Jelly Roll Arbitrage

A jelly roll combines a synthetic long forward (long call, short put, same strike, same expiry) at one expiration with a synthetic short forward at a different expiration, both on the same underlying and same strike. Because a synthetic forward's payoff is just "own the stock forward," the jelly roll's whole payoff collapses to the pure cost of carrying the stock between the two expiry dates — the difference in interest cost and any dividends paid in between — with the stock's actual price movement cancelling out entirely between the long and short legs.

Put-call parity says a synthetic forward at strike $K$ expiring at $T$ is worth $S - K e^{-r(T-t)}$ (adjusted for dividends). Taking the near-expiry version minus the far-expiry version, the stock terms cancel and what's left is purely a function of the two interest-rate discount factors and the dividends paid in the interval — nothing about where the stock actually ends up. If the market price of the jelly roll deviates from that carry cost, an arbitrageur can lock in a riskless profit by trading the actual options against the theoretical fair value.

Worked example. A stock trades at $100, pays no dividends, and the risk-free rate is 5%. A jelly roll long the 3-month synthetic forward and short the 6-month synthetic forward at the same $100 strike should be worth (in present-value terms) the interest saved by delaying the further-out delivery, roughly 100 \times (e^{-0.05 \times 0.25} - e^{-0.05 \times 0.5}) \approx \1.24. If the four options are quoted such that the roll actually costs \2.00, an arbitrageur sells the overpriced roll and replicates the fair-value carry using the underlying and financing, locking in the roughly $0.76 mispricing regardless of where the stock trades at either expiry.