Statistical Consequences of Fat Tails Real World Preasymptotics, Epistemology, and Applications

1 Prologue^*,†
2 Glossary, Definitions, and Notations
	2.1 General Notations and Frequently Used Symbols
	2.2 Catalogue Raisonné of General & Idiosyncratic concepts
		2.2.1 Power Law Class P
		2.2.2 Law of Large Numbers (Weak)
		2.2.3 The Central Limit Theorem (CLT)
		2.2.4 Law of Medium Numbers or Preasymptotics
		2.2.5 Kappa Metric
		2.2.6 Elliptical distribution
		2.2.7 Statistical independence
		2.2.8 Stable (Lévy stable) Distribution
		2.2.9 Multivariate Stable Distribution
		2.2.10 Karamata point
		2.2.11 Subexponentiality
		2.2.12 Student T as proxy
		2.2.13 Citation ring
		2.2.14 Rent seeking in academia
		2.2.15 Pseudo-empiricism or Pinker Problem
		2.2.16  Preasymptotics
		2.2.17  Stochasticizing
		2.2.18 Value at Risk, Conditional VaR
		2.2.19 Skin in the game
		2.2.20 MS Plot
		2.2.21 Maximum domain of attraction, MDA
		2.2.22 Substitution of Integral in the psychology literature
		2.2.23 Inseparability of probability (another common error)
		2.2.24 Wittgenstein's Ruler
		2.2.25 Black swan
		2.2.26 The empirical distribution is not empirical
		2.2.27 The hidden tail
		2.2.28 Shadow moment
		2.2.29 Tail dependence
		2.2.30 Metaprobability
		2.2.31 Dynamic hedging
Fat Tails and Their Effects, An Introduction
	3 A Non-Technical Overview - The Darwin College Lecture ^*,‡
		3.1 On the Difference Between Thin and Thick Tails
		3.2 A (More Advanced) Categorization and Its Consequences
		3.3 The Main Consequences and How They Link to the Book
			3.3.1 Forecasting
			3.3.2 The Law of Large Numbers
		3.4 Epistemology and Inferential Asymmetry
		3.5 Naive Empiricism: Ebola Should Not Be Compared to Falls from Ladders
		3.6 Primer on Power Laws (almost without mathematics)
		3.7 Where Are the Hidden Properties?
		3.8 Bayesian Schmayesian
		3.9 X vs F(X), exposures to X confused with knowledge about X
		3.10 Ruin and Path Dependence
		3.11 What To Do?
	4 Univariate fat tails, Level 1, Finite Moments^†
		4.1 A Simple Heuristic to Create Mildly Fat Tails
			4.1.1 A Variance-preserving heuristic
			4.1.2 Fattening of Tails With Skewed Variance
		4.2 Does Stochastic Volatility Generate Power Laws?
		4.3 The Body, The Shoulders, and The Tails
			4.3.1 The Crossovers and Tunnel Effect.
		4.4 Fat Tails, Mean Deviation and the Rising Norms
			4.4.1 The Common Errors
			4.4.2 Some Analytics
			4.4.3 Effect of Fatter Tails on the "efficiency" of STD vs MD
			4.4.4 Moments and The Power Mean Inequality
			4.4.5 Comment: Why we should retire standard deviation, now!
		4.5 Visualizing the effect of rising p on iso-norms
	5 Level 2: Subexponentials and Power Laws
		5.0.1 Revisiting the Rankings
			5.0.2 What is a borderline probability distribution?
			5.0.3 Let Us Invent a Distribution
		5.1 Level 3: Scalability and Power Laws
			5.1.1 Scalable and Nonscalable, A Deeper View of Fat Tails
			5.1.2 Grey Swans
		5.2 Some Properties of Power Laws
			5.2.1 Sums of variables
			5.2.2 Transformations
		5.3 Bell Shaped vs Non Bell Shaped Power Laws
		5.4 Super-Fat Tails: The Log-Pareto Distribution
		5.5 Pseudo-Stochastic Volatility: An investigation
	6 Thick Tails in Higher Dimensions^†
		6.1 Thick Tails in Higher Dimension, Finite Moments
