surf 2011 final report - eric zhang

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Experimental Verification of the Theory of Dynamic Market Completeness Eric Zhang, Caltech SURF 2011 Prof. Peter Bossaerts, Mentor

Abstract

We sought to experimentally compare prices and portfolio choices for complete and incomplete asset

markets, as explained in “Dynamically Complete Experimental Asset Markets” (Bossaerts, Meloso, and Zame). In this experimental setup, we created an incomplete market by shutting down one asset market and prohibiting trade of that asset. However, there was one additional provision: halfway through the trading process, we would announce that one of three possible future states would not occur. This additional flow of information allows, at least according to the theory, for the incomplete market to fulfill the necessary conditions to be “dynamically complete.” This means that individual portfolios and asset prices should have exactly replicated those of the analogous complete market setup. Using electronic trading software, we tested this theory by implementing experiments among willing participants at Caltech. We found that portfolio choices were in general the same between the incomplete and complete scenarios, as hoped. On the other hand, some of our price predictions were not supported by the experiments. This gives some evidence to support the theory, but more experiments will have to be performed to determine if these results are typical.

Background Information

The purpose of the project was to provide experimental data that would either support or reject a theory around what we call “dynamically complete asset markets.” In the theoretical model, we describe the riskiness of financial assets (such as stocks and bonds) using a set number of possible states of the world. These states can be thought of as possible situations in which the health of the economy and the profitability of the underlying companies vary. Financial assets are distinguished from each other by the range of dividends they pay when a state of the world occurs. Some may pay a more consistent dividend; others may pay a lot in some states and very little in others. These states are drawn at the beginning of trading and then are revealed at the end. Which one of the states occurs at that time is not known beforehand. However, everyone in the market knows that the states are chosen by random draw and what the probabilities of the states are.

The purpose of trading assets in financial markets is to insure against risk. For example, if there is a

significant probability of a state in which most assets do not pay very much, then one might want to buy an asset (unless one maybe already has it) that pays high in that state of the world to protect against the possibility of that state happening. If the state occurs, that asset pays a lot, which partly offsets the low payments from other assets.

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In a complete market scenario, this type of insurance is always possible. One can insure against any one or several states. The technical condition for a market to be complete is that there are as many independent tradable assets as states. By independent, we mean that there is no redundancy; one cannot combine any number of securities to replicate the payouts of any other.

In an incomplete market scenario, this condition is not satisfied - there are not enough independent

securities to match the number of possible states. Consequently, we cannot guarantee that traders will be able to achieve the same range of state payout combinations as in a complete market. Additional conditions are needed to allow this to happen. First, there must be a flow of information over time. This information must allow traders to refine their knowledge of the hidden state. Second, traders must be allowed to trade and re-trade their assets in several trading periods as information is revealed. If these two conditions are fulfilled, then the incomplete market meets the two necessary criteria for being “dynamically complete.” That is, the incomplete market can behave similarly to a complete market if a more dynamic setting of trading and re-trading is allowed. This is the main idea of our theory. Whether this actually happens in practice is what we wish to test.

Now, we have not addressed any assumptions made about the traders in the two market scenarios. The

theory, however, makes the very explicit requirement that all traders be risk-averse. That is, they should be prefer portfolios that pay consistently over all states to portfolios with payouts that swing more wildly over states. The technical condition is that they must have expected utility preferences, or equivalently, mean-variance preferences. Is this a good assumption in practice? It seems so. Some work in economics has suggested that people are risk-averse when making risky decisions about gains, so we will work under this assumption.

Given these assumptions about preferences, the theory predicts that a complete market can achieve an

economic equilibrium. In this equilibrium, since individuals all share the same risk-averse preferences, they (or at least, the group on average) will all trade to what is called the “market portfolio.” The market portfolio is a portfolio that has the same proportion of state payments as the market portfolio (the set of all assets held by all individuals). For example, in the experiments we implemented, there were three states (X, Y, and Z) and three independent assets (Stock A, Stock B, and Notes). The table of state payments is

If the State Is…

X Y Z

Stock A (1 unit) Pays… 10 0 5

Stock B (1 unit) Pays… 0 5 10

Note (1 unit) Pays… 10 10 10

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If the entirety of all assets traded or held is 100 units of Stock A, 50 units of Stock B, 100 units of Notes, and 100 units of cash, the market portfolio pays

in state X. Applying a similar calculation, the market portfolio pays 1350 in state Y, and 2100 in state Z. The consumption in state X compared to consumption to all states, then, is

Similarly, state Y’s relative consumption is 0.243 and state Z’s relative consumption is 0.378. A trader who trades to the market portfolio, then, must end with a portfolio that pays in the same proportions (not necessarily the same amounts). We predict traders should be trading to the market portfolio, at least on average, even when they don’t know what each other’s starting portfolio is. Market forces (supply and demand) and individual preferences should drive them toward such behavior.

Let’s put it in mathematical terms. Let Ci-market be the total consumption of the market portfolio in state i,

that is, the consumption in state i summed over all individuals. Let Cj-indiv be the total consumption of a risk-averse individual in state i. In what we call an Arrow-Debreu Market Equilibrium, individual state consumption satisfies:

In addition to this prediction about portfolio choices, we believe that there is something to be said about

the prices of consumption in each state, i.e., how much traders will value getting a payout in each of the states. According to theory, traders will value the most getting payouts in states where the market payout is small. In other words, they will prefer getting paid in these states because these payouts are rather scarce. Low supply and high demand due to risk-aversion will drive up the price of consumption of these “poor” states. When comparing the prices of the different states, though, we have to adjust according to probability, as outlined below.

We’ll use mathematics again. Again, let Ci-market be the total consumption of the market in state i, that is, the payments in state i summed over all individuals. Let Pi be the price of (a $1 payment in state) i, and let πi be the probability of state i occurring. In Arrow-Debreu Equilibrium, state prices satisfy:

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Notice that probability must be taken into account. If all state probabilities are equal, as we’ll see applies in the complete market scenario, there is no effect. However, in the incomplete market scenario, information will be given in the second period of trading that allows participants to refine their knowledge of the probable states. It will turn out that this information will eliminate one of the states and change the relative probabilities of the remaining states. If we apply the prediction to this market situation, we will have to take account of the relative probabilities when comparing state prices.

What does this mean for dynamic completeness? The main idea of dynamic completeness is that these two

predictions concerning portfolio choices and state prices should apply equally to complete and incomplete markets. This is what we meant when we stated earlier that “the incomplete market can behave similarly to a complete market.”

Methods – Experiment Implementation and Trading Scenarios

We tested the predictions of this theory by implementing electronic financial market experiments. Subjects were obtained from volunteers in the undergraduate and graduate population of Caltech, with recruitment in groups of around 20. At Caltech, these experiments were performed at SSEL, the Social Science Experimental Laboratory. The electronic trading software we used was Flex-e-markets, which had been developed under Peter Bossaerts.

Each experiment took about 2 hours. The first hour was dedicated to ensuring that participants clearly understood their roles as traders, learned how trading various securities would affect their final payouts, and practiced using the trading software Flex-e-markets. This was because the design of the experiment was sufficiently complicated that we needed to ensure that participants fully understood what to do. Otherwise, we might have obtained skewed data because some of the participants understood the instructions better than others or none of them did.

During the second hour, participants traded in two types of market scenarios while we collected data on prices and portfolio choices.

