Chapter 22Market Microstructure

1.Limit Order Markets2.Oligopoly3.The Direction of Trade

1. Limit Order Markets

We now introduce limit order markets as a real world institution describing the financial sector, and as a paradigm for trading mechanisms. We define the general solution to a trading game and, then demonstrate that many auctions, retail sectors and stock markets fit into a limit order framework.

Financial Markets

In the second half of this lecture we examine electronic limit order markets, which are amongst the fastest growing markets within the financial sector.

Instead of dealers mediating between buyers and sellers anyone in good standing can submit and sell limit and market orders.

What can we say about the portfolio management when financial assets are traded on a limit order market?

General description of a trading game

Consider the problem a trader faces in a limit order market. First we describe the trading mechanism in a limit order market.

Then we define his preferences, then the constraints on his trading problem. These constraints determine his choice set.

The description is completed by explaining the trader’s optimization problem, and showing how the solution to the problem is part of a Nash equilibrium.

Trading in limit order markets

Anyone seeking to trade in the market must submit a market order or a limit order.

Each order is for a given quantity, negative quantities standing for units for sale, positive for units demanded.

Limit orders also specify a transaction price.

All market transactions match a market order with one or more limit orders, and take place at the limit order price(s).

Precedence

Market orders to buy are matched against the lowest price limit order(s) to sell.

If two limit orders to buy are submitted at the same price, the order submitted first is matched against a market sell order before the more recently submitted buy order.

Similarly lower priced limit orders to sell have a higher priority than higher priced limit sell orders, and if two bidders seeking to sell a unit at the same price the person who bid first will be matched before his rival seller.

Trading window

The trader in this market has just placed a sell order for 9 units at price 5,800, with an expiry time of 60,000 seconds (that is 16 hours 40 minutes). There are 6 limit orders to buy already in the books (2 at 3800 and 4 at 200), and 4 other limit orders to sell at 6,000.

The Spread

The spread is defined as the difference between the highest priced limit buy order ask price, called the bid price, and the lowest priced limit sell order called the ask.

In the example above, the ask price is 5,800 and the bid price is 3,800, so the spread is 2,000.

Observe that the trader whose display screen is illustrated reduced the spread from 2,200 by placing an order inside the previous bid ask quotes.

Market orders are executed immediately

A market order to buy (sell) one unit is defined by a price which is greater (less) than or equal to the lowest (highest) outstanding limit order to sell (buy).

Therefore market orders transact instantaneously, market buy (sell) orders reducing the number of outstanding limit orders to sell (buy).

Market orders to sell (buy) are matched with the highest (lowest) priced limit order to buy (sell) and executed at the price of the matching limit buy (sell) order.

Limit orders are not always executed

All other orders (sell orders priced higher than the best bid price, buy orders priced lower than the best ask price) are entered in the book as limit orders.

A limit order can be withdrawn at any time before it is matched with an incoming market order as part of a transaction.

The outcomes of the random variables that determine asset valuation, the strategies of all the players, and the past history of the game, determine the execution probability of a limit order.

Examples of limit order markets

In a first price auction, bidders submit a limit order to buy without seeing the trading window and the auctioneer submits a market order once all the limit orders have been placed

In a Dutch auction the auctioneer submits a sequence of increasingly attractive limit orders to sell until a buyer submits a market order.

In an English auction bidders submit compete with each other by submitting limit orders to buy until the bidding stops and the auctioneer submits a market order.

In retail markets stores submit limit order to sell and buyers submit market orders to buy.

A trader’s optimization problem

Consider some markets for financial assets on a stock exchange where traders make bids and offers at prices they choose throughout the duration of the game to maximize their expected subjective value of asset holdings at the end of the game.

It might be convenient to imagine that the length of the game is the difference between the opening and closing time on a typical trading day.

Initial endowment

At the beginning of a game, players receive an initial endowment of : - each stock

- money

The endowment might be: - fixed by the moderator- the realization of a random variable drawn

from a probability distribution that the moderator decides.

All players belonging to the same player type receive the same fixed amount in the first case, and get independent draws from the same distribution in the second case.

