American options under tempered space-fractional diffusion: Riesz scaling, risk-neutral pricing, and exercise boundary analysis

1 Introduction

1.1 Motivation

Empirical asset returns display heavy tails: large log-returns occur more frequently than a Gaussian model predicts, and fitted tail indices are often below 2 in equity, foreign-exchange, commodity, and cryptocurrency markets [1, 2]. Stable laws provide a useful local model for such tails, and the symmetric Riesz derivative is the infinitesimal generator of the corresponding stable motion [3, 4]. For option pricing, however, a pure stable log-price process with index γ < 2 has no finite positive exponential moment. It therefore cannot be inserted directly into an exponential stock model under the usual risk-neutral martingale condition.

Fractional spatial derivatives are introduced here for a modeling reason rather than as a formal generalization of the Black–Scholes equation. The Gaussian diffusion in the classical model produces local second-order smoothing and exponentially small short-time transition tails. By contrast, observed returns often exhibit non-Gaussian scaling, abrupt large moves, and option smiles that are difficult to reconcile with a purely local Brownian generator. A Riesz fractional derivative corresponds to a non-local jump generator and therefore represents the effect of heavy-tailed increments at the level of the pricing equation. This is the mechanism that motivates a space-fractional formulation.

The same modeling step creates a financial constraint. A pure stable process has the desired non-local scaling, but it fails the exponential-moment condition needed for the discounted stock price to be a martingale. The present paper therefore does not price directly under a pure stable law. Instead, it uses a tempered stable process as the risk-neutral model and treats the Riesz operator as the short-maturity principal part that explains the local scaling of the exercise layer.

This paper works with the standard remedy: a symmetric tempered-stable log-price process. Tempering preserves power-law behavior over the observable central range while restoring exponential moments. The resulting pricing equation is a tempered space-fractional Black–Scholes equation in log-moneyness. Its short-maturity principal part is the Riesz equation, so the Riesz homogeneity still determines the leading similarity scale of the American exercise boundary. The distinction between the admissible tempered model and its Riesz asymptotic limit is central to the revised analysis.

American put options under this dynamics introduce a free boundary S_*f*(τ) separating exercise from continuation regions. The interaction between a non-local spatial generator and the free boundary is the focus of this paper.

1.1.1 Empirical motivation

Table 1 reports representative stability-index estimates obtained from symmetric stable fits to daily log-returns. The table is used only to motivate heavy-tailed modeling; the pricing model below uses tempering to ensure risk-neutral admissibility.

Table 1. Representative stability index estimates γ^ for daily log–returns.

1.2 Lie symmetry analysis for space-fractional PDEs

Lie symmetry analysis [5, 6] identifies one-parameter transformation groups that preserve a differential equation. For fractional PDEs the determining equations require fractional prolongation formulae [7–10]. In the Riesz limit, the homogeneity

∂γ∂|x|γ[f(λx)]=λγ(∂γ∂|x|γf)(λx)

provides the scaling relation x ~ τ^1/γ. Tempering breaks this scaling globally, but the break is lower order on the short-maturity length scale ητ^1/γ≪1. We therefore use the Riesz symmetry as an asymptotic symmetry of the tempered equation near expiry.

1.3 Contributions

1.3.1 Admissible model

We formulate the American put problem under a symmetric tempered-stable log-price model with finite exponential moments. The martingale drift is written explicitly in terms of the tempered characteristic exponent, avoiding the invalid compensator formula that would arise from an untempered stable law.

1.3.2 Asymptotic symmetry

We derive the finite-dimensional point symmetries of the constant-coefficient Riesz principal part after removing discounting and drift. The Riesz scaling generator is exact for the untempered, driftless principal part and becomes an asymptotic generator for the tempered pricing equation as τ → 0⁺.

1.3.3 Boundary scaling

The similarity variable ζ = (x−bτ)/τ^1/γ gives the conjectural leading scale

xf(τ)=O(τ1/γLγ(τ)),

where L_γ is slowly varying and includes the known logarithmic correction in the classical case γ = 2 [11, 12]. We retain the explicit coefficient

Bγ=σ2πΓ(1+2/γ)Γ(3/2+1/γ)

as a calibrated Riesz-limit ansatz rather than a proved universal constant.

