A note on the spot-forward parity under stochastic cost of carry

Existing spot-forward parities assume either a predictable cost of carry or a deterministic correlation between returns on the underlying asset price and the best-predicted convenience discount factor. We put forward a new spot-forward parity under any stochastic cost of carry. The forward price equals the spot price times a factor represented by the conditional expectation of the cost of carry accruing over the contract lifetime and computed using the martingale measure associated with the cumulative gain process that results from asset holding. We extend this result and derive a general pricing formula for contingent claims written on the forward price. These results apply to any asset that shows convenience revenue, e.g. a stock share, a coupon bond, foreign currency, and a commodity.