At spot 100, PINN holder pricing misses FDM by 0.167

At spot 100, the neural solver from Yan, Zhu and Li produces a holder-writer interval just 0.146 wide, while its holder price differs from its own finite-difference (FDM) benchmark by 0.167. Reading the interval as a quoting band requires the solver to distinguish prices more finely than that discrepancy. On the holder side at the money, it has yet to do so.

Inside the pricing model

The underlying has two variance factors. The first is instantaneous variance v; the second is its long-run level θ, which is itself stochastic. A liquidity factor L supplies another diffusion term, with volatility βL. Its drift includes the feedback term λ_TC e^{κ_TC} L^ξ. Higher proportional costs therefore raise long-run illiquidity.

Pricing follows Leland and Hoggard-style discrete hedging. With rebalancing every δt = 1/15, expected costs include a term proportional to κ_TC S. Its scale depends on √(2/(πδt)) and the square root of a quadratic form in the option's second derivatives. Subtracting this term gives the holder's lower value; adding it gives the writer's upper value. As the paper puts it, "the model therefore produces a price interval rather than a single arbitrage-free value."

The exercise right belongs to the holder, whose variational inequality determines the exercise boundary. The writer's PDE is solved over the holder's continuation region. Within the holder's exercise region, the writer's value equals intrinsic value.

Two modelling choices support this construction. Auxiliary hedging options trade free of costs, and their hedge ratios must make Δ_i ∂V_i/∂S depend only on time. That restriction removes the price's dependence on the choice of hedging instruments.

The authors call their solver PINN-splitting: Ikonen-Toivanen operator splitting uses a physics-informed network for the inner solve. There are 15 time layers (Δτ = 1/15). At each layer, a 2×64 tanh network solves the elliptic continuation equation using 10,000 Sobol points across S, v, θ and L. A closed-form projection enforces V ≥ payoff, then updates a slack variable. For the writer, the exercise region consists of points where the stored holder value is within 10^-3 of the payoff.

The headline accuracy figures look reassuring. Without costs, differences from FDM range from 0.2173% to 1.0231%; against least-squares Monte Carlo they range from 0.0686% to 1.5842%, for S0 from 80 to 120. With κ_TC = 0.004, the holder differs from FDM by 0.7153% to 1.2814%, and the writer by 0.3941% to 1.7737%. The conclusion says the scheme "agrees closely with FDM and LSMC benchmarks". The paper never compares those errors with the holder-writer spread.

Can it distinguish a 0.146 spread?

At S0 = 100, the PINN prices the holder at 13.8004 and the writer at 13.9467. FDM returns 13.6336 and 13.8682. Its interval is consequently 0.235 wide, versus 0.146 for the PINN. At S0 = 80, the ordering reverses: the PINN interval spans 0.431 (24.3702 to 24.8016), while FDM's spans 0.172. The writer discrepancy at 80 reaches 0.4322, enough by itself to exceed the PINN interval.

At 80, then, the solvers differ on the band's width by a factor of 2.5. At 100, the disagreement runs the other way. They also distribute the cost effect differently. The PINN writer value at 100 is 13.9467, barely above its no-cost value of 13.9432. Almost the entire cost effect falls on the holder: 0.143 out of 0.146. FDM distributes that effect more evenly, with values of 13.6336, 13.8020 and 13.8682.

FDM still needs verification. At S0 = 100 without costs, its value of 13.8020 differs from LSMC's 14.0525 by about 0.25. The PINN's 13.9432 falls between them. It also remains below LSMC from 90 to 120. Disagreement between the benchmarks thus accounts for part of each reported relative difference.

The FDM reference uses a fully explicit scheme, with 40 points in v and 80 each in θ and L. Across every row of all three tables, the PINN exceeds FDM by 0.017 to 0.43. A converged reference would establish where the bias lies.

Exact projection, conditional stability

The design is appealing. Keeping the obstacle constraint out of the network's loss is the authors' way of "avoiding the exposure of the network to the non-smooth obstacle." Projection enforces it exactly. The same collocation points serve every layer, and each layer starts from the previous layer's trained network. At the first layer, Adam reduces the loss from about 2.56×10^4 to below 5.62×10^-2 over 6,000 epochs.

The abstract acknowledges that stability is conditional. It requires Δτ C_Φ < 1, with Φ denoting the nonlinear cost term. The authors describe the sup-norm Lipschitz property of Φ as "a working hypothesis." Since Φ contains second derivatives, that is a demanding assumption.

The authors defend it in two places. Remark 3.1 uses an inverse inequality to argue that the property holds when network weights remain in a compact set Θ_R. Remark 3.2 calls the step condition "comfortably satisfied" at Δτ = 1/15. Yet the inverse-inequality constant depends on R. As R grows, C_Φ therefore lacks uniform control, precisely in the limit required by the approximation argument. The conclusion nevertheless says the scheme "preserves theoretical stability" without attaching a condition. We did not find an error bound for the PINN approximation itself.

Speed remains an open question too. The paper's motivation describes mesh methods as computationally prohibitive in four state dimensions. For its reference, it then runs explicit FDM on a 200×40×80×80 grid over 150,000 time steps. We did not find runtimes for either method.

What the 2018 to 2024 liquidity estimates establish

In the appendix, daily changes in an Amihud-style S&P 500 illiquidity proxy are regressed on L^ξ and L. The source is CRSP, covering January 2018 to December 2024. Estimates are ξ = 0.733 and β1 = 1.281, with t = 3.77 (p = 0.000176) and R² = 0.401. These results support a concave nonlinear drift term for this proxy.

Identifying the transaction-cost channel is a separate matter. The mapping sets β1 equal to κ_L λ_TC e^{κ_TC} Δt. Because κ_TC stays constant throughout the sample, the regression cannot identify the drift's response to costs. The appendix SDE also uses σ_L√L diffusion; the pricing model uses σ_L dW_L. Under the stated mapping, β2 = −2.362 at Δt = 1/252 implies κ_L near 595. Pricing runs instead use κ_L = 2. Only ξ = 0.73 carries over from the regression into those runs.

Our adapted strategy remains untested

We have not run the authors' solver. Nothing in this review tests it. We are evaluating an adapted strategy using end-of-day options, and results are not yet available. Our liquidity proxy draws on daily price and dollar volume, so it cannot reproduce the S&P 500 estimate. We also did not find an observation count for that regression.

End-of-day data permits daily exercise decisions only. It cannot test continuous boundary monitoring or compare the band with executable bid-ask quotes. The model uses frictionless auxiliary options for hedging. Our backtest charges costs on the instruments actually traded and does not interpret the modelled interval as an arbitrage bound.

The hurdle is simple.

For each side, the difference between the PINN and a converged reference must be well below the 0.146 at-the-money width. The holder's current gap to explicit FDM is 0.167. Until that gap shrinks, profits from a backtest of the band measure solver error as much as liquidity.