1.3.4 Numerical benchmark

The numerical section compares the analytical approximation with a Fourier-collocation projected successive overrelaxation (PSOR) solution of the tempered obstacle problem. The benchmark quantifies boundary displacement, pricing errors across moneyness, and grid refinement behavior.

1.4 Related literature

Space-fractional option pricing and fractional diffusion models are discussed in Cartea and del Castillo-Negrete [4], Cont and Tankov [3]. Tempered stable processes were introduced as finite-moment heavy-tail models in Koponen [13], Carr et al. [14]. Jump-diffusion alternatives include Kou's double-exponential model [15]. Lie symmetry analysis of fractional partial differential equations (PDEs) follows [7, 9, 10, 16]. Numerical methods for space-fractional PDEs originate with shifted Grünwald–Letnikov approximations [17, 18], and projected SOR methods for obstacle problems trace back to Cryer [19]. Recent related work includes analytic properties of American options under spatial fractional Black–Scholes equations [20], numerical algorithms for regime-switching fractional American options [21], neural-network approximations for fractional free-boundary problems [22], and double-fractional or fractal-dimension approaches to option pricing and financial modeling [23–25]. More broadly, Riesz and non-local fractional operators continue to be developed for anomalous diffusion and non-local models [26, 27]. The classical American put boundary asymptotics, including the logarithmic correction, originate with Barles et al. [11].

1.5 Outline

Section 2 establishes the tempered fractional model and the Riesz principal part. Section 3 develops the symmetry analysis, similarity scaling, and boundary ansatz. Section 4 presents the Fourier-PSOR benchmark method. Section 5 reports the numerical comparison, and Section 6 concludes. Detailed derivations are collected in Appendix.

2 Mathematical preliminaries

This section fixes notation for the Riesz operator, its tempered counterpart, and the American put problem in log-moneyness. The material in this section is not presented as new; it is included to make the sign conventions, martingale correction, and obstacle formulation explicit before the asymptotic and numerical contributions are developed.

2.1 Riesz and tempered Riesz operators

Definition 2.1 (Symmetric Riesz derivative). For γ ∈ (1, 2], the symmetric Riesz derivative is defined through the Fourier multiplier

F[∂γ∂|x|γf](k)=-|k|γf^(k).(1)

Equivalently, for γ ∈ (1, 2),

∂γ∂|x|γf(x)=-cγp.v.∫ℝf(x)-f(y)|x-y|1+γdy,(2)

with the usual normalizing constant c_γ > 0. With this convention ∂2∂|x|2=∂xx. Equation 2 is the singular-integral representation of the Fourier definition in Equation 1.

Definition 2.2 (Symmetric tempered Riesz generator). Let η > 0. The symmetric tempered Riesz generator Lγ,η is the pseudo-differential operator with Fourier multiplier

Ψγ,η(k)=-(η+ik)γ+(η-ik)γ-2ηγ2cos(πγ/2), γ∈(1,2).(3)

The powers are taken on the principal branch. For real k, Ψ_{γ, η}(k) ≤ 0, and

limη↓0Ψγ,η(k)=-|k|γ.

Proposition 2.3 (Principal properties). The Riesz operator is linear, translation invariant, self-adjoint on suitable domains, and homogeneous:

∂γ∂|x|γ[f(λ·)](x)=λγ(∂γ∂|x|γf)(λx), λ>0.

The tempered operator Lγ,η remains linear and translation invariant but is not homogeneous for fixed η > 0. On the short-maturity Riesz scale x = O(τ^1/γ), tempering is a lower-order perturbation when ητ^1/γ ≪ 1.

2.2 Risk-neutral tempered log-price model

Let X_τ be a symmetric tempered-stable process with characteristic exponent Ψ_{γ, η}, so

𝔼[eikXτ]=exp(τΨγ,η(k)).

