Dear friends, I wonder if someone sees flaws or has some practical disagreement with this method to detect fragility. Don't worry about your comment, I can sift through the more or less intelligent insults in search of substance. Last time I received helpful comments from Traden4AlphaFragility Heuristic

I will read your draft with curiosity and attention.After that I will post comments and questions which I hope would be helpfull (maybe not directly to you but for sure to my understanding) Insults have to be ignored no more no less.

Interesting! I will comment tomorrow.

The fragility of an object seems to be an interaction of the object and it's environment. In your nomenclature, x encodes the distribution of conditions in the environment and f encodes the properties of the object as a mapping from x into payoffs or utility. I like the taxonomy of payoffs but feel that the tail explanation conflates the fragility of the object (i.e., tendency to map x to an extreme negative value) with what might be called the dangerousness of the environment (i.e., heavy tail in x). If the distribution of x has fat tails, would we say that f is fragile if f(x) has fat tails, too? Implicit in Table 1 is that x has thin tails that are fattened (or not) by f. Imagine some wonderful antifragile f_a(X) that maps a nasty Cauchy-distributed x to a pleasant thin-left- and fat-right-tailed outcome. Unless f_a(x)>0 for all x, there exists a distribution, x_dangerous, for which f_a(x_dangerous) has the dreaded fat left tail. My point is that we need to be clear about whether we want a definition of fragility that is intrinsic to the object only (i.e., depends on f only with no dependence on the distributions of x or y) or whether we want a definition that is contextual to the environment (i.e., assumes some x and considers the tails in y=f(x)).Minor comments:1) E1-type errors are only avoided in worlds in which the future values or trajectories are drawn from the set of historical values or trajectories. If, instead, the world produces novel values of key variables (e.g., illiquidity of previously liquid instruments, new extremes in correlation, or flash-crash volatilities), then never-before-seen mis-tracking can occur that surprises (and bankrupts) the modeler.2) E2-type errors reveal themselves in the high unexplained variances of the model with out-of-sample data. Overfitting errors, which are a subset of E2 errors, appear as higher unexplained variance in out-of-sample data than in the in-sample data.3) Please label or define "V" in Definition 1a as well as the zeta shortfall function when they are first introduced.4) I found some of the nomenclature a bit confusing. On page 2, x is an input random variable, f is a payoff function mapping from x to y, and y is the outcome. Thus, x has a distribution and y has a distribution defined by f(x). But then definition 1a seems to implicitly use y as the input, not x. To stay consistent with the nomenclature of page 2, shouldn't page 3 be V(x,f,K,∆s) and the zeta shortfall function integrate with f(x), not f(y)? Then on page 4, you refer to "using the wrong distribution f" and contrasting f and f* when perhaps you meant to say "using the wrong distribution x" which leads to contrasting y=f(x) vs. y*=f(x*). There's two cases here: error-in-x and error-in-f. In the second case, the payoff function may not be known exactly so we also have the case of "using the wrong payoff map of f" which leads to contrasting y=f(x) vs. y*=f*(x). Discrepancies in the payoff function don't seem likely in the case of contractually-specified financial instruments, but if one considers either counterparty risk or risk of government interdiction (e.g. nullification of trades or contracts), then f and f* can be very different.5) When you wrote "where f and g are the respective monomodal probability distributions for x and y," between Eqns #1 & #2, did you actual mean "where f and g are the respective monomodal probability distributions for y and z,"? And if you clarify the nomenclature using my comment #4, then you might want to make it "where y and z are the respective monomodal probability distributions for f(x) and g(x),"6) The paper seems to carry a subtle assumption of a monotonic f such that we have a left tail in x of "bad" values and a right tail in x of "good" values mapping to respective bad and good tails in y. This makes sense for simple derivatives that have a monotonic pay-off in the underlying price of the primitive. Yet most engineered systems (and more complex options strategies) show a second-order relationship between independent variables and outcomes such that, for example, both high and low values of the independent variables produce negative outcomes. For aircraft, too high an airspeed is just as bad (structural failure) and too low an airspeed (stall)........... more to come either today or this weekend! ..........P.S. I still worry about the counterparties to antifragile contracts because they have a fat loss domain and a thin gains domain. ========================EDIT==============I may be more confused than I thought. Is f(y) the probability density function for experiencing pay-off = y? Is that the nomenclature for the entire paper? The paper needs some clarifying to more clearly distinguish between the PDF of underlying random variable, the pay-off function that maps an underlying price to a derivative payoff, and the resultant PDF of that pay-off.

It's really funny because at the moment I'm working at semi static hedging strategies for barrier option under volatility uncertainty (the famous sub replication and super replication) and I was a lot concerned by the gamma sign (or the convexity/concavity of your payoff). I feel that there is a VERY STRONG connexion about your concept of fragility and the worst case measure under volatility uncertainty.I just read the paper in a quick way so I will reread it more in detail for further more clever comment but I have two questions :-Are your concepts of fragility and antifragility depending on the time horizon of the process ? -Is it possible to derive hedging strategies based on turning a fragile payoff to an antifragile one ? It's very similar to what I'm doing right now with convexity matching.-And what about model errors if the unknown is the payoff ? (very unlikely to occur in finance but in physics I wouldn't be surprised).Anyway at first sight, it's a very impressive paper, clear, with a lot of pedagogy.

Thanks a milliony is a function of x, or something else, but it has a distribution. So I treat it as a random variable (by convex/concave transformation). x itself might be a function of something else. So it is the payoff taken the end random variable, that matters.T4a isn't it clear enough in the text?I reposted the paper on SSRN with a better, cleaned up version.Thanks. May be mostly away from computers for 2 weeks.

The only confusing bit is that the f on page 2 seems to map input conditions to output payoff (e.g., might be the payoff of an option as a function of price-at-expiration) and the subsequent f's map outcome to probability density. It's the change from "y=f(x)" in Table 1 to suddenly talking about f(y) in equation 3 on page 4 that caught me. Defining f one way and then redefining it another seemed confusing.I'll look at the new version are respond this weekend.

Love your work NT.