Near zero variance, a small estimation error can move a Heston American put exercise threshold sharply. Jeon, Kim and Oh show why. Their threshold b(τ,y) gives the stock price at which exercise begins, for remaining maturity τ and current variance y. The holder exercises at or below b and holds above it. Wherever y > 0, the authors prove that b is C^∞ and strictly decreasing in both arguments, even when the Feller condition fails. If variance can reach zero, b_y tends to minus infinity at the axis. Each boundary slice turns vertical in variance as it arrives there.
The proof in the interior runs through three steps. Price monotonicity in stock and variance gives each contact point a fixed orthant of the contact set. A one-variable lifting absorbs drift and discount into the operator's principal part. The problem then belongs to the Audrito and Kukuljan class of time-monotone parabolic obstacle problems; the contact orthant excludes every singular blow-up. Hopf arguments show that the smooth surface is a graph over the stock direction and is strictly monotone. The monotonicity result alone would leave the graph property unresolved.
At zero variance, the authors change methods. They control the price using the stock's local time at the strike, CIR inverse-moment and Laplace-transform estimates, and a Blumenthal 0-1 entry argument. These are theoretical results. The paper's numbers are synthetic illustrations.
The threshold and its edge
At the boundary, the price meets the payoff with P_s = −1 and P_y = 0. Its gamma on the continuation side is 2(rK−qb) / [y(b² − 2ρσ b b_y + σ² b_y²)]. The expression is positive, and the price is not C² across the boundary.
The axis threshold B(τ) lies strictly between 0 and K. It is locally Lipschitz in maturity, and b converges to it from positive variance. At stock price B(τ), the premium just above the axis is linear in variance: cy ≤ P − K + B ≤ Cy. Although P_y vanishes on the contact set, its limsup along that stock is at least c > 0. Smooth fit therefore fails in the variance direction at the axis. The boundary slices are tangent to the axis: b_y → −∞, while (B − b)/y → +∞. At expiry, the entire surface, including the edge, approaches the strike.
The computed threshold steepens in the same direction. For parameter set A (Table 2), at one year remaining b_y/K = −1.714 when y = 0.04, against −0.324 when y = 0.64. On set A at y = 0.04, a variance error of 0.01 moves the threshold by about 0.017K. The paper also bounds maturity speed down to zero variance by C_I(1+y), though C_I is unspecified.
There is no market data. The authors use two synthetic parameter sets, both with K = 1, r = 0.05, q = 0.02, κ = 2, θ = 0.04, ρ = −0.7 and T = 1. Set A uses σ = 0.3 and satisfies Feller (0.16 > 0.09). Set B uses σ = 0.6 and fails it (0.16 < 0.36). We found no P&L or hedging test. The authors describe the figures as illustrations of Theorem 1.1, not tests of the endpoint theorems.
Does y = 0 need a boundary condition?
An implementer can discretize the axis without imposing a separate boundary condition. The authors use forward differences there for the degenerate transport equation, with nothing imposed beyond the equation itself. Their scheme combines implicit Euler, the Ikonen and Toivanen operator splitting, and a seven-point mixed-derivative stencil. The stencil remains monotone if |ρ|σΔx ≤ Δy ≤ σΔx/|ρ|. On our arithmetic, both grids meet that condition. Set A has Δy = 0.002 within [0.0013, 0.0027]; set B has Δy = 0.004 within [0.0026, 0.0053]. By our count, each grid reaches an upper variance level of 1.2 (600 × 0.002 and 300 × 0.004). We did not find the center of the 705-node log-price grid, so its placement remains a judgment call. Central first differences give way to upwind differences where they would violate the stencil's sign condition. That occurs at only a few nodes near y = 0.
The bounds on the axis gap cannot size it for trading. With ν = 2κθ/σ², our arithmetic gives ν ≈ 1.78 for set A and log(B/b) ≤ C·y^{1/2}. For set B, ν ≈ 0.44 and the bound is C·y^{0.31}. Neither bound specifies C. The authors call them upper bounds and write, "Exact boundary gap equivalents remain open."
A daily exercise rule
The paper's main results require q ≤ r; its illustration uses 0.02 against 0.05. For q > r, the authors retain a local version only where rK − qs > 0. Under the illustrated parameters, failing Feller moves the boundary by at most about 0.04K. Set B lies above set A across the plotted range. The authors make the code available on request only.
The trading rule from here is our construction. It would take four Heston parameters (κ, θ, σ, ρ), plus r, q and a daily variance estimate y, solve for b(τ,y), then compare the close with that threshold. C_I is absent from the paper's maturity-speed bound C_I(1+y), so the bound alone cannot size daily drift. In a US single-stock put book, high-yield names would need separate handling under the q ≤ r restriction.
Costs would also be ours to supply. The paper has no market inputs, and the live choice is between exercising the put and selling it at the bid.
Where the gamma check weakens
The numerical check covers one identity, the closed-form gamma, on set A only. For y ≥ 0.09, finite-difference gamma agrees with the formula within 1.7%. At τ = 1 and y = 0.64, the figures are 0.992 and 0.990. At y = 0.04, the discrepancy rises to between 2.4% and 3.2%; its worst case is τ = 0.50, with 4.549 against 4.407. The authors note that the boundary is steep in y at y = 0.04 and that the quadratic contact region is thin relative to the grid. They write, "The table is an illustration of the identity, not a convergence study."
On our arithmetic from the set A table, y·P_ss falls from about 0.64 at y = 0.64 to about 0.20 at y = 0.04, both at τ = 1. This agrees in direction with the paper's limit of zero, though the authors do not present the table as a test of that limit.
Accuracy is weakest where the exercise decision is most sensitive, a limitation the authors flag. They argue that agreement within 1.7% for y ≥ 0.09 is what one can expect with first-order time stepping and a boundary read from the grid. That accounts for gamma at moderate variance. The low-variance slope remains harder to assess. At every variance level, the boundary comes from extrapolating √w at the first two continuation nodes. At y = 0.04, the quadratic contact region is thin relative to the grid. The reported b_y comes from a quadratic fit across a variance window 0.1 wide, wider than the entire interval between y = 0.04 and y = 0.09. Its low-variance value therefore averages across the region where the boundary bends fastest.
The paper supplies a smooth target for a solver and shows its solver reaches that target to within a few percent. It leaves the accuracy of a threshold driven by a noisy daily variance estimate to an implementation. A threshold error at y = 0.04 smaller than a typical put spread would change our view of how much the low-variance steepness matters.