Its positive exponential moment exists for η>1. Define

ωγ,η:=Ψγ,η(-i)=-(η+1)γ+(η-1)γ-2ηγ2cos(πγ/2), η>1.(4)

Then the log-price process

logSτ=logS0+(r-aωγ,η)τ+Xaτ, a=12σγ,(5)

satisfies 𝔼[e-rτSτ]=S0.

2.3 American put setup

Definition 2.4 (Tempered space-fractional American put). For x = log(S/K) and V(S, T − τ) = Ku(x, τ), the continuation value satisfies

uτ=aLγ,ηu+bux-ru, a=12σγ, b=r-aωγ,η.(6)

The obstacle and free-boundary conditions are

u(x,0)=(1-ex)+, u(xf(τ),τ)=1-exf(τ),ux(xf(τ),τ)=-exf(τ).(7)

Remark 2.5 (Riesz principal part). Replacing Lγ,η by ∂γ∂|x|γ in Equation 6 gives the Riesz principal equation

uτ=a∂γ∂|x|γu+bux-ru.(8)

The Lie analysis in Section 3 applies exactly to the transformed Riesz principal part and asymptotically to Equation 6 near expiry.

3 Space-fractional American options with tempered Riesz dynamics

This section separates the admissible tempered pricing model from the Riesz principal part that controls the near-expiry scaling.

3.1 Model formulation

The risk-neutral continuation equation is Equation 6. Since Lγ,η is not homogeneous, the full tempered equation has no exact Riesz scaling symmetry. The principal Equation 8, obtained by replacing Lγ,η by ∂γ∂|x|γ, is homogeneous and gives the relevant short-maturity scaling when ητ^1/γ≪1.

The stochastic control problem behind Equation 5 is the finite-horizon optimal stopping problem

V(S,t)=sup0≤θ≤T-t𝔼ℚ[e-rθ(K-St+θ)+∣St=S],(9)

where S follows the risk-neutral tempered-stable exponential model in Equation 5. The dynamic programming principle gives the variational inequality

max{uτ-aLγ,ηu-bux+ru,(1-ex)+-u}=0,(10)

with terminal condition u(x, 0) = (1−^{e^*x*)+}. The continuation Equation 6 is therefore the equality part of a stochastic optimal stopping problem, while the exercise boundary is the free boundary induced by the stopping region.

It is convenient to remove drift and discounting by

x~=x-bτ, w(x~,τ)=erτu(x,τ).(11)

For the Riesz principal part, w satisfies

wτ=a∂γ∂|x|γx~w.(12)

3.2 Lie symmetry algebra of the principal part

Proposition 3.1 (Principal symmetry algebra). For γ ∈ (1, 2), the Riesz principal Equation 8 admits the following finite point symmetries inherited from Equation 12:

Y1=∂τ-b∂x+ru∂u,(13)

Y2=∂x,(14)

Y3=u∂u,(15)

Y4[φ]=e-rτφ(x-bτ,τ)∂u,(16)

Y5=γτ∂τ+(x-bτ)∂x-γrτu∂u,(17)

where φτ=a∂γ∂|x|γφ. Thus Y₄ is the usual infinite-dimensional superposition symmetry of a linear equation. If discounting is first removed, Y₅ reduces to the standard Riesz heat scaling γτ∂τ+x~∂x~. Equations 13–17 respectively give time translation in the moving-discount frame, spatial translation, dependent-variable scaling, linear superposition, and Riesz scaling.

Proof.[Sketch] Equation 11 maps Equations 8–12. Translation in τ, translation in x~, multiplication of the dependent variable, and addition of any solution are inherited from linearity and constant coefficients. The Riesz homogeneity in Proposition 2.3 gives the scaling symmetry of Equation 12. Pulling the generators back through Equation 11 gives the stated formulae.

The implementation of the Lie scaling is therefore analytical rather than algorithmic. First, drift and discounting are removed. Second, the tempered operator is replaced by its Riesz principal part on the short-maturity length scale. Third, the scaling generator of the Riesz heat equation,

γτ∂τ+x~∂x~,

is used to identify the similarity variable ζ=x~/τ1/γ. Finally, this variable is pulled back to the original log-moneyness coordinate. The method does not assert that the American obstacle or the tempered generator is invariant under this scaling; it uses the scaling to identify the leading local layer near expiry.