		6.2 Joint Fat-Tailedness and Ellipticality of Distributions
		6.3 Multivariate Student T
			6.3.1 Ellipticality and Independence under Thick Tails
		6.4 Fat Tails and Mutual Information
		6.5 Fat Tails and Random Matrices, a Rapid Interlude
		6.6 Correlation and Undefined Variance
		6.7 Fat Tailed Residuals in Linear Regression Models
	A Special Cases of Thick Tails
		A.1 Multimodality and Thick Tails, or the War and Peace Model
		A.2 Transition Probabilities: What Can Break Will Break
The Law of Medium Numbers
	7 Limit Distributions, A Consolidation^*,†
		7.1 Refresher: The Weak and Strong LLN
		7.2 Central Limit in Action
			7.2.1 The Stable Distribution
			7.2.2 The Law of Large Numbers for the Stable Distribution
		7.3 Speed of Convergence of CLT: Visual Explorations
			7.3.1 Fast Convergence: the Uniform Dist.
			7.3.2 Semi-slow convergence: the exponential
			7.3.3 The slow Pareto
			7.3.4 The half-cubic Pareto and its basin of convergence
		7.4 Cumulants and Convergence
		7.5 Technical Refresher: Traditional Versions of CLT
		7.6 The Law of Large Numbers for Higher Moments
			7.6.1 Higher Moments
		7.7 Mean deviation for a Stable Distributions
	8 How Much Data Do You Need? An Operational Metric for Fat-tailedness^‡
		8.1 Introduction and Definitions
		8.2 The Metric
		8.3 Stable Basin of Convergence as Benchmark
			8.3.1 Equivalence for Stable distributions
			8.3.2 Practical significance for sample sufficiency
		8.4 Technical Consequences
			8.4.1 Some Oddities With Asymmetric Distributions
			8.4.2 Rate of Convergence of a Student T Distribution to the Gaussian Basin
			8.4.3 The Lognormal is Neither Thin Nor Fat Tailed
			8.4.4 Can Kappa Be Negative?
		8.5 Conclusion and Consequences
			8.5.1 Portfolio Pseudo-Stabilization
			8.5.2 Other Aspects of Statistical Inference
			8.5.3 Final comment
		8.6 Appendix, Derivations, and Proofs
			8.6.1 Cubic Student T (Gaussian Basin)
			8.6.2 Lognormal Sums
			8.6.3 Exponential
			8.6.4 Negative Kappa, Negative Kurtosis
	9 Extreme Values and Hidden Risk ^*,†
		9.1 Preliminary Introduction to EVT
			9.1.1 How Any Power Law Tail Leads to Fréchet
			9.1.2 Gaussian Case
			9.1.3 The Picklands-Balkema-de Haan Theorem
		9.2 The Invisible Tail for a Power Law
			9.2.1 Comparison with the Normal Distribution
		9.3 Appendix: The Empirical Distribution is Not Empirical
	B The Large Deviation Principle, In Brief
	C Calibrating under Paretianity
		C.1 Distribution of the sample tail Exponent
	10 "It is what it is": Diagnosing the SP500^†
		10.1 Paretianity and Moments
		10.2 Convergence Tests
			10.2.1 Test 1: Kurtosis under Aggregation
			10.2.2 Maximum Drawdowns
			10.2.3 Empirical Kappa
			10.2.4 Test 2: Excess Conditional Expectation
			10.2.5 Test 3- Instability of 4^th moment
			10.2.6 Test 4: MS Plot
			10.2.7 Records and Extrema
			10.2.8 Asymmetry right-left tail
		10.3 Conclusion: It is what it is
	D The Problem with Econometrics
		D.1 Performance of Standard Parametric Risk Estimators
		D.2 Performance of Standard NonParametric Risk Estimators
	E Machine Learning Considerations
		E.0.1 Calibration via Angles
Predictions, Forecasting, and Uncertainty
	11 Probability Calibration Calibration Under Fat Tails ^‡
		11.1 Continuous vs. Discrete Payoffs: Definitions and Comments
			11.1.1 Away from the Verbalistic
