Kobayashi-sensei's Economic Research Picks

Part Eight: Does the Introduction of Financial Derivatives Invariably Make a Market Incomplete? - "Gödel's Incompleteness Theorem" in Economics

Faculty Fellow

Econo-kun is in his second year of the master's program at a private university, studying hard to become an economist.

Kobayashi Keiichiro's photoKOBAYASHI Keiichiro: Critics have often noted that the development of financial derivatives, such as options, fosters speculation and may destabilize financial markets. Realistically, markets may possibly become destabilized because of the presence of derivatives. However, it is common sense to economists that destabilization could occur because of the presence of distortions, such as an information asymmetry, in the markets. At least in an undistorted market, as assumed in textbook economic theory, an increase in the number of types of derivatives means an increase in the number of risk-hedging methods. Thus, economists ordinarily expect that the presence of derivatives would increase the stability of markets and raise the level of welfare for the society as a whole.

Econo-kun's photoEcono-kun: But in reality this is actually different?

Kobayashi Keiichiro's photoKOBAYASHI Keiichiro: Bowman and Faust (1997) present a remarkable theoretical model in which an increase in the number of types of derivatives turns a complete market incomplete and lowers the level of social welfare. (A complete market refers to a market in which each existing risk has an insurance product, i.e., financial instrument, to hedge against it. When a market is imperfect, certain risks remain that cannot be fully hedged even by mobilizing all the financial instruments available.) The paper, which presents examples of their theoretical idea was published in the Journal of Political Economy (JPE).

Bowman, D. and J. Faust. (1997) "Options, Sunspots, and the Creation of Uncertainty." Journal of Political Economy 105(5): 957-75.

This is a mysterious paper that makes a strong impression. It has been one of the most intriguing papers for me in the past 11 years. Simply put, Bowman and Faust (1997) state that the basic function of financial instruments is an insurance function, that is, dispersal of various risks present in the economy, such as a sudden urge to buy something and a sudden failure in production. However, adding an option, a type of derivative, creates a risk. Uncertainty about whether or not the option will be eventually exercised resultantly poses a new risk to economic agents, in addition to the risks already present.

Econo-kun's photoEcono-kun: So if I understand this correctly, an option that has been introduced as a means to disperse risks brings a new risk (uncertainty of whether or not the option will be exercised) into the economy and as a result the economy, which has been essentially complete without the option, becomes incomplete?

Kobayashi Keiichiro's photoKOBAYASHI Keiichiro: That's right. Let me explain in more detail. In this Bowman-Faust model, two consumers trade in a financial market. There are three periods, described as time 0, time 1, and time 2. Consumption takes place at time 2. However, the economy assumed in this model has a "preference shock" and thus only one of the consumers consumes. (That is, when time 2 arrives, a state in which consumer 1 consumes and consumer 2 does not consume occurs at a certain level of probability p, and the opposite state in which consumer 2 consumes and consumer 1 does not consume occurs at the probability of p-1.) The preference shock hits the economy at time 1 and this reveals which consumer consumes at time 2. In the economy assumed here, there is no distinction between consumer goods and capital goods. With respect to real production activities, production technology capable of generating A units of output in time 2 from each unit of input made in time 0 is assumed to be the only available technology in the economy. Under this environment, consumer 1 and consumer 2 trade financial products (stock, bond, etc.) at time 0 and time 1.

The risk in this economy is whether consumer 1 or consumer 2 consumes, and there are two possible resulting "states" of the economy; one in which consumer 1 consumes and the other in which consumer 2 consumes. Thus, based on the standard theory of financial economics, two financial instruments must exist in order for this economy to be complete. (In financial economics, a market is known to be complete when the number of financial instruments is equal to or greater than the number of possible states of economy.)

Bowman and Faust first considered an economy in which stocks are the only assets. Of course, such an economy is incomplete. So they showed that adding an option can establish an equilibrium characterizing a complete market, as anticipated by the standard theory of financial economics. What is intriguing, however, is that they also pointed to possible emergence of an equilibrium characterizing an incomplete market. That is, Bowman and Faust demonstrated that the economy, which initially had two intrinsic states, may come to have three states when the option is added:

(1) Consumer 1 consumes and the option is exercised
(2) Consumer 1 consumes and the option is not exercised
(3) Consumer 2 consumes and the option is not exercised

Secondly, they considered an economy in which two types of assets exist - stocks and bonds. In this case, the market is complete. According to the ordinary theory of finance, adding an option to such a market will not affect the completeness of the market because options are considered as a "redundant" financial instrument. However, Bowman and Faust showed that adding the option can lead to four states of economy:

(1) Consumer 1 consumes and the option is exercised
(2) Consumer 1 consumes and the option is not exercised
(3) Consumer 2 consumes and the option is exercised
(4) Consumer 2 consumes and the option is not exercised

Against these four possible states, there are only three types of financial instruments (stocks, bonds, and options). Thus, the market is incomplete.