Remark 3.2 (Classical limit). With the convention Equation 1, ∂2∂|x|2=∂xx. At γ = 2, the principal equation becomes the heat equation after the same drift-discount transform. The larger classical heat-equation symmetry algebra is recovered in that limit; for γ < 2, non-locality removes the Galilean and projective generators.

3.3 Free boundary and asymptotic scaling

Proposition 3.3 (Symmetry breaking). The American obstacle (Equation 7) breaks global time translation, spatial translation, and superposition symmetries. The Riesz scaling is not an exact symmetry of the obstacle problem, but it gives an asymptotic scaling of the continuation layer as τ → 0⁺.

Sketch: The payoff (1 − ^{e^*x*)+} fixes both the strike scale and the expiry time, so translations and solution superposition do not preserve the value-matching condition. Near expiry, x_*f*(τ) → 0 and 1-exf=-xf+O(xf2); the linearized obstacle is compatible with the Riesz length scale x = O(τ^1/γ), up to slowly varying logarithmic factors inherited from the classical free-boundary problem.

Proposition 3.4 (Similarity reduction of the Riesz principal layer). For the transformed principal Equation 12, the similarity ansatz

w(x~,τ)=τμG(ζ), ζ=x~τ1/γ,(18)

reduces the principal equation to

a∂γ∂|x|γζG+1γζG′(ζ)-μG(ζ)=0.(19)

The choice μ = 1/γ corresponds to the linearized payoff scale near the exercise boundary.

Proof. The identity ∂γ∂|x|γx~G(x~/τ1/γ)=τ-1∂γ∂|x|γζG follows from the homogeneity of ∂γ∂|x|γ. Substitution of Equation 18 into Equation 12 gives Equation 19.

3.4 Boundary ansatz

Conjecture 3.5 (Tempered space-fractional boundary scale). For the tempered space-fractional American put with γ ∈ (1, 2] and η>1, the near-expiry exercise boundary satisfies

xf(τ)=logSf(τ)K=-Bγτ1/γLγ(τ)+o(τ1/γLγ(τ)),(20)

where L_γ is slowly varying. In the Brownian limit γ = 2, L₂(τ) contains the classical logarithmic correction. In the numerical comparison we use the Riesz-limit coefficient

Bγ=σ2πΓ(1+2/γ)Γ(3/2+1/γ).(21)

The reasoning behind Equation 20 is as follows. In the exercise layer, the payoff satisfies 1−e^*x* = −x+O(x²) as x → 0. The transformed Riesz principal equation has the similarity variable ζ=x~/τ1/γ, so a boundary that remains in the principal layer must have xf(τ)-bτ=O(τ1/γ) up to slowly varying factors. The negative sign is fixed by the fact that the American put boundary lies below the strike, so x_*f*(τ) < 0 for small positive maturity. The coefficient B_γ in Equation 21 is obtained by matching the Riesz similarity scale with the Brownian short-maturity coefficient at γ = 2 and by extending the stable scaling through the gamma-function moment ratio associated with the symmetric Riesz kernel. This gives a practical leading ansatz for numerical assessment.

The statement is deliberately formulated as a conjecture because two effects are not yet controlled by a rigorous matched-asymptotic proof. First, the American obstacle creates a moving non-local half-line problem, so the Riesz heat scaling alone does not determine the full boundary expansion. Second, the Brownian limit already contains a slowly varying logarithmic correction, and the analog of that correction for the tempered non-local problem has not been proved here. The conjecture records the scaling mechanism and coefficient used in the computations without overstating the available theory.

Remark 3.6 (Status of the coefficient). Equation 21 is an explicit, smooth interpolation that recovers σ2/π at γ = 2 and increases as γ decreases. It is used as an analytical ansatz and calibration target, not as a proved universal leading constant. A proof would require matched asymptotics for the tempered stable obstacle problem on a moving half-line.

