Modeling liquidity, risk and transaction costs in the LSE using low intelligence agents J. Doyne Farmer Santa Fe Institute Institute for Mathematics and its Applications May 26, 2004 Research supported by: McKinsey & Company Bill Miller Credit Suisse First Boston Bob Maxfield McDonnell Foundation Prediction Company

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Goals Understand –risk (price volatility) –liquidity (impact of trading on prices) –transaction costs (spread) Find simple laws relating these to measurable properties of markets

Why economics is hard Must model strategic interaction of agents Standard solution – Selfish, rational utility maximizing agents at equilibrium Problems with standard solution –Lack of rationality –Lack of equilibrium –Intractability in non-trivial settings –Parameters cannot be independently measured No easy fix –How to avoid wilderness of bounded rationality?

Alternative Use extremely simple agent model (random behavior) to probe market institutions. Use this to understand what is dictated by market institutions alone. –Benchmark for agent intelligence Gradually introduce simple forms of bounded rationality, using (imperfect) arbitrage as a guide.

A few seminal papers illustrating zero intelligence H.A. Simon, On a class of skew distribution functions, Biometrika, 42, 425 (1955). Gary Becker, Irrational behavior and economic theory, J. of Political Economy 1, February (1962). D. Gode and S. Sunder, Allocative efficiency of markets with zero intelligence traders: Markets as a partial substitute for individual rationality, J. of Political Economy, (1993) I. Domowitz and Jianxin Wang, Auctions as algorithms, J.E.D.C, 18, 29 (1994) P. Bak, M. Paczuski, and M Shubik, Price variation in a stock market with many agents, Physica A 246, 430 (1997).

Three models 1.Analytically tractable zero intelligence model 2.Empirical model based on study of data 3. First principles model, exploiting arbitrage Common features: –Each has as model of order flow –Price formation rules given by double auction –Prices emerge from interaction of price formation rules and order flow model In some sense intermediate between first principles and purely econometric

Continuous double auction Price adjustment in orders both to buy and to sell Market operates continuously Execution 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 (BEST) BID (BEST) ASK VOLUME SELL BUY VOLUME LIMIT ORDERS

price ( $ ) BID ASK VOLUME Patient trading Patient traders place non-marketable limit orders that do not lead to an immediate transaction Non-marketable limit orders accumulate Limit order book is a storage device NEW ASK Limit Order BUY / SELL # OF SHARES LIMIT PRICE Patient trading Patient traders place non-marketable limit orders that do not lead to an immediate transaction Non-marketable limit orders accumulate

price ( $ ) Impatient trading Market order: An order to buy or sell up to a given volume No limit price is defined Executed immediately Often causes unfavorable price impact Market Order BUY / SELL # OF SHARES BID ASK BID NEW ASK VOLUME Impatient trading

Order cancellation price ( $ ) Limit order cancellations: Limit orders can be cancelled by the owner Market defined expiration price ( $ ) VOLUME

Giulia Iori Eric Smith Laszlo Gillemot Supriya Krishnamurthy Limit order collaborators 1 Marcus Daniels Continuous double auction model collaborators

ZI model (Unrealistic but somewhat tractable) u Limit order arrival: Poisson process in time & price; Market order arrival: Poisson process in time; Cancellation: random in time (like radioactive decay); Separate processes for buying and selling, with same parameters. Depth profile n p,t : Number of shares in limit order book at price p, time t. BID SELL LIMIT ORDERS ASK BUY LIMIT ORDERS SELL MARKET ORDERS BUY MARKET ORDERS 0

Achievements of ZI model so far Dimensional analysis –(price, shares, time) –Rough predictions, simplified analysis Mean field theory and simulation –scaling laws relating order flow and prices A few good results in tests with real data

Ilija Zovko Paolo Patelli Comparison to LSE data

London Stock Exchange data set London screen Analysis based on automated (on book) orders only

Upstairs, downstairs Off-book (upstairs) trades account for 35% of transactions, about 50% of volume. Off-book trades sizes are power law distributed, on-book (downstairs) trade sizes are not. On-book trade sizes long-memory; off-book not* NYSE data (with on/off book mixed together) show same independence of order volume. Problems with mixing on and off book trades due to reporting time lag for off book trades. Which is more important for price formation?

