Definition
A class of no-arbitrage asset pricing formulas in which the price of a claim is
the expected value of an exponential of a quadratic function of an underlying
Gaussian state vector. For a state x_t evolving as a (regime-conditional)
linear-Gaussian VAR, the price-earnings ratio of a perpetuity of cash flows
whose log growth rate is linear in x_t takes the form
Q_t = Σ_τ exp(A_τ + B_τ' x_t + x_t' C_τ x_t)
where the loadings (A_τ, B_τ, C_τ) satisfy a Riccati-style recursion in τ
derived from the Gaussian moment generating function. Compared to the
exponential-affine case (which suffices for ZCB pricing), the quadratic term
appears whenever the cash-flow log growth rate is itself a non-trivial linear
function of x_t and prices reflect the variance of accumulated growth (a
Jensen’s inequality term).
Intuition
A zero-coupon bond has flat cash flow, so its log payoff conditional on the
state path is −Σ r_s, which is linear in the Gaussian state. Its price is
exp(linear) — the affine case. A real-estate-like claim, by contrast, has a
log payoff Σ (ν_s − r_s) that is itself linear in the state, so the
exponentiated quantity is a sum of linear-in-state Gaussians. The expectation
of exp(linear-in-Gaussian) involves the moment generating function of a
Gaussian, which carries a ½ x' Var x term — the quadratic part. Iterated
forward, this turns the recursion for the price into a Riccati equation for
the quadratic loadings.
Formal notation
For a regime-conditional Gaussian state x_t | S_{t:T}, the present value
operator on a Q-measure linear functional δ' x_s + b_s (with δ a
1 × dim(x) row vector) is
E^Q[exp(Σ_{s=0}^{τ−1} (δ_s' x_s + b_s)) | S_{0:τ}] = exp(A_τ + B_τ' x_0 + x_0' C_τ x_0)
with the Riccati-type recursion (schematically)
C_{τ+1} = Φ^Q(S_τ)' C_τ Φ^Q(S_τ) + symmetric correction in δ_τ
B_{τ+1}' = δ_τ' + B_τ' Φ^Q(S_τ) + 2 m^Q(S_τ)' C_τ Φ^Q(S_τ)
A_{τ+1} = b_τ + A_τ + B_τ' m^Q(S_τ) + m^Q(S_τ)' C_τ m^Q(S_τ) + ½ tr(Σ' (...) Σ)
(see the project’s riccati_equations_leather_sagi.md derivation for the
exact form including the regime-conditional Σ Σ’ contribution and the
Jensen-inequality term). The infinite-horizon perpetuity price requires the
Riccati to converge as τ → ∞, which is in turn a no-bubble / spectral
condition on the regime-switching second-moment operator.
Variants
- Affine special case (
C_τ ≡ 0): bond pricing in a Gaussian or Cox- Ingersoll-Ross-style affine model. No Riccati onC; just an affine recursion on(A, B). - Quadratic Gaussian term structure (Ahn–Dittmar–Gallant 2002,
Leippold–Wu 2002): single-regime case with quadratic loadings on
x_tfor bond prices. - Markov-switching exponential-quadratic (Leather–Sagi): adds a finite Markov chain over the VAR coefficients, so the Riccati is taken conditional on the regime path and the price is a finite mixture across paths.
Comparison
- vs. exponential-affine pricing: the affine case is closed-form in a fixed number of state-vector multiplications per maturity; the quadratic case requires iterating a matrix Riccati and is not closed-form.
- vs. lognormal Gordon–Williams: GW is a one-period perpetuity formula with a constant discount rate; the exponential-quadratic case is the dynamic generalization that lets the discount rate co-move with cash flow.
- vs. Bansal–Yaron long-run-risk approximation: log-linearization around the deterministic steady state gives an exponential-affine approximation that under-prices the Jensen term — the project found a 67–72% Jensen gap at 50-year horizons in the exact-grid comparison.
When to use
Whenever the priced cash flow has growth-rate exposure to a Gaussian (or regime-conditional Gaussian) state vector and the horizon is long enough that Jensen’s inequality matters (project finding: matters at 50-year horizons; for short ZCBs the affine approximation is fine).
Known limitations
- Long-run convergence is not free. The Riccati
C_τmay diverge if the spectral radius of the regime-switching second-moment operator exceeds 1; in that case the perpetuity price is infinite and the parameter point is infeasible. The project’s “absorbing NBC” feasibility gate (≈99% accurate) is the production response. - Apartment is the slow asset in the project: needs
T_bar ≈ 50KRiccati iterations to converge at the new MAP, vs. the production default ofT_bar = 1400. - Numerical fragility. The exact Riccati is sensitive to roundoff
near the convergence boundary; the geometric tail operator
η_H · r/(1−r)with a single globalris the project’s leading hypothesis (Exp 12 follow-up N3) for the operator-floor null bottleneck in the smart-tail-cutoff experiments. A → −∞is correct at convergence (becauseexp(A) → 0in the per-period contribution) — this looks like a numerical failure but is the right answer; project gotcha.
Open problems
- A formal Hansen–Scheinkman-style spectral characterization of the regime-switching exponential-quadratic Riccati operator.
- Whether the geometric-tail closed form can be replaced by a state-dependent operator without destroying the structure that makes the Riccati feasible inside an RBPF inner loop.
- AD through the Riccati: ForwardDiff works but is slow (~158 s/eval at the project’s settings); Enzyme is the leading candidate but not yet validated.
Key papers
- leather-sagi-markov-switching-cre-asset-pricing (the project’s central derivation; the appendix is reproduced internally as the Riccati equations note)
My understanding
This is the technical innovation that makes the central paper hard. ZCB
prices in the same model are exponential-affine in a few lines; CRE
prices require a Riccati per regime path per evaluation, and the
infinite-horizon tail is a separate fixed-point problem that interacts
with the no-bubble feasibility check. The project’s T_bar debate, the
adaptive-tail-cutoff experiment (PARTIAL PASS, R-conditional), the
hybrid-tail-cache experiment (KILL_NEGATIVE_LEVERAGE), and the
online-asymptotic-onset experiment (KILL_DETECTOR_FAILS) are all
ultimately about finding a numerically honest way to evaluate this
operator inside the likelihood loop.