WTI's implied imbalance of 0.87 depends heavily on the benchmark price Zimbidis and Sircar chose. Their mathematics holds up: a lognormal benchmark confined above zero can produce a front-month future at minus $37.63. They prove it in closed form up to one scalar root. The event studies give us much less reason to trust the inferred positions.
How the price crosses zero
The paper divides open interest between licensed traders, who can make or take physical delivery, and unlicensed traders, mostly financial participants who must close before the cutoff. Their net position divided by open interest is ε, the imbalance. The observed front-month price combines (1−ε) times the licensed price with ε times a distorted price. For net-long unlicensed traders, the distorted price subtracts a roll option from the licensed price (Case 1, WTI). For net shorts, it adds one (Case 2, nickel).
The roll option values the cost of a forced exit. A trapped long receives (D·F_next − F_front)^+, capturing the loss on selling the front month and buying the next contract in contango. The payoff uses the observed front price, which already includes the option. That feedback makes the pricing PDE nonlinear.
Under bivariate GBM, the front-to-next price ratio reduces the problem to one variable. A further change of variables turns it into the Margrabe exchange-option equation. The resulting valuation uses a Black-Scholes put or call and one monotone scalar root. For Case 1, the solution runs below zero to a boundary at −εD/(1−ε) times the next-month price. Diffusion variance remains strictly positive at zero, allowing paths to cross. The observed price stays positive in Case 2.
The feedback is large at ε = 0.95: it nearly quadruples the at-the-strike value. Set D = 0.90, σ_M = 0.50 and expiry four months away. The no-feedback value is about 3.10, against 11.25 in Case 1 and 8.06 in Case 2. In the authors' Monte Carlo, ε = 0.80 and a starting price of 10 against 25 for the next contract give a 61.0% chance that a risk-neutral path dips below zero at least once. Drift changes little: the frequency is 62.9% at μ = −0.3 and 59.0% at μ = +0.3.
Why 0.87 depends on the starting price
For WTI, the authors use the April 20, 2020 settlements of −37.63 for May and 20.43 for June. Their licensed benchmark is 13.14, or 0.64 times the June price. With σ_M = 0.52 and two trading days to expiry, the closed-form inversion returns ε ≈ 0.87.
The following arithmetic is ours. At a ratio of 0.64 with two days remaining, the Margrabe put is deep in the money and worth almost its intrinsic value. The inversion becomes a ratio of price gaps: (13.14 + 37.63) / (20.43 + 37.63) = 50.77 / 58.06 ≈ 0.874. The volatility estimate comes from 5,974 five-minute return differences and is annualized to 0.5203; it contributes close to nothing here. Move the benchmark and ε moves with its gap, divided by the fixed 58.06 May/June spread. Hold the benchmark fixed and changes to volatility or horizon barely register.
The benchmark takes the May/June ratio at midnight on April 20 and holds it through the day. As the authors state, that choice sets the initial imbalance to zero by design. The resulting intraday series reaches 0.87 at 15:30, with May at −$38.96, then finishes around 0.6182. Every point represents a separate static valuation. The authors also acknowledge that inserting a time-varying ε into their constant-ε formula does not solve the dynamic model.
Nickel's constructed fifteen-month price
Nickel supplies the mirror image with thinner inputs. The $101,365 three-month high at 06:08 on March 8, 2022 was a reported print from trades the London Metal Exchange later cancelled. The fifteen-month price, 60,195, is a scenario value; the paper describes that path as illustrative. Its 1.04 benchmark ratio comes from a chart in the independent review of the March 2022 nickel events commissioned by the LME (Oliver Wyman, 2023). The paper reports a gap of 38,501.59 and an imbalance of 0.9346.
We could not reproduce 0.9346 with the stated inputs. Using ratio 1.044, D = 1, σ_M = 0.5 and a quarter-year horizon, we get a Margrabe call worth about 0.125 per unit of the fifteen-month price. That is roughly 7,500 in dollars and puts ε near 0.84. To obtain 0.9346, the call would have to be worth little more than its 0.044 intrinsic value, requiring a much shorter horizon than the table's 1/4 year. Our check uses the same formula that reproduced WTI's 0.87 exactly. The authors should address the gap.
As ε approaches 1, the price gap scales like ε/(1−ε). Quite different option values then produce implied imbalances between 0.84 and 0.93; the inversion squeezes ε toward 1 and yields little information. The authors put the limitation plainly: "This value depends on the assumed fifteen-month path and benchmark; it is not an independent estimate of historical position imbalance." We agree. The compression makes their caveat stronger.
What could test the event numbers?
The authors say their illustrations "do not establish that feedback alone caused either event, or that the implied imbalance equals the actual historical position imbalance." Their stated purpose is to quantify the imbalance required under the chosen assumptions. They also acknowledge that benchmark prices, the later-maturity futures price and the relevant horizon "all matter". Position data, they say, would be needed to turn the illustrations into tests. That defence is fair. Our objection is sharper for WTI: with 2/252 years remaining, the put is effectively at intrinsic value. The benchmark does nearly all the work; the horizon does almost none. Some ε in [0, 1) can fit any gap, so neither event tests the model, as the authors themselves say.
The open-interest evidence is more useful to a desk. May 2020 open interest dropped from 108,593 to 13,044 contracts on April 20, with about 88% closed in a day. On the penultimate day, residual open interest was 2.06% of the April 2 peak of 634,727. That May 2020 reading was the lowest among 198 front-month cycles from 2010 to 2024; the median was 5.55%. The authors warn that aggregate open interest cannot reveal who was trapped.
We did not attempt a backtest. It would require position-level data linking each holder to delivery capability at Cushing or to LME-deliverable Class I metal. Without those data, ε can only be inferred from prices, leaving any signal circular. The paper has aggregate open interest as its only position data, and the authors say it cannot identify constrained holders.
The Margrabe reduction is a real contribution. It offers a clean price for the inability to take delivery. Contract-level positions tagged by delivery capability would change our view of the event numbers. Until then, WTI's 0.87 mostly reflects a benchmark set at 0.64 times the June price.