Parameters of model Order flow rates Discreteness parameters Three fundamental dimensional quantities: shares, price, time

Testing prediction of spread From mean field theory:

Predicted price diffusion rate

Top 10 Russian jokes, Oct. 23, 2003 с сайта "Немецкая волна" Ученые-экономисты давно стараются понять закономерности, которым подчиняются биржевые курсы, и используют для этого математические модели. На протяжении многих десятилетий такие модели исходили из представлений о брокерах как об аналитиках с выдающимися умственными способностями, обладающих исчерпывающей информацией о рынке и действующих исключительно рационально. Однако удовлетворительно описать реальные изменения биржевых курсов эти модели оказались не в состоянии. Значительно успешнее справляется с этой задачей новая модель, предложенная Дойном Фармером (J. Doyne Farmer), сотрудником Института Санта-Фе в штате Нью-Мексико. Она базируется на предположении, что брокеры Ц полные Ђидиотыї, действующие совершенно случайно и к тому же лишенные какой бы то ни было информации. Сравнив данные, рассчитанные на основе этой модели, с реальными курсами лондонской фондовой биржи за период с 1998-го по 2000-й годы, ученые выявили очень высокую степень совпадения

Volatility autocorrelation

Fat tails in prices

Price fluctuations have fat tails Mandelbrot (63), Fama (64), … –Power law tails? Many theories: –Excessively rapid learning (SFI Stock market) –Heterogeneous beliefs (Lux & Marchesi, …) –Herding behavior (Cont and Bouchaud) –Minority game dynamics (Challet & Zhang, …) –Large market orders (Gabaix, Gopikrishnan, Plerou, Stanley) –…

Investigation of fat tails Szabolcs MikeAnindya Sen Fabrizio Lillo Laszlo Gillemot

Price changes are almost independent of volume

A typical large price change

Gap distribution vs. price distribution

Tail exponent of rtns. Vs. gap

Return vs. number trades

Building a better empirical model for real order flow Real order flow is not random –Unconditional order placement distribution –Correlations in order flow –Dependence on spread –…

Autocorrelation of orders Prices are roughly uncorrelated Order sign, order volume, liquidity all long-memory Anti-correlations between signs vs. volume or liquidity Fabrizio Lillo

Unconditional order placement distribution Ilija Zovko

Distribution of relative limit price conditioned on spread Cumulative probability distribution, relative to opposite best price

Limit order placement conditional on spread Probability density, relative to opposite best price

Empirical model Whether limit or market order depends on spread. Relative limit price depends on spread Lifetime depends on where order is placed. Market orders remove all volume at best with 50% probability Long-memory order flow Return distribution: red = real, black = model

Ecology of arbitrage Jim GirardJim Rutt

Market ecology As in biology, financial strategies have –Variation (innovation) –Selection –Propagation Agents form a specialized and diverse ecology. How to use this quantitatively?

Building a market ecology Exploit arbitrages of ZI model Liquidity demand is food for arbitrageurs. Agents in ecology of arbitrage. –Market maker –Order imbalance trader –Technical trader (not implemented yet)

Two low intelligence agents Market maker –Makes simultaneous bid and offer –Never goes below a minimum spread –Inventory limit Order imbalance trader –Places orders based on imbalance in order book –Threshold depends on (expected return - spread)

Profits vs. agent trading volume

The road to efficiency is not straight Efficiency comes about indirectly, and only through interaction of diverse specialized agents Order imbalance trader: –Exploiting order imbalance only makes it worse –Strategy is limited by widening spread Market maker: –Feeds off of order imbalance trader –Improves order imbalance MM + OI do not remove price anti-correlations –Need technical trader to make market efficient!

Practical implications Trading tactics Automated market making Market design: –To reduce volatility and spread: encourage limit orders, discourage market orders –Understanding determinants of large risks –Order granularity vs. tick size –Detecting poor markets

Conclusions ZI/LI models give benchmark for agent intelligence. Divide and conquer strategy –First understand how order flow affects market –Then understand what determines order flow Parameters are directly measurable*. To tractably model market institutions, it may be necessary to drop heavy baggage of equilibrium –ZI captures feedback between price formation and order flow –Empirical model does this more realistically Noise traders are economic users of market, supporting ecology of arbitrage

Upstairs, downstairs Off-book (upstairs) trades account for 35% of transactions, about 50% of volume. Off-book trades sizes are power law distributed, on-book (downstairs) trade sizes are not. On-book trade sizes long-memory; off-book not* NYSE data (with on/off book mixed together) show same independence of order volume. Problems with mixing on and off book trades due to reporting time lag for off book trades. Which is more important for price formation?

Price changes are almost independent of volume (NYSE)

price ( $ ) Market (price) impact Market impact is change in price caused by market order arrival. Instantaneous market impact is: price immediately after arrival - price immediately before –Closely related to supply and demand BID ASK BID NEW ASK VOLUME

Market impact fn- dim units Market impact function (standard units)

Market impact fn- non dim units Market impact function (non-dimensional units)