The end-of-month option in futures contracts and the Gittins index of holding the underlying

The end-of-month option allows the short in a futures contract to choose when to deliver during a window in which the delivery price is fixed. We characterize optimal delivery through a universal index obtained from the Bank--El Karoui representation theorem. In the semimartingale setting, this is the Gittins index of holding the underlying and collecting its convenience yield, and the futures price is the retirement reward for which the value of this project equals the spot price. This characterization yields price bounds and a criterion for immediate delivery. Dinkelbach's iteration reduces the price computation to a sequence of standard optimal stopping problems, with an explicit contraction bound. We apply the theory to geometric Brownian motion and the Gibson--Schwartz model with stochastic convenience yield. For geometric Brownian motion, we characterize when the futures price equals spot and derive a square-root boundary asymptotic near expiry. For the Gibson--Schwartz model, we establish monotonicity in the yield and characterize the critical yield below which the price equals spot. Numerical results illustrate the price iteration and delivery regions.