Strategic Corporate Management

45-975

Professor Robert A. MillerJune 2009

Teaching assistant:Chen Li: cli1@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 objectives

1. 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 activities where your firm might have a strategic 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 a limit order market and demonstrates that many bargaining games, auctions, retail sectors and stock markets fit within a limit order framework. We then define a portfolio strategy, depict the optimization problem a day trader might solve, and derive some properties of equilibrium prices that arise from arbitrage considerations.

Lecture 1Limit Order Markets

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?

General description of a trading game

First we describe the trading mechanism in a limit order market.

Then we define a set of preferences for a typical trader, and the constraints on his trading problem.

The constraints determine what he can have, his preferences what he wants.

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 to buy.

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.

However once a limit order buy has been matched against a market order sell (or vice versa), it is too late to withdraw the limit order, and a transaction takes place at the price of the limit order.

Examples of limit order markets

In a bargaining game between two players, the seller submits limit order asks, and the buyer submits limit order bids, until one of them submits a market order by accepting the terms of trade offered by the other player’s most attractive limit order.

In a first price sealed bid 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 retail markets, stores submit limit order to sell and buyers submit market orders to buy.

A day trader’s problem

Consider 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 the day, each trader (player) receives an initial endowment of each stock, and some money (or some other liquid asset for settling up).

The endowment might be: - fixed by the firm (game moderator)- his assets from yesterday (the

realization of a random variable drawn from a probability distribution that the moderator decides).

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

Depending on the focus of the study or application, money might be regarded purely as a medium of exchange (having no intrinsic value), or alternatively stand for generalized purchasing power as well.

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.

Subjects 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).

Payoff equivalence

Some features of the solution to a market game are evident without explicitly solving the game.

Perhaps the most important one comes from the notion of arbitrage, which is based on payoff equivalence.

Two bundles of securities are payoff equivalent if they have the same probability distribution determining the payoffs at the end of the game.

Arbitrage pricing

The optimal exploitation of arbitrage opportunities puts bounds on the best prices quoted in the limit order book of payoff equivalent portfolios.

Loosely speaking, arbitrage compels payoff equivalent securities to trade at the same price.

More precisely, it should not be possible, by means of market orders alone, to sell one bundle of securities and purchase another payoff equivalent bundle and make a net profit.

Maximizing expected value

For the purposes of this lecture segment we shall assume that traders maximize their expected value, an assumption that can plausibly applied to firms.

We investigate conditions under which the competitive equilibrium price follows a random walk.

Currency exchange

Suppose U.S. export companies sporadically receive Euro and yuan injections from demanders for their goods in the E.U. between dates 0 and T.

Similarly European (and Chinese) exporters sporadically receive injections of dollars and yuan (euros) for their sales in the U.S. and China (Europe).

Export firms in each country also purchase domestic currency on the foreign exchange market between date 0 and T. At date T all export companies are liquidated and no further value is placed on holding foreign currency.

We assume the U.S. dollar is a dominant currency, meaning all currency prices are quoted in dollars.

Efficient markets hypothesis

We assume each export firm maximizes its expected dividend payments paid in domestic currency before the liquidation date T.

The liquidation value is unknown at all dates t < T, but as new information arrives about foreign trade throughout the trading phase, the traders become more informed about the value of foreign currency.

In competitive equilibrium the price of each exporter is the expected value of its dividend flow plus its liquidation value.

Therefore the exchange rates follow a random walk.

Proving the efficient markets hypothesis

Suppose the dollar price of yuan is lower in date t than its expected price in date s > t.

Chinese exporters buying yuan at date t are not maximizing their value, because the expected value of their companies would be higher if they postponed yen purchases until date s.

A symmetric argument applied to U.S. exporters explains why the the dollar price of yuan is not higher in date t than its expected price in date s > t.

Similar arguments apply to the dollar euro exchange market.

Market liquidity

The hypothesis that asset prices follow a random walk might be regarded as a test of liquidity.

In the previous example we may assume without loss of generality that there is continuous trading in the asset up until a common liquidation date T when the capitalized value of all the firms are recognized.

How would prices behave if consumers had limited opportunities to enter and exit the market, effectively segmenting the market into different time markets?

Illiquid markets

Suppose exporters face the threat of their foreign earnings being confiscated, or there are incomplete markets that limit savings and borrowing opportunities in domestic markets.

Then exporters might immediately capitalize their foreign earnings by converting them to domestic dividends and distributing them as dividends.

In this case successive prices in the foreign exchange market would exhibit mean reversion.

At the other extreme to the random walk observed in perfectly liquid market, prices in disconnected markets are independently distributed, and in a stationary environment, have the same conditional mean.

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.