Three vanilla options remove 96% to 99% of the error left by a stock-only variance-swap hedge in Di Tella, Keller-Ressel and Ongarato's rough Heston model. In the long-only setting, the best three-option basket removes 96.22% to 99.29% of that residual across H=0.1, 0.3 and 0.5. The low end belongs to the roughest case, H=0.1; the high end to classical Heston, H=0.5, computed in closed form. Even one well-chosen option removes 85.52% to 91.95%. Those are useful model benchmarks for compressing a variance hedge. They have never been tested in a trade.
What the options hedge
The target is a variance swap struck at its model fair rate, paying integrated variance minus k. Stock trading hedges the variance risk that moves with the stock. At correlation ρ=-0.7, the fraction 1-ρ² of variance shocks remains. The authors call this purely dynamic error A: 4.8549e-4 at H=0.1, 3.8056e-4 at H=0.3 and 2.8754e-4 at H=0.5.
Static option positions can take on some of that remaining variance exposure. The authors minimise expected squared hedging error under a risk-neutral measure. Given a basket, a Galtchouk-Kunita-Watanabe decomposition determines the stock strategy by separating each claim into a stock-replicable part and an orthogonal residual. The option weights solve the quadratic ν* = C⁻¹B, where B and C are expected covariations of those residuals. Rough volatility is non-Markovian. To handle it, the authors use the forward variance curve, the expected variance E[V_u | F_t] at each future date u, as a Markovian state. Fourier representations of calls and puts complete their semi-explicit formulas.
There are 21 candidate options, around a spot of 100: puts struck from 50 to 95 and calls from 100 to 150, in steps of 5. For each basket size m, the authors search every subset, allowing short positions in one search and restricting the other to long-only positions. The headline table uses long-only baskets.
Three strikes and a continuum
Neuberger's formula exactly replicates variance through dynamic stock trading and a continuum of out-of-the-money options weighted by K⁻². A finite basket approximates that integral. Here, even a coarse selection captures much of its benefit. Small baskets sit near the money; as m increases, they reach into the wings. Puts enter first and receive larger weights. The K=50 put appears at m=3 for H=0.1, m=4 for H=0.3 and m=5 for H=0.5.
For m=3, the authors plot weights against strike on log-log axes and describe a slope close to -2. We did not find a fitted slope. Three points per regime leave that plot suggestive at best. The stronger evidence is η(m), the share of the 21-option error reduction achieved with m options. At m=3, it reaches 99.06% for H=0.1, 99.11% for H=0.3 and 99.32% for H=0.5. Larger baskets contain smaller ones, so adding strikes cannot increase the error in this calculation.
Where must an implementer make a choice?
The paper supplies many settings: λ=0.3, ρ=-0.7, σ=0.3, V0=0.02 and θ=0.066. It simulates 10⁴ paths with 250 Euler steps and a 500-factor sum-of-exponentials kernel approximation, using √max(0,·) truncation. The call contour is at Re u=2 and the put contour at -0.125; each is truncated to ±150 with 100 points. Computing A, B and C reportedly takes about an hour.
We did not find a stated maturity. The strategy plots span 0 to 1.0, making T=1 the natural guess. The authors say they approximate the forward variance curve from simulated paths and store it as an (N+1)×(N+1)×M tensor. We could not find how they form that conditional expectation. An implementer must choose a method, and the choice affects every entry of B and C.
Two procedural issues bear directly on the reported errors. The first is that the comparison across roughness levels mixes calculation methods. The paper discloses closed-form Heston formulas for H=0.5, without discussing their effect on the comparison. H=0.1 and H=0.3 instead use Monte Carlo on the multifactor approximation. At m=12, the error is 1.3933e-5 for H=0.1 versus 9.5131e-8 for H=0.5, roughly 146 times larger.
A has a different footing. The authors calculate it by deterministic quadrature on the Mittag-Leffler function, without Monte Carlo. Its rise from 2.8754e-4 at H=0.5 to 4.8549e-4 at H=0.1 therefore has no sampling noise and supports their conclusion that residual error grows as the regime gets rougher. They interpret the larger errors at lower H as consistent with variance becoming more irregular over short horizons. That may well be right. For m≥1, though, errors depend on Monte Carlo estimates of B and C from 10⁴ paths. Table 3 reports no standard errors. We cannot separate the size of the m=12 gap, or a rough-case floor near 1e-5, from estimation error.
The second issue is subset selection. The authors choose subsets using the same estimated B and C that produce the reported errors. When those estimates are noisy, taking the minimum across every subset pulls the reported error downward. At m=10, the search covers 352,716 subsets. Exact H=0.5 inputs avoid this bias. For H=0.1 and 0.3, the Monte Carlo inputs both select the winning basket and measure its error; the paper does not estimate the resulting downward bias.
The unconstrained solution also contains short positions, which the authors set aside as "mainly due to numerical artifacts." They say a numerical check found that the shorts do not greatly reduce error, although we did not find its comparison figures. Their headline is long-only, so this point does not affect the 96.22% figure.
Would the hedge survive a listed chain?
The setup assumes continuous stock trading and variance monitoring, zero rates, no spreads, and a swap rate equal to the model expectation. All calculations use a risk-neutral measure. The authors say moving to the physical measure makes the formulas "considerably more involved". Their L2 errors assume continuous rebalancing.
Roughness also changes the dynamic leg. Across three paths, the optimal stock position for H=0.1 differs from Heston's by roughly 7% to 110%; for H=0.3, the difference is 4% to 50%. An error in H would reach the delta before the option weights.
We could not test the trade ourselves. We have no quotes for a tradeable variance swap, and realized variance calculated from price bars would supply only a hypothetical payoff against which no position could be held. The paper establishes a model range of 96.22% to 99.29% and a basket shape in which out-of-the-money puts carry the larger weights. It supplies no execution test. Before trusting three strikes on a listed chain, we would want to see the result after charging option spreads.