1. 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 2 of Limit Order Markets Arbitrage Pricing

2. 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.

3. Arbitrage opportunities 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.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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?

9. 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.

10. Summary We 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.