# Fragility Book

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THE TECHN ICAL INCERTOLECTURES ON R ISK AND PROBAB I L I TY, VOL 2

A MATHEMAT ICAL FORMULAT ION OF FRAG I L I TYnassim nicholas talebraphael douady

2015Peer-Reviewed MonographsDESCARTES

descartes peer-reviewed monographsDescartes Monographs accept only manuscripts that have been rigorously peer-reviewed or contain material that has appeared in peer-reviewed journals or hasbeen sufficienctly cited to merit inclusion in the series.

Its aim is to "berize" academic publishing by cutting the middleperson andproducing books of the highest scientific quality at the most affordable price.Descartes engages in ethical publishing, avoiding double-charging the publicsector for books produced on university time.

EDITORIAL BOARD

This format is based on Andr Miedes ClassicThesis.

N. N. Taleb and R. Douady: A Mathematical Formulation of Fragility, c 2015

SUMMARYWe propose a mathematical definition of fragility as negative sensitivity in the tails toa semi-measure of dispersion and volatility (a variant of negative or positive "vega" inoption theory) and examine the link to nonlinear effects. We also define the oppositeof fragile "antifragile" as producing a positive response to increase in stochasticity.

We integrate model error (and biases) into the fragile or antifragile context. Unlikerisk, which is linked to psychological notions such as subjective preferences (hencecannot apply to a passive object such as a coffee cup) we offer a measure that concernsany object that has a probability distribution (whether such distribution is known or,critically, unknown).

The connection between fragility and convexity is established via the fragility trans-fer theorems.

We have recourse to integral transforms to map discrete payoffs (say coffee cup) toa continuous Schwartz functions and apply convexity analysis.

We analyze the effect of size and concentration and scaling on fragility. We explaindiseconomies of scale as concave exposure to shocks (much like "short volatility").

We fold "material fatigue" (or accumulated harm) into our approach by mappingit to nonlinearity when observed at a specific time scale.

Using our convexity/concavity approach we propose a potential source of thin-tailedness in nature from exposures have capped upside and limited downside torandom variables that can be extremely fat-tailed.

We link dose-response to such general problems as climate risk-management.We make a connection between fragility and survival of objects that have a tail risk,

though smaller one in the "body".

Detection Heuristic We propose a detection of fragility, robustness, and antifragilityusing a single "fast-and-frugal", model-free, probability free heuristic that also picksup exposure to model error. While not flawless, the heuristic lends itself to immediateimplementation, and uncovers a large amount of hidden risks related to companysize, forecasting problems, country risk, and bank tail exposures (it explains someforecasting biases).

Incerto This is the second volume of the Technical Incerto, a mathematical formula-tion of the Incerto, the first authors multi-volume treatise on uncertainty.

iii

CONTENTS1 an uncertainty approach to fragility 1

1.1 A Review of The General Notion of "Robustness" 11.2 Introduction 2

1.2.1 Intrinsic and Inherited Fragility: 41.2.2 Fragility As Separate Risk From Psychological Preferences 51.2.3 Beyond Jensens Inequality 5

2 the fragility theorems 112.1 Tail Sensitivity to Uncertainty 11

2.1.1 Precise Expression of Fragility 142.2 Effect of Nonlinearity on Intrinsic Fragility 152.3 Fragility Drift 19

2.3.1 Second-order Fragility 192.4 Expressions of Robustness and Antifragility 19

2.4.1 Definition of Robustness 192.4.2 Antifragility 212.4.3 Remarks 222.4.4 Unconditionality of the shortfall measure 23

3 applications to model error 253.0.5 Example:Application to Budget Deficits 253.0.6 Model Error and Semi-Bias as Nonlinearity from Missed Stochas-

ticity of Variables 263.1 Model Bias, Second Order Effects, and Fragility 27

4 the fragility measurement heuristics 294.0.1 The Fragility/Model Error Detection Heuristic (detecting A

and B when cogent) 294.1 Example 1 (Detecting Risk Not Shown By Stress Test) 294.2 The Heuristic applied to a stress testing 30

4.2.1 Further Applications Investigated in Next Chapters 304.3 Stress Tests 314.4 General Methodology 32

5 fragility and economic models 335.1 The Markowitz Inconsistency 335.2 Application: Ricardian Model and Left Tail Exposure 34