In the first scenario (analogous to a complete market setting), they were given starting portfolios with cash, Stock A, Stock B, and a bond called Notes. Before trading started, one of the states X, Y, and Z was drawn at random with probability 1/3 and kept hidden. This drawn state would determine the payouts of Stock A,

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Stock B, and Notes at the end of trading. Then, during trading, participants were able to trade (buy, sell, or short-sell, the last of which means to sell shares of stock that one doesn’t own. Short-selling initially pays you for the sale of the stock, but you must later pay out a dividend) Stocks A and B and Notes for cash. Finally, the hidden state was revealed; traders received dividends from their shares of Stock A, Stock B, and Notes (or paid them out, if any had been shorted); and the net dividends added to the traders’ cash balances. We refer to this scenario as a complete market setting because there were 3 assets and the same number of states, so one could create any combination of state payouts using the available assets.a

In the second scenario (analogous to an incomplete market setting), nearly the exact same conditions held, except for the restriction that Stock B could not be traded. This is an incomplete market because now there were only 2 tradable assets and 3 states. However, this market was possibly “dynamically complete” because we allowed for trading over two periods and provided information halfway through. The information allowed traders to narrow down the list of possible states that were drawn. If the hidden state was X the announcement would be that the state was “NOT Z.” If the hidden state was Z, then the announcement would be “NOT X.” If the hidden state was Y, the announcement would be “NOT X” or “NOT Z” with equal probability.

b

This additional information allowed for the incomplete market to be dynamically complete, at least in theory.

In each experiment, we ran two replications of the first scenario and four replications of the second scenario. At the end of each experiment, we chose one of the trading scenarios at random and paid each participant based on their portfolio choices and the hidden state that was drawn. Each participant was also subject to a budget constraint - if a participant would ever go bankrupt in one of the possible states for that particular scenario replication, then he/she would be subject to a penalty. The reason for this rule is that the predictions of the dynamic completeness theory hinge on traders never going bankrupt in any state. If an announcement such as “NOT X” eliminates the state X, however, the bankruptcy rule would not apply to that particular state.

a Note that this scenario keeps the probability of all the states equal. Therefore, when we later apply the prediction relating to

state prices, we can essentially ignore the effect of the state probabilities.

b Note that this announcement changes the probabilities of the states – after the announcement, the probability of Y is now half the probability of X or Z, depending on which state was eliminated. This is because the initial probability of Y is equally split between the two announcements whereas the probability of X or Z is not. Since the three states were equally likely to begin with, after the announcement, Y is half as likely to occur as the remaining state (X or Z). So, when we apply the prediction relating to state prices to the second period of trading, we should keep in mind that the probability of Y is half that of the probability of the remaining state.

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Methods – Participants and Starting Portfolios

We were able to run 2 experiments during the SURF period. In both of these experiments, we had 2 types of starting portfolios. Type 1 started each trading scenario (complete and incomplete market scenarios) with 10 (Stock) A, 0 B, -1 Notes (shorted 1 Note), and 10 Cash. Type 2 started with 0 A, 10 B, -2 Notes, and 20 Cash. Notice that the Notes and Cash exactly cancel each other out – this is the equivalent of Notes being initially shorted at $10 to borrow the initial Cash balance.

In each experiment, we tried to split the participants into Type 1 and Type 2 so that there was twice as many of Type 2 as of Type 1 (or as near as possible to this). The reason is that we wanted to easily test the prediction relating to state prices. According to the theory [see “Background” section], the more the market portfolio pays in a state (all probabilities being equal), the less the price of consumption in that state should be. For the complete market scenario, all states are equally likely, so we don’t have to account for probability; for the incomplete market scenario, the probability of state Y is half

as much as that of state X/Z (depending on the announcement), so we will have to take this into account. Now, consider the complete market scenario. Recall that A pays 10 in state X and 0 in state Y and B pays 0 in state X and 5 in state Y. If there are 2 times as many of Type 2 as of Type 1 (and therefore 2 times as much of Stock B as of Stock A), then the market portfolio will pay exactly the same in states X and Y. Consequently, the theory predicts that the state prices of X and Y will be equal. This can be easily looked for during analysis of data.

In the 1st experiment, 14 volunteers showed up to participate. Of these 14, 5 were given “Type 1” initial portfolios and 9 were given “Type 2” initial portfolios. This is very nearly a 2:1 ratio of Type 2 to Type 1 portfolios. In the 2nd, 21 volunteers showed up. Since this number is a perfect multiple of 3, we were able to have an exact 2:1 ratio of Type 2 to Type 1. 14 were Type 2 participants and 7 were Type 1.

Predictions – State Prices

For the 1st experiment, the market portfolio is 50 A, 90 B, -23 Notes, and 230 Cash, which pays 500 in

state X, 450 in state Y, and 1150 in state Z. In the complete market scenario where all probabilities are equal, consumption in state Z should be the cheapest, followed by consumption in state X, which should be closely followed by consumption in state Y. In the incomplete market scenario where the probability of state Y is split in half (after announcement, probability of Y is 1/3, probability of X or Z is 2/3) , if the announcement is “NOT X” the price of (consumption in state) Y should be greater than half the price of Z, whereas if the announcement is “NOT Z” the price of Y should be greater than half the price X (See “Background” Section for predictions concerning state prices and how we account for different probabilities).c

For the 2nd experiment, the market portfolio is 70 A, 140 B, -35 notes, and 350 cash, which pays 700 in state X, 700 in state Y, and 1750 in state Z. In the complete market scenario, the price of Z should be the lowest, while the price of X should be equal to the price of Y. In the incomplete market scenario, if the

c Here’s a sample calculation. Take “NOT X” to be the announcement. Then, the market portfolio pays 450 in state Y with probability 1/3 and 1150 in state Z with probability 2/3. Since the market pays more in state Z, the state price prediction tells us that the price of Y divided by the probability of Y should be greater than the price of Z divided by the probability of Z. This gives an inequality

. Multiplying both sides by 3, we get , which says that the price of Y should be greater than ½ the price of Z.

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announcement is “NOT X” the price of Y should be higher than half the price of Z, whereas if the announcement is “NOT Z” the price of Y should be equal to half the price of X.

Predictions – Portfolio Choices

For the 1st experiment, the market portfolio pays 500 in state X, 450 in state Y, and 1150 in state Z. For

the incomplete market, if the announcement “NOT X” is called, the market pays 450 in state Y and 1150 in state Z, which translates to 450/(450 + 1150) = 0.28 for the proportion of payments in state Y and 1150/(450 + 1150) = 0.72 for the proportion of payments in state Z. If the announcement “NOT Z” is called, the market pays 500 in state X and 450 in state Y. This translates to 500/(500 + 450) = 0.526 for the proportion of payments in state X and 450/(500 + 450) = 0.474 for the proportion of payments in state Y.

For the 2nd experiment, the market portfolio pays 700 in state X, 700 in state Y, and 1750 in state Z. For the incomplete market, if the announcement “NOT X” is called, the market pays 700 in state Y and 1750 in state Z, which translates to 700/(700 + 1750) = 0.286 for the proportion of payments in state Y and 1750/(700 + 1750) = 0.714 for the proportion of payments in state Z. If the announcement is “NOT Z” instead, the market pays 700 in state X and 700 in state Y, which translates to 0.50 for the proportion of payments in state X and 0.50 for the proportion of payments in state Y.

