QuoteOriginally posted by: KurtosisOne point: if you take 1+(a(1) 1+a(2)) etc. it would be another calibration of a(n) but then you would need to increase a(2), then a(3), etc. Very true! Yet, unless a(i)=a(j), we need a higher order calibration to estimate the higher order vol of vol of vol of .....I think the difference between your figure 2 and my 1+(a(1) 1+a(2)) is the difference between progression dispersion (e.g., vol of vol in the markets in which volatility progressively evolves over time) and regression dispersion (i.e., the true sigma never was the prior/estimated sigma due to errors in models of errors in models of errors in ... of errors in models such as with Fukashima's reliability never ever being it's assumed a priori reliability). Perhaps we are speaking of nearly orthogonal phenomena. Vol of vol might be more about how the world changes over time and error on error might be more about how the world never was what we thought it was. I realize these two categories of dispersion may be hard to separate in the financial markets where change in price conflates both change in the world (e.g, a company announces new revenue & profit numbers) and change in our model of the world (e.g., we realize that constant IV models don't work).What might help solidify your paper is a better distinction between time-rate change in error vs. regressed discrepancy between true and estimated values.QuoteOriginally posted by: KurtosisIt seems that the end result would not change (under another calibration), but the math would be messy. When ever I hear "but the math would be messy" my mind fills with red flags and sirens. Making an assumption to make the math easy seems to be what got us in trouble in the first place. That said, I can appreciate the necessity of forcing a natural physical/economic system into an artificial mathematical framework because some answers are better than no answers. This situation also highlights the differences between epistemological limits driven by lack of data (which may be resolved with future events, samples, experiments) versus epistemological limits driven by lack of machinery (which may be resolved with future theories, theorems, and algorithms).Perhaps what would help is some justification for using a time-progressing vol of vol framework versus model-regressing errors on errors framework. For example, what is the physical meaning of a(1) or of a(i) vs. a(j)?QuoteOriginally posted by: KurtosisINDEPENDENCE: in a sampling framework, it matters. Here you can play with calibrations of a(n): the a(n) are the opinions.I was wondering about either ∂a(i)/∂a(j) ≠ 0 or CORR(a(i),a(j)) ≠ 0 for realized values of a(i) and a(j). Perhaps some exploration of the signs on those terms might make a nice sequel to this paper.QuoteOriginally posted by: KurtosisIn the end, what I am doing is set which assumptions that allow for thin-tails, etc. In other words, what are the conditions that allow us to suspend skepticisms, and at what point.Exactly! Your paper provides an interesting model for the time-rate of change of error and the bounds on error progressions that are necessary (but not sufficient) to ensure a non-fragile future. I look forward to draft 2.

QuoteOriginally posted by: KurtosisI have another question: can we define fragility as negative skewness? It would come from a concave function of a random variable.Fragility from ConcavityYes, but only for narrowly scoped units of analysis (e.g., a position, a trader, or a bank). The problem is that one person's positive skewness (antifragility) is their counterparty's negative skewness (fragility). At the systems level, non-zero magnitude of skew can cause one party or the other to fail.

My point is that a fragile (physical) package can be mapped into a very negatively skewed outcome (no upside sharp downside).

QuoteOriginally posted by: KurtosisMy point is that a fragile (physical) package can be mapped into a very negatively skewed outcome (no upside sharp downside).Agreed. If I have negative skew, then I am clearly fragile.But the converse is not true because my positive skew may imply that my counterparty is fragile. If I have positive skew (sharp upside, no downside) I may think that I am not fragile. Yet my upside profits must come from someone (for many types of financial instruments) and that someone may have a negative skew that matches my positive skew (unless I've synthesized my positive skew through a portfolio of instruments that individually have zero skew but collectively create positive skew).Many U.S. homeowners in non-recourse states seemed to have a positive skew. With near-zero equity in their homes, their downside was limited but their upside was not. For the banks, the skew on mortgages was the inverse -- the banks had bounded upside and unlimited downside.In a long-term iterated system (e.g., an economy) my own fragility can't be separated from the fragility of those around me. My positive skew can turn negative if my counterparty fails (i.e., the upside disappears) and my nice positive skew asset becomes illiquid and toxic.

So far,Left skew ->fragile, from concave payoff. (this applies to any object)Right skew ->antifragile, from convex payoff, benefits from increase in vol.Robust: small tails (left and right), mostly from lower vol of total.Category of both tails: fragile but not antifragile. does not benefit from volatility. (the equivalent of short OTM puts long calls).So for the category fragile, a left tail necessary, but can have right tail. In other words, a mixed payoff is fragile, not antifragile.

QuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: KurtosisMy point is that a fragile (physical) package can be mapped into a very negatively skewed outcome (no upside sharp downside).Agreed. If I have negative skew, then I am clearly fragile.But the converse is not true because my positive skew may imply that my counterparty is fragile. If I have positive skew (sharp upside, no downside) I may think that I am not fragile. Yet my upside profits must come from someone (for many types of financial instruments) and that someone may have a negative skew that matches my positive skew (unless I've synthesized my positive skew through a portfolio of instruments that individually have zero skew but collectively create positive skew).Many U.S. homeowners in non-recourse states seemed to have a positive skew. With near-zero equity in their homes, their downside was limited but their upside was not. For the banks, the skew on mortgages was the inverse -- the banks had bounded upside and unlimited downside.In a long-term iterated system (e.g., an economy) my own fragility can't be separated from the fragility of those around me. My positive skew can turn negative if my counterparty fails (i.e., the upside disappears) and my nice positive skew asset becomes illiquid and toxic.That's a good point. Also trying to achieve anti-fragility at the individual level can lead to fragility of the whole system. If everyone makes robust decisions using worst case scenarios that could make the whole system unstable. I remember seeing a nice paper illustrating this recently.

QuoteOriginally posted by: KurtosisSo far,Left skew ->fragile, from concave payoff. (this applies to any object)Right skew ->antifragile, from convex payoff, benefits from increase in vol.Robust: small tails (left and right), mostly from lower vol of total.Category of both tails: fragile but not antifragile. does not benefit from volatility. (the equivalent of short OTM puts long calls).So for the category fragile, a left tail necessary, but can have right tail. In other words, a mixed payoff is fragile, not antifragile.Thinking from a system perspective, if concavity applies to any object, then one can think of risk aversion as concave evaluation outcomes, e.g. my utility to outcomes would be mapped as a concave function. I guess a left skew in utility distribution would mean that I'd be moderately happy most of the time with frequent bouts of depression. If I am risk seeking I'd have frequent moments of joy. Evolution should favor the latter but that can't be good for the whole system.

QuoteOriginally posted by: gardener3QuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: KurtosisMy point is that a fragile (physical) package can be mapped into a very negatively skewed outcome (no upside sharp downside).Agreed. If I have negative skew, then I am clearly fragile.But the converse is not true because my positive skew may imply that my counterparty is fragile. If I have positive skew (sharp upside, no downside) I may think that I am not fragile. Yet my upside profits must come from someone (for many types of financial instruments) and that someone may have a negative skew that matches my positive skew (unless I've synthesized my positive skew through a portfolio of instruments that individually have zero skew but collectively create positive skew).Many U.S. homeowners in non-recourse states seemed to have a positive skew. With near-zero equity in their homes, their downside was limited but their upside was not. For the banks, the skew on mortgages was the inverse -- the banks had bounded upside and unlimited downside.In a long-term iterated system (e.g., an economy) my own fragility can't be separated from the fragility of those around me. My positive skew can turn negative if my counterparty fails (i.e., the upside disappears) and my nice positive skew asset becomes illiquid and toxic.That's a good point. Also trying to achieve anti-fragility at the individual level can lead to fragility of the whole system. If everyone makes robust decisions using worst case scenarios that could make the whole system unstable. I remember seeing a nice paper illustrating this recently.Exactly! What's interesting is the extent that so many modern financial organizations are a kind of pass-through entity. Their assets are someone else's liabilities, their liabilities are someone else's assets, their borrowing costs are someone else's lending revenues, their lending revenues are someone else's borrowing costs, and their profits become someone else's capital gains. In a fractional reserve economy, the buck doesn't stop anywhere it just keeps recirculating.In this type of recursive system, systemic fragility comes from how skews (of either sign) propagate across boundaries and whether some entities accumulate negative skews (creating systemic fragility) or whether all entities have sufficient positive skews to offset their negative skews (avoiding systemic fragility).

Hi, thanks for the comments.The point is settled: there is a trade-off of convexity between entities (managers have the options, investors sold -unwittingly). The aggregate systemic effect...(nature achieves overall antifragility by having fragile --but uncorrelated -- components. the aggregate is information, that is antifragile, the gene is not perishable, it exploits the fragility of individuals...)The solution is to have a system robust to systemic blowups, low debt, low leverage, artisanal, etc.But do you believe we can define fragility for an object (say a glass on a table, your car, a place) in the same way? in terms of negative skewness? This is what is puzzling me as I cannot see exceptions.