6 the origin of thin-tails 396.1 Properties of the Inherited Probability Distribution 416.2 Conclusion and Remarks 44

v

Contents7 small is beautiful: risk, scale and concentration 45

7.1 Introduction: The Tower of Babel 457.1.1 Only Iatrogenics of Scale and Concentration 48

7.2 Unbounded Convexity Effects 487.3 A Richer Model: The Generalized Sigmoid 50

7.3.1 Application 538 why is the fragile nonlinear? the arcsine distribution of life 57

8.1 The Coffee Cup 578.1.1 Illustration Case 1: Gaussian 608.1.2 Illustration Case 2: Power Law 60

Bibliography 63Index 65

vi

NOTES ON THE TEXTincomplete sections(mostly concerned with building exposures and convexity of payoffs: What is andWhat is Not "Long Volatility")

i A discussion of time-scale transformations and fragility changes with lengtheningof t.

ii A discussion of gamblers ruin. The interest is the connection to tail events andfragility. "Ruin" is a better name because the idea of survival for an aggregate.

iii An exposition of the precautionary principle as a result of the fragility criterion.(Integration of the ideas of the paper we are summarizing for Complexity.)

iv A discussion of the "real option" literature showing connecting fragility to thenegative of "real option".

v A link between concavity and iatrogenic risks (modeled as short volatility).

vi A concluding chapter.

acknowledgmentsBruno Dupire, Emanuel Derman, Jean-Philippe Bouchaud, Elie Canetti, Yaneer Bar-Yam, Jim Gatheral (naming such nonlinear fragility the "Tower of Babel effect"), IgorBukanov, Edi Piqoni, Charles Tapiero.

Presented at JP Morgan, New York, June 16, 2011; CFM, Paris, June 17, 2011; GAIMConference, Monaco, June 21, 2011; Max Planck Institute, BERLIN, Summer Instituteon Bounded Rationality 2011 Foundations of an Interdisciplinary Decision Theory- June23, 2011; Eighth International Conference on Complex Systems- BOSTON, July 1,2011, Columbia University September 24 2011; the 2014 NYU DRI Annual Conference"Cities and Development" co-hosted by NYU Marron Institute of Urban Managementon November 18, 2014.

Best Regards,January 2015

vii

1 AN UNCERTA INTY APPROACH TOFRAG I L I TYChapter Summary 1: We provide a mathematical approach to fragility as neg-ative sensitivity to a semi-measure of dispersion and volatility (a variant ofnegative or positive "vega") and examine the link to nonlinear effects. We linkto the litterature on model "robustness" and show how we add nonlinearity tothe conventional approaches.

1.1 a review of the general notion of "robustness"This section is incomplete; it will present a general review the literature and thevariety of definitions of "robustness" in:

Loss (risk) functions in statistical fitting

Loss (risk) functions in risk and insurance

Decision theory (minimax, etc.)

Statistical robustness -

Control theory- Stochastic control

Dynamical systems-

Economic modelling (see Hansen and Sargent) -

We explain why what we do is add a nonlinear dimension to the loss models andsolve some of the issues with loss models since our funtion (nonlinearity of exposure)solves the problem of mini-max with unbounded losses, and how our payoff function("exposure") maps to the loss function.

1

an uncertainty approach to fragility

Breaking Point

Dose

ResponsePayoff of the Coffee Cup

Broken Glass

Higher dispersion of stressor increases probab.of breaking.

Deterministic andinvariantstressor

StochasticStressor

Dose

Response

Figure 1.1: Illustrates why the coffee cup is fragile because it doesnt like variability. Imaginea stressor of intensity k. A deterministic stressor at k will always be more favorable than astochastic stressor with average k. The cup breaks at k + , even if there are compensatingeffects at k that lower the average. The more dispersion around k, given the same aver-age, the higher the probability of breakage. This illustrates the dependence of fragility ondispersion ("vega") and the theorems showing how fragility lies in the second order effect.[INCLUDE STOCHASTIC RESONANCE AS AN OPPOSITE EFFECT]

1.2 introductionIn short, fragility as we define it is related to how a system suffers from the variabilityof its environment beyond a certain preset threshold (when threshold is K, it is called

2

1.2 introductionK-fragility), while antifragility refers to when it benefits from this variability in asimilar way to vega of an option or a nonlinear payoff, that is, its sensitivity tovolatility or some similar measure of scale of a distribution.

K

Prob Density

HK, s- + Ds-L = -

K Hx -WL f Hs_+Ds-L HxL x

HK, s-L = -

K Hx -WL f Hs_L HxL x

Figure 1.2: A definitionof fragility as left tail-vega sensitivity; the fig-ure shows the effect of theperturbation of the lowersemi-deviation s on thetail integral of (x )below K, being a cen-tering constant. Our de-tection of fragility doesnot require the specifica-tion of f the probabilitydistribution.

We are not saying that are no other definitions and representation

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