Individuals in the incomplete market who trade to the market portfolio will have the same proportion of payments, as explained earlier. Moreover, if dynamic completeness holds, individuals in both the complete and incomplete markets will trade to these given proportions on average if “NOT X” or “NOT Z” is the announcement. We compare state Y to state Z or state X because the metric Y/(Y + Z) or X/(X + Z) can be calculated for both the incomplete and complete market scenarios. In contrast, other metrics such as Y/(X + Y + Z) or X/(X + Z) should not be used to compare the complete and incomplete markets because the announcement in the incomplete market takes state X or state Z out of consideration.

Analysis of Data

The data we used for analyzing the experiments was obtained from transaction prices and portfolio choices recorded by Flex-e-markets during trading.

How did we compute prices of state consumption, i.e., how much traders valued receiving payouts in one state relative to another? First, we found the prices set by the market for each of the assets Stock A, Stock B, and Notes. There are several ways to do this, but we chose to use the average transaction price (the price at which an asset was actually bought/sold, as opposed to the prices offered but not acted upon) of each asset. We also considered the effect of adding a buffer of 5 trades to allow for early mistakes in pricing. In most cases, however, the result of excluding the first 5 trades did not differ significantly from the result of using all transactions made. Once the market-determined prices were obtained, we mathematically backed out the

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state prices using the known state payouts of each asset. Since Stock A pays 10 in state X, 0 in state Y, and 5 in state Z, we have the equation

where is the price of (consumption of $1 in state) X, is the price of Y, is the price of Z, and is the

price of Stock A (which we found). Similarly, for Stock B we have

(where is the price of Stock B) and for Notes we have

(where is the price of Notes).

Putting these equations together in matrix form, we have

The coefficient matrix is invertible, so

Since the vector on the right only consists of the prices of the three assets (which we found earlier), we can easily compute the vector of state prices.

The above method can be extended to the incomplete market case, where one of the states is eliminated by the announcement and Stock B cannot be traded (and therefore is not included). For example, if the state is “NOT X,” then all the payoff equations no longer have a term, there is no payoff equation for , and the

matrix equation becomes

Here, we see that the eliminations of one state and one asset balance each other out, leading to a payoff matrix that is still invertible.

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Comparing the portfolio choices of participants was slightly less complicated. We wanted to compare their state consumption choices, which could be easily inferred from their ending portfolios. Applying the state payoff table for each asset, we have

where Cx, Cy, and Cz are the state consumption amounts and NA, NB, NN, and NC are the numbers of units of Stock A, Stock B, Notes, and Cash held, respectively. Relative state consumption is then easily computed from these state consumption amounts.

Results – Prices

The graphs show the time evolution of the prices at which each asset was bought and sold. The average statistics are based on these prices. All figures have been rounded to the nearest cent. Finally, prices in an incomplete session reflect only prices in the second half of trading.

Experiment #1

Complete Market Sessions

Session #1

Trading Activity in Market for Stock A

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High: $5.30

Low: $4.97

Average Price: $5.09

Average Price (excluding first 5 trades): $5.10

Trading Activity in Market for Stock B

High: $5.00

Low: $4.50



Trading Activity in Market for Notes

One single completed transaction for $9.95

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Session #2


High: $5.15

Low: $4.95



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High: $5.05

Low: $4.60




No transactions were completed for notes

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Incomplete Market Sessions

Session #1


High: $5.62

Low: $2.00




Stock B was not allowed to be traded

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High: $9.85

Low: $ 9.18


Average Price (excluding first 5 trades): N/A

Session #2


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High: $5.15

Low: $2.58






High: $9.97

Low: $8.80



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Session #3

Could not be performed due to problems with configuring the flex-e-markets software

Session #4

Could not be performed due to problems with configuring the flex-e-markets software

Experiment #2


Session #1


High: $5.00

Low: $4.68




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High: $5.00

Low: $4.92




No transactions were completed for notes

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Session #2


High: $5.00

Low: $4.62




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High: $5.00

Low: $4.50




High: $9.88

Low: $9.97



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Session #1


High: $6.15

Low: $4.00





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High: $9.88

Low: $9.52



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Session #2


High: $5.00

Low: $2.75





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High: $9.95

Low: $9.18



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Session #3


High: $5.78

Low: $4.53





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High: $9.90

Low: $9.57



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Session #4


High: $6.08

Low: $4.40





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High: $9.92

Low: $9.62



Results – portfolio choices

These graphs show the distribution of portfolio choices among participants, described by the consumption choices they made in one state relative to another. For example, the graph of X/(X+Y) shows the final state payout in state X as a fraction of the total payout in states X and Y. This way of comparing state payouts allows us to compare portfolio choices between the complete and incomplete market scenarios. In the incomplete market scenario, for example, the announcement “NOT X” eliminates state X, so we can only compare consumption in states Y and Z. This is why we compute the ratio of Y/(Y+Z), which compares the

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relative importance placed on consumption in state Y as opposed to in state Z. To allow for comparison between the incomplete and complete market scenarios, we must then use the same ratio.

Each histogram gives the distribution of relative holdings in two states. Individual holdings are sorted into bins of size 0.1. The bin labeled 0.N, for N from 1 to 9, includes all individuals with holdings between 0.(N-1) and 0.N. The left endpoint 0.(N-1) is included in the bin but the right endpoint 0.N is not. So, if an individual held X/(X+Y) = 0.08 then he would be included in the bin labeled 0.1. The last bin marks all individuals who held 0.9 or more, up to 1.0, in a particular state relative to another.

The black bars show the number of agents starting with each type of portfolio (type 1 or type 2) as well as the initial holding of each starting portfolio type. There is also a red bar that shows the holding of the market portfolio and the number of individual portfolios who fell within the same bin as the market portfolio. This is for easy comparison with the black bars.

For the incomplete market sessions, we will only show the relative holdings for states that were not eliminated by the announcement. For the complete market sessions, we will only show relative holdings that can be used to compare with the incomplete market sessions.

Experiment #1


Session #1

State X relative to State Y payouts

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State Y relative to State Z payouts

Session #2


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Session #1 (NOT X was the announcement)


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Session #2(NOT X was the announcement)


Experiment #2


Session #1


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Session #2


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Session #1(NOT Z was the announcement)


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Session #2(NOT X was the announcement)




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Results – Problems during Experiment and Observations Experiment #1

• Was not able to run the last 2 incomplete market scenarios because they were not configured correctly. The problem was that participants were not supposed to be able to trade Stock B, but the configuration allowed them to. We could not fix the problem during the experiment by reconfiguring markets because Flex-e-markets would give an error message.

Experiment #2

• There was confusion about how the bankruptcy rule would be applied in the incomplete market setup. One participant asked if he would be penalized for going bankrupt in a deleted state. The answer is NO, because the bankruptcy rule only applies to possible states. However, this detail was not stated in the instructions. Some traders might have thought that they should not go bankrupt even in the deleted state, which could have affected portfolio choices and prevented traders from obtaining the market portfolio. This issue was only clarified while the experiment was taking place.

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• In the instructions, a trading example for the incomplete market scenario gave incorrect calculations of the expected payout of Stock A after each of the announcements. The calculations assumed state Y has a 2/3 probability instead of a 1/3 probability. Consequently, the trading strategy advocated by the instructions of shorting A in the first period and buying it back after either announcement to take advantage of a drop in the expected payout of A was misleading. After the announcement “NOT X”, the expected payout of A drops; after the announcement “NOT Z,” the expected payout of A rises instead. Therefore, one can take advantage of an expected payout drop for only ONE of the announcements, but NOT BOTH as assumed by the instructions. This faulty calculation may have caused traders to misprice Stock A after each announcement, which would affect not only state prices, but also the range of portfolios each trader could obtain by buying and selling Stock A.

