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THE OPTION ON THE PLOT

Convexity: When Volatility Starts Paying You

Option contract with a fixed price, six-month expiry and a right to decline.The contract shows strike K, fee p and two available branches: buy the land or decline. The illustration assumes the fee is the only liability; neither branch records an executed decision. Option K6 months p Buy Decline
Illustration: K is the fixed purchase price; p is the option fee. The choices remain available until expiry. This is not a reported land transaction.

In this illustrative case, a builder pays a fee for a real option to buy land at a fixed strike price within six months. If the purchase is unattractive, the builder can decline. The fee is the maximum loss only if the contract creates no further liability.

THE HOCKEY STICK

A call bends at the strike

Piecewise-linear call payoff at expiry, net of a fixed premium.The payoff is flat at minus the premium below strike K, then rises one for one with underlying value S. A chord joins two points on the payoff and lies on or above it. Premium shifts the curve vertically. PayoffS at expiry −p0 K Chord
Illustrative expiry payoff, with a true chord between two points. Subtracting a fixed premium shifts the graph vertically without changing its curvature.

At expiry, the simplified net payoff is max(S − K, 0) − p: land value S, fixed strike K and premium p, excluding financing and transaction costs. The slope changes from zero to one. A function is convex on an interval when every chord lies on or above it; for a differentiable function, its slope does not decrease. Capped loss and open upside describe this option, but do not by themselves prove convexity.

THE CHORD TEST

The payoff at the mean is no greater than the mean payoff

Two equally likely states separate around a fixed mean on a square-function payoff.The horizontal axis is the state and the vertical axis is payoff. Each endpoint has probability one half. Their mean state μ remains fixed; the mean endpoint payoff E[f(X)] rises above f(μ), the payoff at the mean. The accent segment is their Jensen gap. This strictly convex example does not exclude equality for linear payoffs. Payoff½ per stateState μ E[f(X)] f(μ)
Jensen, Acta Mathematica 30 (1906), 175–193, doi:10.1007/BF02418571. Diagram: f(x)=x², equal weights and a fixed mean. E[f(X)] labels mean payoff; f(μ) labels payoff at the mean state.

Jensen's inequality states f(E[X]) ≤ E[f(X)] for convex f, with X in its domain and the relevant expectations finite. Here each state has probability ½; moving them equally apart preserves their mean μ. The square function creates a positive gap. A straight segment gives equality. Comparing arbitrary distributions requires a mean-preserving spread in the convex-order sense; a larger variance alone is insufficient. See Jensen's original paper and the convex-order definition.

PRICING VOLATILITY

Positive vega is a conditional pricing result

A European call's model value increases with volatility while all other inputs stay fixed.The calculated curve uses S equal to K, one year to expiry, and zero interest and dividend rates. The horizontal axis is annual volatility; the vertical axis is call value divided by S. Fixed markers compare 20% and 80% volatility. These are model inputs, not observed market data. Call value / SVolatility 00.20.4 0%100% 20% 80%
Model illustration: European call, S = K, one year remaining, zero interest and dividend rates. C/S = 2Φ(σ/2) − 1; Φ is the standard normal CDF. The curve varies annual volatility only. Sources: Black and Scholes (1973), eq. 13, doi:10.1086/260062; Merton (1973), doi:10.2307/3003143.

For standard European calls and puts, Black–Scholes–Merton vega is positive before expiry when the underlying price and volatility are positive, with other inputs fixed. Vega is a sensitivity derived from the model. Its assumptions include a tradable underlying, continuous hedging and frictionless markets. A land option may not meet them; a higher model value also says nothing by itself about whether the purchase price is attractive. See When Uncertainty Pays You.

THE THIRD RESPONSE

The resilient resists shocks and stays the same; the antifragile gets better.

Nassim Nicholas Taleb, Antifragile (2012), Prologue, section II

Taleb's distinction asks what variability does to a particular exposure. His formal treatment with Raphaël Douady studies sensitivity to dispersion in specified tails and links it to nonlinear response. A favorable response to one stressor does not make a person or business immune to another. Sources: Antifragile, Prologue and Taleb and Douady's final manuscript.

