### Discussion Paper

No. 2015-41 |
June 04, 2015

Radical Uncertainty: Sources, Manifestations and Implications

(Published in Special Issue
Radical Uncertainty and Its Implications for Economics)

## Abstract

This paper argues that radical uncertainty is the outcome of standard market activity. The theoretical findings are corroborated with empirical analyses. The model example is applied to asset pricing and radical uncertainty is found a solution to various asset pricing “puzzles”. In conclusion, radical uncertainty should form the basis of economic analysis.

## Comments and Questions

Danke, Christian, for a stimulating but difficult paper. The following clarifications would help. What is the connection between the general price as the average of all prices and the properties of the additive stochastic term introduced in the last paragraph on page 5? How do prices get more volatile ...[more]

... (a stochastic element) as N tends to infinity (the arithmetical mean element)? Finally, on page 14, what is µ? How does it differ from p?

Thanks a lot for your insightful comments.

Your first question addresses the relation between the prices and the additive stochastic term (p.5). This relationship essentially establishes that the individual (subjective) prices can either be averaged to obtain an "objective" price or not. The former is the standard assumption. The ...[more]

... latter applies to uncertainty and implies an increasing variance which finds empirical support. This increase in variance is NOT owed to time non-stationarity but to non-stationarity caused by subjectivity (which in turn constitutes prices rendering the market price uncertain).

The second question points to the same issue. The \mu represents the attempt to average out individual deviations from a supposed "true" or "objective" price. Under risk, this averaging would lead to the objective price for increasing N, under uncertainty it doesn't (see eq. 6 through 8).

Thanks again for your remarks and questions.

Kind regards, Christian Mueller

The paper is very interesting. I need more time to fully understand the proposals, but I hope these minor comments are helpful.

Page 2, para 3,TYPO: ‘and the former risk…’

Bottom of page 2, references: can I Savage (1954) for an early paper on subjective expected utility and unique ...[more]

... subjective probability measures. Also, try Gilboa and Schmeidler (1989) and Bewley (1986) for a discussion of multiple priors.

Page 3. The assumption that price processes are objective is widely contested within the field of behavioural finance, eg: the wide literature on noise trading and the behavioural asset pricing model (BAPM). I can provide more references for these on request.

Bewley, T. F. (1986). Knightian decision theory: Part I. Cowles Foundation Discussion Paper 807.

Gilboa, I., and D. Schmeidler (1989). Maxmin expected utility with non-unique prior. Journal of Mathematical Economics, 18, 141–153.

Savage, L. J. (1954). The foundations of statistics. John Wiley and Sons.

Dear Neill

Thank you very much for your references, hints and pointing out the typo.

There is one important and only one aspect that does make or does not make a difference between my approach and those you mention. This aspect can be cast as a question: Do those alternative ...[more]

... approaches establish the actual price on the basis of purely subjective evaluations or do prices derive from subjective deviations from an otherwise objectively given benchmark? I firmly hold that such a benchmark does not exist and is not needed theoretically. Instead, subjectivity determines prices and thus is one root of (fundamental) uncertainty.

Thanks again for your comments. They as well as Romar's above will certainly help improve the paper.

Kind regards, Christian Mueller

This submission has two aims. First to discuss and give examples of what the author means by uncertainty, and then to attempt to show that financial markets are "radically uncertain".

There are several fundamental problems with this submission.

First the statistics about the number of time "uncertain" appear ...[more]

... in economics preprint is meaningless: the number of papers submitted has certainly increased along the years. Only the proportion of papers including the word "uncertain" may be relevant and then the difference may not be significant. This is, however, the least of the problems.

Next, I do not agree with the disinction made between risk and uncertainty. In some circles, this word probably has a very strict definition, but this does not matter to others. For example, the meaning of "model" has a very different meaning in physics, computer science and economics: a much more feeble one in the two latter cases, yet everybody have the right to use it provided that its meaning is clear to the circle of readers. On the other hand, risk is not well defined, even in Finance. To make such strong dichotomy bewtween risk and uncertainty is wrong. Where does risk come? From uncertainty.

Now, this is the crux of the matter. There are several kinds of uncertainties. First, model uncertainty. Within the stochastic price evolution model (and framework), model uncertainty is null. Then the price is uncertain (i.e., not certain) at a given time because of the stochastic nature of the model. This has nothing to do with risk, but with stochastic processes. True, and this is where economists and all scientists alike usually are way too optimistic, model uncertainty is certainly much more important and much worse, because people tend to forget this part of uncertainty and trust too much frameworks.

Finally, the example of voting is flawed.

First of all, the notations are rather random (or totally uncertain): is $\epsilon_i(t)$ the same thing as $\epsilon(t)_i$?

Then it is based on two very strong assumptions: that each price quote is equal to a vote and that each trade is the vote of a different agent. The problem here is synchronicity vs asynchronicity. The author assumes that Walras theories are also valid in an asynchronous setting, which is wrong. I can advise http://arxiv.org/abs/1506.03758 for an nice attempt at describing the two limits of asynchronicity (limit order books) and Walras auctions (synchronicity).

Then the author should look into market subordination, an idea that dates from the 1970s. For more recent related results, I can advise e.g. figure 3 of http://arxiv.org/pdf/cond-mat/9912051.pdf and figure 4 of http://www.knf.pw.edu.pl/projekty/wirtschaftsphysik/sources/doc20.pdf , and the more recent http://arxiv.org/pdf/1009.2329.pdf . The thing is that the fits of the author have the wrong ansatz, as indeed, non-integer power relationship do exist.

Then the calculus is either wrongly explained (and certainly totally obscure), or simply wrong. What he has in mind is in fact closer to extremal statistics, i.e., the pdf of the min or max of a N random variables.

I would first like to thank the referee for his / her careful reading of the paper and then take this opportunity to respond to the voiced concerns.

Summary

The referee first challenges the terminology of the paper.

A generally accepted terminology for uncertainty and risk is ...[more]

... still emerging. This paper can at best contribute to this ongoing process.

Second, the referee thinks that the model's calculus is either wrongly explained or simply wrong.

Faced with these two options, it must be maintained that the first holds as there is no proof (yet) of the latter.

Several minor issues relating to the notation, wording and the motivation will be dealt with in a revised version.

The detailed response is provided in the below document.