Remark 3.7 (Boundary behavior under the ansatz). This observation is not a new theorem; it is a direct consequence of the ansatz. Ignoring the slowly varying factor, the displacement scale τ^1/γ is smaller than τ^1/2 for γ < 2 and 0 < τ < 1. Thus, the short-maturity boundary lies closer to the strike than the Brownian boundary at the same τ, provided the coefficient variation does not dominate the exponent effect.

3.5 Analytical approximation

Proposition 3.8 (Short-maturity premium approximation). Let Pγ,ηEur denote the European put price computed from the tempered characteristic exponent and martingale drift. A short-maturity American approximation is

Pγ,ηAm(S,τ)≈max{K-S,Pγ,ηEur(S,τ)+Aγ,η(τ)(S/Sf(τ))qγ,η},(22)

where A_{γ, η} is fixed by value matching at S_*f*(τ) and q_{γ, η} < 0 solves the stationary characteristic equation.

Remark 3.9. The approximation in Equation 22 is obtained by writing the American value as a European value plus an early-exercise premium and approximating the premium locally by the admissible exponential mode of the stationary continuation operator. Substitution of e^*qx* into aLγ,ηu+bux-ru=0 gives the characteristic equation for q_{γ, η}. Value matching at S_*f*(τ) fixes A_{γ, η}(τ), while smooth pasting is used only through the boundary ansatz. This is a reduced analytical approximation, not a substitute for solving the full linear complementarity problem when high accuracy is required. Its purpose is to expose the boundary scale and provide fast indicative prices.

Conjecture 3.10 (Approximation error). For fixed admissible parameters and sufficiently short maturities, the error of Equation 22 is expected to be controlled by the next term in the boundary expansion and by tempering corrections of order ητ^1/γ. Establishing a uniform bound remains open.

3.6 Fourier-based European pricing

Let x = log(S/K) and let α_*d*>0 be a damping parameter for which the damped payoff is integrable. The European put can be computed by Fourier inversion in the form

Pγ,ηEur(S,τ)=Ke-αdx2π∫ℝe-ikxp^αd(k)exp(τ[aΨγ,η(k+iαd)+i(k+iαd)b-r])dk,(23)

where p^αd is the Fourier transform of eαdx(1-ex)+ on x < 0. This damped representation replaces the formal, non-integrable payoff transform. Equation 23 is the damped Fourier pricing formula used for the European component.

3.7 Connection to tempered stable processes

The tempered model has algebraic behavior over intermediate ranges and exponential cutoffs in the far tails. The martingale correction Equation 4 is finite for η>1, so the discounted stock price is a martingale under Equation 5. In the formal limit η↓0 the Riesz operator is recovered, but the exponential stock model ceases to be risk-neutral for γ < 2.

3.8 Greeks and calibration

Differentiating Equation 22 gives analytical Greeks away from the exercise boundary. For example,

Δγ,η≈Δγ,ηEur+qγ,ηAγ,η(τ)S(SSf(τ))qγ,η,(24)

Γγ,η≈Γγ,ηEur+qγ,η(qγ,η-1)Aγ,η(τ)S2(SSf(τ))qγ,η.(25)

Equation 24 and 25 give the corresponding Delta and Gamma approximations.

The fractional parameters can be estimated in two complementary ways. Under the historical measure, γ and η may be fitted to log-return increments by maximum likelihood or by matching the empirical characteristic function of a tempered-stable law. For pricing, the risk-neutral parameters should instead be calibrated to option quotes by minimizing a weighted loss

minσ>0,1<γ≤2,η>1∑iwi(Pγ,ηmodel(Ki,Ti)-Pimarket)2,

where the weights may reflect bid-ask spreads or vega scaling. Identifiability of (σ, γ, η) requires strikes and maturities that are rich enough to separate volatility level, tail thickness, and tempering; otherwise only effective combinations of these parameters can be inferred.

The same generator can be used for other contracts after changing the payoff and exercise/monitoring constraints. European options require only the Fourier pricing step. American calls without dividends usually have no early exercise in the classical setting, but dividends, carrying costs, barriers, or regime switching can reintroduce non-trivial stopping regions. Multi-asset or path-dependent options require new state variables and, in general, a different free-boundary geometry.

