Convexity + Anti-fragility (Resources)
What rings true in life also does in the markets
Overview
This pre-read section prepares you to explore how convex payoff structures and antifragile positioning can protect and profit from uncertainty. You’ll build intuition for asymmetric risk-reward profiles and understand why traditional “balanced” approaches often fail during regime shifts.
1. Conceptual Foundations: Antifragility and Uncertainty
Nassim Nicholas Taleb, Antifragile: Things That Gain from Disorder
Required Chapters:
Book I, Chapter 1: “Between Damocles and Hydra” - Introduction to fragility, robustness, and antifragility as a triad
Book III, Chapter 13: “Teaching Birds How to Fly” - On lecturing birds how to fly and Aristotle’s practical wisdom
Book IV, Chapter 14: “When Two Things Are Not the Same Thing” - The barbell strategy and bimodal approaches
Book V, Chapter 18: “On the Difference Between a Large Stone and a Thousand Pebbles” - Why small is beautiful and the logic of fragmentation
Book VI, Chapter 20: “Time and Fragility” - Optionality and how time interacts with convexity
Book VI, Chapter 22: “To Live Long, but Not Too Long” - The benefits of volatility and variability
Key concepts to extract: Barbell thinking (combining extreme safety with extreme risk), optionality as free or cheap convexity, and why volatility benefits antifragile systems.
Nassim Nicholas Taleb, The Black Swan: The Impact of the Highly Improbable
Required Sections:
Prologue: “On the Plumage of Birds” - The Turkey Problem and inductive reasoning
Part One, Chapter 1: “The Apprenticeship of an Empirical Skeptic” - Understanding uncertainty
Part Three, Chapter 11: “How to Look for Bird Poop” - Finding what we need to know vs. what we already know
Part Four, Chapter 15: “The Bell Curve, That Great Intellectual Fraud” - Mediocristan vs. Extremistan and fat tails
Part Four, Chapter 17: “Locke’s Madmen, or Bell Curves in the Wrong Places” - The Gaussian blindness
Key concepts to extract: Fat-tailed distributions, why standard deviation misleads in Extremistan, and the dominance of rare events.
Nassim Nicholas Taleb, Fooled by Randomness: The Hidden Role of Chance in Life and Markets
Required Sections:
Part I, Chapter 3: “A Mathematical Meditation on History” - Alternative histories and path dependence
Part I, Chapter 5: “Survival of the Least Fit” - Why the best performing may simply be the luckiest
Part II, Chapter 6: “Skewness and Asymmetry” - Why outcomes matter more than frequency
Part II, Chapter 8: “Too Many Millionaires Next Door” - Rare events and Russian roulette economics
Part III, Chapter 11: “Randomness and Our Mind” - Behavioral biases in assessing probability
Key concepts to extract: How we underestimate the role of luck, why asymmetric payoffs matter more than win rates, and our cognitive failures with probability.
2. Practical Applications: Tail Risk and Convex Hedging
Mark Spitznagel, Safe Haven: Investing for Financial Storms
Required Chapters:
Chapter 1: “The Tao of Risk” - Introduction to the cost-benefit paradox of risk mitigation
Chapter 2: “Risk Mitigation and Cost-Benefit” - Why “expensive” protection can be cheap geometrically
Chapter 3: “The Safe Haven Strategy” - Framework for tail risk hedging
Chapter 4: “The Cost of Insurance and the Benefit of Loss Mitigation” - Understanding drag vs. protection
Chapter 5: “Local versus Global” - Local arithmetic returns vs. global geometric returns
Chapter 7: “Fragility and Antifragility” - Spitznagel’s interpretation of Taleb’s framework applied to portfolios
Key concepts to extract: The geometric argument for paying up for convex hedges, how compounding changes when drawdowns are mitigated, and the mathematics of tail risk protection.
