Ambiguity Measures: Limits of Volatility-Based Proxies
Summary
The document distinguishes risk, uncertainty in realized returns under a known distribution, from ambiguity, uncertainty about the return probabilities themselves. It reviews volatility of volatility and volatility of the mean as proposed proxies, then reports the claim that both can miss ambiguity when two equities have the same return volatility or the same mean but differ in their probability distributions. A volatility-of-probabilities measure is presented as avoiding those specific shortcomings.
The discussion gives conceptual examples and cites a finance paper, but does not explain how the probability-volatility measure is estimated or provide empirical tests. A follow-up suggests variance in a time-varying beta as another possible proxy; the document does not establish whether it captures ambiguity. The material is therefore a framing of measurement issues rather than a practical estimation guide, and its definitions of risk and ambiguity reflect the cited discussion rather than a universal convention.
Key ideas
- Ambiguity concerns uncertainty about the probabilities governing returns, while risk concerns uncertainty in returns under those probabilities.
- Volatility of volatility can fail to distinguish assets with different ambiguity if their return volatility is constant.
- Volatility of the mean can fail to distinguish assets with different ambiguity if their mean is constant.
- The cited authors argue that volatility of probabilities avoids these particular limitations.
- Variance in a time-varying beta is suggested as an alternative, but its suitability is left unresolved.
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Full text
# On measurements of ambiguity and their shortcomings # On measurements of ambiguity and their shortcomings 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. ## Answer by Kevin (score 2, accepted) https://quant.stackexchange.com/a/77915 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. ## Answer by michaelcarniol (score 2) https://quant.stackexchange.com/a/77966 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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