Banking and FinanceóG63.2751

Courant Institute of Mathematical SciencesóMasters of Finance in Mathematics Program


Lecture 2: Stock Markets


Additional Reading Reference: Investment Management, Frank J. Fabozzi, Prentice Hall

Other references:


Topics to be Covered




How the Stock Market Works, chapters 5,6,7,10

Handout on international stock exchanges and efficient market hypothesis from Houthakker and Williamson

Types of Stock and its Supply in the Market


There are two main types of corporate stock:



Common stock represents ownership in a corporation and holders of the stock retain residual rights to the company in the event it is liquidated. Liquidation can occur if the companyís interest payments cannot be met at any time. In the even of such a liquidation all debtors claims on the company are senior to stock holders. As such, the stocks price tends to suffer in periods of low profits (as the likelihood of liquidation rises) and tends to fair better in periods of high profits. Partly as a consequence of this, stock prices are more volatile than bond prices. Holders of common stock are also entitled to stock dividends. Dividends are period cash (or stock) payments made to holders as a form of profit sharing. Dividends are paid to all owners of record of the stock on a certain announced by the corporation ahead of time. On a particular day, the stock begins to trade ex-dividend, that is, it trades without the dividend. Investors who purchase the stock on or after the ex-dividend date are not entitled to receive the dividend payment. As a consequence, in theory the stock price should drop an amount exactly equal to the dividend amount on the ex-dividend date.


Corporation may alter the supply of their stock in the market place in one of four ways. To understand how this is possible, it is important to note that when a company goes public, its corporate charter states the total number of shares that the corporation is authorized to issue. These shares, called authorized shares, are usually not issued all at once. The number of shares that are issued, are called the outstanding shares. Corporations can change the number of outstanding shares in one of four ways:



The total number of outstanding shares times the current market price of the stock is called its market capitalization, and represents the total market value of the stock. The above four items are discussed in How the Stock Market Works, but it is interesting to make a note about stock splits.


One reason a company might institute a split is to increase demand for a stock by lowering its price. As stocks tend to trade in even lots of 100 shares, a lower price can make purchasing the issue more attractive to small investors. Also, some corporate boards feel that a stock split sends a positive signal to the market about the future growth prospects of the firm. This has been tested in the finance literature: In a paper, Eugene Fama, et al, called "The Adjustment of Stock Prices to New Information" (International Economic Review (February, 1969) conducted what is known as an event study to test whether abnormal price moves are typical around the time of a stock split. The results of the paper were negative.

Overview of Secondary Equity Markets


There are two primary concepts we will discuss: brokers and liquidity. We will start with brokers.


In the abstract, a broker is nothing more than an intermediary between buyers and sellers. A brokerís primary function is to gather information about the supply of and demand for products. In the case at hand we are interested in securities brokers, and here supply refers to what securities are being offered, at what volumes and what prices. Later we will talk concretely about what sorts of brokers exist and what business structures surround them, but for now, we will keep the things in the abstract.


Brokers are able to exist because there are necessarily costs associated with buying and selling securities, which include the seeking out of counterparties, the setting of prices and the determination of the credit-worthiness of the counterparty. A broker can spread these costs out among many counterparties and thus offer lower per head search costs to individual buyers and sellers while creating profit for himself. This lowering of costs increases the pool of available investors and in turn the liquidity of securities. This has the positive effect of increasing interest in the purchase of primary offerings as the likelihood of negotiability is increased.


Brokers play very concrete roles in the buying and selling (that is, in the trading) of securities in a variety of ways. We will discuss these roles in this lecture, starting with a discussion of the New York Stock Exchange (NYSE). The NYSE is a physical trading place where stocks and to a lesser extent bonds are traded. It is the largest exchange in terms of volume in the world, with record volumes in stock trading increasing every year, including a new record set in late August 1998. Year to date through August 1998, the New York Stock Exchange had an average daily trading volume of over 630 million shares, up from around 520 million in 1997.