3.9 Perpetual limit

For a perpetual put, the stationary continuation equation is

aLγ,ηu+bux-ru=0.

If an exponential mode u = e^*qx* is admissible, its exponent solves

aΨγ,η(-iq)+bq-r=0, q<0.(26)

The characteristic exponent is therefore determined by Equation 26.

The value-matching and smooth-pasting calculation then gives the familiar algebraic relation

Sf*=Kqq-1.(27)

At γ = 2 and in the Brownian limit this recovers the classical perpetual put threshold [28].

3.10 Extensions

Asymmetric and multi-asset extensions require separate treatment because skewness and dependence alter the martingale correction and the boundary geometry. The Riesz scaling remains a useful local guide, but the explicit coefficient and the exercise surface must be rederived for each extension.

4 Numerical methods

This section describes the numerical benchmark used to assess the short-maturity analytical approximation. Since the tempered generator is a pseudo-differential operator with an explicit Fourier symbol, a Fourier collocation discretization is a natural reference method on a sufficiently large log-price interval. The American early-exercise constraint is imposed by PSOR.

4.1 Fourier collocation of the tempered generator

Let x_*j* = x_min+jΔx, j = 0, …, M−1, on a truncated log-moneyness interval [x_min, x_max]. Denote by k_*m* = 2πm/(x_max−x_min) the corresponding Fourier modes. The spatial generator in Equation 6 is represented by the multiplier

A(k)=aΨγ,η(k)+ibk-r,(28)

Equation 29 defines the linear complementarity problem, and Equation 30 gives the PSOR update. Where Ψ_{γ, η} is defined in Equation 3. Applying the inverse discrete Fourier transform to A(k)u^(k) gives the collocation approximation to aLγ,ηu+bux-ru.

Remark 4.1 (Truncation). The interval is chosen so that the left boundary lies deep in the exercise region and the right boundary lies in the far out-of-the-money region. The tempered tails reduce wrap-around error relative to an untempered stable operator, but the interval length remains part of the numerical discretization and must be checked in convergence studies.

4.2 Implicit obstacle step

Let A_*M* be the dense collocation matrix associated with Equation 28. An implicit Euler step from τ_*n* to τ_*n*+1 gives the linear complementarity problem

(I-ΔτAM)un+1≥un, un+1≥g,[(I-ΔτAM)un+1-un]j[ujn+1-gj]=0,(29)

where gj=(1-exj)+ is the put payoff in log-moneyness. The system is solved by PSOR:

uj(ℓ+1)=max{gj,(1-ω)uj(ℓ)+ωujn-∑m<jBjmum(ℓ+1)-∑m>jBjmum(ℓ)Bjj},B=I-ΔτAM,(30)

with relaxation parameter ω ∈ (0, 2). Iteration stops when the infinity norm of successive iterates falls below a prescribed tolerance.

4.3 Implementation and stability controls

The computation starts from the terminal payoff uj0=gj at τ = 0 and marches forward in time to the required maturity. The dense matrix A_*M* is formed by applying the inverse discrete Fourier transform to the multiplier (Equation 28); equivalently, each column is the collocation response of the tempered pseudo-differential generator to a grid basis vector. The previous time level is used as the initial PSOR iterate at the next time level. In the reported experiments the stopping tolerance is 10⁻⁸ for the benchmark tables, with at most 1,000 PSOR sweeps for the largest comparisons. The relaxation parameter is chosen in the admissible range (0, 2); values near 1.35 were used in the benchmark runs.

The implicit Euler step is unconditionally stable for the dissipative linear part in the Fourier representation because ReΨ_{γ, η}(k) ≤ 0 and the discount term contributes −r. The projection enforces the obstacle at each sweep. Since the Fourier collocation matrix is dense and non-local, a discrete maximum principle is not claimed. Instead, numerical stability is assessed by PSOR residual tolerance, by the absence of oscillations in the exercise region, and by the grid refinement study in Table 2. The truncation interval [−4, 4] was selected so that the lower endpoint is far inside the exercise region and the upper endpoint is far out of the money for the parameter values considered.