Analysis - prices Experiment #1


Session #1

Using average transaction prices, the price of (stock) A is $5.09, the price of B is $4.86, and the price of notes is $9.95. Backing out the state prices using previously mentioned methods, the price of (consumption in state) X is 0.359, the price of Y is 0.336, and the price of Z is 0.300. As expected, consumption in Z is cheapest. However, the price of X is higher than that of Y, contrary to expectation.

If we exclude the first 5 transactions when taking averages, we get the price of (consumption in state) X is 0.349, the price of Y is 0.324, and the price of Z is 0.322. Here, we see that Y should not be cheaper than X and that Y and Z are much closer in price than would be expected.

Session #2

Using average transaction prices, A is $5.05 and B is $4.91. Since no transactions were recorded for notes, we set its price as $10 because it pays this amount in all states. Backing out state prices, the price of X is 0.343, the price of Y is 0.333, and the price of Z is 0.325. As expected, Z is cheapest; however, X should be cheaper than Y.

If we exclude the first 5 transactions when taking averages, we get that the price of X is 0.343, the price of Y is 0.335, and the price of Z is 0.321. Similar conclusions result.

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Session #1 (“NOT X” was the announcement)

Backing out state prices, the price of Y is 0.391 and the price of Z is 0.604. Half the price of Z is 0.302, which is less than the price of Y. This agrees with theoretical predictions (see “Set Up” section).

If we exclude the first 5 transactions, the price of Y is 0.375 and the price of Z is 0.625. We had to use 10, the expected payout of Notes, as the price of Notes when calculating the state prices due to lack of data. Taking half the price of Z, we get 0.313, which is less than the price of Y. Again, the data agree with predictions.


Backing out state prices from average prices, the price of Y is 0.416 and the price of Z is 0.558. Half of the price of Z is 0.279, which is less than the price of Y. This agrees with predictions.

Excluding the first 5 transactions and assuming the price of Notes is 10 (due to lack of data), the price of Y is 0.403 and the price of Z is 0.597. Half of the price of Z is 0.299, which is less than the price of Y. Again, we find agreement with predictions.

Experiment #2


Session #1

Backing out state prices, we get: the price of X is 0.335, the price of Y is 0.339, and the price of Z is 0.327. As expected, Z is cheapest and the prices of X and Y are (very nearly) equal. The data agree with our theoretical predictions.

The same state prices are obtained when the first 5 transactions for each asset are ignored.

Session #2

Backing out state prices, we find: the price of X is 0.336, the price of Y is 0.339, and the price of Z is 0.319. As expected, the price of Z is lowest and the prices of X and Y are (almost) equal. The data agree with theoretical predictions.

If we exclude the first 5 transactions when calculating asset price averages, we find: the price of X is 0.339, the price of Y is 0.351, and the price of Z is 0.309. Again, we had to use 10 for the

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price of Notes dues to a lack of trading data. Interestingly, the prices of X and Y do not agree so closely.


Session #1 (“NOT Z” was the announcement)

Backing out state prices from average transaction prices, we get: the price of X is 0.511 and the price of Y is 0.464. Half the price of X is 0.256, which is not equal to the price of Y. The data do not agree with predictions, which would have a higher price of X than what we found.

If we exclude the first 5 transaction prices of each asset, the price of X is 0.512 and the price of Y is 0.463. Again, we obtain the same result.


Backing out state prices from average prices, we get: the price of Y is 0.448 and the price of Z is 0.511. We see here that the price of Y, 0.448, is higher than half the price of Z, 0.256. The data agree with predictions.

If we exclude the first 5 transaction prices of each asset, we get that the price of Y is 0.442 and the price of Z is 0.518. Again, the same conclusion follows.


Backing out state prices from average asset prices, we find: the price of X is 0.512 and the price of Y is 0.464. Half the price of X is 0.256, which is not equal to the price of Y. This does not agree with the theoretical predictions.

If we exclude the first 5 transactions for each asset, we find that the price of X is 0.513 and the price of Y is 0.468. Again, the same conclusion follows.


Backing out state prices, we find: the price of X is 0.526 and the price of Y is 0.449. Half of the price of X is 0.263, which is not equal to the price of Y. This data does not agree with the theoretical predictions.

If we exclude the first 5 transactions for each asset, we find that the price of X is 0.528 and the price of Y is 0.448. Again, we get a similar conclusion.

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Analysis – portfolio choices Experiment #1


Session #1

Looking at the graphs of relative holdings, we see that trading toward the market portfolio is occurring. Many of the traders are obtaining portfolios with relative holdings near the market portfolio ratios. On the other hand, there is also a significant portion (about 5 out of 14, or around 1/3) of participants who did not change their portfolios very much. Some had started out with type 1 portfolios, others with type 2 portfolios. This leads me to wonder whether participants were given a strong enough incentive to trade. However, this was the first trading session to occur in the experiment, so perhaps participants were unsure of what to do.

Session #2

Here, we see that our previous concerns relating to incentive to trade were maybe mistaken. In this trading session, there was a much stronger movement toward the market portfolio holdings, with participants obtaining portfolios that were closer to the market portfolio than in the first session. Fewer participants remained at or near their initial portfolio positions. This leads me to believe that participants were indeed sufficiently motivated by the instructions to trade. The results of the previous session, then, were probably due to participants being unsure of what to do or afraid of making drastic changes to their portfolios.


Session #1

In this session, we only compare the relative holdings of Y and Z, as the state X was eliminated by the announcement. Here, we see that 6 out of 14, or about 43%, of participants traded to positions at or near market portfolio relative holdings. 4 participants did not change significantly from their starting positions, and 2 traded father from the market than they started. Overall, then, we see significant trading toward the market portfolio.

If we compare the graph of Y/(Y+Z) of this incomplete session to the graph of Y/(Y+Z) of the complete sessions, we see that similar behavior is occurring in terms of trading to the market. This supports the idea of dynamic completeness, that is, that incomplete markets can replicate the trading behavior found in complete markets.

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Session #2

Again, here we are comparing relative holdings of Y and Z. In this session, we see that even more individuals had traded away from the market holding than in the 1st incomplete market session. This is unusual, as individuals should have been trading toward portfolios that paid more in state Z than in state Y. This is because Z was more likely to occur than Y after the announcement “NOT X.” More experiments will have to be run to determine if this behavior is a result of a flaw in the instructions, confusion about probabilities, or something else. We see from the graph that this behavior of trading away from the market was significant enough to shift the center of the distribution away from the market holding.

Experiment #2


Session #1

In this trading session, we see movement to the market portfolio relative holdings. For X/(X+Y), there were relatively few who remained at their initial holdings, but also few who traded very close to the market. The distribution is relatively spread out. On the other hand, the distribution is centered on the market holding, so, participants have traded, on average, to the market portfolio. For Y/(Y+Z), more individuals are trading very close to the market holding, but this may be because the initial relative holdings and the market relative holding are somewhat close together. Again, though, the distribution of holdings is centered on a mode at the market holding, so traders are trading to the market portfolio on average.

Session #2

Here, we see strong movement toward the market relative holdings for both X/(X+Y) and Y/(Y+Z). Both distributions are centered on the market holding and there are relatively few individuals who remained at their initial holdings.


