A Delta-only hedge leaves most of this liability's risk untouched. The one-year root-mean-square hedging error is 35.825 under the model-consistent Delta-only hedge, against 7.019 for a stock-call-bond portfolio matched to fund, variance and rate exposures. Same liability, same model, same date. Adding the other two factors cuts the error by 80.4%.
Everything else in Molent and Gaudenzi's paper builds the machinery required to calculate that comparison.
Why the contract defeats a lattice
The paper prices a twenty-year equity-linked policy funded by annual premiums. At each anniversary, D = 100 enters a fund that subsequently tracks an equity index. The policy provides death and maturity guarantees with an accumulation rate 2%, plus a Bermudan surrender right exercisable at each anniversary. The policyholder is 50 at issue. Mortality comes from the 2025 ISTAT table, while discounting uses the ECB euro-area AAA Svensson fit dated 15 July 2026.
The financial model combines Heston variance, with V0 = 0.04, kappa 2.00 and sigma_V 0.30, and a CIR++ rate factor. The latter is a square-root short-rate process shifted deterministically to reprice the initial curve exactly. Correlation runs through all three factors: rho_SV = -0.70, rho_Sr = -0.20 and rho_Vr = 0.02.
The fund creates the computational difficulty. Variance and the rate factor are Markovian square-root processes that fit recombining lattices. The accumulated fund does not, since a deterministic 100 is added once a year. Two paths can arrive at the same equity level at year 10 yet carry different fund values because their contributions entered at different index levels. The fund lattice therefore spreads rather than recombines.
Regression Monte Carlo is the usual way around this problem. Its flexibility comes with three stacked errors: sampling, basis approximation and a fitted stopping rule that is systematically suboptimal.
Molent and Gaudenzi discard the fund grid. At every joint variance-rate node, they represent contract value as a one-dimensional function of the fund, stored through a sorted knot list and a cubic interpolant on each interval. For an interval from a to b, the four interpolation points are a, a+(1-1/sqrt2)(b-a), a+(1/sqrt2)(b-a) and b. An annual contribution simply translates the argument from F to F+D.
Surrender is imposed directly on the stored curve as the obstacle max{continuation, payoff}. When continuation crosses the payoff, the crossing becomes a new knot. Working backward, the algorithm generates knots as inverse images of child knots under the affine branch maps. A Douglas-Peucker-type compression rule then removes any knot whose deletion keeps the curve within the certified tolerance |U - Utilde| <= eps_abs + eps_rel*F.
Compression carries the method, and the reported results are stronger than I expected. At production tolerances (1e-4, 1e-6), the maximum retained fund-knot count at any factor node is 187 for the trinomial scheme and 215 for the binomial. Reducing eps_abs by a decade changes the twenty-year fair annual premium by 6.213e-5 while adding three knots. Moving eps_rel to 1e-7 shifts the premium by roughly 5.2e-4 and requires 302 extra knots. My reading of the 6.213e-5 change for three knots is that, away from its kinks, the value function behaves almost like a piecewise cubic.
Fewer than two hundred numbers capture a state variable expected to explode.
Competitive accuracy
The independent benchmark uses 100 replications of a million antithetic full-truncation Euler paths, sampled at 256 steps per year. It gives P_MC = 111.6586729 with standard error 0.0012648. Against that benchmark, the deterministic method converges cleanly.
At N_yr = 32, the direct trinomial scheme produces 111.653842, an error of 0.004830, in 191.3 seconds on twelve threads. The binomial scheme with Richardson extrapolation reaches 0.008857 in 116.2 seconds, including both required grids.
Moment matching matters. Even at N_yr = 4, the direct trinomial value of 111.594769 lies closer to the benchmark than the direct binomial result at N_yr = 32, which is 111.512365. Matching both exact CIR moments at each active node, instead of matching the mean and a leading-order variance, is worth eight refinements.
The Delta results are credible as well. Pathwise Monte Carlo estimates inception Delta at 0.810044207 with standard error 1.70e-4. Richardson at N_yr = 16 misses by 4.166e-5, within one standard error. At N_yr = 32, the trinomial error is 4.162e-4. This remains outside the 95% half-width of 0.000333, although it decreases monotonically rather than entering the band.
The theoretical argument uses a Markov-inequality bound equal to 18/(b-a) times the supremum value error on a cell containing no preserved singularity. Actual performance is much tighter. At policy year 10, a 1e-4 value tolerance yields a local Delta error of 1.162e-6, three orders below what the bound permits.
The proved rate stops short of production
The paper draws the theoretical boundary carefully. Weak convergence through Kushner-Dupuis applies to both strict chains. First-order weak accuracy, together with the Talay-Tubaro expansion supporting Richardson, applies only to the strict, unpruned binomial scheme under smoothness conditions (R1) to (R5). The authors make no global first-order claim for the production adaptive-trinomial scheme.
