Across the six three-asset standard baskets, roughly nine tenths of the accuracy gain arrives before the fourth moment is matched. Mean absolute error against Monte Carlo is 0.0437 for the direct three-moment shifted lognormal. Rewriting the price as a combination of probabilities lowers it to 0.0059, and adding the fourth moment takes it to 0.0012. Maximum absolute errors move the same way: 0.1028, then 0.0152, then 0.0026.
Section 5.1 gives the authors' own ranking: "These results identify two distinct gains. The exact probability reformulation produces the largest improvement, and matching the fourth moment then reduces the remaining error further." The order flips for the crack spread. A mixed-sign basket offers no reformulation, and the fourth moment by itself reduces absolute error from 0.3752 to 0.0583, a reduction of 84.5%.
The authors explain the ranking immediately. "The gain therefore does not arise only from adding one moment; it also depends on approximating the positive lognormal sums that appear in the exact pricing identity." Their second answer is the crack spread, and it carries more weight.
Most of the gain comes from the change of measure
A weighted sum of correlated lognormals has no tractable law. The exercise region of a basket call also has a curved boundary, which blocks an extension of the Black-Scholes argument.
The paper delays approximation. Using two Gaussian shift lemmas, it absorbs each asset's lognormal martingale factor into a shift of the correlated normal vector. Applied asset by asset, this turns the basket call into a weighted sum of Gaussian probabilities, one per asset plus one for the discounted strike. The identity is exact.
For what the paper calls a standard basket, one positive weight with all the rest negative, each event is divided through by the positive asset's term. Every probability then becomes the CDF of a positive correlated lognormal sum at the single threshold 1. The fall from 0.0437 to 0.0059 comes from fitting those positive sums instead of fitting a mixed-sign region bounded by a nonlinear curve.
Approximation enters after that step. Each normalised sum is replaced by W = exp(s·sqrt(Y)·N + m) + tau. Y equals alpha with probability p and beta otherwise, with the first four moments matched. Lemma 2.7 expresses the CDF as a two-component normal mixture. The final price therefore requires only a handful of Phi evaluations.
A crack spread contains two positive weights and one negative weight, which rules out the positive-sum construction. The paper instead uses the signed proxy pi = A(exp(sN+m) + tau) with P(A = 1) = p. Moment matching proceeds through a hyperbolic-form Cardano root, while Theorem 3.5 supplies a closed-form call.
The evidence comes from disjoint test sets
The reformulation result belongs entirely to the six synthetic standard baskets (T = 1, r = 0.03, weights (2/3, -1/3, -1), alpha = 0.9, beta = 1.1, strikes 10 to 40). The empirical exercise uses a normalised 3:2:1 crack spread with weights (2/3, 1/3, -1). Its inputs are Yahoo continuous futures RB=F, HO=F and CL=F from January 2015 through July 2026, sampled daily and annualised on 252 days. Pricing uses the direct signed proxy, with no CDF reformulation available.
At the one-year at-the-money baseline (K = B0 = 63.2358), the four-moment price is 14.8775. Monte Carlo gives 14.8192 with standard error 0.0275. The absolute error is 0.0583, or 2.12 MC standard errors. The three-moment price reaches 15.1945, leaving an error of 0.3752 and 13.64 standard errors. The reported 84.5% reduction compares those two errors. The authors describe the four-moment figure as closer to the benchmark rather than indistinguishable from it.
Keep the benchmark in view. The volatilities and correlations are historical, and the paper says explicitly that the exercise "measures the approximation error of the analytical formulas; it is not a calibration to observed option prices."
Its window study shows the practical cost. Re-estimation on trailing 1-, 3-, 5- and 10-year samples (251, 755, 1256 and 2511 valid returns) shifts the Monte Carlo price from 11.5865 to 14.7830. Under all four windows, the four-moment method has the smaller error: 0.0265, 0.0525, 0.0330, 0.1093 against 0.3874, 0.1345, 0.1309, 0.4332. The authors' conclusion is appropriate: calibration uncertainty is economically important and separate from approximation error. The benchmark moves by 3.20, from 11.5865 to 14.7830, while the approximation error is 0.3752.
Why does the fourth moment stall at 30 days?
