Bond and dividend-strip prices in regime-switching affine models admit closed-form Riccati recursions conditional on a regime path

Statement

In a regime-switching essentially-affine asset pricing model with Gaussian innovations and a finite-state Markov chain S_t, both zero-coupon bond prices and dividend-strip prices admit closed-form path-conditional exponential-affine representations:

• For bonds, B_n^π(x_t) = exp(a_n + b_n' x_t) where (a_n, b_n) satisfy a linear recursion driven by the regimes along the path π.
• For dividend strips on an augmented state z_t = (x_t', ν_{j,t})', D_n^π(z_t) = exp(α_n + β_n' z_t) with an analogous recursion.

The unconditional price is recovered by averaging over the S^n regime paths weighted by Markov transition probabilities.

Evidence summary

• Theoretical (strong): Direct derivation in riccati-equations-leather-sagi Sections 2-4, using the standard guess-and-verify argument for affine pricing combined with the Gaussian moment generating function. The recursion is mechanical once the path is fixed: at each step (α_n, β_n) updates linearly in α_n and quadratically in β_n (the quadratic part being the Jensen ½ β' Σ Σ' β correction).
• Implementation (strong): The recursion is the basis of the production compute_quadratic_pricing_factors_msvar function in the project’s SimMdlPrices module. It is the analytic comparator for the project’s Monte-Carlo strip pricer (mc_strips.jl) and matches it to within MC noise.
• Lineage (strong): The same structure appears in the affine term-structure literature (Duffie-Kan, Dai-Singleton) and explicitly in Bikbov-Chernov 2013 (whose paper this note extends). The contribution of Leather-Sagi is the explicit augmented-state version for CRE income strips.

Conditions and scope

See conditions in the frontmatter. The claim does not assert that the unconditional price is finite — that requires a separate no-bubble check (see asset-pricing-no-bubble-condition-regime). It also does not assert a particular accuracy of the path-pruning approximation beyond Bikbov-Chernov’s empirical claim of accuracy out to ~40 quarters; the formal closed-form is only at the path-conditional level.

Counter-evidence

None at the level of the conditional recursion itself — this is a textbook calculation. Practical caveats:

• Direct iteration loses precision near the no-bubble boundary because ½ β' Σ Σ' β grows rapidly. This is a numerical, not theoretical, limitation.
• Path pruning bias when summing over regime paths is a one-sided bias that has not been bounded analytically.
• Path enumeration cost scales as S^n, which is why the project uses pruning + Monte Carlo / cumulant approximations for the long-horizon tail.

Linked ideas

(Tracker: any future work that proposes a non-Gaussian-innovation extension, a higher-than-quadratic Riccati, or a non-heuristic path-pruning scheme would link here as extends or addresses_gap.)

Open questions

• Is there a closed-form characterization of the unconditional price (i.e., post-averaging over regime paths) beyond the cumulant expansion of Section 7.3?
• Can the path-pruning error be bounded analytically as a function of regime persistence?
• What is the right generalization for non-Gaussian innovations (jump-diffusion, stochastic vol)? The general transform-analysis machinery of Duffie-Pan-Singleton applies in principle.