Session #1(“NOT Z” was the announcement)

The distribution of holdings seems to center on the market holding position, however, rather few individuals actually traded to the market position. The spread of holdings is rather diffuse. Can we then say that individuals traded on average to the market? Yes, even though rather few individuals actually traded to or near the market.

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If we compare the graph of X/(X+Y) for this session to the complete market sessions, we see that the complete market sessions have stronger convergence to the market holding – more individuals in these sessions are picking portfolios with relative holdings very close or at the market holding. Even so, both the complete and incomplete sessions have trading behavior that is on average obtaining the market holding.


The graph of Y/(Y+Z) for this session shows very strong convergence to the market holding. 10 individuals, nearly half of all participants, traded to or very near to the market holding. In addition, very few participants remained at their initial holdings. The distribution has a clear mode at the market holding, with the vast majority of other holdings positioned around this center.

Comparing this graph of Y/(Y+Z) to the graphs of Y/(Y+Z) in the complete market sessions, we see very similar distributions. All three graphs have modes at the market holding, few individuals who kept their original relative holdings, and strong clustering around the center. This supports our idea that complete markets and incomplete markets can yield similar types of trading patterns.


The graph of X/(X+Y) relative holdings shows rather strong movement by the traders toward the market holding. Very few traders remained at initial holdings and the rest either moved toward or obtained the market holding. This result supports our predictions more strongly than the other incomplete market session in which state Z was eliminated by the announcement.

Comparison of this session’s results with the X/(X+Y) results of the complete market sessions tells us that very similar trading behavior occurred in the two types of markets, as the distributions of holdings are very similar in center and spread. This provides some evidence for the thesis that complete and incomplete markets can behave alike.


In this session we obtain very similar results to the one above. This gives further support for the predictions previously made.

Conclusions/Extensions

According to the data gathered from the two experiments, the theory dynamic completeness may be applicable to real life trading situations. Much support has been given for the predictions concerning portfolio

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choices, as we have seen in the analysis portion of the report. On the other hand, we also saw that prices showed more mixed results, with some trading sessions supporting but others not. More experiments will have to be done to gather data, as two experiments provide too small a base of data. In addition, more work needs to be done to ensure that directions are given clearly to all participants, to experiment with ways to motivate participants to trade, and to correct misleading sections of the instructions. Therefore, we cannot make any serious conclusions as to the general validity of dynamic completeness, at least until further exploration and testing has been done.

References

“Dynamically Complete Experimental Asset Markets.” Bossaerts, Peter, Debrah Meloso, and William Zane. California Institute of Technology: March 5, 2008. Unpublished.

Acknowledgements

There were many people who helped me find and carry out this research project. I owe thanks to Nilajan Roy and Marius Stan, would recommended me for the SURF program. Thanks of course to my mentor, Peter Bossaerts, for the opportunity to learn about this subject and for his mentorship and financial support. Thank you also to the SFP office for the opportunity and financial support to conduct research through SURF. I also owe thanks to Walter Yuan at SSEL for his technical and programming advice, Wendy Shin for her invaluable aid with writing the Matlab program we used to check participants for bankruptcy during experiments and for teaching me how to configure Flex-e-markets, Nilajan Roy for help with logistics and preparation, Barbara Estrada for helping me obtain the funding for each experiment, and Elizaveta Bradulina for her help with trying to make sense of all the theory. Finally, thanks to my family for helping me through it all.

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Appendix – 1st Set of Participant Instructions

(Note: these were the first set of instructions used in an experiment. They were later altered to deal with problems that were discovered during this first experiment.)

Web Address: filagora.caltech.edu/fm/

Username:

Password:

Plan

First, we’ll explain how everything works (30 min), hold several practice sessions where you can get acquainted with using the software and trading securities (30 min), and then start actual trading sessions (total: 50 min).

General Instructions

The experiment will consist of several sessions of trading. These sessions are independent of each other – your performance in any one will not affect any other. Each session, which will last a specified amount of time, will begin with one of two trading scenarios. These two scenarios will differ in three ways: the amount of total time you have to trade, what type of trading is allowed, and how much information you are given. Everyone will be trading at the same time and all information we give you will be available to everyone.

At the beginning of each scenario, you will start with some cash, a certain number of units of risky securities (Stock A and Stock B), and a certain number of units of a risk-free security (Notes). You will be able to trade (buy, sell, or short-sell) or keep (hold) any of these in a public market. At the end of the scenario, trading will stop and the securities will pay out dividends according to a randomly drawn event (state). These dividends will add to your cash balance. Your cash balance will determine your payment.

Good so far? Make sure you understand everything.

2 Scenarios

Here are the two types of scenarios:

1) You are given a starting portfolio (not the same for everyone) of Stock A, Stock B, Notes, and

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Cash. One of three states X, Y, or Z is drawn at random (with equal probability) and kept

hidden. Then, the market opens for 5 minutes of trading, after which the market closes.

2) You are given a starting portfolio (not the same for everyone) of Stock A, Stock B, Notes, and

Cash. One of three states X, Y, or Z (with equal probability) is drawn at random and kept

hidden. However, here, you can’t trade Stock B. In other words, if you start with 7 shares of

B, you can’t sell them off or buy any more. Trading will happen for 5 minutes. Then, we will

provide a piece of information about the state: either “The State is NOT X” or “The State is

NOT Z.” Trading will happen for another 5 minutes. Then, the market will open for a final 5

minutes, after which trading stops.

After each scenario is played out, the hidden state will not be revealed. Instead, at the end of the experiment, we will pick one of the sessions at random and reveal the state that was hidden for that particular scenario. This will decide how much you are paid for your participation in this experiment. We will mention this later on.

Dividends

After each scenario ends, markets will close. At this time, every share (unit) of a security will give a specific payout, depending on the type of security as well as the randomly drawn state. These states are equally likely to be drawn. Payouts according to the state drawn are shown below.

If the State Is…

X Y Z




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Scenario #2 Probabilities

In scenario 1, finding out the probability of each state is pretty straightforward (1/3).

For scenario 2, the announcement halfway through of either “NOT X” or “NOT Z” makes probabilities more difficult. Here’s why:

The picture above shows the relative probabilities of the two remaining states when the announcement eliminates one of them. Notice that when, for example, NOT Z (Z with a bar) is called, the probability that the revealed state will be X is twice the probability of Y being the revealed state.

Why? Suppose the state is Y. You would expect that the announcement could be NOT X or NOT Z with equal probability. Remember that the initial probability of Y is 1/3. Then, since NOT X and NOT Z are equally likely, this probability is split between the two announcements (divided by ½). But the probabilities of X and Z are both 1/3. So, given either announcement, the probability of Y is half the probability of the other remaining state.

Pay attention to this when you’re trading in scenario #2. When you’ll trading, you’ll want to pick portfolios that pay less in state Y and more in state X or state Z (depending on the which announcement is given).

Example: Suppose you have 1 unit of Stock A and are trading in the second half of scenario 2. The information is “NOT Z.” What do your payoff possibilities look like? What is the probability of state X? Of state Y?

0

0

0

0 0

Compare probabilities for NOT Z

Compare probabilities for NOT X

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Short-selling

Most people, when they trade, they will only think of buying and selling if they can afford to do so. However, in this experiment, you can take advantage of short-selling, that is, you can sell shares of a security (A, B, or Notes) that you don’t actually own. If you do so, you receive money from the sale, but instead of receiving a dividend, you will have to pay out the dividend at the end when the state is drawn. Note that this is exactly the opposite of buying and owning a security.

