Certified Joint Pricing Errors in Rough Heston

I. Introduction

At a fixed model parameter and maturity, every strike is priced from the same Fourier transform values, so the error at each frequency is one common complex number. Bounding each price separately discards that constraint. We retain these common variables in the price-error set and ask whether they can change a financial certification decision. The resulting dependence is deterministic and comes from the pricing formula.

Write $c^*$ for the model-price vector, $c^{\rm fast}$ for the stored pricing output, and $\bar c$ for an independent reference centre. The complete inclusion is

Each component of $\delta$ is a rigorously enclosed common Fourier error. Finite omitted frequencies stay in $\delta$, which lets the remainder $r$ retain strip quadrature, the true infinite tail and arithmetic without charging those finite nodes twice. When the reference is refined, $d$ and its certified uncertainty change together. We identify neither the reference price nor the fast output with the exact model price.

A. Contributions and dependencies

Our main analytical result takes a complex Riccati residual to the Heston exponent through a finite-history kernel. Theorem 3.1 gives both the nonexpansive curve bound and the positive-dissipation envelope, with the initial curve term and the physical factor $\nu^{-1}$ included. We compose the underlying comparison, which uses established Caputo convexity and positive-resolvent results, with the model's pricing functional, certify the history weights, and put them into the complete output budget.

The second result certifies the actual pricing output. We translate the shared transform disks to that output (Theorem 5.1) and evaluate them in portfolio and finite-objective directions (Section 5.2, including Theorem 5.3), working with support functions, convexity and the sorting of a fully known unit-cost menu. A matched quarter-year comparison isolates what the shared errors contribute to the budget decision, while the ablation separates propagation, time-local envelopes, omitted-frequency certification and recentering.

The third result, Theorems 6.1 and 6.2, certifies a structural domain for the Gatheral–Radoicic rational construction, and its all-frequency statement carries explicit restrictions on correlation, fractional order and mean reversion. Prices use a different object, the residual certificate from an independently generated reference field. Nearby candidates and output controls then test the financial decisions and the reuse of that certificate.

In the matched experiments, finite-history propagation reduces the complete six-month spread bound by about 72.91% with output and centre unchanged (Table 3). At a 0.5-point budget and alpha=.52, finite-history joint bounds certify 28/28 fixed portfolio directions, compared with 13/28 for signed marginal bounds (Table 11). For the quarter-year spread, the joint bound is 0.233318843 points against 0.252393939 points for signed marginal aggregation with every other input matched, so only the joint certificate passes a 0.25-point budget (Table 10). On a prespecified five-point roughness grid under the original quotes, joint bounds strictly separate 10/10 candidate pairs at N_t=2048, against 7/10 for matched marginal bounds, and alpha=0.520 is the strict minimizer of the midpoint objective on that grid (Tables 5 and 6). All five candidates remain incompatible with at least one original bid/ask row, so this result is a certified midpoint-loss ranking of a finite set and it identifies no market-calibrated model.

Table 1. Dependencies of the rational-output and independent-reference proof chains.

The two proof chains share the model but certify different objects. The algebraic signs settle the structural properties of the rational output, the continuous residual settles the reference error, and output translation connects the latter to the stored Padé price.

B. Related work

Gatheral--Radoicic [GR2019, GR2023] construct two-end rational approximations, and Jeng--Kilicman [JK2020, JK2021] analyse global Padé approximations and public SPX data. The formula and endpoint-matching method used here are theirs. What we add is a structural statement about the raw matching system throughout its explicitly certified domain.

Abi Jaber–El Euch [AbiJaberElEuch2018] give the Volterra model and affine transform. Li–Liu [LiLiu2018] establish Caputo convexity through regularization, Kopteva [Kopteva2021] propagates a time-dependent residual with a positive fractional inverse, Simon [SimonCM2015] gives Mittag–Leffler positivity, and Trefethen–Weideman [TrefethenWeideman2014] give strip trapezoidal analysis. We prove in Appendix C the regularity statements needed to apply these results to the pricing functional.

