I struggle to understand the difference between Knightian uncertainty versus Black Swan event. If I understand at least the basic premises, both views say that uncertainty is different from risk, and that uncertainty becomes risk only with hindsight, i.e. after the event has happened.

Is this correct?

This question borders on 'philosophy' perhaps, so I'll fully understand if it is downgraded or moved to another forum.

  • $\begingroup$ What source introduced Knightian uncertainty $\endgroup$
    – develarist
    Commented Jan 27, 2021 at 13:49
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    $\begingroup$ University of Chicago economist Frank Knight (1885-1972), introduced a distinction between "risk" and "uncertainty" in 1921 in his book Risk, Uncertainty and Profit. $\endgroup$
    – nbbo2
    Commented Jan 27, 2021 at 13:51
  • $\begingroup$ how much of Knightian uncertainty falls under the label of "risk" as opposed to "uncertainty"? And how much of the meaning of "black swan" would fall under the category of "risk" and "uncertainty" or neither? i.e. black swan is 20% risk, 70% uncertainty and 10% itself distinguishable from both risk and uncertainty $\endgroup$
    – develarist
    Commented Jan 27, 2021 at 18:14

2 Answers 2


I'm sure this falls short of proper philosophical precision, but here goes. Hark back to a slightly modified rehash of Donald Rumsfeld's infamous:

Reports that say that something hasn't happened are always interesting to me, because as we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns—the ones we don't know we don't know. And if one looks throughout the history of our country and other free countries, it is the latter category that tends to be the difficult ones.

Seen thus, one can think of traditional risk as an "unknown known". That is, I can look at the price of an OTM put option, CDS etc. and work out the probability of a given crash or event of default (for any given recovery rate). I don't know whether the event in question will happen or not; but I can clearly see if the market is pricing this at 1% vs 2% or 5%.

Knightian Uncertainty is a "known unknown". I know that crashes and defaults can happen; but I can't be sure what the actual probability is (and thus whether the market's estimate of this is fairly priced or not).

While the Black Swan is that cliche of the "unknown unknown". It transcends Knightian Uncertainty, usually because the event in question is thought to be (almost) impossible beforehand so isn't priced at all. As opposed to my Knightian ignorance how it should be priced. Suppose for instance that the market does not crash at all; but the NYSE just shuts up shop for a 6 month period like it did on the outbreak of WWI in 1914. What then?

So to recap, I can manage my price risk with those puts. I have Knightian uncertainty over the actual value of this insurance. The Black Swan risk is that the nature of the event risk is so unique and/or mystifying that I cannot be sure that my insurance actually protects me in the first place.

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    $\begingroup$ Can I rephrase this, and in doing so most likely do injustice to your good explanation, as follows: 1. risk = I have a model, 2. Knightian uncertainty = I have model risk, 3. Black Swan = Shoot, I didn't know I needed a model for this? $\endgroup$
    – user34971
    Commented Jan 27, 2021 at 11:38
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    $\begingroup$ A nice answer $-$ though I would not be surprised if some authors actually conflate the two concepts. I've always thought Rumsfeld's quote is a good one to explain these concepts (you might want to check the sentence in the middle of the 5th para though, there might be a typo). Also, I think the rephrasing of risk as "probability model" and Knightian uncertainty as "model risk" is a good one. $\endgroup$ Commented Jan 27, 2021 at 11:57
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    $\begingroup$ Agreed - the "probability model" vs "model risk" analogy is a good one. And nicely concise to boot. You could then classify the BS as "I can't model it (because the available sample size is zero)"? And thanks Daneel for catching the typo. $\endgroup$
    – demully
    Commented Jan 27, 2021 at 12:08

Actually, the issue is fundamental to quantitative finance, even though most people never go deep enough to notice. It has to do with the axiomatic foundations of Frequentist and Bayesian statistics. The Black Swan is a different thing and is not related to Knightian uncertainty.

Frequentist methods are founded on aleatory risk, while Bayesian methods are grounded on epistemic uncertainty. It can matter quite a bit which you use.

Black Swans are really about model misspecification.

First, what is aleatory risk? It is risk from the effect of chance alone. In Null Hypothesis based methods such as Pearson and Neyman's Frequentist and Fisher's Method of Maximum Likelihood, the probabilities are conditioned on the model chosen and the loss function.

What is epistemic uncertainty? It is the uncertainty from lacking complete information. Knightian uncertainty is a subset of this form of uncertainty.

Frequentist models are "complete" models. They are built on top of a null hypothesis, which is a probabilistic form of modus tollens. A decision is either in an acceptance region or a rejection region subject to some chosen cutoff, usually called $\alpha$. If the null is true, then there can be no uncertainty, just chance. If the null hypothesis is rejected, then the decision is already determined, and so things like Black Swans only matter emotionally. That assumes you have correctly specified your model, of course.

Bayesian models are "incomplete" models. They cannot have a null hypothesis. There should be one hypothesis for every possible true construction of the world. The problem is the thing that you do not know.

