Liquidity feedback lifts swap strikes much more than direct liquidity exposure in Fu and Wang's baseline, but that ranking depends on χ = 1. The authors added that coefficient; they inherited the other baseline values from He et al. Their sensitivity table shows the feedback effect rising almost in proportion to χ.
The rest of the paper offers careful, reproducible numerical work. That combination deserves scrutiny before the model gets reused.
How χλ² changes the variance target
Most of the model comes from He et al., the predecessor paper. Its price process is dS/S = r dt + √η dW1 + σλ dW3. The signed Ornstein-Uhlenbeck factor λ represents liquidity stress; η is a Cox-Ingersoll-Ross-type variance factor. Liquidity reaches returns directly through σλ, adding σ²λ² to instantaneous return variance. A two-state Markov chain sets the baseline variance target at 0.08 or 0.12.
Fu and Wang put χλ² into that target. Squared stress raises the level toward which variance reverts. Returns therefore feel this effect later, through η, giving the authors their delayed feedback channel. In the transform, this introduces the forcing terms εχξ2 and εχξ6 into the λ² Riccati equations. A time-dependent backward ODE handles the chain: the Riccati coefficients change over time, making a single matrix exponential incorrect. The authors describe the addition candidly: "The contribution is this particular liquidity-feedback specification and its analysis, not a new general transform class."
No market data enters the exercise. The baseline combines the predecessor's numerical example with χ = 1, which the authors say "is not an estimate of a particular market." The calculations are entirely risk-neutral. They estimate no liquidity premium and claim no source of returns.
What does the absolute-return 'volatility swap' pay?
Squared simple returns form the variance leg, with scaling m²/T and m = 100. Denote that realized measure RVA. The "volatility swap" instead sums absolute simple returns and normalizes them by m√(π/(2NT)); denote this RVO. With N = 252, the variance strike is 1606.2044. We calculate its square root as about 40.08 in volatility units, against an absolute-return strike of 39.7207. The resulting 0.36 gap, also our arithmetic, is between √E[RVA] and E[RVO]. The paper distinguishes both from E[√RVA], the expected square root of realized variance used for a conventional volatility-swap strike.
The authors make this distinction in the abstract and repeat it in the body. Their method cannot directly price either a conventional square-root realized-variance swap or a volatility option. Those payoffs require the joint path of all returns, calling for path simulation or an enlarged joint transform. They also identify simple returns as their own convention, without claiming that traded variance swaps avoid log returns.
The channel ranking at χ = 1
Monthly monitoring (N = 12) gives a clean nested comparison. The variance strikes are 1619.9988 for the full model, 1295.7203 with feedback removed, 1580.7111 without the direct channel, and 1256.5192 with neither channel. Taking out feedback reduces the strike by 324.2784; taking out direct liquidity reduces it by 39.2877. For the absolute-return leg, the respective changes are 4.1035 and 0.4646.
The variance leg has a tiny interaction: +0.0866, or 0.024% of the two-channel increment. The absolute-return interaction goes the other way, at −0.0921, or −1.976%. The description "Approximately additive" fits the variance leg much better.
Against χ = 0, variance strikes increase by 12.4952%, 25.0269%, 37.5951% and 50.2001% as χ takes values of 0.5, 1, 1.5 and 2. Removing direct liquidity trims the full strike by 39.2877. This is about 3% of the 1295.72 no-feedback strike used as the denominator in the χ table; the 3% is our arithmetic. Our linear interpolation of that table puts the point where feedback matches this effect near χ ≈ 0.12. We assume linearity between χ = 0 and 0.5. We also assume an unchanged direct-channel effect across χ, although the paper measures the interaction only at χ = 1. On that interpolation, feedback becomes the larger channel above the matching point.
Fu and Wang acknowledge the dependence. The abstract limits the conclusions to those that "concern the stated illustrative parameter settings," and the body describes the decomposition as "is parameter-dependent." Their warning goes further: "Neither channel ranking nor parameter magnitudes should be generalized to market data without estimation and identification checks."
That leaves a question unanswered. If the ranking cannot carry over to market data, why lead the abstract with it? Why does the economic summary say "delayed feedback dominates direct return liquidity"? The result depends on χ = 1, the baseline value the authors supplied themselves. It measures the model's response to a coefficient nobody has estimated.
Another sensitivity barely registers within this baseline. As ρ2 runs from −0.75 to 0.75, the absolute-return strike moves from 39.7745 to 39.7746 and the variance strike changes by 0.076%. Price-liquidity correlation has almost no effect here across the tested range of −0.75 to 0.75.
Numerical traps the authors caught
The numerical settings are precise: DOP853 with rtol 1e-10 and atol 1e-12. Fourier integration uses Gauss-Legendre nodes starting at zero. Its adaptive upper bound, U0 = max{40, 8√(N/[T(η0+σ²λ0²)])}, is refined until prices agree to 2e-6. For Monte Carlo, the authors publish seeds and use 300,000 paths arranged in 150,000 antithetic pairs. Sufficient admissibility checks pass for 147 of 147 production rows, each at T = 1.
Three issues shifted prices by more than the solver tolerance.
First came the stage junction. An earlier version took the chain ODE through that junction in a single adaptive pass. Tightening the tolerance from 1e-7 to 1e-8 moved the N = 12 price from 1619.943884 to 1620.053829. The split-integration value, near 1619.9988, falls between them: the error did not decline monotonically with tolerance. After finding the problem, the authors split the integration and revised the N = 252 strike from 1606.510890 to 1606.204436. All other figures in this note use the corrected version.
The Fourier upper bound also needs more attention than its fixed floor of 40 might suggest. For N = 52, setting U = 40 leaves an absolute-return pricing error of 1.23. Raising it to U = 160 cuts the error to 2.30e-08.
A further discrepancy appears in the comparison with the predecessor's graph. The example page in He et al. leaves the initial regime unstated; Fu and Wang condition on X0 = e1. Their values miss that graph by 1.86 (0.131%) at N = 1 and by about 0.4 at several higher frequencies. The missing regime specification could explain the gap. The paper leaves its cause unresolved without the predecessor's code and complete settings.
Simulation agreement is good where it should be. For N = 252, the transform price of 1606.2044 compares with Monte Carlo at 1606.42 [1605.55, 1607.29]. With N = 1, the simulated mean is 1817.71, or 12.8 above the transform, within a wide interval. These intervals capture sampling error only, as the authors state. Pricing both legs by transform at N = 252 takes 0.1754 seconds. The 300,000-path simulation, with 1008-step paths, takes 47.19 seconds.
Without a traded strike
We could not test the model on our data. We have neither variance-swap nor absolute-return-swap quotes, leaving us without an executable entry strike. Nor does the paper provide a calibrated risk-neutral state (η, λ, regime) for any historical date. Realized payoffs alone cannot supply a P&L.
An estimate of χ would change the verdict on the channel ranking. It would require a measured liquidity proxy alongside joint option and liquidity data, the missing step the authors themselves identify. Until then, the 324-versus-39 split describes χ = 1.