EXPOSURE AND SURVIVAL

Size for survival; test the response separately

Concave, constant and convex response profiles under the same equal-weight spread.Each pair of endpoint states moves equally apart around a fixed mean. The average response falls for the concave profile, stays fixed for the constant profile and rises for the convex profile. These are conditional mathematical illustrations, not evidence of the safety or return of a reserve, portfolio or biological system.Fragile Robust Antifragile Equal-weight spread
Conditional response profiles: higher vertical position means a larger response. These are not measured returns or recommended allocations. Taleb, Antifragile (2012), Appendix I, figs. 19–27; Taleb and Douady (2013), doi:10.1080/14697688.2013.800219.

These schematic profiles compare the same equal-weight spread around a fixed mean. Their average responses fall, stay level or rise. Taleb's barbell heuristic combines a protected reserve with a limited speculative allocation. Its safety depends on the risks being assessed: liquidity failure, shared liabilities or joint losses can reach both parts. Predetermined affordable loss limits exposure; it does not certify convexity.

CONCENTRATED RETURNS

The top tenth captured 48–93% within the samples

The top tenth of sample members captured 48–93% of returns across eight innovation datasets.One of ten equal decile markers connects to a fixed interval from 48% to 93% on a zero-to-100% return-share axis. The endpoints are the reported minimum and maximum across datasets. This is not a fitted distribution, confidence interval, pooled-sample share or prediction.Top tenth 48%93% 0%100%Return share
Source: Scherer and Harhoff, Research Policy 29 (2000), 559–566, doi:10.1016/S0048-7333(99)00089-X. Reported abstract-level range across eight datasets. Ten equal marks denote deciles, not ten observations; individual dataset values are not shown.

Scherer and Harhoff report eight invention and innovation datasets from firms and universities. The interval shows the top tenth's share of returns across those samples. It is neither a confidence interval nor a pooled share, fitted power law or forecast. The result motivates careful corpus comparison; it does not establish a land-option base rate or show that a model can identify opportunities. The authors' institutional abstract reports the summary.

Use the terms to test an uncertain estimate

Use probabilities where the reference class supports them. Under Knightian uncertainty, inspect the enforceable loss limit, possible upside restrictions, total price and expiry alongside the estimate. A favorable curve cannot compensate for an unaffordable commitment.

WHAT KILLS THE CURVE

A ceiling introduces a second kink

A cap flattens the rising call payoff and introduces a downward change in slope.The payoff retains the original loss floor and rising straight arm, then becomes flat at the ceiling. A chord spanning the upper kink lies below the payoff, demonstrating the failure of convexity over that interval. This does not imply that every smaller interval is non-convex. PayoffS at expiry −pK Ceiling
Same expiry payoff as the earlier call example, with an added upper cap. The dashed chord demonstrates failure of convexity across the upper kink. The lost uncapped arm remains dashed in grey.

Capping the rising arm changes its slope from one back to zero. This call-shaped payoff is no longer convex across the capped region: a chord spanning the upper kink falls below it. A cap can still leave smaller intervals convex. Price and time are separate checks; paying too much or reaching expiry can make a convex position unprofitable.

WHERE IT PAYS

Check the curve, then floor, ceiling, price and time

  • Floor: identify every liability that could escape the fee.
  • Ceiling: identify who or what can limit the gain.
  • Price: include the premium and other committed costs.
  • Time: establish when the right ends and what must happen before then.

Convexity concerns curvature over the relevant domain. The option's terms determine practical exposure; probabilities and price determine its attractiveness. Carry those separate checks back to When Uncertainty Pays You.

Read the transcript

01 · THE OPTION ON THE PLOT

A builder pays a fee for the right to buy a plot at a price fixed today. The right lasts for six months. This is an illustrative contract, not a reported transaction. Suppose a new road makes the site more valuable. The builder can buy at the agreed price. If the purchase is unattractive, the builder can let the right expire and the fee is lost. That limit holds only if the agreement creates no further liability. The road changes the value of the land. The contract changes how much of that change reaches the builder. Its strike, expiry and right to decline are the starting point for the payoff.

02 · THE HOCKEY STICK

At expiry, compare the land's value with the strike price. This simplified diagram excludes financing and transaction costs. Below the strike, the right is unused and the net payoff stays at the negative premium. Above the strike, each extra unit of land value increases the payoff one for one. Subtracting the same fee from every outcome moves this graph down without changing its shape. Join two points on the payoff with a straight chord. Every such chord lies on or above the payoff. That is the convexity test over the stated interval. This call passes: its slope changes from zero to one at the strike. A loss cap and room for upside are useful terms to inspect, but they do not establish this property for an arbitrary business opportunity.

