Static arb-free constraints: butterfly, calendar, no-negative-density
Three inequalities every option surface must satisfy: butterfly (positive density), calendar (calls monotone in maturity), and Lee-bound wings. Violate one and you've found a free trade.
A vol surface that satisfies smile fitting on every individual slice can still be arbitrageable across strikes or maturities. The static arbitrage constraints are three inequalities the surface must respect simultaneously: the implied risk-neutral density must be non-negative (butterfly arb-free), call prices must be non-decreasing in maturity at every strike (calendar arb-free), and the wing slopes must satisfy Roger Lee’s moment-formula bound. This tutorial walks through detecting each violation directly from call prices.
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Worked example
You see three Black-Scholes-implied call prices on a non-dividend stock with $S = 100$, $T = 1$, $r = 5\%$: $C(95) \approx 13.35$, $C(100) \approx 10.45$, $C(105) \approx 8.02$. (a) Verify butterfly arb-free via the discrete second derivative. (b) Compute the implied risk-neutral density at $K = 100$ using Breeden-Litzenberger with $h = 5$.
β Worked example Β· expand
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Reflection
These constraints are *necessary* for no-arbitrage but not sufficient. What kinds of inter-strike or inter-maturity arbitrage do the three inequalities together fail to detect, and what classes of model can produce them?