Fit two skew-sticky thresholds to a price series and the paper gives you a hedge. Whether it gives you a price depends on one exact equality at every threshold: 2rξρ_ξ = 2κ_ξ - 1. The skew at threshold ξ is κ_ξ, while ρ_ξ controls stickiness. Every threshold imposes its own equation, all of which must accept the same interest rate r. Consider the paper's Skew-Sticky 3, with κ = 0.7 at both thresholds, ρ = 1, and interfaces at -1 and +1. One interface requires r = 0.2. The other requires r = -0.2. No equivalent local martingale measure exists for any r.
Anagnostakis, Criens and Urusov state the issue in the abstract: "When the NFLVR condition fails, the framework may produce multiple hedging equations corresponding to non-minimal strategies, whose associated prices can exceed the minimal hedging capital." NFLVR means no free lunch with vanishing risk. Under the fundamental theorem of asset pricing, it is tied to the existence of an equivalent local martingale measure (ELMM).
The abstract then promises evidence: "We illustrate both the effectiveness and limitations of the approach through numerical experiments involving diffusion models with irregular features." Section 4.4.3 supplies it. Skew-Sticky Model 3 is the setup without an ELMM. With 32 rebalances, its mean tracking error is 0.005 ± 0.009 and its tracking-error standard deviation is 0.201 ± 0.010. At 4096 rebalances, the corresponding figures are 0.001 ± 0.002 and 0.041 ± 0.002. Bachelier has an ELMM and records 0.192 ± 0.009 and 0.019 ± 0.001 at those same frequencies. The hedge continues to track when NFLVR fails. Theorem 2.12 supplies a testable condition for deciding whether the initial value qualifies as a price. An exact equality makes the treatment of a fitted model that misses it a practical question.
We ran no backtest of this. Daily bars cannot identify the scale function and speed measure that determine the model, especially sticky or singular behaviour at an interface. An implementation would need a tractable parametric substitute and would calibrate that instead. The paper also assumes a continuously self-financing hedge. End-of-day option chains and daily rebalancing would measure discretization error in a substitute model, rather than replicate the paper. No figure below comes from a run of ours.
Scale, speed and the interest rate
The market contains one risky asset and one bank account paying the constant rate r. The asset is a regular continuous strong Markov process on an open interval. Its specification consists solely of a scale function s and a speed measure m. Both s and its inverse are differences of convex functions. The price need not follow an assumed SDE. This class allows partially reflecting thresholds (skew), points where the price spends positive time (stickiness), and fractal slowdowns.
The authors use (s, m, r) to build an auxiliary diffusion with characteristics (G, m̄). Two objects drive the construction. The first is the signed measure ν(dx) = -r x s'₊(x) m(dx). The second is a stochastic-exponential-type function g, obtained from an integral equation. It transforms the speed measure according to m̄(dx) = s'₊(x)/g(x) m(dx), while G is the integral of g.
Stieltjes derivatives against this auxiliary pair define the hedging PDE. The paper calls a solution with the one-sided regularity in Definition 2.5 a "good solution". From it, the strategy holds ∂⁻ₓu shares and leaves the balance in cash. Theorem 2.6 establishes that the strategy is self-financing and ends at h(S_T) for bounded continuous payoffs. Theorem 2.12 gives necessary and sufficient conditions on (s, m, r) for an ELMM. When one exists, Theorem 2.14 identifies the initial value as the minimal hedging capital. Without an ELMM, the paper establishes only that the value may be strictly higher. Example 3.3 exhibits such a case.
The numerical work studies a two-threshold diffusion on R, with skew-sticky interfaces at ±1. There are four parameter sets: Bachelier and three versions of the same family, labelled Skew-Sticky 1, 2 and 3. The direction of skew at each interface and the interest rate distinguish them. The traded instrument is a bear spread with strikes -2 and 2, over horizon T = 10.
The authors use implicit finite differences on the truncated domain [-50, 50], with 500 time steps and up to 4000 space steps. Homogeneous Neumann conditions apply at the edges. Around the interfaces, the grid uses finer spacing, h² rather than h. Paths are generated through the space-time Markov chain approximation of Anagnostakis, Lejay and Villemonais. The experiment contains 2000 paths and rebalancing counts from 32 to 4096.
Table 1 gives mean tracking errors at 4000 space steps for all four models and all eight rebalancing counts. Their absolute values are typically below 0.07. Dispersion declines monotonically. For Bachelier, it falls from 0.192 ± 0.009 at 32 rebalances to 0.019 ± 0.001 at 4096.
There is no market data in the paper at all.
When does hedge cost become price?
Three conditions are jointly necessary and sufficient for an ELMM in the skew-sticky market. They are the non-explosion condition (2.11), equality κ_ξ = κ̃_ξ at each threshold, and local square integrability of x ↦ (rx - b(x))/σ²(x). The threshold equality creates the difficulty. Written in the input parameters, it becomes 2rξρ_ξ = 2κ_ξ - 1 at every threshold ξ. Each threshold contributes an equation, while the same r must solve all of them.
