Paper: arXiv 2610.21196

Authors: Erhan Bayraktar

Abstract

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.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a highly mathematical and theoretical framework for pricing futures contracts with an end-of-month option, utilizing advanced concepts like the Gittins index and Bank–El Karoui representation theorem. While the empirical validation is limited to numerical examples, the rigorous theoretical development and application to specific models demonstrate a strong foundation. The novelty lies in connecting this specific futures option to the Gittins index and providing a convergent iteration for pricing.

Research Flowchart

  flowchart TD
    A[Research Goal: Characterize Optimal Delivery in Futures with End-of-Month Option] --> B{Methodology: Gittins Index & Optimal Stopping};
    B -- Applies to --> C[Inputs: Futures Contract Specs, Spot Price, Convenience Yield];
    C --> D{Computational Process: Bank--El Karoui, Dinkelbach Iteration};
    D --> E[Outcomes: Optimal Delivery Rule (Gittins Index), Price Bounds, Immediate Delivery Criterion];
    E --> F[Specific Applications: GBM, Gibson--Schwartz Model (Stochastic Yield)];
    F -- Leads to --> G[Key Findings: Futures Price = Spot Conditions, Square-Root Boundary Asymptotic, Yield Monotonicity, Critical Yield Characterization];