			11.1.2  There is no defined "collapse", "disaster", or "success" under fat tails
		11.2 Spurious overestimation of tail probability in psychology
			11.2.1 Thin tails
			11.2.2 Fat tails
			11.2.3 Conflations
			11.2.4 Distributional Uncertainty
		11.3 Calibration and Miscalibration
		11.4 Scoring Metrics
		11.5 Scoring Metrics
			11.5.1 Deriving Distributions
		11.6 Non-Verbalistic Payoff Functions and The Good News from Machine Learning
		11.7 Conclusion:
		11.8 Appendix: Proofs and Derivations
			11.8.1 Distribution of Binary Tally P^(p)(n)
			11.8.2 Distribution of the Brier Score
	12 Election Predictions as Martingales: An Arbitrage Approach^‡
		12.0.1 Main results
			12.0.2 Organization
			12.0.3  A Discussion on Risk Neutrality
		12.1 The Bachelier-Style valuation
		12.2 Bounded Dual Martingale Process
		12.3 Relation to De Finetti's Probability Assessor
		12.4 Conclusion and Comments
Inequality Estimators under Fat Tails
	13 Gini estimation under infinite variance ^‡
		13.1 Introduction
		13.2 Asymptotics of the Nonparametric Estimator under Infinite Variance
			13.2.1 A Quick Recap on -Stable Random Variables
			13.2.2 The -Stable Asymptotic Limit of the Gini Index
		13.3 The Maximum Likelihood Estimator
		13.4 A Paretian illustration
		13.5 Small Sample Correction
		13.6 Conclusions
	14 On the Super-Additivity and Estimation Biases of Quantile Contributions ^‡
		14.1 Introduction
		14.2 Estimation For Unmixed Pareto-Tailed Distributions
			14.2.1 Bias and Convergence
		14.3 An Inequality About Aggregating Inequality
		14.4 Mixed Distributions For The Tail Exponent
		14.5 A Larger Total Sum is Accompanied by Increases in "0362_q
		14.6 Conclusion and Proper Estimation of Concentration
			14.6.1 Robust methods and use of exhaustive data
			14.6.2 How Should We Measure Concentration?
Shadow Moments Papers
	15 On the shadow moments of apparently infinite-mean phenomena ( with P. Cirillo)^‡
		15.1 Introduction
		15.2 The dual Distribution
		15.3 Back to Y: the shadow mean (or population mean)
		15.4 Comparison to other methods
		15.5 Applications
	16 On the tail risk of violent conflict and its underestimation (with P. Cirillo)^‡
		16.1 Introduction/Summary
		16.2 Summary statistical discussion
			16.2.1 Results
			16.2.2 Conclusion
		16.3 Methodological Discussion
			16.3.1 Rescaling method
			16.3.2 Expectation by Conditioning (less rigorous)
			16.3.3 Reliability of data and effect on tail estimates
			16.3.4 Definition of an "event"
			16.3.5 Missing events
			16.3.6 Survivorship Bias
		16.4 Data analysis
			16.4.1 Peaks over Threshold
			16.4.2 Gaps in Series and Autocorrelation
			16.4.3 Tail Analysis
			16.4.4 An Alternative View on Maxima
			16.4.5 Full Data Analysis
		16.5 Additional robustness and reliability tests
			16.5.1 Bootstrap for the GPD
			16.5.2 Perturbation Across Bounds of Estimates
		16.6 Conclusion: is the world more unsafe than it seems?
		16.7 Acknowledgments
	F What are the chances of a third world war?^*,†
Metaprobability Papers
	17 How Thick Tails Emerge From Recursive Epistemic Uncertainty^†
		17.1 Methods and Derivations
			17.1.1 Layering Uncertainties
			17.1.2 Higher Order Integrals in the Standard Gaussian Case
			17.1.3 Effect on Small Probabilities
		17.2 Regime 2: Cases of decaying parameters  a( n)
