Auction and Markets 73-340

Professor Robert A. MillerJanuary 2010

Teaching assistant:Hao XueEmail: hxue@andrew.cmu.edu

Preamble

This course is about creating gains from trade and strategically exploiting their division. We begin by introducing limit order markets, a paradigm for trade and exchange, as well as an institution for trading financial securities. Then we analyze competitive equilibrium, a concept often to determine comparative advantage.The latter parts of the course venture behind competitive equilibrium price determination to market microstructure. We analyze bargaining rules, optimal contracting, several auction formats and their connections to monopoly.This leads to our final topic, the size and scope of firms.

Course objectives1. Develop a working familiarity with limit order

markets as a trader and as an analyst.

2. Increase your intuition for predicting and evaluating competitive equilibrium outcomes, and its implications for your firm.

3. Learn to optimally respond in a bargaining situation, as a contractor and as a bidder in an auction.

4. Design bargaining, contracting and pricing rules to advance your own interests.

5. Identify trading and investment strategies where your firm might have a comparative advantage.

Course materials

The course website is:

http://www.comlabgames.com/45-975M

At the website you can find: the course syllabus power point lecture notes games you can download project assignments the on line (draft) textbook

A limit order market is a real world institution for characterizing the financial sector, and it is also a paradigm for describing trading mechanisms more generally. This lecture defines portfolio strategies for limit order markets and then characterizes the optimization problem of a day trader solve. This yields some properties of equilibrium prices that arise from arbitrage considerations.

Lecture 1 of Limit Order MarketsIntroduction

Financial Markets

Electronic limit order markets are amongst the fastest growing markets within the financial sector.

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

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

Trading in limit order markets

Traders submit a market order or a limit order.

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

Limit orders specify a transaction price, market orders a reservation price.

Market transactions match market orders with limit orders, and take place at the limit order price(s).

Thus market orders are executed instantaneously, but limit order might never be executed.

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.

Tradingwindow

The difference between the highest priced limit buy order ask price (the bid price), and the lowest priced limit sell order (the ask price) is called the spread, here 2,000.

The trader just placed a sell order for 9 units at price 5,800, with an expiry time of 60,000 seconds, reducing the spread from 2,200 by placing an order inside the previous bid ask quotes.

A day trader’s problem

Consider a stock exchange where traders make bids and offers at prices throughout the duration of a game to maximize their expected subjective value of asset holdings at the end of the game.

For example suppose the length of the game is the difference between the opening and closing time on a typical trading day.

At the beginning of the day, each trader (player) begins with an initial endowment of each stock, and some money (or some other liquid asset for settling up), fixed by the firm or his assets from yesterday.

Asset value

J

j jTjTT xcmw1

For now, suppose that all the trader cares about is wealth at the end of the day, which we denote by w. We further assume that wealth is a weighted sum of his liquid and illiquid assets.

Let mT denote money held at the end of the game, and let xjT denote the amount of the jth asset that the trader holds at the close of trading at time T. Then

where cjT denotes the subjective value of the trader for each unit of asset j at the end of the trading day.

Preferences

J

j jTjTT xcmu1

We assume the trader has utility function u(w), which we assume is concave increasing in wealth w.

Then the utility the trader realizes at the close of day T is:

Information

Information about the factors affecting returns, the stocks and the state of each market is determined for each player type:- Information could differ across player types at the start of the game - The way information is updated could differ

across player types too.

Players might be differently informed about: -asset valuations -total supply of each asset -contents of the order book -the history of transactions.

Uncertainty

Recall cjT is a random variable that represents the ex-post return on the jth asset at the end of the game.

Trading decisions taken at time t < T cannot be based on cjT because cjT is unknown at time t.

Denote by t the trader’s position at time t. This includes her assets and outstanding orders, along with 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 her expected utility at time t may be expressed as:

tJ

j jTjTT xcmuE |1

jTjT xc

where:

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

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:

01

k jktjt sx

,,, 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 At each successive instant t [1, T] the trader

may do nothing, or take an action in any 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 qjkt in market j at price pk, where qjkt < 0, by reducing djkt or sjkt.

2. Submit a market buy or a market sell order for quantity qjt. When qjt > 0

the trader buys the stock, and when qjt < 0 he sells it.

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

where qjkt > 0, by increasing djkt or sjkt.

The strategy space

Let qjk(t) denote the quantity the trader would pick for the jth asset at the kth price at time t given his state t (where negative quantities indicate sell orders),

Note that if pk is above the spread (the lowest ask), and and qjkt(t) > 0, then the order is a market buy. If pk is below the spread (the highest bid), and qjkt(t) < 0, then the order is a market sell.

If qjk(t) is defined for every asset j, every price pk and for each t [1, T], it is called a portfolio strategy.

The trader’s optimization problem

The trader sequentially makes the choices that at each successive instant t [1, T] to 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, given his current endowment of the J assets, plus his outstanding limit orders.

tJ

j jTjTT xcmuE |1

The valuation function of trader n

Let qjk(t) denote a portfolio strategy, and define the value of using this strategy at time t by:

tJ

j jTjTTqtq xcmuEW |1

Then for all t < r < T, the law of iterated expectations implies the expected utility Wq(t) solves the recursion:

This equation says that the change in expected utility from trading throughout the day should behave like a random walk.

trqtq WEW

Interdependence between players

The actions of all the players affect the trading opportunities of each other.

Consequently the probability distributions any 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 do not fully describe how to take the expectations over future trading opportunities.

Equilibrium in a limit order market game

Suppose each trader picks a strategy to solve his own optimization problem, and calculates the expectation knowing the strategy the other traders picked.

If every player followed such a strategy, then we say that the players are in a Bayesian Nash equilibrium for the limit order market game.

A Bayesian Nash equilibrium exists for this game (although a proof is beyond the scope of this course).

SummaryWe discussed the portfolio management problem traders face in limit order markets, and how their trading strategies are resolved in the Bayesian equilibrium solution .

One implication of the equilibrium is the theory of arbitrage pricing, which shows that securities with the same probability distribution of dividends and capital gains must trade at the same price.

We showed that prices should follow a random walk in liquid markets when traders are risk neutral. The result does not hold when markets are not liquid. We might observe mean reversion.