Days to Expiry — Options Portfolio Tracker logo
Days to Expiry
Options Portfolio Clarity
←Back to Courses

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.

book near glass windowbook near glass window

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:

  1. 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

  2. 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

  3. Liquidity Stress and Forced Selling

    • When diversification fails: all assets become correlated to liquidity
    • Margin calls, redemptions, and cascade effects

  4. The 60/40 Portfolio Illusion

    • Why the classic balanced allocation is fragile to regime shifts
    • Historical backtest limitations and survivorship bias

  5. 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:

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:

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:

  1. How does a barbell strategy differ from diversification? What are you giving up, and what are you gaining?
  2. Why might an expensive tail hedge be geometrically “cheap” over time? (Reference Spitznagel Chapter 2)
  3. What makes an options payoff convex? How does this relate to antifragility? (Reference option payoff diagrams)
  4. When do correlations between assets increase, and why does this matter for portfolio construction? (Reference the regime shifts essay)
  5. How does Jensen’s Inequality explain why volatility can be valuable rather than merely risky? (Reference your course handout)
  6. 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