The New York Stock Exchange is structured around a set of rules determining who may trade securities on the exchange and what securities may be traded there. Thus rules place requirements on both individuals wishing to trade and on companies wishing to be traded. Letís begin with who may trade. In order to trade on the New York Stock Exchange you must be a member of the exchange. Membership carries privilege and prestige. The privilege is the exclusive right to conduct business on the exchange floor. The prestige is partly due to the fact that there are at all times a limited number of memberships available on the NYSE. In particular, there are currently 1,366 active members representing only 430 firms (these numbers change! The NYSE posts them regularly on the web at The number of memberships have not changed since 1953. There are four types of membership:



There are also rules governing the listing of a stock on the New York Stock Exchange. Only listed stocks may be traded on the exchange, and the listing requirements are as follows (the complete listing requirements may also be looked up on the NYSE Web Page:



The New York Stock Exchange trades the most volume of any exchange in the world. In addition, it has a large turnover rate, that is, the ratio of shares traded to total shares outstanding is high as well. The most current figures for turnover rate for the NYSE are approximately 70%.


The Mechanics of NYSE Trading


When an individual wants to buy or sell a stock they must communicate certain information to a broker. The broker, who usually works with brokers who trade directly on the floor of the exchange (e.g., with floor brokers of their own companies) needs to know:



The first to items are straightforward to understand, but the third needs some clarification. Because of the high volume of trading, and intra-day volatility of markets, it is not always convenient or optimal to place an order at a precise price (although this is certainly possible). As a consequence, there are a variety of ways of expressing how an order is to be regarded and also the amount of time the broker has to execute the order. Here are the major types of orders:



Notice that limit orders and stop orders are only executed upon a certain condition being met. As a consequence, the customer must specify ahead of time how long the order shall remain active. There are several options: day orders, open orders and good till canceled. Complete descriptions of these may be found in How the Stock Market Works In addition, a customer may enter market-on-close (MOC) orders, specifying that the order is to be executed at the marketís close.


The Specialist


The New York Stock Exchange, like many US stock exchanges, operates on what is known as the specialist system. This system has a particular way of handling the flow of buy and sell orders and maintaining liquidity. It is by no means the only possible system for handling a large volume of trades, and in fact it stands in opposition to another important system embodied by the NASDAQ exchange (more later).


The New York Stock Exchange operates as a "continuous auction" in which brokers constantly make bids to buy stocks and offers to sell stocks. Whenever a bid to buy by one broker is met by an offer to sell at the same price by another, the trade can be executed directly between the two brokers or by means of the specialist. The specialistís role, however, is special because he is charged with continuously offering a bid and offer in the stock that he specializes in. For example, Here is a detailed example (modeled after Houthakker and Williamson, p. 117):


A client contacts his broker in order to sell 100 shares of XYZ. The broker will first check the current market quotation: this is the specialists bid and offer price for the stock. The specialists has posted 66 bid and 66 ¼ ask. That is, the specialist is willing to buy the stock for $66 per share and sell it for $66 ¼ per share. How many shares? The specialist is willing to buy at least 100 shares and sell at least 100 shares. Given this quote, the broker knows that, at least for the moment, the client may sell his shares for at least $66 per share. Given this information, the client places a market order with his broker to sell 100 shares of XYZ at $66 per share. The broker then calls his or her floor broker with the order. The floor brokerís job is now to sell the 100 shares at the best possible price. The limits of this price are that the sale can go off right now for at least $66 per share, but no more than $66 ¼ per share. Why? Because the specialist is willing to sell at $66 ¼ per share, so it is likely that the specialist is waiting to execute unexecuted limit orders to sell at $66 ¼ per share. Conclusion: there are no buyers currently willing to pay $66 ¼ per share. The question remains if the possibility exists of selling the shares at $66 1/8 per share. The floor broker, owing to his position on the floor of the exchange can physically call out "100 at 6 1/8", meaning "I have 100 shares of XYZ for sale at 66 1/8." Two important points arise here: first, the stock for sale is implicit in the trading post at which the trader is posted (see How the Stock Market Works). Second, if the broker finds a buyer at 66 1/8, then the trade gets executed between the bid and the ask. This illustrates a second important point: the bid-ask price represents at a point in time the worst case prices for buying (one buys at the bid) and selling (one sells at the ask). However, due to the possibility of simultaneous arrival of orders, it is possible to market execute orders between the bid-ask price. There is a second way this can happen, known as hidden limit orders that we will deal with in a moment.


