Some Aspects Of A Theory Of Finance

I consider this post to be a complement to some recent comments by Lars Syll on this year's Nobel prize. I here outline an (unoriginal) theory to contrast with the Efficient Market Hypothesis.

Keynes described the prices of bonds, shares, and other financial instruments as, at any time, reflecting a balance of Bulls and Bears. As G. L. S. Shackle points out, if this is an equilibrium, it is an "inherently restless" equilibrium. If a price continues unchanged, Bulls, who expect the price to rise, will eventually be disappointed. Likewise, a symmetrical condition is true for Bears. Furthermore, news from outside the stock market will be changing market expectations, causing some Bulls to become Bears and vice versa.

For this account to make sense, it seems to me, the market must be populated with decision makers who have vastly different ontological and epistemological beliefs. It is not a case of all agents having one model, with different parameter estimates being updated in light of experience. Shackle, in modeling such decision makers, introduces the concept of focal points. If profits exceed the upper focus point at some time, the decision maker will realize that more profit can be gained than is accounted for in his theory. And so he will adopt some other theory, in some sense. Likewise, losses or too little profit will, outside of some range, lead to another change of mind, with consequent changes in plans and actions.

Stock prices are numbers. One might initially think they could be described by probability distributions. One could think of the price of a stock at a given time as a random variable. And, in this special case, perhaps Shackle's model of decision-making reduces to the theory of sequential statistical hypothesis testing.

A random variable is a function from a sample space to the real numbers, where a sample space is a set of all possible outcomes of a random experiment. An event is a (measurable) subset of the sample space. So a stock price is not an event in the sample space. For example, the recent brouhaha in the United States over the debt limit is presumably an aspect of an event that has had some impact on stock prices. Can one assume that all possible events are known beforehand? That all agents who might bet on stock prices know all events? If not, how does it make sense to model stock prices as coming from a probability distribution? Paul Davidson suggests extending the concept of non-ergodicity to cover these cases where the complete sample space is not known, and the mere possibility of some future events are capable of surprising someone:

"In expected utility theory, according to Sugden..., 'A prospect is defined as a list of consequences with an associated list of probabilities, one for each consequence, such that these probabilities sum to unity. Consequences are to be understood to be mutually exclusive possibilities: thus a prospect comprises an exhaustive list of the possible consequences of a particular course of action... An individual's preferences are defined over the set of all conceivable prospects.' Using these definitions, an environment of true uncertainty (that is, one which is nonergodic) occurs whenever an individual cannot specify and/or order a complete set of prospects regarding the future, because the decision maker either cannot conceive of a complete list of consequences that will occur in the future; or cannot assign probabilities to all consequences because 'the evidence is insufficient to establish a probability' so that possible consequences 'are not even orderable' (Hicks)... Hicks associates a violation of the ordering axiom of expected utility theory with 'Keynesian liquidity' ..., since, for Hicks..., 'liquidity is freedom' to delay action that commits claims on real resources whenever the decision maker is ignorant regarding future consequences." -- Paul Davidson (1991, p. 134)

In the above quote, Davidson is contrasting what he calls the True Uncertainty Environment with the Subjective Probability Environment. Davidson argues that my favorite definition of ergodicity, as applying to stochastic processes in which time averages converges to ensemble averages, characterizes the Objective Probability Environment.

One might argue that a more mainstream approach to finance is more empirically applicable than the Post Keynesian approach outlined above. It seems to me that if one wants to argue this, one needs to establish some reason why a diversity of agent views could not persist. I am unaware of any such argument, and I think the literature on noise trading should lean economists to think otherwise.

Traders in stock and bonds constitute one audience for theories of finance. I suppose they prefer mathematical theories that they can use to price financial instruments. I am not sure that the above theory can be cast into mathematics further than Davidson and Shackle have already accomplished. I will note that when Dutch Shell found themselves wrong-footed in the 1970s by the formation of OPEC and the oil crisis, they set up a group to do scenario planning. These are the sort of events whose existence in the sample space might not have been long foreseen. As I understand it - I forget why, some members of that group explicitly took inspiration from Shackle's work.

References

• Davidson, Paul (1991). Is Probability Theory Relevant for Uncertainty? A Post Keynesian Perspective, Journal of Economic Perspectives, V. 5, No. 1: pp. 129-143.
• DeLong, J. Bradford, Andrei Shleifer, Lawrence H. Summers, and Robert Waldmann (1990). Noise Trader Risk in Financial Markets, Journal of Political Economy, V. 98, Iss. 4: pp. 703-738.
• Hogg, Robert V. and Allen T. Craig (1978). Introduction to Mathematical Statistics, Fourth edition, Macmillan.
• Shackle, G. L. S. (1940). The Nature of the Inducement to Invest, Review of Economic Studies, V. 8, N. 1: pp. 44-48.
• Shackle, G. L. S. (1988). Treatise, Theory, and Time, in Business, Time and Thought: Selected Papers of G. L. S. Shackle, New York University Press.
• Shackle, G. L. S. (1988). On the Nature of Profit, in Business, Time and Thought: Selected Papers of G. L. S. Shackle, New York University Press.