Market access

In the market module there is only one medium of exchange called money: all trades involve a transaction between money and a stock. (For example trading in more than one currency are excluded.)

A player might be allowed to submit limit and/or market orders to buy and/or sell in an asset market.

Depending on the game, money might be regarded purely as a medium of exchange, or stand for generalized purchasing power as well.

Preferences

J

j jTjTjT xcbmu1

T

j jTjTjT xcbmu1

Let u(c) denote a concave increasing utility function.

If bj and cjt enter multiplicatively in the trader’s preferences for the stock, bjcjT denotes the subjective value of the trader for asset j at the end of the game. Then the overall objective that the trader uses to assess her trading and portfolio decisions is:

Alternatively if (bj + cjT) denotes the subjective value of the trader for asset j at the end of the game the objective is:

Information

Information about the factors, the stocks and the state of each market is determined at each player type:

- information at the start of the game

- how the information is updated.

Subjects might have different information about:

-valuations, - assets, -order book, - history of transactions.

Uncertainty

When the jth asset is a stock, cjT is a random variable that represents the ex-post return on the common component at the end of the game,

It is therefore impossible to base trading decisions at time t < T on cjT because it is unknown at time t.

Accordingly denote by t the information available to the trader at time t.

Assume that at each instant t the trader maximizes the expected value of the utility at T from her portfolio.

Expected value

Then in the multiplicative case her expected utility may be expressed as:

tjTjT

J

j jT xcbmuE |1

jTjTj xcb

where:

is the liquidation value of her stock holdings in the jth stock at time T.

The additive case is similar.

Solvency

To survive, a financial institution must enforce traders to honor contracts between themselves.

While the fiduciary rules vary across institutions, the market module captures the essence of many, if not most.

In the market module the trader is constrained at each point in time by how much she can offer for sale within each market and how much she can buy in total.

This implies there are J constraints for placing sell orders but only 1 constraint for placing buy orders.

Constraints on sell ordersWe denote the set of buy and sell prices by

For convenience we assume there are no short sales. Therefore the total amount of each asset up for sale cannot exceed her holdings.

Let sjkt denote the quantity of the jth asset for sale at price k at time t. We require:

jtk jktk xsp 1

,,, 3211 pppp kk

Constraint on buy orders

An overall budget constraint on buy orders prevents the trader from placing orders that exceed her money holdings.

It effectively constrains the seller from exchanging (selling) more money for assets than she holds.

Let djkt denote the quantity of the jth asset demanded at price k at time t. We require:

J

j tk jktk mdp1 1

Choices at time t Players exploit their trading opportunities

defined in the choice set to make decisions throughout the game.

At each successive instant t [1, T] the trader may do nothing, or take an action in one market j {1, . . . , J} subject to the constraints defined in the three previous slides:

1. Delete an existing limit order to buy or sell a quantity q in market j.

2. Submit a market buy or a market sell order for quantity q.

3. Submit a limit buy or a limit sell order for quantity q in market j at price pk.

er

The trader’s optimization problem

In the multiplicative case the trader sequentially makes the choices that at each successive instant t [1, T] maximize:

subject to the rules prescribing the orders he is permitted to place, the J budget constraints preventing short sales, and 1 overall budget constraint preventing borrowing.

tjTjT

J

j jT xcbmuE |1

Interdependence between players

The actions of the other players affect the trading opportunities of the nth player.

Consequently the probability distributions the nth player uses to take expectations over future events are partly determined by the trading strategies of the other players.

Therefore the previous two slides provide an incomplete description of the trader’s optimization problem, because they does not fully describe how to take the expectations over future trading opportunities.

The strategy space

Let dj(nt) be an indicator function showing which market the nth trader will choose to act in at time t when his information is nt, where dj) =1 if he picks market j and dj(nt) = 0 otherwise.

Let q(nt) denote the quantity he picks (where negative quantities indicate sell orders), and p(nt) the price (which is constrained to be the market price if the trader is permitted to make only market orders are in this market).

Then a strategy for the nth trader is the vector function s(nt) = (dj(nt), q(nt), p(nt)) for each t [1, T].