The specialistís role can be described in more detail as follows. When a stock becomes listed on the NYSE it is assigned a specialist. The specialists may buy and sell stock executing orders for other brokers or for their own accounts. Their stated purpose is to maintain orderly flow in the market. But what does this mean in particular? Letís look at the mechanics:



The specialists book is closely held and its contents are only known by the specialist. An example of what a specialists book may look is as follows (from Houthakker and Williamson, p. 119). Suppose at 11:00 am the book looks like this:


Buy 1,000 limit 88 Sell 300 limit at 91 ¼
Buy 200 limit 88 ½ Sell 800 limit 92
Buy 100 limit 89 Sell 800 limit 92
Buy 500 limit 90 Sell 800 limit 92 ½
Buy 200 limit 90 ¾ Sell 200 on stop 89
Buy 300 on stop 93 Sell 200 on stop 87


Letís examine this: the highest bid limit order is 88, that is, the highest price anyone is willing to pay right now is $88 per share. The highest buy limit order is 90 ¾ while the lowest sell limit order is 91 ¼. Thus, the bid-ask spread is currently at most ½ point; that is, the specialist will post a quote of no worse that 90 ¾ bid and 91 ¼ ask. Moreover, the quote will appear with a bid size and ask size, wherein the bid size will be 200 and the ask size will be 300. Now suppose the last trade took place at $91 per share at 11:00 am, between the last bid and the ask. Now suppose a market order arrives at 11:05 am to sell 1,000 shares. What happens?


The specialists job is to execute the order while attempting to satisfy all buy limit orders first. If the specialist sets the price at $89 per share, then all of the limit orders to buy at $89 or above would be triggered: there are a total of three, at $89 per share, $90 per share and $90 ¾ per share for a total of 800 shares. Suppose the price is set to $88 ½ per share. Then 1000 share from limit buy orders would be triggered, but the price would then have fallen below $89 per share (because a single order exists at $89 per share), and the sell stop order would be triggered. Since the limit order has to be satisfied before the market order, the 200 shares in the stop order would have to be "covered" by some of the 200 shares at $88 ½ per share. Thus, the only way to satisfy the complexity of the book is to reduce the price all the way to $88 per share.


If this were the case and the orders matched up and the new price was set at $88 then the trade price would drop precipitously from $91 to $88 in a matter of few minutes. This is problematic for the market because it would create the public perception that there had been a rather large move in the price over a short period of time. This in turn might create a spate selling, due to the erroneous perception that something fundamental had happened to the value of the company XYZ that would make it worth suddenly less.


In order to avoid such market disorder, the specialist is charged with meeting the temporary demands placed on the market, such as in the example above by buying or selling where appropriate to create the appearance of an orderly market whose published price fluctuations reflect the actual level of price volatility. In the case of the above example, letís see how the specialist would go about this.


First, the specialist is the unique person who as access to the complete book of orders, and he knows that there is a fairly balanced set of orders above and below the market. As a consequence, the specialist can buy some shares below the market by setting the price at $90 ¾. Then the specialist will buy 800 shares at $90 ¾ while assigning the other 200 to the limit order. This is clearly good for the market because now the last trade price is $90 ¾ as opposed to $88. In addition, the specialist has just bought the stock at the bid price. This is an advantage, because now if another situation arises where a market order to buy comes in, the same logic is likely to allow the specialist to sell at the ask price. If the bid-ask spread remains stable, then the specialist can "make the spread" by buying at the bid and selling at the ask.


This said, we have to ask are the any risks that the specialist undertakes for the reward of buying low and selling high? The answer is yes: if the market happens to move away from the specialist, that is, if the market moves up while the he is short (i.e., has sold stock that he does not own, more on that later) or moves down while he is long, he can be stuck holding inventory that has radically changed in value.


