A short choosing the delivery day earns at most 0.085bp across the paper's sampled one-week constant-yield cases, with r = 5% and σ = 30%. Bayraktar shows why the yield assumption matters. Under a stochastic yield, the discount reaches 2.4bp a week and 22bp a month at δ0 = r. For a desk, the useful result is a yield threshold.

What can the short choose?

The end-of-month option sets the delivery price m when the window [0, T] opens. The short chooses a delivery time τ and receives m minus the spot price at τ, discounted. Because the contract starts at zero value, its futures price is the smallest ratio, across delivery rules, of expected discounted spot to expected discount factor. Lindensjö had given that formula but saw no general way to compute it. Björk put the question to Bayraktar in 2007.

Both spot and the terminal futures price E[X_T] bound the ratio from above. The futures quote stays fixed at m through the window. All information about the option therefore comes from the deliverable's spot path; a stitched futures series or commodity ETF loses it.

The delivery index and the price

Bayraktar decomposes discounted spot into X0, a martingale and a process A that accumulates discounted convenience yield. With geometric Brownian motion (GBM) and yield q, A grows at e^{-rt}qX. The short holds a project paying A until delivery, when it retires for m. Through the Bank-El Karoui representation and Bank and Föllmer's bandit view, that project has a Gittins index K. Delivery occurs when K first falls to m*, or at T if it stays above. At inception, immediate delivery is optimal exactly when the index is at most spot.

Dinkelbach's device turns pricing into repeated ordinary optimal-stopping problems: solve at a trial price, take the ratio under the resulting rule, then repeat. The error contracts by at most 1 − E[e^{-rT}] each step. For a one-week window and r at or below 5%, that bound is under 10^-3. In the test case (r = 5%, q = 4.5%, σ = 30%, T = 1 year), one step starting at spot gives c* = 0.999312 to six digits. Observed contraction is 7×10^-4, against a guaranteed 0.049. A Monte Carlo check using 4×10^5 paths values the contract at 0.0004 ± 0.0003. For a one-week window, one American-style solve at spot gets within 10^-3 of the distance to the price. The one-year test case matched six digits, though its per-step guarantee is only 0.049.

Constant yield leaves little to collect

Under GBM, the futures price equals spot when q ≤ 0. It also equals spot for 0 < q < r if the window does not exceed a critical horizon T_c. With r = 5% and σ = 30%, T_c is 0.0111 years at q/r = 0.98, 0.301 at 0.90 and 20.8 at 0.40. More volatility cuts that horizon: at q/r = 0.90, T_c is 1.195 years with σ = 15% and 0.109 with σ = 50%. For a one-week window at σ = 30%, q must exceed 0.974r before the price drops below spot.

Even then, the discount is small. It is 0.085bp over a week at q = r and 6.9bp over a full year at q = 4.5%. In the sampled weekly cases with q ≥ 5.5%, c* is within 0.01bp of the terminal-delivery ratio e^{(r−q)T}. The author notes that carry can account for much of a positive discount. Bayraktar's conclusion is that constant yield makes this chiefly a bet on the sign of r − q, apart from a band of width of order σ√T around q = r.

The proved square-root result puts the initial delivery boundary at r/q, falling as (r/q)(1 − Cσ√s), where C = 0.638833. It implies T_c ≈ ((r/q − 1)/(Cσ))². At σ = 30%, the estimate is within 2% at q/r = 0.98 and 4% at 0.96. Accuracy falls away farther from q = r: at q/r = 0.90 it gives 0.336 years versus 0.301, and at q/r = 0.40 it gives 61.3 years versus a numerical 20.8, again with σ = 30%.

Use it near q = r and nowhere else.

Borrowed oil parameters, larger discounts

The Gibson-Schwartz calculations use Schwartz's oil parameters (κ = 1.876, α = 0.106, σ = 0.393, σ_δ = 0.371, ρ = 0.766, r = 6%). Their critical initial yield is 2.5% for a week and −1.6% for a month. The short still waits a month when the initial yield lies between −1.6% and 0. Yield volatility and mean reversion toward α = 10.6%, above r = 6%, can each make waiting pay. At δ0 = r, the discount is 2.4bp over a week, about thirty times the GBM figure, and 22bp over a month. Bayraktar assigns the gap to volatility in δ rather than volatility in price.

At high yields, carry again dominates. With δ0 = 20%, the weekly discount is 26.8bp; frozen-yield carry is 26.9bp. At 10%, those figures are 8.33bp and 7.7bp. The paper cautions that the frozen-yield comparison leaves out mean reversion and yield uncertainty. Its 0.6bp gap is therefore an upper reading of timing value at best.

The numerical grids converge: weekly δ_c comes out at 2.75, 2.44, 2.45 and 2.46% across four grids. Bayraktar acknowledges that the scheme's edge closure is not monotone. His heuristic formula for δ_c is unproven and only asymptotic; for the month it gives −0.8% against −1.6%. More consequentially, every Gibson-Schwartz figure depends on κ, α, σ, σ_δ and ρ from a 1997 calibration. The negative monthly threshold is the paper's most interesting finding, and its force rests on those borrowed 1997 parameters, including the mean-reversion level α = 10.6%.

What would a market test require?

We did not run a backtest. Our data lack the deliverable's spot path within the window, and stitching a series removes the option altogether. Under the oil parameters, the model implies that deliveries in a weekly window should cluster on day one when yield is below about 2.5%. Late deliveries at yields well below δ_c would put the omitted quality and cheapest-to-deliver choices, or the borrowed 1997 parameters, first in line for scrutiny.