−− calibration with state-dependent rough drift

running: Fit stage 3: global_paired_search

As-of 2026-10-30; Fitting 3 expirations: 2026-11-30, 2026-12-01, 2026-12-02. 19 earlier expirations frozen; 22/39 included so far.

Error scope: active fitting expirations — 3 expirations (2026-11-30, 2026-12-01, 2026-12-02).
Current error: 3.250593640581046 | Retained error: 3.250475086636927 | Stage evaluations: 161/no cap | Cumulative optimizer attempts: 17299
Latest confirmation attempt: 3.2527687956040983 (improved) | Confirmed error: 3.2527687956040983 | Added expiry confirmed error: 4.199021449670266 | Unsuccessful confirmations: 1
Total pricing evaluations: 17496 (17295 fitting; 201 confirmation). Current paths: 524288; confirmation paths: 524,288.

Preserved through 2026-11-27. The 2 quality limit applies to the new continuation. Unchanged earlier expirations above that limit: 2026-11-03, 2026-11-09, 2026-11-11, 2026-11-12, 2026-11-13, 2026-11-16, 2026-11-17, 2026-11-18, 2026-11-19, 2026-11-20, 2026-11-23, 2026-11-24, 2026-11-25, 2026-11-27. Their plots and individual errors remain visible; their quotes do not enter the active fitting error.

Two-state reciprocal model with state-dependent rough drift

There are two states, each represented in smooth and rough coordinates:

$$ \large \begin{gathered} \zeta_i=\mathord{\text{כ}}_i\xi_i^{\lambda_i},\\[6pt] \mathord{\text{כ}}_i>0,\qquad\lambda_i\ne0,\qquad j=3-i. \end{gathered} $$

Both signs of the coordinate powers are defined; this calibration uses the $(-,-)$ regime. Each rough coordinate evolves on the boundary clock of the other smooth coordinate. Its coefficients use $\xi_i$, the state other than the clock driver $\xi_j$.

Term structure of measure-dependent coefficients

Each expiry date $T_m$ labels the preceding interval $(T_{m-1},T_m]$. Parameters transformed by the cash-to-asset numeraire change have independently fitted values at every tenor, including inside an active bootstrap fitting window:

$$ \kappa_i(t)=\kappa_{i,m},\qquad \vartheta_i(t)=\vartheta_{i,m}. $$

These are precisely the transformed coefficients of the current parameterization:

$$ \kappa_i^S=\kappa_i-\rho_i\sqrt{H_i},\qquad \vartheta_i^S=\vartheta_i+\mathord{\text{מ}}_i\sqrt{\mathord{\text{ש}}_i}. $$

Their descriptor entries are drift[expiration_date][parameter_name]. The other parameters—including $\theta_i$, $\mathord{\text{ר}}_i$ and $\mathord{\text{נ}}_i$—retain shared fitted values across the active bootstrap block. Classification follows the transformation formula even if a correlation happens to be zero at the current trial. The optional inverse-stock quote convention is a separate operation and does not change this classification. States and accumulated clocks carry across boundaries without resetting.

Processes, clocks and correlations

For the discounted stock under the domestic measure:

Calendar-time state increments

$$ \large \begin{aligned} \mathbb E_{\rm d}[d\xi_i]\bigr|_{dt} &=(\theta_i-\kappa_i\xi_i)dt,\\[8pt] d\langle\xi_i\rangle\bigr|_{dt} &=2\xi_i\,dt, \end{aligned} $$

Rough-clock state increments

$$ \large \begin{aligned} \mathbb E_{\rm d}[d\zeta_i]\bigr|_{d\mathord{\text{ז}}_j} &=\left[\vartheta_i+\mathord{\text{ר}}_i\xi_i^{\aleph_i} -\mathord{\text{נ}}_i\xi_i^{\gamma_i}\zeta_i\right]d\mathord{\text{ז}}_j,\\[8pt] d\langle\zeta_i\rangle\bigr|_{d\mathord{\text{ז}}_j} &=2\zeta_i\,d\mathord{\text{ז}}_j, \end{aligned} $$

Stock drift and state independence

$$ \large \begin{aligned} \mathbb E_{\rm d}[dS]&=0,\\[6pt] d\langle\xi_1,\xi_2\rangle&=0. \end{aligned} $$

Stock variance

$$ \large \begin{aligned} \frac{d\langle S\rangle}{S^2} &=\frac{H_1\xi_1+H_2\xi_2}{2}dt\\[6pt] &\quad+\frac{\mathord{\text{ש}}_1}{2\zeta_1}d\mathord{\text{ז}}_2 +\frac{\mathord{\text{ש}}_2}{2\zeta_2}d\mathord{\text{ז}}_1, \end{aligned} $$