Proposition 4.10 provides the paper's most candid result. Matching the means forces the signed CDF difference to integrate to exactly zero over the real line. Global moment matching therefore says nothing about the one-sided region that generates the payoff. The authors put it plainly: "four matched moments do not, by themselves, guarantee an accurate option value."
The short-maturity results bear this out. At 30 days, mean absolute error is almost unchanged, at 0.0094 for three moments and 0.0093 for four. Three moments also produce the lower RMSE, 0.0113 versus 0.0121. Four moments win at only 44.4% of strikes.
The winning share climbs to 55.6% at 90 days, 62.5% at 180 days and 75.0% at one year. Over the same progression, basket skewness rises from 0.3548 to 1.0833 and kurtosis from 3.3073 to 6.3262. Across all 34 strike-maturity cases, MAE declines from 0.0973 to 0.0720.
Four moments lower the mean discounted integrated-CDF bound at every maturity, by 8.4%, 23.2%, 35.6% and 45.5%. Its correlation with realised absolute error is 0.883, compared with 0.824 for three moments. At the baseline strike, however, the three-moment CDF is locally closer. The paper shows why: price depends on the integrated CDF discrepancy above K.
What the implementer must decide
The formulas are complete. Four parts of the fitting procedure remain open:
- The variance states alpha = 0.9 and beta = 1.1 are fixed by choice. The paper also proposes the symmetric parameterisation alpha = 1 - delta, beta = 1 + delta, with a fifth-moment selection rule. Remark 4.14 says to read the reported tables as the baseline fixed-(alpha, beta) specification unless refined delta values appear. Mixing the two specifications in one comparison would be wrong. Following the delta route requires recomputing the moment equations and every reported price.
- Root selection requires searching all roots over the relevant domain, rejecting inadmissible candidates, retaining the smallest normalised four-moment residual, breaking ties with the fifth moment, and falling back to three moments otherwise. The paper gives no solver, bracket or tolerance value. Its "prescribed numerical tolerance" is left to the implementer.
- Parameter recovery excludes p = 1/2, so near-symmetric baskets need separate treatment. The fitted sign probability is 0.999992 at 30 days and 0.9961 at one year, effectively at the boundary. The negative branch is almost vestigial, although reproducing the fourth moment still requires it. The stated mitigation is the hyperbolic Cardano form.
- Data vintage and roll convention affect the crack-spread inputs. B0 = 63.2358 is the latest aligned value from a continuous front-month series. Returns that touch the nonpositive April 2020 WTI print are deleted without winsorisation. The authors describe that treatment as transparent. They separately acknowledge that continuous front-month histories include roll effects and do not represent one fixed contract tenor. The word discretionary is our interpretation rather than theirs.
The guarantees leave another opening. For the signed proxy, call prices inherit nonnegativity, decreasing and convex behaviour, a Lipschitz bound and put-call parity. Remark 4.6 observes that every fitted probability is valid and falls in [0, 1]. Global monotonicity and convexity of the assembled standard-basket price do not follow automatically because the probability terms are fitted separately with strike-dependent parameters. The remark also recommends checking nonnegativity and put-call parity across the full strike grid. The crack-spread grids contain no violations of monotonicity, discrete convexity or the forward lower bound. Each of the six standard baskets has one strike, preventing a strike-grid check there.
From analytical benchmark to desk use
Contract-matched risk-neutral inputs and errors against observed spread-option prices would move the method closer to desk use. The authors identify exactly that next step. Their reported comparison covers the three-moment shifted lognormal and Monte Carlo. Kirk, Carmona-Durrleman, Bjerksund-Stensland, Curran and the Fourier approach appear in the introduction but are not run. We found no runtimes, no delta or vega, and no hedging errors, although strike derivatives are available in closed form and hedging remains listed as future work.
We could not test this ourselves. The empirical application requires RBOB, ULSD and WTI futures series, which we do not hold. Observed crack-spread option quotes are also unavailable to us for a pricing comparison. A portfolio of listed single-name options does not reproduce a call on a weighted basket payoff.
The exact identity is the paper's real contribution. The fourth moment works as a long-maturity refinement, moving error from 0.0059 to 0.0012 on the six standard baskets and from 0.3752 to 0.0583 (84.5%) on the one-year mixed-sign crack spread. In either case, approximation error remains small beside the error in the volatilities supplied to the model.