Skew-Sticky 3 places κ = 0.7 at both thresholds, sets ρ = 1, and locates its interfaces at -1 and +1. The model therefore has no ELMM for any interest rate. Its thresholds simultaneously require r = 0.2 and r = -0.2. When the equality fails, the market permits increasing profits. The authors say these can be constructed explicitly and refer to their companion paper in Remark 3.1(c).
The restriction appears elsewhere too. For J = (0,1), the paper says (2.11) is "always violated when r > 0, which intuitively means that the positive drift pushes the process towards the boundary 1". The conclusion does not depend on the diffusion coefficient. The sticky Black-Scholes model from the earlier Anagnostakis paper cited here also satisfies NFLVR only at r = 0.
A model fitted with two or more interfaces therefore supplies a hedge and its cost. Calling that cost a price requires the calibration to meet the equality.
An arbitrary boundary extension
PDE uniqueness requires J* = J, meaning that the auxiliary diffusion has no accessible boundary. If an accessible boundary exists, the authors direct the reader to extend m̄ there "in an arbitrary manner". Example 3.4 constructs two bounded continuous good solutions with identical terminal conditions. One lies strictly above the other. The selected solution depends on whether m̄({0}) = ∞, which makes the boundary absorbing, or finite, which makes it reflecting.
Overpricing arises through a separate failure. The example follows Delbaen and Schachermayer and uses BES₃, the three-dimensional Bessel process. Its auxiliary diffusion has 0 as a regular boundary. Thus J* is not J, and no ELMM exists. Example 3.3 shows, in the authors' words, that "without NFLVR our hedge does not necessarily achieve the minimal hedging capital". For a constant payoff h, the fundamental value function is e^{-rT}h. Using the strict martingale density Z instead gives e^{-rT}h·E^P[Z_T], where E^P[Z_T] < 1. The PDE overprices even a constant payoff.
An implementer receives this boundary choice without a financial principle for making it. Discussion 2.7(i) gives the authors' answer. Boundary classification for the auxiliary diffusion does not affect the hedging result, and each choice produces a well-defined hedge. The strategy evaluates u and ∂⁻ₓu only at the price, which remains inside J. Changing the boundary changes the value function while leaving the hedge valid. The paper says this plainly.
Tracking error versus premium
Dispersion declines monotonically in all four configurations. Skew-Sticky Model 3, despite having no ELMM, moves from 0.201 ± 0.010 at 32 rebalances to 0.041 ± 0.002 at 4096. On the first two log-log points, the slopes are -0.48 for Bachelier and then -0.31, -0.51 and -0.42 for the three skew-sticky cases. The paper compares these with the classical N^{-1/2} result of Bertsimas, Kogan and Lo.
The reported bear-spread premiums put those dispersions in perspective. Bachelier has dispersion of 0.019 against a premium of 2.0, below 1%. For Skew-Sticky 2, the figures are 0.016 against 14.778, around a tenth of a percent. Its dispersion is also lower than the smooth benchmark at every one of the eight rebalancing counts in Table 1. The comparison is 0.115 versus 0.192 at 32 rebalances, and 0.016 versus 0.019 at 4096. The authors tentatively attribute this to the negative rate r = -0.2, which reduces effective risk-neutral volatility.
Skew-Sticky 1 is at the opposite extreme. At 4096 rebalances, its dispersion is 0.131 on a premium of 0.271, roughly 48%. Its mean error at 4096 is 0.013 ± 0.006, above the 0.008 ± 0.008 recorded at 1024. That comparison is ours, and the authors do not discuss it. Their discussion instead concerns dispersion, "particularly for outward-pointing skew (Model 1)", and slower log-log decay, "which we attribute to the non-smooth behaviour at the thresholds".
The experiment includes no transaction costs. Algorithm 1 accrues interest on cash and funds share trades at the simulated price, without a spread or commission.
The authors describe the numerical section as an examination of "the qualitative behaviour predicted by the theory, not on a rigorous convergence analysis". Its confidence intervals "are used only to indicate sampling variability". They also solve the transformed PDE (4.2), rather than the hedging PDE (2.6), because the finite-difference method yields "less biased value fields" for the transformed version. We did not find the metric used for that comparison. A replicator would have to revisit the choice.
The single-path results contain labelling conflicts as well. Plot legends report 128 and 4096 hedges, whereas the text reports 4096 and 256. In a figure concerning Model 1, Table 3's fourth row is labelled Skew Sticky Model 2 at 4096 hedges and gives tracking error -0.09.
A calibration that recovered κ_ξ, ρ_ξ and threshold locations from an observable price series would change my reading. It would also need to come close enough to 2rξρ_ξ = 2κ_ξ - 1 before the PDE value could be quoted as a price. The paper does not demonstrate how those quantities can be estimated from price data. Sections 3.1 and 4.1 provide the two-threshold parametric family itself, without the estimation step. As it stands, the paper proves a clean and genuinely new hedging theorem for a model class it does not show how to calibrate. Its pricing claim remains conditional on an equality.