hard · Quantitative Finance derivatives
A dealer fits a parametric implied-volatility surface to a single maturity and notices that the calibrated smile, when converted to a risk-neutral density via the Breeden-Litzenberger relation, produces negative probabilities in the left wing.
What is the precise no-arbitrage condition being violated, and where does it bite?
- Butterfly (convexity) arbitrage: the call price is not convex in strike there, equivalently the second strike-derivative of the call price has gone negative
- Calendar arbitrage: total implied variance is decreasing in maturity at those strikes, so the surface is not monotone increasing in time to expiry
- Vertical-spread arbitrage: the call price is increasing in strike there, so the first strike-derivative of the call price exceeds zero, which is forbidden
- Put-call parity arbitrage: the quoted put and call implied vols differ at the identical strike and maturity, which breaks the parity-implied density relationship
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