03 · THE CHORD TEST

Jensen's inequality compares a payoff at an average state with the average of the payoffs. Start with two equally likely states and a square-function payoff. Move the two states equally apart while holding the same mean. The average payoff sits on the midpoint of their chord, above the payoff at that mean. That vertical difference is the Jensen gap. The general statement allows equality: the payoff at the mean is no greater than the mean payoff. Both points on a straight segment give the same result. The relevant expectations must exist and be finite, and the payoff must be convex over the states being considered. For a comparison between arbitrary distributions, a mean-preserving spread gives the ordering we need. An increase in variance alone does not.

04 · PRICING VOLATILITY

The expiry payoff describes the contract's eventual result. A pricing model asks what the right is worth before it expires. Black and Scholes, and separately Merton, published their option-pricing work in nineteen seventy-three. The illustrated call has a strike equal to today's underlying price, one year remaining, and zero interest and dividend rates. Increase only volatility from twenty to eighty percent. This raises the model value. The sensitivity is called vega, derived from the pricing formula. Standard European calls and puts have positive vega before expiry under the model's conditions. Those conditions include a tradable underlying and continuous hedging without transaction costs. A land option can depart sharply from them. Even a correctly valued convex contract can be a poor purchase at the price offered.

05 · THE THIRD RESPONSE

Taleb puts the distinction this way in Antifragile: The resilient resists shocks and stays the same; the antifragile gets better. The question concerns a particular exposure to a particular source of variability. With Raphaël Douady, he formalizes sensitivity to dispersion in specified tails and studies its connection to nonlinear response. Their treatment allows mixed curvature and thresholds. It is more specific than saying that a person or a business likes uncertainty. Exposure that benefits from one kind of variation can still be damaged by another.

06 · EXPOSURE AND SURVIVAL

Give each profile the same equal-weight spread around a fixed mean. The average response falls for the concave profile, remains level for the constant one, and rises for the convex one. These are schematic responses over a stated range. They do not certify the behavior of an actual asset. Taleb's barbell heuristic pairs a protected reserve with a speculative allocation small enough to lose. Protection must be assessed against named risks. A common creditor, a liquidity failure or a shared liability can reach both parts. The builder's fee also belongs in the total exposure across all commitments. Affordable-loss sizing helps preserve the ability to continue. The separate curvature test asks how the payoff changes when the underlying states spread.

07 · CONCENTRATED RETURNS

Scherer and Harhoff report eight datasets of invention and innovation returns attributable to firms and universities. They examine the top tenth of cases in each dataset. That group captured between forty-eight and ninety-three percent of total sample returns. The endpoints describe a range across samples, not a confidence interval or a single pooled return share. The data support concentration. They do not supply a fitted power law, the probability of success for this land option, or evidence that a model can find the next opportunity. The paper also warns that substantial variation can remain even in large portfolios. Spreading commitments does not ensure that the builder will capture a large winner.

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09 · READ THE SHAPE

A probability estimate is useful when its reference class supports it. Under Knightian uncertainty, a precise-looking number can hide how little is known. The contract offers a different set of things to inspect: the fee, the liabilities, the price at which the land can be bought, and the date the right expires. Read those terms alongside the estimate. An enforceable loss limit reduces one exposure; it leaves the purchase price and the uncertain benefits to be judged. A curve that passes the mathematical test can still describe a commitment the builder cannot afford.

10 · WHAT KILLS THE CURVE

Add a ceiling to the rising arm and the payoff becomes flat. The slope now changes from one back to zero. A chord across that kink lies below the payoff, so convexity fails over that interval. A capacity limit or contractual quota could create such a cap in a business exposure. The claim is local to the range under examination; smaller intervals can still be convex. Premium and time are separate issues. A fixed premium shifts the payoff down without changing curvature. Waiting reduces the remaining exercise window, and an option can expire unused. Neither the chord test nor the absence of an upper cap proves that the position will earn a profit.

11 · WHERE IT PAYS

Before holding the right, identify every liability that could escape the fee. Check what can limit the gain, count the premium and other committed costs, and establish when the right ends. These are the floor, ceiling, price and time checks. Then test the payoff over the relevant range. Convexity describes its curvature; the contract sets practical exposure; probabilities and price determine whether the position is attractive. Those questions travel together back to When Uncertainty Pays You.

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