies in ourmodel. In the future, the market may offer explicit flowequilibrium trading mechanisms.


Flow Equilibrium and Real Markets

• Our model predicts vanishingly small market depth availableat any point of time. Market depth is predicted to beavailable only over time.

• In real markets, market depth available on the “top of thebook” is influenced by rules of time and price priority. Thesmaller the minimum tick size, the smaller is the externalitycreated by those rules and the less of liquidity is expected tobe available.


Black (1995)

We formulate in a precise mathematical model ideas that FischerBlack has formulated intuitively in his last paper called“Equilibrium Exchanges,” where he outlined his thoughts abouthow would people trade in the equilibrium, if there are norestrictions on trading strategies or on exchanges.

There is no conventional liquidity available for market orders andconventional limit orders. Traders use indexed limit orders atdifferent levels of urgency. Price moves by an amount increasing inlevel of urgency. In our model of smooth trading, we formallyprove the intuition of Black (1995).


Conclusions

This paper is meant to be a realistic model of the assetmanagement industry, not a “behavioral finance” model.

• We presented a dynamic oligopoly model of informed tradingin continuous time with imperfect competition and agreementto disagree.

• In flow equilibrium, speed of trading is derived endogenously.

• Transaction costs depend on the speed of trading.

• Agreement to disagree generates momentum, mean-reversion,and other anomalies.


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