In that the specialist has been assigned the duty of keeping an orderly market, there are official rules governing the trading behavior of the specialist. One class of rules ensures that the specialist will never trade ahead of customers. That is, that all prevailing limit orders will be executed first. Another class of rules ensures the fair handling of limit orders: they govern which limit orders will be executed ahead of others. For example, if two limit orders of the same size are in the specialists book, there is a specific rule deciding which one will be executed first.


Clearly having privileged knowledge of the specialists book could be a large advantage to traders. The book, however, is closed to the public and only the specialist knows for sure its exact contents. The public, however, is not without some information about the specialists book. The public, after all, knows the specialists quote. Every trade that takes place and every quote the specialist prints is available from a variety of live feeds. The data available is often referred to as time-and-sales data and includes



It does not state, however,



Because of the value inherent in knowing the specialists book, some market participants have devised algorithms that attempt to deduce from the tape what the contents of the book might be. This is called book abduction.


A sample tape may look as follows:

100b 400 100¼a 500 2:01pm
100 ¼ 200 2:01 pm
100 ¼ 600 100½a 400 2:01 pm
100 ½ 500 2:02 pm


In the above sample tape, the first line says the bid price was $100 per share and there were 400 shares available at that price. That is, there are customers with limit orders totaling at least 400 shares to buy at 400. That means, you could sell up to 400 shares at $100 per share. On the other hand the ask price is 100 ¼ and there are investors with orders to sell up to 500 shares at this price. On the next line, a trade is indicated: 200 shares were traded at 100 ¼.


Assignment: (a) Starting with the specialists book above (omitting the stop orders) create the arrival of at least 12 new orders, at least 4 of which are market orders. Among the limit orders, try to display the following possibilities:



After each new order, show how the specialistís book would change, and try to predict how the specialist might or might not trade in the face of each new order. After each new order also show the next line of the tape. If the new order is a limit order that comes in above or below the market, you do not have to show a new quote. If the new order comes inside the market, then the bid-ask will change. Sometimes just the bid or ask size will change. Try to illustrate as many of these possibilities as possible.

  1. Now using (a) try to devise a simple algorithm for deducing the contents of the specialists book from the tape. The algorithm should be rule based and should update its "guess" as each new line of the tape gives more information.
  2. BONUS! If you want, write a computer program to simulate the random arrival of new orders and keep track of the specialists book. Have the program output a new line of the tape after each arrival. Then as a next step, implement your algorithm in part (b) and feed the output of your "tape" into your algorithm and see the extent to which the code correctly deduces the book.

The Opening Auction


The market open is an interesting aspect of the NYSE. The market is officially open only during the hours 9:30AM to 4 PM. Before 9:30AM brokers may submit pre-opening market on open orders (MOO) and various limit orders. The specialists job is to find an opening price that executes as many market orders and existing limit orders as possible. This is usually straightforward, but if for some reason there is a serious imbalance between buy orders and sell orders, it can present a problem. If, for example, there has been extremely bad economic news, or trading was halted on the previous day, there may be significantly more sell orders than buy orders. In such an event, the specialist is not absolutely bound to meeting the imbalance with his own inventory. Instead, he may delay the opening of the stock and post bid and ask prices that attempt to bring in buyers to meet the excess selling demand. The specialist may adjust the quoted bid price downward to attract more buyers until the imbalance of buyers and sellers is reduced sufficiently to commence trading.


Important topics not covered in these notes: The designated order turnaround (DOT) electronic system.


Over-the-Counter Markets


The New York Stock Exchange is a centralized exchange. All trading passes through the same place and there is a specialist handling each stockís market. An alternative type of stock exchange is the over-the-counter market. Instead of having a central trading place, participants all view the same communications network known as NASDAQ. The key to the NASDAQ market is a collection of dealers known as market makers who continuously offer bid and ask quotes for the stocks they make markets in. A key difference between NASDAQ and the NYSE is that NASDAQ market making is competitive, with many dealers simultaneously making markets in the stocks.