Autocallable Pricing as a Latent Computation: Spectral Methods with Formal Error Bounds

We show that the price of a worst-of autocallable with step-down barriers, discrete monitoring, and stochastic volatility is computable as a composition of seven Latent operations (Nagy 2026). The multi-asset density is a grade-\(d\) Latent \(\Lambda \in \mathcal{H}^{\otimes d}\); eigenvalue conditioning reduces it to a rank-\(K\) approximation; COS backward induction performs grade-1 coordinate updates at each observation date; and the price is the projector \(\langle \Lambda, G_{\text{payoff}} \rangle\). The total error decomposes as \(\varepsilon \leq \varepsilon_{\text{eigen}} + \varepsilon_{\text{cos}} + \varepsilon_{\text{barrier}} + \varepsilon_{\text{discrete}} + \varepsilon_{\text{sv}}\), with all components nonneg and controlled by the analyticity parameter \(\rho\) via the Latent Theorem (Theorem 1). Complexity is \(O(Q \cdot M \cdot N)\) — independent of the number of assets \(d\).

We prove that the single-factor spectral pricer is a systematic lower bound for path-dependent autocallables, with the gap arising from Brownian bridge fluctuations between observation dates. The gap is bounded by \(M \cdot P_{\text{cross}}^{\max} \cdot c_{\max}\) where \(P_{\text{cross}}\) is the reflection-principle crossing probability, and vanishes in all degenerate limits (\(\sigma \to 0\), \(M = 1\), \(B \to 0\), \(B \to \infty\)). Peak gap is 5.27% at ATM barriers, declining to <1% at deep OTM.

We further show that multi-factor temporal conditioning — eigendecomposing the Brownian motion covariance \(\Sigma_{ij} = \min(t_i, t_j)\) — reduces the gap from 3.26% to 0.11% with a single temporal factor (\(K_t = 1\)), and closes it entirely at full rank (\(K_t = M\)).

Greeks (Delta, Vega, Rho, barrier sensitivity) are derivatives of the Latent projector — themselves Latents — computed analytically via adjoint backward recursion at the same \(O(Q \cdot M)\) cost as pricing, with zero Monte Carlo noise.

All results are backed by 98 formally verified theorems and a dual Python/Rust implementation with Monte Carlo verification (200k+ paths, 11 Rust + 9 Python tests).