Convexity and Jensen's Inequality
How nonlinear payoffs interact with fat tails.
Convexity and Jensen's Inequality are central to Taleb's analysis of risk and decision-making under uncertainty. When payoffs are nonlinear — which they almost always are — the relationship between expected inputs and expected outputs is fundamentally different than intuition suggests.
Convexity and Concavity
Convex Function
A function is convex if for any two points and , and any :
Geometrically: the function lies below the line connecting any two points. The function "curves upward."
Concave Function
A function is concave if for any two points and , and any :
Geometrically: the function lies above the line connecting any two points. The function "curves downward."
Common Examples
Convex functions:
- (parabola opening upward)
- (exponential)
- (call option payoff)
Concave functions:
- (square root)
- (logarithm)
- (capped payoff)
Interactive: Geometric Definition of Convexity
g(x) curve
Chord connecting points
x₁
1.00
g(x₁) = 1.00
x₂
4.00
g(x₂) = 16.00
λx₁ + (1-λ)x₂
2.50
weighted avg
λ
0.50
weight on x₁
Function at weighted average
g(λx₁ + (1-λ)x₂) = 6.250
Weighted average of function values
λg(x₁) + (1-λ)g(x₂) = 8.500
g(λx₁ + (1-λ)x₂) ≤ λg(x₁) + (1-λ)g(x₂)
Convex: The function lies below the chord. The value at the weighted average (6.25) is less than the weighted average of values (8.50).
Jensen's Inequality
Jensen's Inequality
For a convex function and random variable :
For a concave function:
Why This Matters
Jensen's Inequality says you cannot just take the expected input and apply the function to get the expected output. The nonlinearity creates a systematic gap between and .
Interactive: Jensen's Inequality in Action
Distribution of X
Dashed line: E[X] = 5.00
Transform x²
Function is convex
E[X]
5.000
g(E[X])
25.000
Transform applied to mean
E[g(X)]
28.996
Average of transformed values
Jensen's Gap: E[g(X)] − g(E[X]) = 3.996
For this convex function, E[g(X)] ≥ g(E[X]). The average of the outputs (28.996) exceeds the output of the average input (25.000).
The Gap Explodes Under Fat Tails
Taleb's key observation: the gap between and can be enormous when has fat tails.
The Quadratic Case
Let (a convex function). Jensen's Inequality tells us:
The difference is the variance:
Now consider a Pareto distribution with :
- may be finite
- (variance is infinite!)
The gap is not just large — it is infinite. You cannot approximate by computing .
Read: “For Pareto with alpha less than or equal to 2, the expected value of X squared is infinite, while the square of the expected value is finite”
The average of squares vastly exceeds the square of the average
Interactive: How the Gap Scales with Variance
Distribution
Dashed line: E[X] = 5.00
Gap vs σ
Gap grows as σ² (quadratically)
E[X]
5.000
(E[X])²
25.000
E[X²]
25.999
Gap = Var(X)
0.999
Theoretical variance: 1.000
For Gaussian, Var(X) = σ² = 1.000
The Fat Tail Catastrophe
For Gaussian distributions, the Jensen gap is well-behaved: it equals σ² and grows quadratically with spread. You can estimate E[X²] reliably because variance is finite and stable.
Implications for Real-World Payoffs
Most payoffs in the real world are nonlinear. This nonlinearity interacts critically with fat tails.
Convex Payoff
A payoff that accelerates as the input grows. You benefit disproportionately from positive deviations and are relatively protected from negative ones.
Examples: call options, upside exposure, systems that improve under stress.
Concave Payoff
A payoff that decelerates as the input grows. Negative deviations hurt you more than positive deviations help.
Examples: bounded profits with unlimited losses, insurance sellers, systems that degrade under stress.