Why short at all? Shorting can be useful as a way of borrowing money to make additional purchases, taking advantage of overpricing, or changing the amount of risk of your portfolio.

Example: Suppose you are trading in scenario 1 with no securities and $10 cash. You sell 1 unit of Stock B @$5. What do your payoff possibilities look like?

Limits on Trades

The trading software we’re using (Flex-e-markets) will automatically prevent you from buying securities if you don’t have enough cash to commit to that purchase. So, don’t worry about accidentally overbuying.

However, it will not prevent you from making short-sales that may leave you bankrupt if particular states are drawn.

For example,

Suppose you have 3 Stock A, 2 Stock B, 1 note, and $0 cash.

If you sell 10 shares of A @ $3 each, then your net portfolio is – 7 Stock A, 2 Stock B, 1 note, and 10 x $3 = $30. Since you only had 3 of A, you end up selling 3 units of A and shorting the rest (7 units). If state X is called at the end, your net payout is: -7 x $10 + 2 x $0 + 1 x 10 + $30 = - $30. You’ve gone bankrupt!

Try at all costs to end up with a portfolio that will never go bankrupt in any state. Your experiment payment will depend on this.

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To help you avoid bankruptcy, we will provide you with a simple program that will check your portfolio for possible bankruptcy in each state. Just input the number of each security in your portfolio and then push “Compute!” The program will give you the payout of the portfolio in each state, i.e., the final cash holding in states X, Y, and Z. Remember to make sure that before time is called, your ending portfolio has no chance of going bankrupt.

How you will be paid

Your pay in this experiment will be based on 2 things:

1) $5 for participating

2) Half of your earnings (cash + dividends) from one of the sessions. Which one we pick will be decided randomly. For this chosen session, we will reveal the hidden state. So, most of your pay will depend on your performance in this session. You can earn as little as $5, but we expect you to earn $30 on average for 2 hours of participation, and you could earn as much as $50. If you go bankrupt in this randomly chosen session, then you will only receive the $5 participation fee.

All accounting (for earnings, dividends, and cash) is done in USD.

Trading Sessions

In the practice part, we will run 1 round of scenario #1 and 2 rounds of scenario #2.

In fleximarkets, the names of the markets where we will run these (in order) are:

Exp aug122011 – C – Practice

Exp aug122011 – I – Practice

Exp aug122011 – I2 – Practice

In the main part of this experiment, we will alternate between 1 round of scenario #1 and 2 rounds of scenario #2.

In fleximarkets, the names of the markets where we will run these (in order) are:

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Exp aug122011 – C

Exp aug122011 – I

Exp aug122011 – I2

Exp aug122011 – C2

Exp aug122011 – C3

Expaug122011 – C4

Your Starting Personal Portfolio

You will start both sessions with ____ shares of Stock A, ____ shares of Stock B, ____ shares of Notes, and ____ units of Cash (negative shares mean short-sales). Note that others may start with different initial portfolios.

Examples (for illustration only)

Tables 1 and 2 give two sample outcomes for scenario #1, where the initial portfolio is 10 A, -1 Note, and 10 Cash. These examples are not here to tell you how you should trade, but rather to give you a feel for how trading affects your ultimate payout in each state.

In the first table, the trader does nothing.

In the second, he sells 5 of A and buys 5 of B. Trades for each security are assumed to take place at expected payouts: $5/unit A, $5/unit B, and $10/unit Note. Notice that the expected payouts in the two tables are the same, but the state payouts are more evenly distributed in the second. This is because Stock B pays in state Y, whereas Stock A does not. Notes and Cash do not contribute because their effects cancel. Also, if prices are not as assumed (and this is perfectly possible), then payoffs will change accordingly.

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Tables 3 and 4 give two sample outcomes for scenario #2. (Red highlight means you can’t trade B)

In the first, the trader does nothing.

In the second, he sells 20 of A (resulting in short of 10) and buys 2 Notes. Then, if the announcement is NOT X, he buys back 10 of A. If the announcement is NOT Z, he buys back 10 of A. Why? In period 1, A has an expected payout of 5. In period 2, given NOT X, A has an expected payout of 0*2/3+5*1/3 = 5/3. In period 2, given NOT Z, A has an expected payout of 10*1/3+0*2/3 = 10/3. Either way, A is more valuable, given the information available, in period 1 than in period 2. Therefore, it makes sense to sell many of A at a high price in period 1 and then buy it back at a lower price in period 2.

Table 2

Period 1

Initial Holdings A 10 B 0

Notes -1 Cash 10

Dividends State X 10*10 - 1*10 + 10 = 100 State Y 10*0 - 1*10 + 10=0 State Z 10*5 - 1*10 + 10 = 50

Trade A -5 B +5

Notes 0 Cash +5*5-5*5 = 0

Final Holdings A 5 B 5

Notes -1 Cash 10

Final Dividends State X 5*10 + 5*0 - 1*10 + 10 = 50 State Y 5*0 + 5*5 - 1*10 + 10 = 25 State Z 5*5 + 5*10 - 1*10 + 10 = 75

Expected Payout (50 + 25 + 75)/3=50

Table 1

Period 1


Notes -1 Cash 10

Dividends State X 10*10 -1*10 +10 = 100 State Y 10*0 - 1*10 + 10 = 0 State Z 10*5 - 1*10 + 10 = 50

Trade A 0 B 0

Notes 0 Cash 0


Notes -1 Cash 10

Final Dividends State X 10*10 - 1*10 +10 = 100 State Y 10*0 - 1*10 + 10 = 0 State Z 10*5 - 1*10 + 10 = 50


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Table 3

Period 1 2 2

Initial Holdings NOT X NOT Z

A 10 10 10 B 0 0 0

Notes -1 -1 -1 Cash 10 10 10

Dividends State X 10*10 - 1*10 +10 = 100 NOT X 10*10 - 1*10 + 10 = 100 State Y 10*0 - 1*10 + 10 = 0 10*0 - 1*10 + 10 = 0 10*0 - 1*10 + 10 = 0 State Z 10*5 - 1*10 + 10 = 50 10*5 - 1*10 + 10 = 50 NOT Z

Trade A 0 0 0 B 0 0 0

Notes 0 0 0 Cash 0 0 0

Final Holdings

A 10 10 10 B 0 0 0

Notes -1 -1 -1 Cash 10 10 10

Final Dividends

State X 10*10 - 1*10 + 10 =100 NOT X 10*10 - 1*10 + 10 = 100 State Y 10*0 - 1*10 + 10 = 0 10*0 - 1*10 + 10 = 0 10*0 - 1*10 + 10 = 0 State Z 10*5 -1*10 + 10 = 50 10*5 - 1*10 + 10 = 50 NOT Z

Expected Payout 0*1/3 + 50*2/3 = 33.33 100*2/3 + 0*1/3 = 66.66

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Table 4

Period 1 2 2

Initial Holdings NOT X NOT Z A 10 -10 -10 B 0 0 0

Notes -1 1 1 Cash 10 90 90

Dividends State X 10*10 – 1*10 + 10 = 100 NOT X -10*10 + 1*10 + 90 = 0 State Y 10*0 - 1*10 +10 = 0 -10*0 + 1*10 + 90 = 100 -10*0 + 1*10 + 90 = 100 State Z 10*5 -1*10 + 10 = 50 -10*5 + 1*10 + 90 = 50 NOT Z