			17.2.1 Regime 2-a;``Bleed'' of Higher Order Error
			17.2.2 Regime 2-b; Second Method, a Non Multiplicative Error Rate
		17.3 Limit Distribution
	18 Stochastic Tail Exponent For Asymmetric Power Laws^†
		18.1 Background
		18.2 One Tailed Distributions with Stochastic Alpha
			18.2.1 General Cases
			18.2.2 Stochastic Alpha Inequality
			18.2.3 Approximations for the Class P
		18.3 Sums of Power Laws
		18.4 Asymmetric Stable Distributions
		18.5 Pareto Distribution with lognormally distributed
		18.6 Pareto Distribution with Gamma distributed Alpha
		18.7 The Bounded Power Law in Cirillo and Taleb (2016)
		18.8 Additional Comments
		18.9 Acknowledgments
	19 Meta-Distribution of P-Values and P-Hacking^‡
		19.1 Proofs and derivations
		19.2 Inverse Power of Test
		19.3 Application and Conclusion
	G Some confusions in behavioral economics
		G.1 Case Study: How the myopic loss aversion is misspecified
Option Trading and Pricing under Fat Tails
	20 Financial theory's failures with option pricing^†
		20.1 Bachelier not Black-Scholes
			20.1.1 Distortion from Idealization
			20.1.2 The Actual Replication Process:
			20.1.3 Failure: How Hedging Errors Can Be Prohibitive.
	21 Unique Option Pricing Measure With Neither Dynamic Hedging nor Complete Markets^‡
		21.1 Background
		21.2 Proof
			21.2.1 Case 1: Forward as risk-neutral measure
			21.2.2 Derivations
		21.3 Case where the Forward is not risk neutral
		21.4 comment
	22 Option traders never use the Black-Scholes-Merton formula^*,‡
		22.1 Breaking the Chain of Transmission
		22.2 Introduction/Summary
			22.2.1 Black-Scholes was an argument
		22.3 Myth 1: Traders did not "price" options before Black-Scholes
		22.4 Methods and Derivations
			22.4.1 Option formulas and Delta Hedging
		22.5 Myth 2: Traders Today use Black-Scholes
			22.5.1 When do we value?
		22.6 On the Mathematical Impossibility of Dynamic Hedging
			22.6.1 The (confusing) Robustness of the Gaussian
			22.6.2 Order Flow and Options
			22.6.3 Bachelier-Thorp
	23 Option Pricing Under Power Laws: A Robust Heuristic^*,‡
		23.1 Introduction
		23.2 Call Pricing beyond the Karamata constant
			23.2.1 First approach, S is in the regular variation class
			23.2.2 Second approach, S has geometric returns in the regular variation class
		23.3 Put Pricing
		23.4 Arbitrage Boundaries
		23.5 Comments
	24 Four Mistakes in Quantitative Finance^*,‡
		24.1 Conflation of Second and Fourth Moments
		24.2 Missing Jensen's Inequality in Analyzing Option Returns
		24.3 The Inseparability of Insurance and Insured
		24.4 The Necessity of a Numéraire in Finance
		24.5 Appendix (Betting on Tails of Distribution)
	25 Tail Risk Constraints and Maximum Entropy (w. D.& H. Geman)^‡
		25.1 Left Tail Risk as the Central Portfolio Constraint
			25.1.1 The Barbell as seen by E.T. Jaynes
		25.2 Revisiting the Mean Variance Setting
			25.2.1 Analyzing the Constraints
		25.3 Revisiting the Gaussian Case
			25.3.1 A Mixture of Two Normals
		25.4 Maximum Entropy
			25.4.1 Case A: Constraining the Global Mean
			25.4.2 Case B: Constraining the Absolute Mean
			25.4.3 Case C: Power Laws for the Right Tail
			25.4.4 Extension to a Multi-Period Setting: A Comment
		25.5 Comments and Conclusion
		25.6 Appendix/Proofs
Bibliography and Index