Futures, Forwards, and Spot: Measure Changes and Their Consequences

Abstract

These notes develop a unified treatment of measure change in FX and commodity derivatives, using the pricing of futures, forwards, and spot as the unifying problem. Four transitions are treated as first-class objects: the domestic–foreign (quanto) change between risk-neutral measures associated with two currencies; the spot–forward change between the bank-account numeraire and a zero-coupon bond; the futures–forward convexity adjustment induced by daily margining under stochastic rates; and the rolling-spot / stochastic-discount picture relevant when a freely tradable spot does not exist. The exposition is formal. The Girsanov theorem and the numeraire-change theorem of Geman, El Karoui and Rochet are taken as the engine, and every subsequent measure change is derived from an explicit Radon–Nikodym derivative. Each transition is then paired with a product chapter: vanillas, digitals and forward-starts; single and double barriers under both continuous (American) and fixing-based (European) monitoring; arithmetic Asians; and the canonical leveraged accumulator — twelve months of daily fixings, client long one call and short two puts per fixing at a common strike, with an upper knock-out on the fixing. The Greeks are treated twice: once as transformations of the standard sensitivities between measures, and once as genuinely new sensitivities that emerge only under a measure change — basis delta and vega, quanto-correlation sensitivity, funding delta, and the Greeks of the convexity adjustment. The hedging chapter confronts the practical question of what a desk actually trades once a pricing measure has been chosen: the basis risk opened by pricing in the spot measure but hedging with listed futures, and the rolling-hedge attribution for the canonical accumulator under both choices. Two final chapters address markets in which a freely tradable spot does not exist. Non-deliverable forwards are developed as a generic framework and illustrated on USDBRL, with settlement mechanics — PTAX, CDI, and the T+2 / T+1 split — deferred to an appendix. Commodities are treated as the practical limit of this picture: prompt-month futures as a proxy for spot, convenience yield, and the date-structure subtleties of agricultural, energy, and LME metal markets. Worked numerical examples run throughout on EURUSD, USDJPY, USDMXN and USDBRL for FX, and on sugar, WTI and LME copper for commodities, using a Black–Scholes baseline with an SVI smile slice. Cross-currency basis, multi-factor HJM convexity and collateral effects are flagged as extensions.