With lambda_V = 1, the minimal-span stencil search may choose a one-sided triplet whose limiting standardized support is (-2, 0, 1), with probabilities (1/6, 1/2, 1/3). Its third standardized moment is -1. The resulting raw third moment is O(h^{3/2}) rather than O(h^2).
The most accurate configuration therefore lacks a proved rate. It also runs more slowly. On the mortality-extended no-surrender premium at N = 640, direct trinomial takes 191.3 seconds, versus 116.2 seconds for binomial Richardson. The configuration carrying the proved rate requires extrapolation, yet the authors decline to claim its second-order theorem for the kinked payoff actually being priced. Their phrase is "theoretically motivated and numerically validated accelerations". It is accurate. Production accuracy ultimately rests on the refinement table.
Numerical screening selected the grid pair (lambda_V, lambda_X) = (1, 1.4) for the best end-to-end pricing accuracy on the central parameterization. The reported results use that same configuration. Readers should know that the stencil scale was fitted to the answer it generated.
How thin is the external check?
Surrender premiums are compared with cross-fitted two-fold least-squares Monte Carlo. Continuation values are regressed on simulated paths, using one fold for fitting and the other for pricing. The calculation uses one million paths and 256 steps per policy year.
The deterministic prices exceed LSMC by 0.1554 for fund-based surrender, 0.1026 for guarantee-based surrender and 0.2986 for mixed surrender. The largest relative gap is 0.253%. Its direction is consistent with a fitted stopping rule. The paper also states that the LSMC intervals "quantify held-out sampling uncertainty only; regression and time-discretization bias are not included", leaving the gap undecomposed.
Precision differs sharply across the two sides. The no-surrender benchmark, based on 100 replications of a million paths at 256 steps per year, has a standard error of 0.0012648. For the surrender products, the held-out LSMC intervals are about 0.16 wide for P_FB ([111.6279, 111.7902]), 0.24 for P_GB and 0.19 for P_MX. Those checks are roughly two orders of magnitude coarser than the no-surrender benchmark.
The surrender-option components show internal stability over refinement. Across the five finest trinomial grids, H_FB stays in [0.2201, 0.2209], H_GB in [5.7745, 5.7792] and H_MX in [6.4243, 6.4287].
Two hundred knots
Exercise diagnostics assume optimal risk-neutral surrender. The paper states the limitation directly: "These results are descriptive risk-neutral policy diagnostics, not behavioral lapse predictions." Under those policies, the mixed contract surrenders on 89.7464% of paths, at a mean surrender year of 9.001. The guarantee-based contract is exercised much less often, yet its mean surrender year is earlier at 7.871.
For the mixed payoff, the fund branch creates a second continuation region at intermediate fund levels. That feature pushes average exercise later. The surrender premiums should therefore be read as upper bounds on economic value under an optimally exercising policyholder. H_GB = 5.7763 on a base premium of 111.64 illustrates the scale.
The hedge experiment uses the no-surrender liability, so this result does not rely on lapse behavior. Its message about factor exposure is direct: rho_Sr drives the economics. Moving equity-rate correlation by 0.1 changes the fair annual premium by about 0.217. Comparable changes are 0.035 for rho_SV and -0.017 for rho_Vr. The equity-rate effect is about six times the 0.035 equity-variance result.
The authors instead flag variance-rate correlation as the commonly neglected input, observing that it is "often set to zero in hybrid stochastic-volatility/stochastic-rate models". The model decomposition also shows that enrichment depends on the payoff. Moving from Black-Scholes with deterministic rates to full Heston-CIR++ reduces P_NS by 0.553 and P_FB by 0.793, while increasing P_GB by 0.569 and P_MX by 0.170.
The authors describe their hedge experiment as a static, frictionless factor-exposure diagnostic rather than a complete asset-liability-management study. It covers one period at a single date. The call and bond incur no transaction costs, mortality is treated as diversified, and the physical drift appears only as "a stated drift scenario". The comparison of 7.019 with 35.825 should be read at exactly that strength: a factor-exposure diagnostic.
Nothing here was reproduced independently, because there is nothing empirical to reproduce. The paper reports no historical or market-calibrated measurement. Its inputs consist of one ECB curve dated 15 July 2026, one ISTAT mortality table and a single parameter set. Every reported figure, including the hedging RMSEs, is mark-to-model output from the authors' own model.
Our own platform contains stock, ETF, futures and end-of-day option prices. It has no policy-level premium records, guarantee schedules, observed surrender behavior for equity-linked contracts or actuarial mortality table. Without the contract cash flows, there is no liability available to hedge.
A hedge test rebalanced annually across the full twenty years, with a spread charged on each leg, would change my view of the method's practical reach. The single-period experiment already establishes a real finding: vega and rho exposures dominate Delta for these liabilities. The computational idea travels further. A path-dependent state variable receiving only deterministic additions can be handled by translating the value function's argument, and that construction generalises past insurance.