Trade A -20 +10 +10 B 0 0 0

Notes +2 0 0 Cash 20*5-2*10 = 80 -10*5/3 = -50/3 = -16.66 -10*10/3 = -100/3 = -33.33

Final Holdings A 10 – 20 = -10 -10 + 10 = 0 -10 + 10 = 0 B 0 0 0

Notes -1 + 2 = 1 1 + 0 = 1 1 + 0 = 1 Cash 10 + 80 = 90 90-16.66 = 73.33 90-33.33 = 56.66

Final Dividends State X -10*10 + 1*10 + 90 = 0 NOT X 56.66 + 1*10 = 66.66 State Y -10*0 + 1*10 + 90 = 100 73.33 + 1*10 = 83.33 56.66 + 1*10 = 66.66 State Z -10*5 + 1*10 + 90 = 50 73.33 + 1*10 = 83.33 NOT Z

Expected Payout 1/3*83.33 + 2/3*83.33 = 83.33 1/3*66.66 + 2/3*66.66 =

66.66

Remember, these are just examples. There may be strategies that give higher payouts in state Y or even higher expected payouts. Make sure you follow these examples completely. They are here so you can get a better feel for what prices you may see and how trading a security affects your final state payoffs and expected payout.

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Appendix – 2nd Set of Participant Instructions

(Note: a different set of instructions were used for the first experiment. The instructions were later changed to adapt to problems and confusion that resulted from that experiment. )

Web Address: filagora.caltech.edu/fm/

Username:

Password:

Outline

Here’s the basic outline for how this experiment will run:

1. Instructions and practice trading sessions – 1 hour

2. Trading sessions – up to 1 hour

General Instructions

The experiment will consist of independent sessions of stock trading. Each session will begin with a particular trading scenario. At the beginning, you will start with some cash and a portfolio of Stock A, Stock B, and Notes. Stock A and B give randomly determined payouts. Notes always pay the same amount. Cash is cash. After you get this starting portfolio, you will be able to buy and sell stocks and notes for cash. In this stock market, all buy and sell orders are public - when you make an offer, anyone can accept it, and the other way around. You can see all offers being made and accept any of them. At the end of trading, the securities will pay out dividends. These dividends add to your cash. Your ending cash balance is what matters.

There are two types of trading scenarios.

Scenario 1:

All stocks and notes can be bought and sold for cash

The trading period will be 4 minutes long

Scenario 2:

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You will not be able to buy or sell Stock B. If you have any, you can’t get rid of it. If you don’t have any, you can’t buy any. You can trade Stock A and Notes, though.

There will be two trading periods: one for 3 minutes, one for another 3 minutes. In between these periods, we will give you more refined information about the random payouts of the stocks.

We’ll go over all of this again during the practice.

Dividends

As previously mentioned, the Stocks A and B give random payouts. Dividends depend on the drawing of a random event. We will call these random states. There are 3 states: State X, State Y, and State Z. They are equally likely to be drawn (probability 1/3). Below is a table that lists how much each stock or note pays in each state. Accounting is all done in US Dollars ($ USD).

If the State Is…

X Y Z




Stock A pays the most in State X but nothing in State Y. Stock B pays the most in State Z but nothing in State X. Note pays 10 in all states.

How do we determine the state? At the beginning of a trading session, we will pick the random state using a random number generator. We will hide it until after all the trading sessions have been completed. Then, we will choose one of the trading sessions at random and reveal the state that was picked. This is for the purpose of determining which session we will base your pay on.

Scenarios and Probabilities

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In scenario 1, there are three equally likely states. Finding out the probability of each state is pretty straightforward (1/3).

In scenario 2, as described earlier, in between the two trading periods we will provide some information about the random payouts. This information will take the form of an announcement that tells you something about the state that was drawn. This announcement will be that the hidden state is “NOT X” (if it is Y or Z) or that it is “NOT Z” (if it is X or Y). After the announcement, the probabilities of the remaining states change. Here’s why.

The picture above shows the relative probabilities of the two remaining states after the announcement eliminates one. Notice that when at t = 1 (1st period), for example, NOT Z (Z with a bar) is called, the probability that the hidden state is X is twice the probability of Y being the hidden state.

Why? Suppose the state is Y. You would expect that the announcement could be NOT X or NOT Z with equal probability. Remember that the starting probability of Y was 1/3. Then, since NOT X and NOT Z are equally likely, this probability is split between the two announcements (divided by ½). But the starting probabilities of X and Z are both 1/3. So, given either announcement, the probability of Y is half the probability of the other remaining state.

Pay attention to this when you’re trading in scenario #2. When you’re trading, you may want to pick portfolios that pay less in state Y and more in state X or state Z (depending on the which announcement is given). This is because, of course, the payout in state Y is the one you’ll less likely get.

0

0

0

0 0

Compare probabilities for X and Y given announcement NOT Z

Compare probabilities for Y and Z given the announcement NOT X

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Short-selling

Most people think of trading as buying and selling using the cash or stocks you have. However, in this experiment, you can take advantage of short-selling. Short-selling is the idea that you can sell shares of a stock (A, B, or Notes) that you don’t actually own. If you do so, you get money from selling it, but instead of getting the dividend at the end, you will have to pay it out. So, this is exactly the opposite of buying and owning a stock.

Why would you ever short at all? Shorting can be useful as a way of borrowing money. If you want to buy more shares of A but don’t have the money to, you can short shares of B or of the notes to get the money. The only two ways to fund purchases for your portfolio are to spend the cash you start out with and to use the money you make from shorting stocks.

It’s also useful to short stocks to take advantage of expected price drops. For instance, if before the announcement A is priced at $5 and you expect after the announcement that the price will drop to $3, it makes sense to short A at the beginning and buy it back after the price drop, making $2 profit per share of shorted stock.

Limits on Trades

It’s natural to wonder what trades you’re allowed to do, or if there are no restrictions. Can I buy $10 worth of A? Can I short 10 shares of Notes? Well, for buying, there are limits. If you can’t commit the cash to the purchase, the trading software will not let you buy a stock. By commit, I mean you’re not allowed to commit this cash to purchasing a share of A or to buying a share of B. If you offer to buy A, you can’t use that money to offer to buy B unless you cancel the first order. Just keep that in mind. The trading software automatically checks to make sure you have the cash.

There is another danger, however: Short-selling. There’s nothing wrong with shorting to borrow money or make a quick profit. However, take it too far, and you will run into trouble. This is because excessive shorting gives negative payoffs in some states, i.e., bankruptcy. We will not allow this. We will charge a penalty if your portfolio generates bankruptcy in some states.

For example,

Suppose you want to short a share of A. You sell it at $5. At the end of the trading period, you will have to pay A’s dividend instead of receiving it. In State X, that’s $10. State Y, $0. State Z, $5. If State X is picked, you got paid $5 at first but now have to pay out $10, for a loss of $5. In the other states, you’re positive. Think

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about this. If you short too many shares of A at this price, you will run up a lot of losses in state X. Then, if X is called, the losses will overtake your cash balance and you may end up bankrupting yourself.

Your pay for this experiment depends on your ending cash balance and whether you could ever go bankrupt. If you go bankrupt in any state, even if it wasn’t the hidden state, you will be penalized.

