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 $100 \times (e^{-0.05 \times 0.25} - e^{-0.05 \times 0.5}) \approx \textdollar 1.24$. 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.