Convexity Of Option Prices In Strike

Look at a listed option chain and check three neighboring call strikes. The middle one's price should never sit below the straight line connecting its two neighbors. This isn't a pattern that happens to hold empirically — it is a hard arbitrage boundary, because if it were ever violated, you could construct a trade that costs nothing, can never lose, and sometimes pays off.

A U-shaped bowl, never a tent

Think of three call option prices, plotted against their strikes, as three points you could connect with straight lines. If the middle point dips below the line joining its neighbors — forming a little valley — then buying the outer two and selling twice the middle one (a butterfly spread) costs you a negative amount of money, i.e. someone pays you to put the trade on, and the position can never be worth less than zero at expiry. Free money with no downside is exactly what "no arbitrage" forbids. So the curve of call prices against strike must always bow outward like the bottom of a bowl, never pinch inward like the peak of a tent — this is convexity in strike.

The formula

$$C(K_2) \;\le\; \lambda\, C(K_1) + (1-\lambda)\, C(K_3), \qquad K_2 = \lambda K_1 + (1-\lambda) K_3$$

In plain English: for any strike $K_2$ sitting between two other strikes $K_1 < K_2 < K_3$ (with $\lambda$ the fraction of the way $K_2$ sits between them), the price of the option at the middle strike can never exceed the straight-line blend of the two outer prices. Restated with equal strike spacing ($K_2 - K_1 = K_3 - K_2$), the condition becomes the familiar butterfly inequality: $C(K_2) \le \tfrac{1}{2}\big(C(K_1) + C(K_3)\big)$ — the middle strike costs at most the average of its two neighbors. This is exactly Breeden-Litzenberger's second derivative, $\partial^2 C/\partial K^2$, which must stay non-negative everywhere — the curve's curvature is literally a probability density, and probability densities cannot be negative.

Worked example 1: spotting a violation and trading it

Calls trade at: $100 strike costs $8, $105 strike costs $4.50, $110 strike costs $2. Check convexity at the middle strike: the straight-line average of the outer two is $(8 + 2)/2 = 5.00$, i.e. $5.00. The actual $105 call costs $4.50, which is below $5.00 — convexity holds here, no arbitrage. Now suppose instead the $105 call is quoted at $5.60, above the $5.00 average. Buy one $100 call ($8), buy one $110 call ($2), sell two $105 calls ($2 \times 5.60 = 11.20$, i.e. $11.20 received). Net cash today: $-8 - 2 + 11.20 = 1.20$, i.e. $1.20 received up front. At expiry this butterfly's payoff is zero everywhere except a small bump near $105, where it is strictly non-negative — you collected $1.20 today for a position that can never cost you anything later. That is riskless profit, and it is exactly why real markets never let this configuration persist.

Worked example 2: reading the shape as a probability

Take three adjacent, tightly spaced strikes 1 point apart: $99, $100, $101, priced at $3.40, $2.55, $1.85. The discrete second derivative approximates the probability density at $100: $C(99) - 2C(100) + C(101) = 3.40 - 5.10 + 1.85 = 0.15$. Dividing by the squared strike spacing ($1^2 = 1$) and by the discount factor (assume $\approx 1$ for a short-dated option) gives an implied probability density of about $0.15$ per dollar around $100 — a small, positive number, exactly as convexity guarantees it must be. If those three prices had instead produced a negative second difference, that would be mathematically equivalent to claiming a negative probability, which is nonsense, and the market would immediately be arbitrageable via the butterfly above.

Drag the strike and premium here to build intuition for the middle leg of a butterfly — then imagine two of these sold against one bought at $100 and one at $110, which is exactly the trade Example 1 executed.

What this means in practice

Every options market maker's quoting system enforces strike convexity automatically before a quote is ever shown, because the alternative is being picked off by a butterfly trade within seconds of posting a bad price. Convexity also underlies how the entire risk-neutral density is extracted from a listed strike chain (see Breeden-Litzenberger Formula) and is the reason volatility surfaces are fit with smooth, arbitrage-consistent interpolation rather than connecting quoted strikes with straight lines.