A clean WTI vanilla fit can leave calendar spread option vols wrong by 37.6% on average. Johansen, Lange and Skovmand get an at-the-money mean absolute error of 0.30 $/bbl a year in implied Bachelier vol from their three-factor model. Yet it overprices one-month calendar spread options at every contract except M1. Anyone validating a curve model on vanillas before using it to mark the roll should pay attention.

An arithmetic model of the curve

The authors use an arithmetic HJM: futures volatility has no price-level multiplier. A futures price moves by a loading times the square root of a Heston-type variance factor times a Brownian shock. Each loading combines an exponential with a linear polynomial, allowing it to decay or hump along the curve. An upper-triangular mean-reversion matrix lets variance factors spill into one another.

This structure makes any linear basket of futures follow the same dynamics as a single future. The same single Fourier integral therefore prices a calendar spread, a strip average or a flow contract, just as it prices a vanilla. The geometric analogue requires approximations for those contracts. The arithmetic model also needs fewer state variables: 3n against 6n for the geometric version with humped loadings.

The quotes support the arithmetic choice. Relative to ATM, the median 20-delta risk reversal is -0.05 in Bachelier vol, versus -0.23 in Black vol. The authors estimate the model with an unscented Kalman filter using NYMEX settlements from January 2013 to February 2024. The panel contains 2,810 days, 36,530 futures, 235,277 vanillas and 70,830 one-month calendar spread options (CSOs). The comparison rests on two variants. V3 uses futures and vanillas alone; VC3 adds CSOs.

Why does the vanilla fit miss the roll?

Vanillas identify each contract's marginal distribution. A CSO also depends on how adjacent contracts move together. The distance from perfect correlation is small: implied one-month decorrelation is about 0.002 at two months.

In this model, decorrelation is a ratio. Its numerator weights each factor's variance by the change in its loading over one month. Total variance forms the denominator, and vanillas see only that denominator. Reallocate the same total variance across factors and vanilla prices stay unchanged, even as CSO prices move.

V3 gets that allocation wrong. Its transitory factor carries 35% of M1 variance, with a loading decay rate of 2.237 a year and a loading that halves within months. The steep loading keeps model decorrelation elevated further along the curve, where market decorrelation falls. V3 overstates it by a factor of 1.2 at M2 and 1.8 at M6. Its rank correlation with observed two-month decorrelation is 0.13.

VC3 raises that correlation to 0.92. It shifts front-end variance to the middle factor, which accounts for 72% of M1 variance, and leaves 9% with the transitory factor. The transitory factor then serves as a curve decorrelation factor. Its variance factor tracks at-the-money CSO variance, allowing VC3's decorrelation to follow the market day by day. The 0.92 rank correlation measures that tracking. CSO vol error drops to 14.5%, with prices now slightly too low: mean error is -0.207 $/bbl a year, versus +0.244 $/bbl a year for V3.

The better spread fit costs something elsewhere. VC3's vanilla MAE increases from 0.79 to 0.86 $/bbl a year; ATM MAE rises from 0.30 to 0.39. Evaluated on futures and vanillas alone, its parameters produce a log-likelihood of 446,720, against 481,426 for V3. Futures fit does improve, with RMSE falling from 0.46% to 0.38%.

Much better marks, a small daily hedge gain

For V3's vanilla book, futures alone remove 9% of variance. Adding three ATM options brings effectiveness to 0.97 and reduces mean absolute daily P&L from 25.38 to 2.93 cents per option. With three options, VC3 performs worse at 3.21 cents (t=12.54).

The CSO book requires a distinction between per-option error and portfolio variance reduction. With two hedge options, VC3 lowers mean absolute daily P&L to 0.89 cents, compared with 0.95 for V3 (t=-3.83). The unhedged figure is 1.59.

Six hundredths of a cent a day per option.

Portfolio variance reduction gives little reason to prefer either model. Variance-based effectiveness is 0.68 for V3 and 0.64 for VC3; the paired test is insignificant (t=0.72). Just the top 1% of dates account for 64% of unhedged variance, so a few days dominate this measure. The authors report that neither model's hedge options improve on the six futures spreads alone under the median-based measure. Over ten days, VC3 does better: 2.50 against 2.65 cents (t=-2.24). We saw no transaction costs in any hedge P&L, and all of these calculations use settlements.

Limits of the evidence

The main models use what the authors call skew forcing. During estimation, this device tilts each day's vanilla smile in proportion to that day's average skewness less its sample mean. It keeps the factors from chasing the skew. The authors compute reported prices and hedges without the forcing.

They also show what happens without it during estimation. V3-U achieves the best vanilla MAE, 0.59, but adding its third hedge option reduces effectiveness to 0.83 and raises P&L to 5.27 cents. The effectiveness decline is insignificant; the P&L increase is significant. Leverage correlations of 0.89 and -0.70 make two of its factors nearly degenerate at the money. The authors conclude that V3-U appears to get better prices by fitting each day's skew at the expense of describing surface movements.

Skew forcing is an in-sample device: it subtracts a full-sample mean skew, and its coefficient is fitted over the whole sample. The smiles reveal an observable cost. When priced without the forcing, V3 barely follows the call skew from 2022 to 2024.

The out-of-sample exercise splits V3's sample into halves and tests in both directions. Vanilla MAE changes little, from 0.53 to 0.56 and from 1.02 to 1.07. Futures RMSE rises from 0.29% to 0.49% in the earlier half (t=5.47). In the later half, it rises from 0.52% to 0.80% (t=1.86). We did not find an equivalent exercise for VC3. The CSO pricing improvement therefore remains an in-sample result.

CSOs are thinly traded, too. Their volume is about 3% of vanilla volume and their open interest about 8%; median-day volume is under 1%. As for the arithmetic-versus-geometric comparison, vanilla errors from the two-factor geometric model are 16% larger (t=2.58, 2.24 excluding March to June 2020). The authors themselves describe the significance as "not by a wide margin".

We could not test any of this. Our options history starts around 2020, while the paper's panel begins in 2013, and we have no confirmed coverage of WTI calendar spread option quotes. Those quotes are the whole experiment. Futures cannot substitute for them.

Validate the spread prices

A vanilla calibration says nothing about how a multi-factor model allocates each contract's variance among factors. That allocation determines every spread price derived from the model. Anyone pricing or risk-managing spreads should validate against spread quotes, while expecting a larger improvement in marks than in daily hedge P&L. Here, CSO vol error falls from 37.6% to 14.5%; CSO hedge P&L falls from 0.95 to 0.89 cents. The case for estimating on spread quotes would be much stronger if a VC3 estimate retained its 0.92 decorrelation tracking on the half it was not fitted to.