For example, using a Frequentist method, you could determine from Mercury's orbit that Newton's Laws are false. If the test was done before Relativity existed, then a Bayesian could not test Relativity as a hypothesis. All a Bayesian could do is test the known possible explanations. Technically speaking, it may be possible to construct a Bayesian nonparametric method that was predictive. Still, it would have the problem of now having a working solution that has no scientific explanation.

When a new idea happens, then a Bayesian can test it. Knightian uncertainty is about the hypothesis that the Model T will put horse and buggy makers out of business one year before the Model T is invented. While a Bayesian could assign a weight to unknowns, there is, in fact, a formal mechanism to do things like that, under Knightian uncertainty, there isn't even a way to guess about it.

Finally, the Black Swan is really about using a misspecified probability model. For example, most standard economic models assume either normality or log-normality. From that point of view, things like the recent GME explosion could never happen. That is because the models are badly misspecified, from a probabilist's point of view. If one assumed a stable model instead without a mean or a variance, then the Black Swans vanish. The extreme events are now a normal part of the probability model. If you live somewhere where it only rains once every ten years and you anticipate that, then you are not surprised when it only rains once every ten years. If you lived on the coast of the United Kingdom and were suddenly dropped in that desert, you would feel surprised a lot.

Now let us get back to how this actually does matter to finance.

There are several ways to define risk, but let's define risk as the exposure to loss. We didn't say, "exposure to missing information." Also, we didn't say exposure to historical loss. Once an event is over, there is no risk anymore. A loss or a gain is now a statement of historical fact. Black Swans only matter to misspecified risk models. It wasn't so much that there was an unknown risk; if there were, we couldn't talk about Black Swans; it is that we ignored them.

Uncertainty has no chance component to it. In Bayesian thinking, chance doesn't really exist. What does exist is a system that is too complicated and complex to know enough to make statements that you can be sure are 100% true.

Imagine that one minute after your read this post, a meteor strikes your location, killing you. From a Frequentist perspective, that was a chance event. From a Bayesian perspective, you were killed for lack of information. Had you had an open window to look at the sky, had NASA been able to find your number to call you and tell you to run, you would not have been there. No chance was involved. That piece of space rock may have formed billions of years ago, and every single movement of it has corresponded to the laws of physics for that entire time.

Barring human intervention, that strike was set in motion essentially just after the Big Bang and always was going to occur. If it was outside your risk model, it is a Black Swan. It isn't due to Knightian uncertainty, however, or we couldn't talk about the possibility.

Now let us go one last level deeper, to calculus, set and measure theory. These differences are due to one mathematical difference between Bayesian and Frequentist methods. Bayesian methods are incompatible with Kolmogorov's third axiom of probability. They use a different set of axioms.

I know you probably had one day on the axioms of probability in a statistics class. It is usually the class that everyone rolls their eyes at and asks, "when am I ever going to use this?" You use them all the time; nobody ever linked them up for you before. Without that one difference, we couldn't discuss any of this.

  • $\begingroup$ Interesting points! Regarding a meteor hitting someone being a chance event from the frequentist viewpoint, I think this might be an oversimplification. See "Deterministic or stochastic universe in frequentist statistics". I also wonder if focusing on hypothesis testing is relevant here. Both a frequentist and a Bayesian model yield density predictions (though the latter does it much more elegantly) and this is what often matters in practice. Unless one is stuck with NHST for some external reason, in spite of NHST's bad reputation. $\endgroup$ Commented May 7, 2021 at 17:35
  • $\begingroup$ Also, are you sure the Bayesian philosophy implies that the universe is deterministic (In Bayesian thinking, chance doesn't really exist)? Would you have a reference? I have posted my question for a broader audience here. $\endgroup$ Commented May 7, 2021 at 17:52
  • $\begingroup$ @RichardHardy Yeah, I have a presentation from the Nuclear Regulatory Commission on it, but it is broadly discussed. nrc.gov/docs/ML1120/ML112000408.pdf $\endgroup$ Commented May 7, 2021 at 22:40
  • $\begingroup$ Regarding chance doesn't really exist, what about Heisenberg's uncertainty principle? (See it mentioned in the relevant context on p. 2 of this randomly chosen document from Cornell.) I am no physicist, so perhaps this has been refuted already? $\endgroup$ Commented May 12, 2021 at 11:21
  • $\begingroup$ @RichardHardy there is a Bayesian construction of quantum mechanics that converts the uncertainty principle to an "uncertainty" principle. It is not my area, however. I have read communications on the topic but not delved into it. I was working on some areas of the discrete calculus behind the physics but I am not a physicist. There is another issue as well related to axiomatization. Cox starts with "plausibility" as a primitive idea. De Finetti whose axiomatization is the oldest argues that probability does not exist. It is a construction of the mind to make sense of the world. $\endgroup$ Commented May 12, 2021 at 14:04

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