Table 2. Grid refinement for the Fourier-PSOR benchmark, γ = 1.7, T = 0.25.

4.4 Boundary extraction and convergence

At each time level, the discrete exercise boundary is estimated as the largest stock value below the strike for which the projected solution coincides with the payoff to numerical tolerance. The benchmark prices in Section 5 are obtained by interpolation on the log-price grid. Grid refinement is assessed by comparing prices at representative strikes against a finer reference grid.

Remark 4.2 (Relation to Grünwald–Letnikov schemes). Shifted Grünwald–Letnikov discretisations remain appropriate for the untempered Riesz limit and for finite-difference implementations of tempered fractional derivatives. The Fourier collocation benchmark used here is chosen because the tempered characteristic exponent is available explicitly and gives a direct, transparent discretization of the corrected risk-neutral model.

5 Numerical results

This section compares the Riesz-limit analytical approximation with the Fourier-PSOR benchmark described in Section 4. Unless otherwise stated, the parameters are

K=100, r=0.05, σ=0.20, η=3, T=0.25.

The benchmark computations use the interval [−4, 4] in log-moneyness. The largest grid reported below uses M = 512 collocation points and N = 160 time steps.

5.1 Boundary constants and exercise boundary

Table 3 reports the constants in the Riesz-limit boundary ansatz. The coefficient is monotone in the stability index and recovers the Brownian value at γ = 2.

Table 3. Boundary constants from Equation 21 with σ = 0.20.

Figure 1 compares the analytical boundary curves with benchmark boundary estimates at T = 0.25. The analytical scale gives the correct qualitative movement of the boundary, but the Brownian and near-Brownian cases show a visible displacement because the slowly varying logarithmic factor is not included in Equation 20. The figure also shows that lower stability indices keep the leading boundary closer to the strike at short maturities, which is the expected consequence of the exponent 1/γ. The benchmark points are therefore used to diagnose the missing higher-order boundary correction rather than to claim that the leading ansatz is exact. Table 4 reports the corresponding numerical exercise-boundary comparison at T = 0.25.

Table 4. Exercise boundary comparison at T = 0.25.

5.2 Price comparison

Table 5 compares benchmark and analytical prices for γ = 1.7. The approximation is accurate for in-the-money and at-the-money options in absolute terms. Relative errors become less informative for deep out-of-the-money options because the benchmark prices are small; this is visible at S = 120, where the absolute error remains below five cents but the relative error is large.

Table 5. American put prices for γ = 1.7, T = 0.25.

5.3 Grid refinement

Table 2 reports a refinement study using the M = 512, N = 160 computation as reference. The error is measured as the maximum absolute difference at S = 100 and S = 110. The decrease under refinement indicates that the Fourier-PSOR benchmark is stable for the parameter range used in the price comparison.

5.4 Computational cost and sensitivity

Figure 2 compares representative runtimes for repeated analytical evaluations and for the Fourier–PSOR benchmark. The analytical approximation is substantially cheaper, while the benchmark provides the reference values needed for error assessment. The outcome is computational rather than theoretical: the asymptotic formula is useful for rapid exploration across parameter values, whereas the PSOR benchmark is needed when the obstacle and tempering corrections must be resolved numerically.

Figure 3 shows the sensitivity of the boundary constant and boundary level to γ, σ, and maturity. The exponent 1/γ controls the principal maturity scaling, while σ enters linearly through B_γ. The first panel isolates the dependence of the coefficient on the stability index. The second panel shows the nearly linear widening of the exercise displacement as volatility increases. The maturity panel confirms that the ansatz is primarily a short-maturity description and should not be extrapolated mechanically to long maturities without solving the full obstacle problem.

5.5 Interpretation

The numerical evidence supports the use of the Riesz-limit ansatz as a short-maturity approximation, especially around the exercise and at-the-money regions where American put valuation is most sensitive to the boundary. The remaining discrepancy in the boundary location, particularly as γ approaches the Brownian limit, is consistent with the absence of the slowly varying logarithmic correction in the leading ansatz. A rigorous matched-asymptotic treatment of this correction remains the main analytical task.

