Static arbitrage: the constraints every quote must respect

A handful of relationships between option prices must hold regardless of any model, because violating them lets someone build a portfolio with a guaranteed non-negative payoff and a negative cost. A market maker checks them continuously, and an interviewer will hand you a quote sheet with one broken in it.

Constraint	Statement	The trade if violated
Monotone in strike	$C(K)$ falls as $K$ rises	Buy the cheap low strike, sell the dear high one
Call spread bound	$0 \le C(K_1) - C(K_2) \le (K_2 - K_1)e^{-rT}$	Buy or sell the spread against cash
Butterfly non-negativity	$C(K_1) - 2C(K_2) + C(K_3) \ge 0$	Buy the butterfly for a credit
Calendar	$C(T_2) \ge C(T_1)$ for $T_2 > T_1$ , same strike	Buy the long-dated, sell the short-dated
Put–call parity	$C - P = S - Ke^{-rT}$	Conversion or reversal
Lower bound	$C \ge \max(S - Ke^{-rT},\, 0)$	Buy the call, short the stock, lend the strike

Table 4.1 · The constraints. Only the calendar row needs a caveat: it holds cleanly for European options without dividends, and a large dividend between the two expiries can invert it legitimately.

Derivation 4.2

Why prices must be convex in strike

The butterfly constraint is convexity, and convexity follows from the payoff.

$\text{payoff of } \tfrac12\text{ at } K_1 + \tfrac12 \text{ at } K_3 \ \ge\ \text{payoff at } K_2$
With $K_2$ the midpoint, the average payoff of the wings dominates pointwise.
$\Rightarrow \tfrac12 C(K_1) + \tfrac12 C(K_3) \ge C(K_2)$
A portfolio that never pays less must never cost less.
$\Rightarrow C(K_1) - 2C(K_2) + C(K_3) \ge 0$

C \text{ is convex in } K

Figure 4.3 · Why a butterfly can never pay out less than nothing. The three legs are drawn faintly and their sum in full. It is zero outside the wings, worth

\$10

at the middle strike, and never below zero anywhere — which is the entire reason being paid to put it on would be an arbitrage rather than a view.

Why these are different from a mispricing. Saying an option is cheap on your model means you and the market disagree about volatility, and you might be wrong. Saying a butterfly trades at a credit means someone will pay you to take a portfolio that can never pay out less than nothing — no model, no view, no volatility assumption. That is why these constraints are checked mechanically and continuously: a violation is free money or, far more often, a sign that one of your inputs is stale.

Example 4.4

Calls on the same expiry are quoted at $\$12.00$ for the $\$90$ strike, $\$7.00$ for the $\$100$ and $\$3.50$ for the $\$110$ . Is anything wrong?

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Worked solution

Formula $C(K_1) - 2C(K_2) + C(K_3) \ge 0$
Substitute $= 12.00 - 2(7.00) + 3.50$
Solve $= 12.00 - 14.00 + 3.50 = 1.50 \ge 0 \quad \checkmark$
$\text{Call spreads: } 12.00 - 7.00 = 5.00 \le 10 \quad \checkmark$
$7.00 - 3.50 = 3.50 \le 10 \quad \checkmark$
Answer $\text{consistent — no static arbitrage}$

Sanity check. Change the middle quote to $\$8.50$ and the butterfly becomes $12.00 - 17.00 + 3.50 = -\$1.50$ : you would be paid $\$1.50$ to hold a structure whose payoff is never negative. That is the violation to look for.

Static arbitrage: the constraints every quote must respect

After this lesson you should be able to

Why prices must be convex in strike

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