Ben Hammouda et al. [BenHammouda2026, Section 3.2] develop error-controlled multilevel rough-Heston quadrature. Their practical tolerance procedure replaces unavailable constants with numerical error indicators and assesses numerically whether the tolerance was attained, whereas our certificate encloses a stored output a posteriori, tails and arithmetic included.

Boyarchenko–de Innocentis–Levendorskii [BL2025] study reliable pricing, numerical bias and incorrect calibration, and we use their modified Adams method as a numerical control. They describe Conformal Bootstrap as an ad-hoc principle; it gives no deterministic interval enclosure. Hager–Kreher [HK2026] study analytic expansions in the Hurst parameter and local convergence, while Bayer–Breneis [BBWeak2023, BBSimulation2023] bound Markovian kernel approximations and investigate low-dimensional simulation. Each comparison below specifies the model, the numerical output and the error guarantee.

Appendices F–G give a separate classical Heston extension based on common occupation probabilities and an inexact trial field. Its terminal witness establishes strict improvement of an outer error set; a complete annual monetary certificate remains open.

II. Model, units and fixed configurations

Our conventions for the fractional integral and Caputo derivative are

Under the regularity of the trajectories used below, $I^\alpha f\in AC$ with zero initial value, so $D_C^\alpha I^\alpha f=f$. The Caputo operator and all time scalings keep their fractional definitions. With $a=u-i/2$, the physical Riccati state $h$ satisfies

Define

Then $D_{C,x}^\alpha H=F(H)$ and $D_{C,t}^\alpha Z=\nu F(Z)$. Dividing the normalised state error by $\nu$ gives the physical $h$ error, and dividing the physical residual $r_t=D_t^\alpha\widehat Z-\nu F(\widehat Z)$ by $\nu$ gives the residual bound for the normalised equation.

For pricing and the numerical example, we fix $\kappa=0$ and the entire forward variance curve

The curve parameter $\lambda_\xi$ and the Riccati parameter $\lambda_R$ are defined separately, and curve (2.4) stays fixed when the candidate $\alpha$ changes. The probability model is

Equation (2.5) is the forward variance representation with zero Riccati mean reversion. For this curve, Appendix A verifies the probability model and the associated affine transform using Theorems 2.1 and 2.3 and Example 2.2 of Abi Jaber and El Euch.

Let $M_T=S_T/F_T$ be the normalised positive martingale and write $\phi_T(a)=\mathbb E M_T^{ia}$. The exact characteristic exponent is

With $m=K/F_T,k=\log m,c=C/(DF_T)$, the model European call price is

The prices considered below are defined jointly by the model specification, the decimal inputs and the normalisation.

All experiments use model parameter $\nu=0.2897$, $\rho=-0.7445$, Riccati mean reversion zero, Fourier step $1/8$, a finite reference/error grid through 128 (1025 nodes), 100-bit outward dyadic arithmetic and 64 forward-moment series terms. The actual Padé output instead uses composite Gauss–Legendre order 8 through 200 (352 nodes) and Jacobi order 256, and its Fourier cutoff is distinct from the certificate cutoff. $N_t$ counts source time cells; we write out the number of frequency nodes explicitly so that it cannot be confused with $\nu$.

Table 2. Fixed configurations: maturity in years; time cells, reference nodes, residual subcells and history bins are counts.

H0 uses uniform-state propagation, and H1 uses finite-horizon global propagation on the same field. H2 uses the low-frequency 128-bin bank. H3/H4 extend the reference through 128, retaining that low bank and using global/128-bin propagation for the high bank. Q0/Q1/Q2 retain the full history from zero and use exact quarter-terminal interpolation of the half-year field. Q0 has 1430/1353/550 time knots and Q1/Q2 have 1430 knots; they reuse the larger-horizon continuous banks and recompute quarter moments, output and tails. The alpha=.9 source field has 1025 time knots and 641 stored frequencies through 80, of which 513 are nonzero reference nodes in H0/H1/Q0, while the .52/.60 fields have 2049 knots and 1025 stored frequencies through 128.