Stock–state covariances

$$ \large \begin{aligned} \frac{d\langle\xi_i,S\rangle}{S}\Bigr|_{dt} &=\rho_i\sqrt{H_i}\xi_i\,dt,\\[10pt] \frac{d\langle\zeta_i,S\rangle}{S}\Bigr|_{d\mathord{\text{ז}}_j} &=\mathord{\text{מ}}_i\sqrt{\mathord{\text{ש}}_i}\,d\mathord{\text{ז}}_j. \end{aligned} $$

All coefficients are state indexed. The parameters $\rho_i$ and $\mathord{\text{מ}}_i$ are component Brownian correlations in $[-1,1]$. In particular, $\mathord{\text{מ}}_i$ is the correlation of the $\zeta_i$ rough channel with its stock channel. The rough $\xi_i$ correlation is $\operatorname{sign}(\lambda_i)\mathord{\text{מ}}_i$. Positive $\mathord{\text{מ}}_i$ gives negative rough-variance/stock leverage. These component correlations need not equal correlations with the aggregate stock innovation.

The clocks retain their normalization:

$$ \large \begin{aligned} \mathord{\text{ז}}_i(t)&=\lim_{\epsilon\downarrow0} \frac{\theta_i(1-\theta_i)}{\epsilon^{\theta_i}} \int_0^t\mathbf1_{\{\xi_i(s)<\epsilon\}}ds,\\[8pt] \mathbf1_{\{\xi_i>0\}}d\mathord{\text{ז}}_i&=0. \end{aligned} $$

The parameter domain includes $0<\theta_i<1$ and $\theta_1+\theta_2>1$, intended to exclude simultaneous zeros. The independent-affine argument alone does not prove this property for the full coupled boundary-clock system. Two contacts at different times inside one numerical interval are not simultaneous contacts. There is no magic clock.

On the positive interior, the full smooth-coordinate equations are

$$ \large \begin{aligned} \mathbb E_{\rm d}[d\xi_i] &=(\theta_i-\kappa_i\xi_i)dt\\ &\quad+\left[ \frac{\vartheta_i+\mathord{\text{ר}}_i\xi_i^{\aleph_i}-1+1/\lambda_i} {\lambda_i\mathord{\text{כ}}_i}\xi_i^{1-\lambda_i} -\frac{\mathord{\text{נ}}_i}{\lambda_i}\xi_i^{1+\gamma_i} \right]d\mathord{\text{ז}}_j,\\ d\langle\xi_i\rangle &=2\xi_i\,dt+ \frac{2}{\lambda_i^2\mathord{\text{כ}}_i} \xi_i^{2-\lambda_i}d\mathord{\text{ז}}_j,\\ \frac{d\langle\xi_i,S\rangle}{S} &=\rho_i\sqrt{H_i}\xi_i\,dt+ \frac{\mathord{\text{מ}}_i\sqrt{\mathord{\text{ש}}_i}}{\lambda_i\mathord{\text{כ}}_i} \xi_i^{1-\lambda_i}d\mathord{\text{ז}}_j. \end{aligned} $$

The induced smooth drift and variance in rough coordinates are

$$ \large \begin{aligned} \mathbb E_{\rm d}[d\zeta_i]\Bigr|_{dt} &=\left[\lambda_i(\theta_i+\lambda_i-1)\frac{\zeta_i}{\xi_i} -\lambda_i\kappa_i\zeta_i\right]dt,\\ d\langle\zeta_i\rangle\Bigr|_{dt} &=2\lambda_i^2\frac{\zeta_i^2}{\xi_i}\,dt,\\ \frac{d\langle\zeta_i,S\rangle}{S}\Bigr|_{dt} &=\lambda_i\rho_i\sqrt{H_i}\zeta_i\,dt. \end{aligned} $$

Numeraires, initialization and dimensions

The same drift family is preserved by the local stock-numeraire change:

$$ \large \begin{aligned} \kappa_i^S&=\kappa_i-\rho_i\sqrt{H_i},\\[8pt] \vartheta_i^S&=\vartheta_i+\mathord{\text{מ}}_i\sqrt{\mathord{\text{ש}}_i}. \end{aligned} $$

The parameters $\mathord{\text{ר}}_i$ and $\aleph_i$ are unchanged. All model coefficients remain fitted: four measure-dependent coefficients per expiry interval, plus twenty-four shared parameters including both initial states and all four correlations. The editable descriptor initializes them from the retained global fit. Expiry speeds start at one, with the first speed fixed to anchor the time scale. The stochastic-dividend extension is absent; market forwards and discounts supply deterministic carry.

