Certified Joint Pricing Errors in Rough Heston

I. Introduction

At a fixed model parameter and maturity, different strikes use the same Fourier transform values. Bounding each price separately discards a constraint: the error at each frequency is one common complex number. 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.

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

Each component of $\delta$ is a rigorously enclosed common Fourier error. Finite omitted frequencies remain in $\delta$; the remainder $r$ retains strip quadrature, the true infinite tail and arithmetic, without charging those finite nodes twice. A reference refinement changes $d$ and its certified uncertainty together. Neither the reference price nor the fast output is identified with the exact model price.

A. Contributions and dependencies

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

The second result is a certificate for the actual pricing output. Shared transform disks are translated to that output and evaluated in portfolio and finite-objective directions. A matched quarter-year comparison isolates the contribution of shared errors to the budget decision. The ablation separates propagation, time-local envelopes, omitted-frequency certification and recentering. Support functions, convexity and sorting a fully known unit-cost menu are the tools used in this construction.

The third result certifies a structural domain for the Gatheral–Radoicic rational construction. The all-frequency statement has explicit restrictions on correlation, fractional order and mean reversion. An independently generated reference field supplies the residual certificate used for prices. Nearby candidates and output controls test the financial decisions and the reuse of that certificate.

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 establish the rational output's structural properties; the continuous residual establishes the reference error. Output translation connects the latter to the stored Padé price.

B. Related work

Gatheral--Radoicic [GR2019, GR2023] construct two-end rational approximations. Jeng--Kilicman [JK2020, JK2021] analyse global Padé approximations and public SPX data. The formula and endpoint-matching method used here are theirs. Our structural statement concerns 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. Appendix C proves 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 tolerance attainment numerically. Our certificate instead encloses a stored output a posteriori, including tails and arithmetic. This comparison concerns the tolerance procedure in the cited paper.

Boyarchenko–de Innocentis–Levendorskii [BL2025] study reliable pricing, numerical bias and incorrect calibration. We use their modified Adams method as a numerical control. They describe Conformal Bootstrap as an ad-hoc principle; it does not supply a deterministic interval enclosure. Hager–Kreher [HK2026] study analytic expansions in the Hurst parameter and local convergence. Bayer–Breneis [BBWeak2023, BBSimulation2023] bound Markovian kernel approximations and investigate low-dimensional simulation. Each comparison below specifies the model, numerical output and 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 retain their fractional definitions. Set $a=u-i/2$. 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. Likewise, 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 Riccati parameter $\lambda_R$ are defined separately. Curve (2.4) remains fixed when the candidate $\alpha$ changes. The probability model is

Equation (2.5) is the forward variance representation with zero Riccati mean reversion. Appendix A verifies the probability model and affine transform associated with this curve 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 model specification, decimal inputs, and normalisation jointly define the prices considered below.

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 uses composite Gauss–Legendre order 8 through 200 (352 nodes) and Jacobi order 256; its Fourier cutoff is distinct from the certificate cutoff. $N_t$ counts source time cells. The number of frequency nodes is written explicitly to avoid confusing it 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; 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; 513 of these are nonzero reference nodes in H0/H1/Q0. 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). N2L is a separate descriptive 128-bin reuse of N2 at alpha=.52. The nearby-grid objective experiment uses N1/N2. The H low and high banks each have 8189 closed cells (four nonstartup subcells), for 513 and 512 frequencies respectively. Thus the 0.115215934-point H4 certificate and 0.394999331-point N2L certificate use different reference centres, node coverage and residual banks; 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 use a uniform auxiliary grid $y_j$ from zero to $\nu T^\alpha$, followed by $x_j=y_j^{1/\alpha}$ and $t_j=x_j/\nu^{1/\alpha}$. In exact arithmetic this is the graded rule $t_j=T(j/N_t)^{1/\alpha}$. Generation uses binary64 arithmetic and forces the endpoints to zero and one half; certification uses the saved nodes as exact dyadics, rather than identifying them with the analytical rule. For quarter-year restriction it keeps every original knot strictly below $T_q=1/4$, appends the exact endpoint, and encloses the linear interpolation of the original bracketing field coefficients in rational interval arithmetic. All preceding history is retained.

III. Finite-history propagation: nonexpansion and dissipation

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

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. An integration bin takes the maximum over every intersecting closed residual cell, including boundary cells. 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 admits a direct interval calculation. For $\sigma\ge0$, set $\lambda=\nu\sigma$ and

The assumptions and scaling are those of Theorem 3.1. 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 provides the absolute series tails and interval-difference proof. The implementation uses twelve outer terms, sixty-four inner curve terms and 100-bit outward dyadic primitives. It 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 $\widehat Z=I^\alpha\bar G$, with recorded coefficients interpreted as exact dyadics. 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. The reference is this continuous field, not the nominal nodal solver states.

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

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

The strip proof and the exact-solution high-frequency comparison are in Appendix D.2. A finite omitted reference value is explicitly charged, 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 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 supplies bounds $|z_n|\le\epsilon_n$; at a deliberately omitted finite node $\bar\phi_n=0$, the true-transform envelope supplies the radius. 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. A finite reference sum enclosed by interval arithmetic contributes its centre uncertainty when a numerical representative of $\bar c$ is used. Omitted finite nodes remain in (5.2); they are not also charged as an infinite tail. No stochastic covariance is used in this representation.