Put-Call Symmetry

Put-call parity relates a call and a put with the same strike on the same underlying. Put-call symmetry is a different, less-taught relationship: it relates a call struck at $K$ on an underlying currently at $S$ to a put struck at $S$ on the same underlying, now imagined to sit at $K$ — the strike and spot swap roles. Formally, ignoring dividends and financing, a call struck at $K$ is worth the same as $\frac{K}{S}$ puts struck at $\frac{S^2}{K}$, a relationship that comes directly from the symmetry of the risk-neutral lognormal price distribution around the forward.

The practical use is in constructing hedges and replicating exotic payoffs: barrier options, in particular, can often be statically hedged using a symmetric position in vanilla puts and calls, because the symmetry lets you swap a hard-to-hedge barrier feature for a portfolio of plain options that automatically has the matching value at the barrier level. This is why symmetry shows up constantly in FX options trading, where strikes are naturally quoted relative to a moving spot and the swap-strike-and-spot relationship maps cleanly onto delta and risk-reversal conventions already in use.

Put-call symmetry assumes zero drift (or that spot equals the forward) — with nonzero rates or dividends the exact relationship needs an adjustment, which is why traders usually apply it around the forward price rather than raw spot.