6 Conclusion

This paper has formulated the American put problem under a risk-neutral tempered space-fractional model. The tempering restores the finite exponential moments required for an arbitrage-free exponential stock price, while the Riesz operator remains the short-maturity principal part of the generator. This separation clarifies the role of stable scaling in option pricing: the pure Riesz dynamics provide the local asymptotic structure, whereas the tempered process supplies the financially admissible risk-neutral model.

After removing drift and discounting, the principal equation reduces to a Riesz heat equation and admits the scaling x~τ^1/γ. The American obstacle and the tempering parameter break this symmetry at the global level, but the scaling remains relevant in the near-expiry continuation layer. On this basis, the exercise boundary has been described through a conjectural asymptotic scale with a slowly varying factor. The coefficient B_γ gives a useful Riesz-limit approximation and recovers the Brownian scaling constant in the limiting case γ = 2.

The Fourier-PSOR benchmark confirms that the analytical approximation is most reliable in the in-the-money and at-the-money regions over short maturities. In these regions the absolute pricing errors are small, while relative errors in far out-of-the-money options are amplified by the small benchmark prices. The grid refinement study indicates stable convergence of the benchmark computation, and the boundary comparison shows the expected limitation of the leading ansatz: near the Brownian limit the missing slowly varying logarithmic factor produces a visible boundary displacement.

Further analytical work is needed to establish the matched asymptotic expansion rigorously, including the slowly varying logarithmic factor suggested by the classical American put boundary. Market calibration also remains to be developed on documented option panels, with uncertainty estimates for (σ, γ, η) and a careful assessment of identifiability across strikes and maturities. These extensions would strengthen the case for the tempered space-fractional framework as a practical model for American option pricing under heavy-tailed returns.

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Appendix A. Detailed derivations

A.1 Tempered martingale correction

For the symmetric tempered generator (see Equation 3),

𝔼[ezXτ]=exp(τΨγ,η(-iz))

whenever |z| < η. Taking z = 1 gives the finite moment correction

ωγ,η=Ψγ,η(-i).

Therefore,

𝔼[Sτ]=S0exp((r-aωγ,η)τ)𝔼[eXaτ]=S0erτ,

which proves the martingale property of e-rτSτ.

A.2 Pullback of the scaling generator

Starting from the Riesz principal equation

uτ=a∂γ∂|x|γu+bux-ru,

set x~=x-bτ and w = e^*rτ*u. Then

wτ=erτ(uτ-bux+ru)=a∂γ∂|x|γx~w.

The Riesz heat equation is invariant under

(x~,τ,w)↦(eϵx~,eγϵτ,w),

because ∂_τ and ∂γ∂|x|γ both scale by e^−γϵ. Pulling this transformation back to (x, τ, u) yields the generator Y₅ in Proposition 3.1.

A.3 Characteristic exponent for the perpetual ansatz

If u = e^*qx*, then

Lγ,ηeqx=Ψγ,η(-iq)eqx

whenever the exponential mode lies in the domain of the generator. The stationary continuation equation becomes

aΨγ,η(-iq)+bq-r=0.

The negative root, when it exists and is unique, gives the decaying continuation branch used in Equation 27. For γ = 2 this reduces to the classical quadratic exponent for the perpetual Black–Scholes put.

A.4 Stable tail used in the Riesz ansatz

The symmetric stable density associated with the untempered Riesz operator satisfies

pγ(ζ)~Γ(1+γ)sin(πγ/2)π|ζ|1+γ, |ζ|→∞,

for γ ∈ (1, 2) [3]. The ansatz (Equation 21) uses this Riesz profile only as a local, short-maturity reference for the tempered model.

Footnotes

Edited by: Said Hamadene, Le Mans Université, France

Reviewed by: Rami Ahmad El-Nabulsi, Czech Education and Scientific Network, Czechia

Abdelmjid Qadi El Idrissi, Hassan Premier University, Morocco