Econo-kun's photoEcono-kun: So an initially complete market becomes incomplete when an option is added. Why does this happen?

Kobayashi Keiichiro's photoKOBAYASHI Keiichiro: This is just my intuitive guess and something of a wild idea, but I suppose it may be related, probably in a fundamental aspect, to Gödel's incompleteness theorem, which is well-known in the world of mathematics.

It seems to me that the key derived from the findings in Bowman-Faust lies in the structure in which the option exercise decision is contingent on the stock price (a call option is to be exercised when the stock price falls below the strike price).

If we refer to the set of financial products in the second economy considered above as F:
F={stock, bond, option}
The proposition that the option exercise decision is contingent on the stock price can be described as a structure in which the payoff of F, (financial instruments as a whole), is determined by the price (vector) of F, that is, the payoff of F is determined by F's own price.

Gödel's incompleteness theorem shows that a proposition G cannot be proved as true even when it is a true proposition if it refers to itself. (See Raymond M. Smullyan, Gödel's Incompleteness Theorems, etc. for the substance and logic of Gödel's theorem.)

Econo-kun's photoEcono-kun: That is, in the Bowman-Faust economy, the payoff of the F set of financial instruments is determined by F's own price and this may be why the state in which F is not complete can occur despite F being a consistent set of financial instruments. Do I have that right?

Kobayashi Keiichiro's photoKOBAYASHI Keiichiro: By replacing the notion of "provable" in Gödel's incompleteness theorem with the notion of "completeness" in economics, it may be possible to show that the findings in Bowman-Faust are the results derived by applying Gödel's theorem to economics.

Incidentally, the introduction of many options has also been proven to, by necessity, make the market complete (see Kajii, A., 1997, "On the Role of Options in Sunspot Equilibria," Econometrica). This may seem to contradict the findings in Bowman-Faust but that is not the case (or at least not in my opinion). The key difference between the two papers is closely tied to our earlier discussion (on the link between the findings in Bowman-Faust and Gödel's theorem).

Kajii's model consists of period 0 and period 1, in which both payoff determination and option exercise occur in period 1. Because of the way this model is constructed, it is impossible for the stock to be traded after the option is exercised. Therefore, the notion of a "stock price after the option is exercised" does not exist. In the Bowman-Faust model, the option exercise decision is made in time 1 and the stock payoff is determined in time 2. This means the stock is traded in the market only after the option is exercised (or is confirmed as not exercised). Notions such as a "stock price after the option is exercised" and a "stock price after the option is confirmed as not exercised" therefore naturally exist. This difference is believed to have led to the incompleteness result in the Bowman-Faust model. My intuitive guess is that Kajii's findings hold only for special cases in which the payoff of a certain financial instrument is determined without trading in the market after the option is exercised, whereas Bowman-Faust captures the more general nature of financial markets.

Finally, let me offer one prediction or conjecture. In Bowman-Faust, the possibility of an otherwise complete market becoming incomplete with the addition of an option is presented only as a unique case. However, if the structure dealt with in the paper is closely related with Gödel's incompleteness theorem, it should be possible to prove a theorem (economic version of Gödel's theorem) more generally applicable in economics (or in financial economic models). That is, to show that an innately complete market invariably becomes incomplete with the introduction of a certain derivative, as a theorem generally applicable to any financial market model. This is a much more powerful result than what was shown in Bowman-Faust and the theorem, if proven, would overturn the very foundation of the idea that a continuous increase in the number of types of financial products will eventually make any market complete; a belief widely held in economics and by market participants. Gödel's incompleteness theorem overturned mathematicians' belief that the continuous development of mathematics will eventually enable determination of the truth or falsehood of any proposition in the world and I wonder if something similar might happen in the world of financial economics.

Econo-kun's photoEcono-kun: This could turn into one of the great discoveries of our time. As a student of economics, this is going to keep me up at night. Thank you very much.

June 13, 2007

June 13, 2007

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