An eight-point cliquet spread across models fitted to the same five SPX smiles belongs in reserves. Beiglböck, Hasenbichler and Pammer also report a fast way to calibrate one of those models: calibration and pricing together take 0.273 seconds on their SPX example. Heston SKR prices the three-year reverse cliquet at 122.000; QuantLib's Heston SLV, using identical Heston parameters, prices it at 122.139. Calibration takes SKR 0.197 seconds and SLV 18.802. The close prices are striking, though the spread across reference models is the number a desk should carry before trying to trade it.
Five smiles leave room between dates
Vanilla smiles fix the price distribution at each maturity. They leave open how those distributions connect through time, which matters for forward-starting products. Dupire local vol supplies one connection. SLV scales a stochastic-vol reference through a leverage function, usually calibrated with a McKean-Vlasov particle scheme. The authors describe that scheme as somewhat delicate in both well-posedness and numerics.
Their Stochastic Knothe-Rosenblatt construction, or SKR, proceeds from one maturity to the next. Its name refers to the classical transport method that builds a joint coupling through conditional steps. At each maturity, SKR retains the joint law of the calibrated price and the reference's latent factor, such as Heston variance, then restarts the reference from that state. It chooses a martingale that reaches the next smile exactly while minimizing the expected quadratic variation of its difference from the reference. The resulting optimizer is an R-Bass martingale: it shifts the reference's terminal value according to the starting state, applies one increasing map and takes conditional expectations. Martingale Sinkhorn finds the shift and map by alternating a monotone rearrangement onto the target smile with an update that restores the martingale condition. With a Brownian reference, this becomes the Conze and Henry-Labordère scheme for Bass LV.
As the grid gets finer, SKR should approach SLV with leverage λ(t,x) = σ_Dup / sqrt(E[σ_R² | X_t = x]). The main text gives a formal argument. An online supplement proves strong order one half, E sup |X^h - X|² ≤ C_T h, for one autonomous factor, constant correlation strictly inside (-1, 1), and a regular initial joint density of price and factor.
The experiments constrain the models with only five SPX smiles: 0.25, 0.5, 1, 2 and 3 years, from 23 June 2020, using ESSVI fits published by Farkas, Ferrari and Ulrych. The exotics observe quarterly. Seven of their twelve observation dates fall between the fitted smiles, where values come from the calibrated bridge's conditional expectations. The authors acknowledge that those dates have no smile constraint.
What does the cliquet spread measure?
For a cliquet with notional 100, a 50% coupon budget and a 7.5% quarterly loss cap, Bass LV gives 113.987. The prices rise to 116.991 under 3/2 SKR, 120.144 under Bergomi-1F SKR and 122.000 under Heston SKR. Standard errors range from 0.035 to 0.039. The 8.0-point spread comes from the models' different connections between the same vanilla smiles, subject to Monte Carlo and fit error.
The authors trace the direction of the difference to fewer negative quarters and smaller clipped losses in the SKR models than in Bass LV. A smaller contribution from the aggregate floor partly offsets those effects. Prices for the memory autocallable follow the same order over a narrower range, from 104.899 with Bass LV to 106.834 with Heston SKR.
The paper gives no market quote for either product, leaving the right price unresolved. Its fitted 3/2 correlation also reaches the boundary at -1.0000. Implied-vol plots with 95% Monte Carlo bands show the vanilla fit; we did not find a numeric repricing error. The table gives a trader a measure of reference-model dispersion on a single date.
The speed comparison uses different code
On an Intel i5-12500 with 32 GiB, Heston SKR's total valuation takes 0.273 seconds, versus 25.778 for Heston SLV. The implementation details matter. Dupire LV and Heston SLV run in QuantLib's C++ with 64 time steps per year, and SLV calibration uses 250,000 particles. Bass LV and SKR run in the authors' own C++. All four models are priced on 10^5 paths.
Dupire calibrates in 0.005 seconds against Bass LV's 0.079 because its calibration is analytic. Its pricing takes 4.290 seconds, versus 0.018 for Bass LV; we attribute most of that difference to time-stepping overhead in QuantLib's code. Heston SLV pricing takes 6.976 seconds against Heston SKR's 0.076. The calibration difference has a structural explanation: five one-dimensional fixed points replace a particle McKean-Vlasov solve. We expect some of that gap to remain in a same-code comparison, which the paper does not provide. Bergomi-1F takes longer than the other SKR references, with 0.9204 seconds of fixed-point iteration and 1.0074 seconds to price the cliquet.
The 0.139 price gap between Heston SKR and Heston SLV amounts to about 2.5 combined standard errors. SKR's five-maturity grid could contribute, as could SLV's discretization; we did not find a decomposition. The Sinkhorn convergence proof applies to compactly supported targets under strict irreducibility and a support condition on the reference's terminal laws. Elsewhere, the authors report linear convergence observed numerically.
On one SPX date, Heston SKR comes within 0.139 of Heston SLV at about 1/95 of its calibration time (0.197 s against 18.802 s).
Our SPY run is still ahead
We cannot trade SPX index options, so we are building the overlay on listed SPY options instead. We will extract marginals from end-of-day chains and fit a stochastic-vol reference to them. The paper's SPX prices and timings do not transfer to SPY. None of its numbers is a backtest: it reports model prices for two exotics on one day, without hedging costs or realized P&L. Our run must establish whether the gap between SKR and LV forward-smile prices appears in hedged P&L after costs. If it does, the eight points could mean more than a reserve. For now, they stay in the model-risk column.