N1/N2 generate each five-point field and continuous residual bank, using two nonstartup subcells (2047/4095 closed cells), and the nearby-grid objective experiment runs on these two. N2L is a separate descriptive 128-bin reuse of N2 at alpha=.52. The H low and high banks have four nonstartup subcells and 8189 closed cells each, for 513 and 512 frequencies respectively. So the 0.115215934-point H4 certificate and the 0.394999331-point N2L certificate use different reference centres, node coverage and residual banks, and their difference includes more than 128-bin propagation. BL-core512/1024 are nominal history-stepping controls whose complete guarantees use the identified N1/N2/N2L reference bank.

The H and N field generators begin with a uniform auxiliary grid $y_j$ from zero to $\nu T^\alpha$ and then set $x_j=y_j^{1/\alpha}$ and $t_j=x_j/\nu^{1/\alpha}$, which in exact arithmetic is the graded rule $t_j=T(j/N_t)^{1/\alpha}$. Generation runs in binary64 arithmetic and forces the endpoints to zero and one half, so certification takes the saved nodes as exact dyadics rather than identifying them with the analytical rule. The quarter-year restriction keeps every original knot strictly below $T_q=1/4$, appends the exact endpoint, encloses the linear interpolation of the original bracketing field coefficients in rational interval arithmetic, and retains all preceding history.

III. Finite-history propagation: nonexpansion and dissipation

We apply the established tools of Caputo comparison and norm convexity [LiLiu2018; Kopteva2021; SimonCM2015], and Appendix C.1 proves the absolute-continuity statement they require here, including almost-everywhere passage and the continuous endpoint. Under the hypotheses below, the complex divided difference has real part at most $-\nu\sigma$, and composing its error bound with the pricing functional gives Theorem 3.1, whose proof is in Appendix C.2.

For $\beta>0$, write $g_\beta(t)=t^{\beta-1}/\Gamma(\beta)$. Define $k_\lambda(t)=t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^\alpha)$, with $k_0=g_\alpha$. All convolutions below start at time zero.

Theorem 3.1 (finite-history pricing envelope). Let $0<\alpha<1$, $\nu>0$, $T>0$, and $Z,\widehat Z\in AC([0,T];\mathbb C)$ have zero initial value. Let

where $F(z)=-b+dz+z^2/2$, $\Re d=-s_0$, $\Re Z\le0$, $\Re\widehat Z\le\epsilon_R$, and $\sigma=s_0-\epsilon_R/2\ge0$. These equations and inequalities hold almost everywhere; the state bounds hold everywhere by continuity. Let $\xi\in AC[0,T]$, $\xi(0)=V_0\ge0$, and assume

Define $L_T$ and $\widehat L_T$ from $\nu^{-1}\int_0^T\xi(T-t)D_C^\alpha Z(t)\,dt$ and the same expression with $\widehat Z$. With $\lambda=\nu\sigma$,

In particular, $R/\nu\le\delta_F$ implies

Proof route. Set $e=Z-\widehat Z$; the physical equation's divided-difference coefficient has real part at most $-\nu\sigma$. The AC convexity lemma applied to a smooth modulus and the positive zero-initial inverse gives the state bound, including its continuous endpoint. Absolute Fubini and AC integration by parts then express the exponent difference as $\nu^{-1}(q_\alpha*e)(T)$, retaining the zero state initial values and the curve term $V_0g_{1-\alpha}$. Positivity of $q_\alpha$ propagates that state bound, and $0\le q_\alpha*k_\lambda\le\xi$ gives the curve envelope. The argument keeps the entire fractional history and removes $\nu^{-1}$ only upon normalizing the physical residual by $\nu$; Appendix C.1–C.2 gives the full proof.

At $\sigma=0$, set $k_0=g_\alpha$; the two exponent envelopes coincide. Positive dissipation permits further reduction. The initial term $V_0g_{1-\alpha}$ and the factor $\nu^{-1}$ remain in both bounds. Stochastic admissibility of the experimental curve is verified separately in Appendix A.

For a full-history residual envelope $R\le R_j$ on each closed cell $[a_j,b_j]$, the computable cell-weight form is

Every weight integrates the history from time zero, and an integration bin takes the maximum over every closed residual cell it intersects, boundary cells included. Section 7 compares curve and dissipative-kernel weights on matched inputs. Appendix C.3–C.4 proves decreasing-curve and constant-curve variants and a complementary global-state bound.