Both signs of $\aleph_i$ are permitted. For $\lambda_i<0$, nonnegative $\mathord{\text{ר}}_i$ and positive $\aleph_i$ increase repulsion from $\zeta_i=0$. A negative exponent instead strengthens the term at large $\zeta_i$; the remaining drift must control its growth. In rough coordinates the two state-dependent terms are

$$ \large \mathord{\text{ר}}_i\mathord{\text{כ}}_i^{-\aleph_i/\lambda_i} \zeta_i^{\aleph_i/\lambda_i} -\mathord{\text{נ}}_i\mathord{\text{כ}}_i^{-\gamma_i/\lambda_i} \zeta_i^{1+\gamma_i/\lambda_i}. $$

Their boundary behavior must be assessed jointly. Neither a constant test on vartheta alone nor finite numerical output proves global existence, martingality or finiteness of inverse moments.

Within a fitted block, $\theta_i$ is shared and fitted. The coefficient $\vartheta_i$ varies by tenor because its cash-to-asset transformation is nontrivial. With $[\xi_i]=T$ and $[\mathord{\text{ז}}_j]=[\zeta_i]=T^{1-\theta_j}$:

The editable descriptor exposes every initial value, change scale and bound; a number fixes a parameter. An active window with $M$ distinct dates has $24+4M$ model entries, plus date-keyed expiry speeds. The first speed is fixed at one. On bootstrap continuation, the frozen prefix retains its coefficients, simulation state and fitting provenance. Each remaining tenor keeps its own four measure-dependent coefficients, even within a multi-expiry fitting block. Shared parameters continue to fit jointly across that active block; the initial states are carried from the frozen prefix rather than refitted there.

Numerical procedure

Each configured outer interval uses the existing touch-conditioned smooth draws in $\xi$, followed by the reciprocal-coordinate rough updates. There are no extra physical-time subdivisions or magic-time substeps. Both touch records are redrawn at the next outer interval.

After the smooth endpoints are drawn, an active rough block starts from $\zeta_i^{\rm sm}=\mathord{\text{כ}}_i(\xi_i^{\rm sm})^{\lambda_i}$ with duration $u=\Delta\mathord{\text{ז}}_j$. Freeze only the new immigration coefficient for this block:

$$ \large A_i=\vartheta_i+\mathord{\text{ר}}_i(\xi_i^{\rm sm})^{\aleph_i}. $$

A zero rough increment is the identity and requires no singular coordinate evaluation. On active blocks the reciprocal inverse-moment calculation requires $A_i>1$ and $A_i+\mathord{\text{מ}}_i\sqrt{\mathord{\text{ש}}_i}>1$. These are requirements of this finite-step approximation, not a boundary theorem for the nonlinear SDE.

For $q_i=\gamma_i/\lambda_i>0$ and $\mathord{\text{נ}}_i>0$, use the existing equilibrium construction with $A_i$ throughout:

$$ \large \begin{aligned} a_i&=\mathord{\text{נ}}_i\mathord{\text{כ}}_i^{-q_i},\\[6pt] \zeta_{*,i}&=(A_i/a_i)^{1/(1+q_i)},\\[6pt] K_i&=A_i/\zeta_{*,i}. \end{aligned} $$

The exact deterministic flow of $K_i z-a_i z^{1+q_i}$ is

$$ \large F_{i,h}(z)=\left[e^{-q_iK_ih}z^{-q_i} +\frac{a_i}{K_i}(1-e^{-q_iK_ih})\right]^{-1/q_i}. $$

Apply this map for $u/2$, take one existing reciprocal affine draw with drift $A_i-K_i z$ and quadratic variation $2z\,du$, and apply the map for $u/2$ again. In other regimes use the existing frozen rate $b_i=\mathord{\text{נ}}_i(\xi_i^{\rm sm})^{\gamma_i}$, with no surrounding maps. Map the resulting endpoint back to $\xi_i$.

The conditional stock factor uses the same intermediate affine draw and the shifted shape $A_i+\mathord{\text{מ}}_i\sqrt{\mathord{\text{ש}}_i}$. Its analytic normalization retains conditional mean one; the affine first-moment match does not assert an exact match for the subsequent nonlinear endpoint map. Residual variance retains the existing reciprocal fair-variance formula and 24-node quadrature. All stock normalizations are analytic, independent of the simulated sample mean. Raw forward errors and their Monte Carlo standard errors remain displayed separately from controlled option prices.

At $\aleph_i=0$, use the scalar coefficient $A_i=\vartheta_i+\mathord{\text{ר}}_i$, retaining the previous arithmetic and random draws. This permits an exact seeded initialization comparison. Nonzero aleph introduces frozen-coefficient error in addition to the existing finite-step composition and drift approximations; convergence requires a separate time-grid study.

Coefficient changes occur at existing expiry grid boundaries. The grid, random draws and number of outer steps are unchanged. A flat curve reproduces the scalar-coefficient model exactly.

Confirmed expiry time scales

Individual expiry errors