Arbitrage-Free Volatility Surfaces

Stretch a bedsheet over a mattress and pull it too tight in one place and it wrinkles somewhere else — the fabric can't be shaped into just anything, because it has to stay a single continuous surface. A volatility surface has the same kind of internal consistency requirement, except the "wrinkles" aren't cosmetic: a surface that violates them is quietly quoting a set of option prices that lets someone else assemble a trade that makes money with zero risk.

The two constraints

Calendar spread arbitrage is about time: total variance must never decrease as maturity increases, for a fixed strike.

$$w(K, T_2) \geq w(K, T_1) \quad \text{for } T_2 > T_1$$

In words: total variance $w$ (implied vol squared times time) at strike $K$, further out at $T_2$, must be at least as large as at the closer maturity $T_1$. If not, the longer-dated option would be cheaper than the shorter-dated one at the same strike, and a trader could buy the cheap one, sell the expensive one, and lock in a profit with no risk.

Butterfly arbitrage is about strikes: within a single maturity, the implied risk-neutral probability density must be non-negative everywhere, which translates into the call price being convex in strike.

$$C(K - \Delta K) - 2C(K) + C(K + \Delta K) \geq 0$$

In words: take three equally spaced strikes, and this specific combination — buy the wings, sell twice the middle, a textbook butterfly spread — must never cost less than zero, because a butterfly's payoff can never be negative. If it goes negative, the market implies a negative probability somewhere, which is nonsense.

Worked example 1 — checking calendar spread arbitrage

A 3-month ATM vol of 25% gives total variance $w_3 = 0.25^2 \times 0.25 = 0.0156$. A 6-month ATM vol of 20% gives $w_6 = 0.20^2 \times 0.5 = 0.02$. Since $0.02 \geq 0.0156$, this pair is fine — even though vol dropped, total variance still rose, because time did more work than vol fell. Now suppose the 6-month vol were instead 15%: $w_6 = 0.15^2 \times 0.5 = 0.01125$, less than $w_3 = 0.0156$. That surface has calendar arbitrage: it's priced as if less could happen by month six than had already happened by month three.

Worked example 2 — checking butterfly arbitrage

Three calls at $95, $100, $105 strikes on the same expiry are priced at $8.20, $5.50, and $3.60. Plugging into the butterfly test: $8.20 - 2(5.50) + 3.60 = 0.80$, positive — no arbitrage; the butterfly (buy the wings, sell two $100 calls) costs $0.80, consistent with a valid density. Now suppose the $100 call were mispriced at $6.10 instead: $8.20 - 2(6.10) + 3.60 = -0.40$. That butterfly would cost negative $0.40 to put on — someone gets paid $0.40 upfront to hold a position that can never pay out less than zero at expiry. Free money, and the $100 strike is quoted too rich relative to its neighbors.

What this means in practice

Anyone fitting or smoothing a volatility surface — pricing library, risk system, research model — has to check both constraints, because a naive smooth curve fit through noisy quotes can easily violate one, especially near the thin, wide-quoted wings. Parametric models like SVI are popular partly because they can be constrained to satisfy both automatically.

Practice

1. A 2-month ATM vol is 30% and a 5-month ATM vol is 24%. Compute both total variances and state whether this pair has calendar arbitrage.
2. Three calls at $48, $50, $52 are priced $3.10, $1.90, $1.00. Run the butterfly test — is this surface arbitrage-free at these strikes?