To help you avoid bankruptcy, we will provide you with a simple program that will check your portfolio for possible bankruptcy in each state. Just input the number of each security in your portfolio. The program will give you the payout of the portfolio in each state, i.e., the final cash holding in states X, Y, and Z.

How you will be paid

Your pay in this experiment will be based on 2 things:

3) $5 for participating

PLUS

4) Half of your earnings (cash + dividends) from one of the sessions. That is, for every $1 you earn in the electronic trading environment, you make $0.50 in real life. Which trading session we pick will be decided randomly. For this chosen session, we will reveal the hidden state. So, most of your pay will come from your performance in this randomly chosen session.

MINUS

5) $5 bankruptcy penalty, if you would be bankrupt in any state of the chosen session, regardless of whether this state was actually chosen.

You can earn as little as $5, but we expect you to earn $30 on average for 2 hours of participation, and you could earn as much as $50.

Your Starting Personal Portfolio

You will start both sessions with ____ shares of Stock A, ____ shares of Stock B, ____ shares of Notes, and ____ units of Cash (negative shares mean short-sales). Note that others may start with different initial portfolios.

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Examples (for illustration only)

Tables 1 and 2 give two sample outcomes for scenario #1, where the initial portfolio is 10 A, -1 Note, and 10 Cash. These examples are not here to tell you how you should trade, but rather to give you a feel for how trading affects your ultimate payout in each state.

In the first table, the trader does nothing.

In the second, he sells 5 of A and buys 5 of B. Trades for each security are assumed to take place at expected (average) payouts: $5/unit A, $5/unit B, and $10/unit Note. Notice that the expected payouts in the two tables are the same, but the state payouts are more evenly distributed in the second. This is because Stock B pays in state Y, whereas Stock A does not. Notes and Cash do not contribute because their effects cancel. Note that if prices are not as assumed (and this is certainly possible), then payoffs will change accordingly.

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Tables 3 and 4 give two sample outcomes for scenario #2. (Red highlight means you can’t trade B)

In the first, the trader does nothing.

In the second, he sells 20 of A (resulting in short of 10) and buys 2 Notes. Then, if the announcement is NOT X, he buys back 10 of A. If the announcement is NOT Z, he buys back 10 of A. Why? In period 1, A has an expected payout of 5. In period 2, given NOT X, A has an expected payout of 0*2/3+5*1/3 = 5/3. In period 2,

Table 1

Period 1


Notes -1 Cash 10

Dividends State X 10*10 -1*10 +10 = 100 State Y 10*0 - 1*10 + 10 = 0 State Z 10*5 - 1*10 + 10 = 50

Trade A 0 B 0

Notes 0 Cash 0


Notes -1 Cash 10

Final Dividends State X 10*10 - 1*10 +10 = 100 State Y 10*0 - 1*10 + 10 = 0 State Z 10*5 - 1*10 + 10 = 50


Table 2

Period 1


Notes -1 Cash 10

Dividends State X 10*10 - 1*10 + 10 = 100 State Y 10*0 - 1*10 + 10=0 State Z 10*5 - 1*10 + 10 = 50

Trade A -5 B +5

Notes 0 Cash +5*5-5*5 = 0


Notes -1 Cash 10

Final Dividends

State X 5*10 + 5*0 - 1*10 + 10 =

50

State Y 5*0 + 5*5 - 1*10 + 10 =

25

State Z 5*5 + 5*10 - 1*10 + 10 =

75 Expected Payout

(50 + 25 + 75)/3=50

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given NOT Z, A has an expected payout of 10*1/3+0*2/3 = 10/3. Either way, A is more valuable, given the information available, in period 1 than in period 2. Therefore, it makes sense to sell many of A at a high price in period 1 and then buy it back at a lower price in period 2.

Table 4

Period 1 2 2

Initial Holdings NOT X NOT Z A 10 -10 -10

Table 3

Period 1 2 2

Initial Holdings NOT X NOT Z

A 10 10 10 B 0 0 0

Notes -1 -1 -1 Cash 10 10 10

Dividends State X 10*10 - 1*10 +10 = 100 NOT X 10*10 - 1*10 + 10 = 100 State Y 10*0 - 1*10 + 10 = 0 10*0 - 1*10 + 10 = 0 10*0 - 1*10 + 10 = 0 State Z 10*5 - 1*10 + 10 = 50 10*5 - 1*10 + 10 = 50 NOT Z

Trade A 0 0 0 B 0 0 0

Notes 0 0 0 Cash 0 0 0

Final Holdings

A 10 10 10 B 0 0 0

Notes -1 -1 -1 Cash 10 10 10

Final Dividends

State X 10*10 - 1*10 + 10 =100 NOT X 10*10 - 1*10 + 10 = 100 State Y 10*0 - 1*10 + 10 = 0 10*0 - 1*10 + 10 = 0 10*0 - 1*10 + 10 = 0 State Z 10*5 -1*10 + 10 = 50 10*5 - 1*10 + 10 = 50 NOT Z

Expected Payout 0*1/3 + 50*2/3 = 33.33 100*2/3 + 0*1/3 = 66.66

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B 0 0 0 Notes -1 1 1

Cash 10 90 90 Dividends

State X 10*10 – 1*10 + 10 = 100 NOT X -10*10 + 1*10 + 90 = 0 State Y 10*0 - 1*10 +10 = 0 -10*0 + 1*10 + 90 = 100 -10*0 + 1*10 + 90 = 100 State Z 10*5 -1*10 + 10 = 50 -10*5 + 1*10 + 90 = 50 NOT Z

Trade A -20 +10 +10 B 0 0 0

Notes +2 0 0 Cash 20*5-2*10 = 80 -10*5/3 = -50/3 = -16.66 -10*10/3 = -100/3 = -33.33

Final Holdings A 10 – 20 = -10 -10 + 10 = 0 -10 + 10 = 0 B 0 0 0

Notes -1 + 2 = 1 1 + 0 = 1 1 + 0 = 1 Cash 10 + 80 = 90 90-16.66 = 73.33 90-33.33 = 56.66

Final Dividends State X -10*10 + 1*10 + 90 = 0 NOT X 56.66 + 1*10 = 66.66 State Y -10*0 + 1*10 + 90 = 100 73.33 + 1*10 = 83.33 56.66 + 1*10 = 66.66 State Z -10*5 + 1*10 + 90 = 50 73.33 + 1*10 = 83.33 NOT Z

Expected Payout 1/3*83.33 + 2/3*83.33 = 83.33 1/3*66.66 + 2/3*66.66 =

66.66

Remember, these are just examples. There may be strategies that give even higher expected payouts. Make sure you follow these examples completely. They are here so you can get a better feel for what prices you may see and how trading a security affects your final state payoffs and expected payout.

Trading Sessions

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In the practice part, we will run 1 round of scenario #1 and 2 rounds of scenario #2.

In flex-e-markets, the names of the markets where we will run these (in order) are:

Exp aug172011 – 1-1– Practice (scenario 1)

Exp aug172011 – 2-1– Practice (scenario 2)

Exp aug172011 – 2-2 – Practice (scenario 2)

In the main part of this experiment, we will alternate between 1 round of scenario #1 and 2 rounds of scenario #2.

In flex-e-markets, the names of the markets where we will run these (in order) are:

Exp aug172011 – 1-1

Exp aug172011 – 2-1

Exp aug172011 – 2-2

Exp aug172011 – 1-2

Exp aug172011 – 2-3

Exp aug172011 – 2-4

surf 2011 final report - eric zhang

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