Fractional memory improves VIX call pricing most when volatility is high. Yang, Su and Horvath report out-of-sample call-price MSE reductions of 13.06% overall and 40.93% for quotes with VIX at or above 25, relative to the Markovian parent of their direct VIX model. The authors lead with this concentration in high volatility, including in their abstract. The question is what the gain measures.

How memory enters the model

Instantaneous squared VIX (ISVIX) is the short-horizon limit of squared VIX. At any horizon, squared VIX equals expected ISVIX averaged over that horizon. In the Markovian model of Luo et al., called LZZ in the paper, ISVIX reverts exponentially toward a stochastic level θ. One exponential therefore carries today's shock into both the 9-day and 1-year indices.

RISVIX replaces the exponential with the fractional kernel t^(α-1)/Γ(α), where α lies between 1/2 and 1. Expected ISVIX then decays according to a Mittag-Leffler function, quickly at first and with a fatter tail. The time-zero curve still has a closed form: VIX² = w(τ)V0 + (1 − w(τ))θ0, with w a Mittag-Leffler weight. Only memory α, reversion speed κ and the two states V0 and θ0 determine that curve. The diffusion scales σV and σθ affect the distribution of future VIX alone. So does a jump with exponentially distributed sizes, which fires rarely: its intensity in the option fit is 0.0178.

This separation allows a two-step fit. In the term-structure comparison, the first step fits α, κ and daily (V, θ) to closing VIX9D, VIX, VIX3M, VIX6M and VIX1Y. The sample covers 1,883 balanced days from January 2018 to June 2025. Curve-level MSE drops from 0.6295 to 0.5639 (10.42%). Slope MSE falls by 17.21%, and curvature MSE by 13.18%.

For the option test, the authors repeat the first step using only 2018 to 2023, obtaining α 0.8937 and κ 7.0357. They hold these fixed in step two and fit the four distributional parameters to call midquotes. Pricing uses a Riccati-Volterra transform and an erfc-based Bromwich integral. Training covers 47,451 option quotes from 2018 to 2023; testing covers 14,543 quotes from January 2024 to June 2025. Test MSE comes in at 0.6293 versus 0.7238, and MAE at 0.7061 versus 0.7537. Across VIX levels, the out-of-sample MSE reductions are 10.0% below 20, 21.5% at 20 to 25, 37.4% at 25 to 30, 45.9% at 30 to 40 and 39.6% at 40 and above.

A precise sample, some numerical choices left open

The call filter is explicit: 30 to 126 days to expiry, K/VIX in [0.7, 1.3], a midquote of at least 0.50 and a spread no greater than 10% of mid. The variance window τ0 stays fixed at 30 days. Both fitting stages minimize ordinary squared errors. Step one uses index levels; step two uses prices in VIX points.

We did not find a stopping tolerance or iteration cap for the alternating updates of α, κ and the states. The paper says the cycle continues until one is reached. Nor did we find a value for η, the damping parameter used in the inversion contour. The theory permits any η within a strip, whose width is never calculated. Readers must also choose how to discretize the fractional Volterra equation and evaluate the Mittag-Leffler function. These are defensible choices to leave open, and they are precisely where separate implementations can diverge.

What is held out?

Six global parameters remain frozen throughout the test window. Every test day, however, the two states are fitted again to that day's five indices. The authors reset each date to local time zero. They describe this as standard practice, since rough models condition on some initial input; RISVIX uses two numbers for that input.

The paper's own Theorem 3.1 shows that VIX² at later dates contains a history integral over the past path. The local model sets that integral to zero. The paper also acknowledges a restriction on the time-zero curve: it can move only monotonically from V0 toward θ0. A humped curve therefore receives the best monotone fit.

The test answers a narrower pricing question: with today's curve given and shape parameters frozen, are the calls priced more closely? Memory starts afresh each test day. The out-of-sample improvement exceeds the in-sample improvement, 13.06% against 9.61%. That is unusual, though eighteen months gives little basis for reading much into it.

Shock days account for much of the edge over LZZ

Term-structure level MSE falls 28.73% against LZZ on shock dates, defined as daily VIX increases at or above the 90th percentile. On the remaining dates, the reduction is 1.18%. Fitting January to June 2020 separately lowers α to 0.7051 (H = 0.2051), compared with 0.8942 (H = 0.3942) across the full sample. The full-sample H = 0.39 implies mild roughness alongside the 0.20 estimated on the same curves by the rough Heston benchmark (YFK).

The gains vary by horizon. Across the full sample, VIX9D error falls 33.30%, but RISVIX's MSE exceeds LZZ's by 1.05% at 30 days and by 18.83% at one year. Calls settle on the 30-day index. In the COVID fit, three-month MSE exceeds LZZ's by 3.33%. Out-of-sample option results also weaken at higher strikes: calls with K/VIX above 1.1 improve only 3.93% in MSE and 0.48% in MAE. The paper's single implied-vol illustration, using 62 quotes from March 17, 2023, places RISVIX implied vols below the market at higher moneyness.

BPS, the rough Hawkes-Heston model, is the nearest outside competitor out of sample. Its test MSE is 0.6757 versus RISVIX's 0.6293. On MAE, 0.7067 versus 0.7061 leaves them nearly tied. The authors attribute BPS's high calibration MSE of 0.9218 to errors during COVID in 2020. We did not find how BPS, HJT or AIL were calibrated, or a significance test for any difference in errors.

The missing residual-fading trade

The spread filter allows up to 10% of mid. For all 61,994 quotes spanning 2018 to 2025, the mean midquote is 4.28 points. At that sample-average price, the widest permitted spread is 0.428 points. RISVIX's mean absolute test error is larger, at 0.7061 points. Those residuals could be executable mispricings or model error. Midpoint MSE cannot distinguish them, and the paper reports no strategy trading the residuals. A residual-fading book filled at bid and ask during the 2024 to 2025 window would settle the question. A positive result would give the valuation tool a claim to being a signal.

We could not run that test.

We lack VIX index option chains, which step two calibrates to directly. VX futures have no strike dimension. Options on volatility ETFs have a different underlying payoff. Neither can reproduce the experiment.

RISVIX deserves a place beside LZZ as a pricer on a VIX desk, chiefly on days when VIX is above 25.