For an American put under Schwartz's two-factor oil model, Saul'yev's sweep gets within 2.1×10^-3 of its own fine-grid value in 0.41 seconds (NY=100, NT=600). A trader still cannot tell from this paper whether the price beats Black's formula against a traded premium. The authors identify that comparison as their next step; they have yet to run it.
The solver and its contract
The starting point is Schwartz (1997). Spot has lognormal diffusion, with risk-neutral drift r minus δ. The convenience yield δ, the return from holding physical barrels, follows a mean-reverting Ornstein-Uhlenbeck process and is correlated with spot. For European options, Black's formula applied to the Schwartz forward gives a closed form with a two-factor variance. An American option adds an exercise boundary that depends on spot and yield. Pricing it calls for a PDE solver or Longstaff-Schwartz least-squares Monte Carlo (LSMC).
Prabhu and Duffy solve the PDE. They use log spot and cut the domain off at 1.2 either side of ln S0 and 0.4 either side of δ0. Their Alternating Direction Explicit method (ADE) marches backward through time, sweeping the grid and using neighbours already updated at the new time level. The diffusion terms gain implicit-style stability without a linear-system solve. The double sweep (Barakat-Clark) averages an upward and a downward pass. Saul'yev's variant makes one pass per step, reversing direction on the next. After each step, a pointwise max against the payoff handles early exercise.
All runs use Schwartz's 1997 WTI estimates: κ=1.49, α̂=0.0161, σ1=0.158, σ2=0.115, ρ=0.189, r=0.05. The contract has K=100 and T=0.5, and the authors test three starting states. No market data enters the tests. The claim concerns numerical accuracy against the closed form and LSMC, at much lower cost.
What do the price checks show?
The European comparison is tight. ADE prices the option at 11.601279 against an analytical 11.601861, a gap of 5.82×10^-4. Plain Monte Carlo returns 11.602859, with a standard error of 0.0164725.
For the American base case (S0=90, δ0=0.1), ADE produces 11.892050. LSMC with 250,000 paths produces 11.888911, leaving a 0.003139 gap inside its 0.014396 standard error. At the money, the gap is 0.003314: 5.020935 against 5.024249.
The high-yield result deserves a closer look. At δ0=0.3, ADE gives 16.952677 and LSMC gives 16.989201. Their 0.036524 difference is about 11 times the other two gaps. The authors describe it as small, about 0.2% of value. We did not find a standard error reported for this scenario; if it resembles the base case's 0.014396, the difference amounts to about 2.5 standard errors.
Its direction raises a further question. In the runtime tests, the paper's LSMC allows exercise on 50 dates. If the scenario runs use that setup too, ADE offers far more exercise opportunities by projecting at every time step. With fewer exercise rights, a Bermudan should have a lower value. Yet LSMC is higher. The at-the-money result runs in the same direction (5.024249 against 5.020935), although it lies inside one standard error. The high-yield gap therefore looks like a larger instance of the same pattern. The paper does not say whether it prices LSMC on the paths used for regression, a choice that would push the estimate up. Another possibility is the solver's yield grid: in this case it extends from -0.1 to 0.7 and uses linear extrapolation at the edges. We cannot tell which accounts for the gap.
One sweep earns its place
The double sweep buys no measurable accuracy here. Across every tested grid, the variants agree within 2.1×10^-5. At NY=100 their estimated temporal convergence orders are 1.65 and 1.64; at NY=200 both reach 1.88. Saul'yev runs 1.80 to 1.89 times faster, near the 2x predicted by the authors' cost model. The authors acknowledge that the extra sweep adds no measurable accuracy in this problem. Their designated primary scheme is thus the one to skip.
At NT=100, spatial refinement backfires. Moving from NY=100 to NY=400 takes the price from 11.8821 down to 11.7705. That 0.11 fall exceeds every ADE-LSMC gap in the paper; the largest is 0.036524. Consistency error scales with Δτ/h rather than Δτ/h². A desk refining space alone while keeping a coarse time step would get a worse price. Time steps must shrink at least as fast as grid spacing, a trap the authors flag themselves.
Where the 210x claim gets thin
At comparable runtimes, the PDE solver wins. With NY=200 and NT=600, the double sweep takes 2.986 seconds and has an error of 1.9×10^-3 against its own fine-grid reference. LSMC takes 3.365 seconds for 250,000 paths, with a standard error of 0.0144. Those are different measures, convergence error for the solver and sampling error for LSMC. In a two-factor problem, the PDE's advantage is plausible, and the measured runtimes favour it.
The larger speedup ratios rest on a weaker comparison. The paper reports roughly 210x for the double sweep and 390x for Saul'yev, using an extrapolated LSMC run of about 1.2×10^7 paths and roughly 160 seconds. That run was never performed. For ADE, the reference is V_ref = 11.892754, obtained from the same scheme on a 400 by 400 grid with 2400 steps. The authors say this measures convergence rather than modelling error, which they assess separately against the closed form and LSMC. But the separate LSMC check has a 0.0144 standard error, about seven times the 1.9×10^-3 convergence error. It cannot establish accuracy at the 2×10^-3 level used for the speedup claim.
Even the reference continues to move: the NY=400, NT=1200 entry is 11.8921, or 6.5×10^-4 below V_ref. On the LSMC side, the count includes sampling error alone, leaving out regression bias and the Bermudan gap. The ratio joins unlike error bars. It is the least secure figure in the runtime results.
Listed futures options remain untested
Prabhu and Duffy call the trader-driven extensions preliminary. The futures-price exercise obstacle matters for listed options, but we did not find prices reported for it. Seasonality increases values by about 2 to 3%, partly because the seasonal curves lift average σ2 by about 0.019. The amplitudes, the authors say, are "illustrative rather than calibrated." Internal Greek checks give an ATM spot delta of -0.520 and a yield delta of 16.814 against an analytic estimate near 17.
Every benchmark comes from the model. Parameters come from a single 1997 estimate, with calibration to a real option chain left for future work.
We could not test the method on our own data. We hold dated CL futures, but have no history of options on crude futures and therefore no traded premiums to compare with the prices. We also lack a physical spot series, and spot is one of the model's two state variables.
The European gap of 5.82×10^-4 supports the solver's construction, and Saul'yev's single sweep is the practical choice. The result that would change its standing is a market comparison. The authors propose fitting the futures curve and implied vols for corn or soybean chains, then checking model prices against Black's formula and market premiums across strikes. Our version would ask whether Saul'yev prices beat Black's formula on a CL options chain.