A. Strict computation of the dissipative pricing kernel

The positive kernel in Theorem 3.1 can be handled by a direct interval calculation. For $\sigma\ge0$, set $\lambda=\nu\sigma$ and

The assumptions and scaling are those of Theorem 3.1, and the positive fractional inverse follows from [Kopteva2021, SimonCM2015]. We enclose the pricing-kernel weights and use them in the same actual-output certificate.

Zero-damping special case of Theorem 3.1. If $\sigma=0$,

The estimate uses $q_\alpha\ge0$ and preserves the physical scaling at zero damping. Appendix C gives the AC regularity and comparison proofs; Appendix A treats the stochastic-model hypotheses.

For the fixed curve in (2.4), write $A_0=\alpha_0$ and $\lambda_\xi$ for its own curve parameter. On the computational domain $\lambda T^\alpha<1$, define

A cell $[a_j,b_j]$ has weight $W_\lambda(T-a_j)-W_\lambda(T-b_j)$; Appendix C gives the absolute series tails and the interval-difference proof. The implementation, with twelve outer terms, sixty-four inner curve terms and 100-bit outward dyadic primitives, is an enclosure of the full Mittag–Leffler kernel functional with explicit truncation and rounding remainders. Ordinary floating-point Mittag–Leffler values do not enter a guarantee.

IV. Continuous certificates and complete Fourier prices

The reference is a fixed $\widehat Z=I^\alpha\bar G$, a continuous field whose recorded coefficients we interpret as exact dyadics; the nominal nodal solver states are not the reference. A startup power expansion and closed-cell derivative bounds certify its full physical residual; Appendix D.1 gives the construction, arithmetic and history integrals. The configuration table distinguishes the four-subcell H/Q banks from the two-subcell nearby banks.

We use the derivative-based reference exponent $\widehat L_T=\nu^{-1}\int_0^T\xi(T-t)\bar G(t)dt$; a substitution-based exponent would require an additional residual-conversion term. Once $\eta_n$ encloses the error of the derivative-based exponent, the exponential difference and the true half-shift modulus bound give the transform disk

Theorem 4.1. Suppose $M_T>0,\mathbb E M_T=1$. For any $0<a_*<1/2,h_*>0$, let $g(z)=e^{-ikz}\phi_T(z-i/2)/(z^2+1/4)$. Replacing the integral in (2.7) by the infinite trapezoidal sum incurs a price error of at most

Appendix D.2 contains the strip proof and the exact-solution high-frequency comparison. A finite omitted reference value is charged explicitly, independently of the true infinite tail.

Suppose the node values $\widehat\phi_n$ have exact-error bounds $\varepsilon_n$. The finite price sum

satisfies

In these equations $N_u$ is the Fourier sum cutoff, whereas $N_t$ denotes time-grid cells. The origin coefficient is the trapezoidal half-weight multiplied by $1/(1/4)$. The exact arithmetic remainder encloses the phase, square root, exponential and summation operations.

V. Shared errors and financial decisions

A. Joint inclusion at the frozen output

Fix one model parameter, maturity, contour and reference grid. Let $c^*\in\mathbb R^p$ be the continuous-model price vector, $\bar c$ the exact finite reference sum, and $c^{\rm fast}$ the frozen rational output. For each included frequency let $\bar\phi_n$ be the reference transform and put $z_n=\phi(u_n-i/2)-\bar\phi_n$. Appendix D.1 gives bounds $|z_n|\le\epsilon_n$, and at a deliberately omitted finite node $\bar\phi_n=0$ the radius comes from the true-transform envelope. The same $z_n$ enters every strike at this parameter and maturity.

For the Lewis rule in Section 4, set $m_i=K_i/F$, $k_i=\log m_i$, and

The half weight at zero is already included in $a_{i0}$. The complete error is

where $\mathcal R$ includes the true infinite-grid tail and analytic-strip discretisation error. When a numerical representative of $\bar c$ is used, a finite reference sum enclosed by interval arithmetic contributes its centre uncertainty. Omitted finite nodes remain in (5.2), so we do not also charge them as an infinite tail. This representation uses no stochastic covariance.