Mark Spitznagel, The Dao of Capital: Austrian Investing in a Distorted World (Optional)
Recommended Sections:
Chapter 2: “The Roundabout” - Austrian capital theory basics and lengthening the production structure
Chapter 4: “The Austrian Advantage” - Patience, asymmetry, and strategic positioning
Chapter 6: “Waiting” - The value of positioning and patience in investing
Chapter 8: “The Homestead” - Antifragility through self-sufficiency and optionality
Key concepts to extract: How roundabout methods create superior long-term results, the role of patience in asymmetric investing, and Austrian economic principles applied to markets.
3. Options Mechanics and Payoff Structures
Recommended Resource: Lawrence G. McMillan, Options as a Strategic Investment (5th Edition)
Required Sections:
Chapter 2: “Covered Call Writing” (pages 31-67)
Focus on: Payoff diagrams, profit/loss at expiration, how upside is capped
Assignment mechanics and early exercise considerations
Chapter 16: “Selling Puts” (pages 258-282)
Cash-secured put mechanics
Strike selection based on volatility environment
Assignment and stock acquisition process
Chapter 25: “LEAPS” (pages 483-512)
Long-dated call options as stock substitutes
Time decay characteristics vs. short-term options
Creating leverage with defined risk
Alternative/Supplementary: Sheldon Natenberg, Option Volatility and Pricing (2nd Edition)
Chapter 6: “Volatility” (pages 87-112) - Understanding how volatility affects option pricing
Chapter 8: “Risk Measurement I” (pages 135-158) - Delta, gamma, and convexity
Online Resources for Visual Learning:
The Options Industry Council (OIC) - www.optionseducation.org
“Covered Calls” interactive tutorial
“Cash-Secured Puts” strategy guide with payoff diagrams
“LEAPS Strategies” comprehensive module
CBOE Learning Center - www.cboe.com/education
“Understanding Option Greeks” - Focus on gamma as the measure of convexity
Interactive payoff diagram tools
4. Regime Shifts and the Fragility of the “Middle”
Homework: “Why Balanced Portfolios Break Under Stress”
Assignment: Come up with your own reflections to the following questions:
Key items:
The Correlation Breakdown
Historical correlation patterns during normal vs. crisis periods
Case studies: 2008 Financial Crisis, March 2020 COVID crash, 2022 bond-stock correlation reversal
Hidden Leverage in “Safe” Assets
Duration risk in bond portfolios
REITs, utilities, and dividend stocks as rate-sensitive bets
How seemingly conservative positions amplify losses
Liquidity Stress and Forced Selling
When diversification fails: all assets become correlated to liquidity
Margin calls, redemptions, and cascade effects
The 60/40 Portfolio Illusion
Why the classic balanced allocation is fragile to regime shifts
Historical backtest limitations and survivorship bias
Barbell Alternative
Contrasting fragile “middle ground” with antifragile extremes
Maximum safety + maximum convexity vs. moderate risk everywhere
Supplementary Reading:
Artemis Capital Management, “The Allegory of the Hawk and Serpent” (2020)
Available free at: artemiscm.com
Section II: “The Dragon Portfolio” - Understanding correlation regimes across inflation and growth scenarios
Section III: “Volatility and the Alchemy of Risk” - How volatility clustering affects portfolio outcomes
Resolve Asset Management, “The Allegory of the Hawk and Serpent: A Summary” (2020)
Shorter 10-page digest if time is constrained
5. Mathematical Intuition (No Advanced Math Required)
Jensen’s Inequality
Primary Source:
Keith J. Devlin, “The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern”
Chapter 6: “The Measure of Uncertainty” (pages 89-112) - Accessible introduction to expected value and nonlinearity
Online Resource:
Khan Academy: “Jensen’s Inequality” module (AP Statistics)
Video: “Convex functions and Jensen’s Inequality” (12 minutes)
Link: khanacademy.org/math/statistics-probability
Course Handout: “Jensen’s Inequality for Investors” (5 pages)
Visual examples with option payoffs
Why volatility increases the value of convex positions
