Market forces II: Liquidity

J. Doyne FarmerSanta Fe Institute

La Sapienza March 15, 2006

Research supported by Barclays Bank

Empirical behavioral model: collaborators

Szabolcs MikeBudapest U. ofTech. and Econ.

Fabrizio LilloUniversity of Palermo

Austin GerigU. Of Illinois

Price impact on longer time scales

What is liquidity

• Roughly speaking, it is the inverse slope of the demand (- supply) curve (price impact).– Large price change for given demand -> high– Small price change for given demand -> low

• Component of supply and demand having to do with “how many people are around to trade with”.– Identified with fluctuating component

• Farmer et al. 2004, Weber & Rosenow 2006– Liquidity fluctuations drive large price changes

– time scale?

What causes volatility?

• Theory: Information• Alternatively, can made an impact theory (Clarke)– Each trade has a price impact– Price diffusion is proportional to trading volume

– Standard dogma in finance literature

• We find liquidity fluctuations are more important– Gillemot, Lillo, and Farmer, “There’s more to volatility than volume”

Volatility at 2 hour timescale - AZN

– Size of standing orders is power law distributed

– Standing orders executed at a fixed rate

– N standing orders, replenished when removed

What are microscopic determinants of

volatility?• Assume random walk model:

• N = number of price changes• N = fn, I.e.

n = number of tradesf = fraction that penetrate

How well does this model explain price changes? How much does each factor account for?

€

variance =σ 2N

€

variance =σ 2 fn

Explaining power law distribution of price

flucutaitons• Power law distribution of price fluctuations is viewed by physicists as a sign that markets are “out of equilibrium”.

• Now many different models:– SFI, Brock and Hommes, minority game, Lux and Marchesi, Iori, ….

– Many of them “explain stylized facts”– Which is right?

• Go next step: Explain distribution in detail

Agent based models

• Most agent-based models suffer from inability to calibrate behaviors of agents.

• Easy to get lost in “wilderness of bounded rationality”.– Too many ad hoc models

• Behavioral economics?

Elements of model

• Assume continuous double auction• Must model people’s actions:

1. Signs of orders (buy or sell)2. Prices where orders are placed3. Cancellation

• Stochastic representative agent model– Model for conditional probability of

above behaviors; art is to find right variables to condition on

– Order placement and cancellation fully determine prices via mechanistic rules of market

(1) Autocorrelation of order signs

Lillo and Farmer (2004)Bouchaud, Gefen, Potters, and Wyart (2004)

Long-memory was (partially) predicted by the ZI model

Long memory raises several questions

• Efficiency paradox– All else equal, long-memory of orders implies strong linear predictability of prices.

– Prices aren’t predictable -- why isn’t this transmitted to prices?

– Exploiting inefficiency does not remove it.

• What causes long memory?

Model of strategic order splitting

Assumptions:• Hidden order size is power law distributed.

• Hidden order arrival is IID• Execution rate is independent of hidden order size.

Matches empirical results based on comparison of upstairs and downstairs markets

Implies lack of market clearing -- slow tatonnement

€

γ=α −1

€

C(τ ) ~ τ −γ

Assumptions imply

€

P(V > v) ~ v−α

Long-memory efficiency paradox

Liquidity imbalance

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E[r(T)ε] = Δr(τ )τ =1

T

∑

= P+(τ )R+(τ ) + P−(τ )R−(τ )τ =1

T

∑

€

Δr(τ ) > 0 implies -R -(τ )

R+(τ )>P+(τ )

P−(τ )

€

Can further decompose Δr(τ ) = ΔrM (τ ) + ΔrQ (τ )R± = M± +Q±

Sign imbalance and liquidity imbalance vs. time

Return decomposition

Price impact appears permanent

Strategic motivation?

• We understand how paradox is resolved: Why is it resolved?

• Our idea: Liquidity matching.• E.g., suppose liquidity provider matches liquidity taker:

• Suggests two way “liquidity matching game”

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L(t +1) = λP+(t)

P−(t)+ (1− λ )L(t) −Cδ(t)

(2) Prices for order placement

• Where is the best price for an order?

• Depends on many factors:– Time horizon of opportunity– Market conditions– Cost of not making transaction– …

• Situation is very different for aggressive vs. non-aggressive orders

Continuous double auctionContinuous: Market operates asynchronously

Double: Price adjustment in orders both to buy and to sellExecution priority: • Lower priced sell orders or higher priced buy orders have

priority• First order placed has priority when multiple orders have

same price.

price ($)

SPREAD

PRIORITY

PRIORITY

(BEST) BID

(BEST) ASK

VOLUME

SELL

BU

Y

VO

LUM

E

LIMIT ORDERS

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p(x) ≈ p(x | s > s1)N(s > s1)

N(s > x)

Cancellation

• Total probability of cancellation is almost independent of number of orders in book

• Implies probability of cancellation per order is inversely proportional to number of orders.

Cancellation

Empirical behavioral model

1. Generate order sign with order splitting model.

2. Generate limit price with unconditional distribution p(x). Assume all orders have same size. If limit price equals or crosses opposite best, generate transaction.

3. Cancel orders based on relative distance to opposite best, order imbalance in book, and total number of orders in book.

Ad hoc: Require at least two orders in book at all times.

The ugly

• Simulation blows up if tick size is too big relative to price.– Due to spread dynamics, when tick size is large, market orders do not remove sufficient orders from book

– Implies we are missing a key element for these stocks, probably in cancellation model

Threshold for model convergence

Comments

Flaws:– High volatility or large tick size stocks– Not enough clustered volatility– Not efficient w.r.t. observing order signs (doesn’t capture liquidity imbalance dynamics)

Promise:– Equation of state linking order flow and prices

– Fundamental implications for price formation

Testing prediction of spread

• Equation of state from mean field theory

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E[s] = μ

αf (σδ

μ)

Conclusions• Regularities in order placement and cancellation.– Strategic equilibria or “behavioral regularities”?

• “Explains” many aspects of price formation– Correctly predicts magnitude, functional form.

• Raises fundamental questions about causality– Information arrival vs. internal dynamics of market

• Regulatory applications– Should markets encourage provision of liquidity?– Screening of specialists

• Intermediate level of modeling– Between econometrics and microeconomics– Divide and conquer strategy