A 4/2 or inverse-CIR model prices the forward below spot when its parameters violate one inequality. Bayraktar and Xing identify that inequality.
For a strict local martingale, the expected discounted terminal price is below today's spot. The paper leaves open whether a market should treat that shortfall as a trade or a model defect. It cites the bubble interpretation of missing mass, then gives a practical reason to care: the forward valuation equation admits two solutions with linear growth. Those solutions are s and s(1 − m), where m is the lost mass. Without a boundary condition at the singular face, a partial differential equation (PDE) pricer can return either.
What is proved, and where does it apply?
The argument starts with an identity for lost mass. The expected terminal value E_y[S_T] equals the probability that a factor diffusion, after a drift shift, remains in its state space at T. The shift adds σθ to the factor drift; θ is the asset's volatility loading. The martingale question thus becomes a question about whether the shifted process exits. The authors credit this classical identity to earlier work.
Their contribution is a set of coefficient tests for the possible exits: a smooth boundary point, infinity, or a singular piece such as an axis or corner. At smooth boundary points the tests are exact. They are also exact at infinity under radial or level-set structure. For singular pieces, they generally give sufficient conditions, with exact tests available in special classes.
At a smooth boundary, the sign of the shifted Fichera drift determines the test. This is the outward normal drift less half the divergence of the covariance. Suppose the boundary and coefficients meet the smoothness hypotheses and there is no explosion at infinity. A nonpositive drift everywhere then prevents boundary exit, subject to a modulus condition called (T) wherever the drift is exactly zero and θ is unbounded. The authors show that (T) recovers Feller's condition for a CIR factor, including critical equality, with constant 2max{κ,0}.
The paper's second organizing principle is explicit: "the Girsanov drift can create an exit only where the coefficient θ is unbounded." Wherever volatility remains bounded, the original and shifted laws are locally equivalent. The boundary therefore behaves alike under both laws.
This is pure theory, without data or calibration.
The inverse-CIR boundary
In the two-factor inverse-CIR model, volatility takes the form g(Y²)/√Y¹, with g positive and bounded. The first factor has drift α₁ + ρη₁g(y₂) − κ₁y₁ under the shifted measure. On either side of the threshold, this drift grows linearly, ruling out explosion at infinity. The distinction here is exit at y₁ = 0, where asset volatility diverges. As the paper observes, Dandapani and Protter's radial tests concern explosion at infinity; this model loses mass through a finite boundary point.
The condition is 2α₁ − η₁² ≥ 2η₁ρ⁻ sup g, with ρ⁻ = max(−ρ, 0). The factors must satisfy Feller's condition 2αᵢ ≥ ηᵢ². Here ρ denotes the correlation of the asset with the CIR factor. Since volatility is the inverse square root of that factor, asset and volatility move together when ρ < 0. The paper states that S is always a martingale for ρ ≥ 0. Negative spot-vol correlation corresponds to ρ > 0 in this parameterisation, and the condition then reduces to Feller's. The restriction binds when calibrated spot-vol correlation is positive.
Correlated 4/2 models: one correlation survives the test
Theorem 6.2 extends the result to d ≥ 2 square-root factors, each satisfying Feller's condition, with an arbitrary positive-definite correlation matrix. Their volatility coefficient is u√y₁ + v·g(y₂,...,y_d)/√y₁. This is the modulated 4/2 form: the other factors alter the inverse-square-root component through g. Its test is 2α₁ − η₁² ≥ 2η₁(ρ₁v)⁻ḡ, where ḡ = sup g.
Only the asset's correlation with the first factor appears. Correlations among factors, and the asset's correlations with factors two through d, affect the dynamics without changing the threshold. The Heston loading u drops out too. Its contribution u√y₁ remains bounded near zero and cannot create an exit. Failure of the inequality gives E_y[S_T] < 1 at every horizon and from every starting point.
Example 6.6 shows how demanding a global condition can be. Set ρ₁ = −1/2 and the factor correlation to 3/5, with sup g = 2 never attained. The model is a martingale iff α₁ ≥ 3/2, including equality. Throughout 1/2 ≤ α₁ < 3/2 it is strict, even when the second factor's bad region lies far into its tail. The paper's absorbed-factor example makes the converse point about the starting value. With c(Z₀) = 3/5, the density remains strict because the infimum is 2/5. As the authors put it, "a favourable initial value does not replace the global coefficient condition."
Coupling is the advance over prior work. Grasselli's one-factor 4/2 criterion leaves the shifted factor CIR, while the multifactor 4/2 of Gnoatto, Grasselli and Platen separates across independent factors. When g is nonconstant and ρ₁v ≠ 0, the shifted first factor depends on the others. The shifted system becomes non-affine, and the remaining factors acquire a drift that blows up like 1/√y₁.
The authors acknowledge that correlated square-root factors do not meet the regularity requirements of their general smooth-boundary theorem. They give Theorem 6.2 a separate proof, controlling the singular drift with an occupation-time bound. They also acknowledge that (T) is not necessary. Their logarithmic family contains martingales for which (T) fails for every c > 0.
Two side results deserve attention. For Wishart factors, the shifted Fichera drift recovers the known non-attainment threshold ΩΩ⊤ ⪰ (n+1)Q⊤Q. The cone's curvature supplies the n+1. Freezing the direction and treating the smallest eigenvalue as CIR instead yields 2Q⊤Q, an incorrect result for n ≥ 2.
The other concerns Dandapani and Protter's strict example. Its published comparison functions give v = ∞ and therefore fail that paper's own explosion criterion. Bayraktar and Xing repair the argument using exponent (3+√5)/4, establishing strictness for ϵ ∈ (0,1).
An untested screen for ETF options
The paper specifies no market. We would use its conditions for US ETF options; the trading rule is our construction, and the authors do not propose it. We propose a pair of relative-value books whose sole difference is that one calibration must satisfy 2α₁ − η₁² ≥ 2η₁(ρ₁v)⁻ḡ. Both would trade on end-of-day quotes, net of costs. We are still building that experiment and have no figures to report. Whatever it finds, it would neither confirm nor refute the theorems, which rest on their proofs.
End-of-day data also restrict the conclusions available. They support daily valuation and discrete hedging, without verifying continuous replication or whether a quoted mispricing could have been executed. US ETF options are American, their underlying ETFs pay distributions, and rates are stochastic. The model must accommodate all three before it can price anything. The paper itself cites earlier work connecting strict local martingales with American valuation.
We expect this screen to be cheap hygiene. We applied a similar parameter gate to signature volatility (/articles/martingality-in-efm-signature-volatility-models). This one costs a single inequality.