The Option Seller vs. Option Buyer
Option buyer (convex payoff):
- Pays a small premium
- Gains are unlimited if the underlying moves favorably
- Under fat tails, can be much larger than expected
Option seller (concave payoff):
- Receives a small premium
- Losses are unlimited if the underlying moves unfavorably
- Under fat tails, expected losses can be catastrophic
Taleb's Point About Option Mispricing
If markets price options assuming thin-tailed distributions, but actual returns have fat tails:
- Out-of-the-money options are systematically underpriced
- Option sellers are taking much more risk than they realize
- Option buyers are getting much more value than models suggest
This is one of Taleb's trading insights: buying cheap out-of-the-money options can be profitable precisely because the market underestimates fat tail probabilities.
Interactive: Option Pricing Under Fat Tails
Price Distribution (Current: $100)
Dashed line: Strike K = $105
Payoff Functions
P(in the money)
37.1%
Price > $105
Option Buyer E[payoff]
$3.81
Convex payoff
Option Seller E[payoff]
$-3.81
Concave payoff
Thin-Tailed Scenario
Under Gaussian assumptions, extreme price moves are very rare. The option buyer's expected payoff is $3.81. Markets typically price options assuming something close to this.
👆 Switch to "Student's t" to see how fat tails change the picture
Convexity and Decision Making
Jensen's Inequality has profound implications for how we should make decisions under uncertainty:
Seek Convex Payoffs
Under uncertainty, especially with fat tails, convex payoffs mean you benefit from volatility. The expected outcome is better than what you get from the average scenario.
Structure your exposure so that extreme events help rather than hurt.
Avoid Concave Payoffs
Concave payoffs mean volatility hurts you. Under fat tails, the expected outcome is much worse than what you get from the average scenario.
The worst cases are worse than you think, and they matter more than you think.
Interactive: How Volatility Affects Different Payoff Shapes
Distribution of X (μ = 4)
Dashed line: E[X] = 4.00
Expected Outcome vs Volatility
E[g(X)] increases with σ
E[X]
4.000
(constant = μ)
g(E[X])
16.000
(constant)
E[g(X)]
16.999
(varies with σ)
Convex: Volatility is Your Friend
With a convex payoff, more spread (volatility) increases your expected outcome. E[g(X)] grows with σ while g(E[X]) stays constant. You benefit from uncertainty.
Connection to Antifragility
This is the mathematical foundation of Taleb's concept of antifragility:
- • Antifragile = convex payoff → benefits from volatility
- • Fragile = concave payoff → harmed by volatility
- • Robust = linear payoff → neutral to volatility
Under fat tails, where extreme events are more likely, this distinction becomes even more critical. Seek convex exposure!
The Treadmill vs. Barbell Strategy
Treadmill (neutral): Consistent moderate exercise, moderate diet, moderate everything. Linear relationship between input and output.
Barbell (convex): Extreme safety (very low risk most of the time) combined with extreme exposure to positive outcomes. Very convex payoff structure.
Taleb argues the barbell approach — combining extreme caution with aggressive upside exposure — exploits convexity under uncertainty.
The Mathematical Perspective
For the mathematically inclined, here is why convexity matters so much under fat tails:
Read: “The gap is approximately half of g double-prime at the mean times the variance”
The convexity of g (measured by the second derivative) times the spread (variance) determines the gap
This approximation (from Taylor expansion) shows:
- If (convex), the gap is positive:
- If (concave), the gap is negative:
- The gap scales with variance — and under fat tails, variance can be infinite!
When Variance is Infinite
The approximation above breaks down when . In this case, the gap between and can be infinite too.
This is why fat tails matter so much for nonlinear payoffs: the standard intuition completely fails.
Key Takeaways
- Convex functions curve upward; concave functionscurve downward
- Jensen's Inequality: for convex , ; inequality reverses for concave
- The gap scales with variance — under fat tails, this gap can be enormous or even infinite
- Convex payoffs benefit from volatility; concave payoffs are hurt by it
- Under fat tails, seek convex exposure (like options) and avoid concave exposure (like selling options)
- You cannot evaluate expected payoffs by applying the payoff function to the expected input — the nonlinearity matters