The valuation function of trader n

Let sn0(nt)) denote the optimal order strategy,

and define the value of correctly solving at time t by:

o

ntnjTnjT

J

j njnTnt sxcbmuEW ,|1

Then for all t < r < T, the value function W(nt) solves the recursion:

This equation says that optimized value should behave like a random walk throughout the game.

ntnrnt WEW

The strategic solutionto limit order market game

Suppose each trader n = 1, . . . . ,N picks a strategy to solve their own optimization problem, and calculates the expectation knowing the strategy the other traders picked.

The resulting strategy choice se(nt) for each trader n = 1, . . . . ,N is a symmetric Nash equilibrium for the limit order market game. It corresponds to the solution of the strategic form for this game.

Summary

In the second half of this lecture we defined a limit order market, and described the maximization problem of a day trader, nesting it within the solution to the game.

Then we showed that in simple trading settings, limit order markets have a rich enough strategy space for players to communicate their valuations to each other and exhaust almost all the gains from trade.

2. Oligopoly

How is the solution to a trading game affected by relaxing the assumption that there is only a single supplier in each market? We explore the extent to which monopoly power is diluted by rival producers.

Resale as a potential source of competition with the monopolist

Some economists believe that the possibility of resale by consumers limits the monopolist’s ability to engage in price discrimination.

They say a low valuation consumer can buy the good at a low price and then compete with the monopolist to supply a high valuation demander.

Price competition between the monopolist and low valuation consumers supposedly drives down the price that high valuation consumers pay.

Primary versus secondary suppliers

Similarly lawyers for Alcoa once argued that although the company had a monopoly on the supply of primary aluminum ingots, it did not have a monopoly on the market, because there were secondary supplies from scrap.

Judge Learned Hand rejected Alcoa’s arguments in 1945, finding them guilty of violating the Sherman Act which prohibits collusion. He noted that resale by secondary suppliers does not prevent Alcoa from restricting the total supply of aluminum

Do secondary suppliers, such as consumers of primary ingot reduce Alcoa’s profits from selling secondary or aluminum ingot as scrap?

Monopoly with re-tradingWe consider the three cases, where trading occurs:

1. on a half open time interval of [0,1) with consumption at t =1

2. on a closed time interval of [0,1] with consumption at t =1.

3. on a closed interval of [0,1] with consumption from the services of a durable good over the time phase [0,1].

Now assume consumers may themselves sell the product they purchase from the monopolist.

Competition from repairs and maintenance

The monopolist faces “extra supply” from consumers lengthening product life to postpone the cost of replacement.

One way of mitigating this competition is to void all product warranties if an unauthorized mechanic maintains it.

Companies such as Xerox have circumvented this competition in the past by leasing its goods rather than selling them outright.

Collusion as a contract between rivals

Collusion is outlawed by the Sherman Act.

It restricts opportunities to establish a long term relationship of mutually sustaining collusive prices with the threat of a price war.

Furthermore the larger the number of rivals the more difficult it is to reach and enforce tacit agreements on price.

Thus increasing the number of suppliers tends to foster competition between them.

Price competition from rival suppliers

If demand for a product is a linear function of price then: q = - p

Suppose rivals compete on price. As a function of (p1,p2), the net profit to the first firm is:

The profit function for the second firm is derived in a similar fashion.

p1

p1p1 c if p1 p212 p1p1 c if p1 p2

0 if p1 p2

Quantity competition by rival suppliers

Rather than compete on price, both firms could choose their own production and let the market price adjust to clear the market.

For example suppose that in a two stage game, firms construct capacity for production in the first stage, and market their produce in a second stage.

Accordingly denote by q1 and q2 the quantities chosen by the firms.

The industry price is derived from the demand curve as: p = ( - q1 – q2)/

Net profit to the ith firm

Net profit to the ith firm:

ii qcqq

qq

21

21 ),(

The marginal profit with respect to qi:

cqq

q

qqi

21

1

21 2),(

The competitive limit

If firms compete on price then (in a one period model where there are no opportunities for colluding), then in the absence of monopoly, price equals marginal cost.