Simple numerical examples without calculus
Kelly Criterion
Primary Source:
William Poundstone, Fortune’s Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street
Chapter 7: “Entropy” (pages 109-128) - Kelly’s original insight
Chapter 10: “Beating the Market” (pages 163-185) - Application to investing
Chapter 16: “Amazon.con” (pages 265-282) - When Kelly sizing goes wrong (over-betting)
Technical Paper (Optional):
Ed Thorp, “The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market” (2008)
Available at: edwardothorp.com
Read: Introduction and Section 2 “The Kelly Criterion” (pages 1-6)
Course Handout: “Kelly Sizing and Geometric Growth” (8 pages)
Derivation in plain English
Why fractional Kelly is often better
Position sizing examples for options strategies
Expected Value vs. Path-Dependence
Primary Sources:
Ole Peters & Murray Gell-Mann, “Evaluating Gambles Using Dynamics” (2016)
Published in Chaos journal, available at: arxiv.org/abs/1405.0585
Read: Abstract, Introduction, and Section II “Ensemble and Time Averages” (pages 1-4)
Skip the heavy math; focus on conceptual examples
Nassim Taleb, “Ergodicity” Technical Incerto essay
Available at: fooledbyrandomness.com (Technical Papers section)
Or in Skin in the Game, Chapter 19: “The Logic of Risk Taking” (pages 189-208)
Course Handout: “Why Good Bets Can Still Ruin You” (6 pages)
Coin flip examples with absorbing barriers
Sequence-of-returns risk in retirement
Visual demonstrations of path-dependence
Why time averages ≠ ensemble averages for most real processes
Additional Math Resources:
3Blue1Brown YouTube Channel
“Visualizing the chain rule and product rule” - Understanding derivatives of payoff functions
“But what is a convex function?” - Visual intuition for convexity
Link: youtube.com/c/3blue1brown (search for “convexity” and “derivatives”)
Course Video Lecture: “Gamma and Convexity” (20 minutes)
Will be posted on course portal
Covers how second derivatives create antifragility
6. Preparation Questions
As you complete these readings, consider:
How does a barbell strategy differ from diversification? What are you giving up, and what are you gaining?
Why might an expensive tail hedge be geometrically “cheap” over time? (Reference Spitznagel Chapter 2)
What makes an options payoff convex? How does this relate to antifragility? (Reference option payoff diagrams)
When do correlations between assets increase, and why does this matter for portfolio construction? (Reference the regime shifts essay)
How does Jensen’s Inequality explain why volatility can be valuable rather than merely risky? (Reference your course handout)
What’s the difference between a strategy with positive expected value and one that’s geometrically optimal? (Reference Kelly Criterion readings)
7. Learning Objectives
By completing this pre-read, you should be able to:
Explain the difference between fragile, robust, and antifragile systems using Taleb’s framework
Identify convex and concave payoff structures in options strategies through payoff diagrams
Understand why traditional balanced portfolios can be fragile to regime shifts
Apply basic mathematical intuition about convexity, dispersion, and path-dependence
Articulate the philosophical case for barbell positioning in uncertain environments
Calculate basic position sizing using Kelly principles
Recognize when correlation assumptions break down
8. Reading Schedule Suggestion
Week 1: Conceptual Foundation (8-10 hours)
Days 1-2: Taleb’s Antifragile chapters
Days 3-4: The Black Swan sections
Day 5: Fooled by Randomness sections
Week 2: Practical Applications (6-8 hours)
Days 1-2: Spitznagel’s Safe Haven
Days 3-4: Options mechanics (McMillan selections or OIC modules)
Day 5: Review options payoff diagrams and practice drawing your own
Week 3: Regime Analysis and Math (6-8 hours)
Days 1-2: Regime shifts essay and Artemis Capital paper
Days 3-4: Mathematical intuition readings (Jensen, Kelly, path-dependence)
Day 5: Watch supplementary videos and review course handouts