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Ambiguity in quant finance is defined as the uncertainty in the probabilities of the return distribution, whereas risk is defined as the uncertainty in the returns of the asset.

There are various measures of ambiguity such as the volatility of volatility (historical, IV), volatility of the mean, volatility of probabilities etc.

In Brenner and Izhakian (2018, JFE), the authors state that a shortcoming of each of the ambiguity measured by 2 other measures (as compared to theirs which is the volatility of probabilities):

  • Volatility of Volatility: Two equities with different degrees of ambiguity but constant volatility.
  • Volatility of the Mean: Two equities with different degrees of ambiguity but constant mean.

The authors state that their ambiguity measure (measured by the volatility of probabilities) does not share the same shortcomings as the 2 measures above. What exactly do they mean?

Please let me know if you need more details, I understand that this question might not be worded the best.

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    $\begingroup$ I would like to add that Knightian uncertainty and ambiguity are the same thing. $\endgroup$
    – KaiSqDist
    Commented Jan 5 at 4:55
  • $\begingroup$ Thanks for your inputs @RichardHardy, I guess Knightian uncertainty is a much broader concept (beyond the field of finance), but my work follows the research of Izhakian from Baruch quite closely and at least he defines the two to be the same. This is my first time hearing about Knightian risk actually, and it sounds more like a case where we are able to at least quantify a probability distribution for a set of outcomes, whereas ambiguity is something higher-order such as being uncertain about the probability distribution and/or risk. $\endgroup$
    – KaiSqDist
    Commented Jan 5 at 12:48

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Just wanting to add to the discussion in the comments on risk, uncertainty, and ambiguity but it's too long for a comment.

The terms risk, uncertainty, and ambiguity are certainly sometimes messed up. Here's a great explanation from Stanford's Nick Bloom on Econofact (3:47) who is probably the world's leading expert on economic uncertainty and made many major contributions to the field:

So Frank Knight back in 1921 said, look, there are two types of things you can think about. There is known uncertainty, which he called risk, which to be honest nowadays is just basically called uncertainty, which is when you know the distribution of something. So good example, if you're flipping a coin, if it's fair, we know it's 50% heads, 50% tails. Then he called something back then uncertainty, which is now generally called Knightian uncertainty, is when you don't know the distribution of something. So an example might be if I asked you to say, "how many coins have ever been minted in the history of human civilization?" And you're going to be like, "how on earth do I know, that's like everything from Romans to..." It's just impossible to figure it out. And that's now called Knightian uncertainty, and that is kind of, I know there's unknown unknowns or known unknowns, but it's something that we can't put a probability distribution on. It's also being called ambiguity as well sometimes in the literature.

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    $\begingroup$ My favorite treatment of risk, uncertainty and such is Hansson "Decision Theory: A Brief Introduction" (1994/2005). He does not use the term "ambiguity" at all. His terminology is the following: certainty ~ deterministic knowledge, risk ~ complete probabilistic knowledge, uncertainty ~ partial probabilistic knowledge, ignorance ~ no probabilistic knowledge. Details are available on p. 27-28 in the reference. $\endgroup$ Commented Jan 6 at 15:38
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    $\begingroup$ @RichardHardy That's certainly the original Knightian classification I still learnt in textbooks when I did my undergraduate. But as I go through modern papers (including my own :p), I do tend to equate risk and uncertainty and tend to treat ambiguity (or Knightian uncertainty) as the other category. I think that's nicely reflected in Bloom's quote who explains how the meaning has shifted in over 100 years. $\endgroup$
    – Kevin
    Commented Jan 6 at 16:45
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    $\begingroup$ I am fairly sure that is not the original Knightian classification, as indicated explicitly on p. 27. The 4 categories of Hansson are a finer scale than that of Knight or Luce & Raiffa. Otherwise, Hansson would not have to write the following about these older categorizations: Many – perhaps most – decision problems fall between the categories. And if you are equating risk and uncertainty, you seem to be going back to the coarser scale of Knight, just with new labels of the same categories. Though again that depends on what you mean by risk and uncertainty. What a mess!.. $\endgroup$ Commented Jan 6 at 17:06
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    $\begingroup$ @RichardHardy I agree with the second half of your comment. It's somewhat close to the original Knightian classification, just with different labels (uncertainty and ambiguity) instead of the old-school ones (risk and uncertainty). It seems to me that the literature has evolved (unsurprisingly) and terminology has changed with it. $\endgroup$
    – Kevin
    Commented Jan 6 at 17:49
  • $\begingroup$ Thanks both for the interesting and fruitful discussion. It seems to me that the foundation of working with these kind of problems regarding risk, uncertainty and Knightian concepts is to have a consensus on the terminology and what both parties are referring to (and mean) at the least :) $\endgroup$
    – KaiSqDist
    Commented Jan 7 at 3:23
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How about just estimating a time-varying beta model (e.g., Bollerslev, Engle, and Wooldridge 1988) and then using the variance of the beta as the measure of ambiguity? Would this measure capture the characteristics you described?

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  • $\begingroup$ Thanks, I will take a look! $\endgroup$
    – KaiSqDist
    Commented Jan 12 at 2:33
  • $\begingroup$ Hi Michael, I just got a look at the paper. Why do you suggest the variance of the beta to be a measure of ambiguity? Are you suggesting that since the beta is based on the covariance, the time-varying nature of the covariance (and therefore "risk") can be taken as ambiguity? Interested to hear your thoughts. $\endgroup$
    – KaiSqDist
    Commented Jan 28 at 17:20
  • $\begingroup$ Yes, exactly. The properties of the variance of covariance are (I think) what one would want from a measure of ambiguity, too, e.g., expected returns and the risk premium could be increasing or decreasing in ambiguity (Armstrong Banerjee Corona RFS). $\endgroup$ Commented Jan 29 at 18:22

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