If there are N firms which compete on quantity, then in a symmetric equilibrium each firm produces:

As N diverges, the industry quantity converges to the point where price equals marginal cost.

q N 1 1 c

The strategy space

If firms compete on price, then two firms suffice to reduce price to marginal cost.

If firms compete on capacity quantity, then as the number of firms increase, price converges to marginal cost.

But what happens if each firm can credibly commit to match the lowest competing offer?

These scenarios illustrate the importance of the strategy space in determining the final allocations and the realized gains from trade.

Monopolistic Competition

Monopolistic competition describes a product differentiated market in which each supplier sells a good with no perfect substitutes, but several imperfect substitutes.

To what degree do features of monopoly appear in differential product markets?

Brand loyal customers can be charged higher prices that deter the switchers. Each firm delineates its customer base partly through its price strategy.

Free entry

When marginal costs are constant or increasing, entry occurs sequentially until it is unprofitable to do so. This corresponds to a competitive market where rival suppliers compete for demanders.

When marginal are declining, or when there are fixed costs and constant marginal costs, the result is harder to predict.

One possibility is a trigger strategy equilibrium in which a monopolist or cartel maximizes producer surplus by threatening to compete by dropping the price to zero if another firm enters: this may be illegal under the Robinson-Patman Act.

3. The Direction of Trade

In the experiments above we have taken as given which traders are buyers and which ones are sellers. This is not a useful assumption to make in all markets, particularly markets for financial securities. What happens when the direction of trade for each player is determined by the terms of trade? Can we devise trading games that have desirable properties in which the identities of buyers and sellers depend on the configuration of valuations?

A trading game

Consider the following two stage game of price determination for three (or any odd number of players), each of whom can buy or sell at most one unit of a good.

Every player knows their own valuation for trading a unit, but not necessarily the valuations of anyone else. Then players simultaneously or sequentially indicate a reservation price at which they are indifferent between buying or selling a unit.

In the second stage of the game every player places limit or market orders but only at the median reservation price announced in the first stage. They cannot place sell (buy) orders if their announcement is higher (lower) than the median price.

Incentive compatibility

We now prove that the weakly dominant strategy is for each player to truthfully announce his valuation.

Note first that only lies which affect whether the player becomes the median valuation are payoff relevant.

Suppose he would have been the median price if he had been truthful (and not traded) but instead he announces a higher valuation. Consequently a lower valuation becomes the trading price. The lying median valuation player now must buy at a higher price than his true valuation, and thus incur a loss.

If a person becomes the median valuation by lying about his type then he foregoes gains from trade.

Summarizing the properties of the trading mechanism

The trading mechanism is:

1. self financing because it requires no outside funding to implement.

2. satisfies incentive compatibility because each player has a weakly dominant strategy to truthfully announce his valuation.

3. Satisfies the participation constraint because each player prefers to participate in the trading game rather than consume his own endowment

4. Efficient, as all the gains from trade are exploited.

A generalizationWhen there are an even number of players the trading mechanism is modified slightly.

The first stage proceeds exactly as before. In the second stage the players announcing the two median valuations are excluded from participating in the second round of trading, and must consume their respective endowments. The other players are permitted to trade any single price selected between the two valuations

This mechanism is self financing, satisfies the participation and incentive compatibility constraints, but is not fully efficient because the two median players are prevented from trading with each other and must consume their endowments instead.

Limit orders as language

We revisit the simple trading game where each trader is endowed with one unit, only wants one extra unit at most, valuing each of the first two units the same.

All players with valuations above the median submit a limit buy order at their valuation less the median, and all players with valuations below the median submit a sell order at their valuation plus the median.

Then all players withdraw their valuations and submit bids only at the revealed median valuation.

Can this collective behavior be rationalized?

SummaryResale does not limit monopoly power unless the secondary producers can affect the characteristics of the product.

A competitive fringe typically reduces but does not necessarily eliminate monopoly rent.

When there are several comparably sized suppliers, the strategies space in which rivals compete can greatly affect the rent they collectively make.

When the direction of trade is endogenous, and players have private valuations, there are trading games whose solution outcomes converge to the efficient allocation as the number of traders increases.