Hmmm... The crucial question is: are two objects equally fragile if they have identical skews? An object with a equally-likely payoffs of {-9, -2, 2, 4, 5} has the same negative skewness as an object with a equally-likely payoffs of {-18, -4, 4, 8, 10} yet the latter seems more fragile than the former due the -18 versus -9 worst-case payoff. And an object with a equally-likely payoffs of {-12, -7, -5, 9, 15} has positive skewness, but also seems more fragile than the first object due to the -12 vs. -9 worst-case and the 60% vs. 40% chance of a negative outcome. And if we define some alternative measure of skew (e.g., rawskew = E[(Xi - E[Xi])^3 ]) then I'm sure we can create payoffs with identical rawskew values but different downside extrema.Isn't fragility related to some measure of downside extrema, normalized in some way? Perhaps we can find the fragility function by listing it's properties. These might include some kind of independence from the upside structure, some kind of independence of modest downsides, and some kind of independence of extreme downsides far worse than the breaking point of the system. What do you think?You also mentioned physical fragility. Material scientists determine the brittleness (= tendency to shatter = fragility) of a material using the Charpy impact test. This test swings a pendulum axe at a standard-sized notched sample. The test measures the energy left in the swinging axe after it shatters the sample which lets one infer the energy required to shatter the sample. Brittle (= fragile) materials require very little impact energy to break. In a financial context the analog would be the amount of downside the object can absorb without breaking which would seem to be related to the reserves in the system (or the probability of exceeding those reserves). Another facet of the Charpy test is the rapid application of overwhelming force. Other materials tests measure a material's strength and plastic deformation under more slowly-applied forces. Not all brittle materials are weak. Similarly, a bank may be able to handle a slow bleed of losses that accumulates to a very high level over time much better than an instantaneous loss of the same magnitude. Or a dynamic hedging strategy may be able to handle arbitrarily large slow movements in the markets but not handle the same large movement in one jump. Thus fragility also seems to have a time-rate or path dependent component.

The idea of fragility as convexity. You can map it into potential energy, but this representation uses a simple generator (at the top of the stairs you release energy by coming down, at the bottom you need it to climb). T4A seems to look at how mush energy is needed to push the object to fall, not what it releases by falling. In terms of finance (and thermodynamics), convexity ->theta is the price of gamma off that convexity (to keeps things balanced); but let's ignore finance for now, just focus on objects. The 3rd graph focused on convexity but concavity is more intuitive. (On Thom's Catastrophe theory, later)

Interesting graphs.First, I sense there's an implicit ∆x/∆t and ∆f(x)/∆x dynamic in figs 11 & 12 and that fig 12 approaches the logistic function. Is a large possible potential energy value -- a high f(+∞ ) - f(-∞ ) -- sufficient to say we have a fragile function? Doesn't the maximum slope matter? If ∆f(x)/∆x were small everywhere as opposed to having a large jump, would we still say that the low side of f is "antifragile" and the high side of f is "fragile"?Second, how does one convince investors, managers, and regulators that we have a function of a certain "fragile" shape and that we are in the "fragile" regime/end of the function? Prior to 2007, many would have proclaimed that housing was a "big gains, small losses" system when the opposite was true. Can we measure the potential energy in the system and the latent headroom of f so we can ascertain whether we are on the left-hand, antifragile side or right-hand, fragile side of f(x) in figure 12?

All nonlinearities can be expressed in terms of logistic functions.I am puzzled as to for an individual organism (no aggregate, forget macro) whether my mapping is right.

The mapping seems right for a fixed and known world. It's just a matter of properly relating "fragility" to the amplitude, scale, phase, and offset of the logistic function = f(x). We might also want to think about how volatility of x or the rate of motion in x (= dx/dt) affects things (e.g., high volatility in x or high dx/dt can imply high fragility).My two remaining concerns are: 1) whether we can know the parameters of f and our current x on f; and 2) whether innovation, fraud, regulatory/political dynamics don't markedly change our f in crucial ways (e.g., a run-away feedback loop or unintended consequence might converts an antifragile position of x1 in f1 into a fragile position of x2 in f2. That is, what if the world is unknown or complexly dynamic?

I don't think one-dimensional skewness/asymmetry is the best description for the fragile/robust/antifragile triangle ... the cost of becoming antifragile (generating a steep gain function in one dimension) may be paid by generating a steep loss function in an unrelated dimension.The evolution of the brain and cooperation contribute to create an individual that will be considerable stronger when part of a group, but taken alone the individual's fragilities will be apparent.The Hydra picture would be helpful here, describing the multiple dimensions of fragility/antifragility in the same individual/object.Even in objects where creating a strong structure will lead to a resonant frequency or a specific direction on which cutting is easier.Basically what I'm trying to say is that the antifragile will be fragile on the dimension of the cost paid to raise the step (too heavy a cost and it'll die, too strong and it'll be brittle, ...).