Objective
Systematically disentangle four bottlenecks in the regime-switching pricing recursion used inside MLE: (1) bad local surrogate family, (2) bad regime-mixture collapse, (3) strip compounding, (4) inefficient deep-tail summation. Test Methods 0-6 plus a tail shortcut, each against exact one-step and MC multi-step truth proxies.
Setup
- Core problem: after exact conditional-on-j propagation, the regime-mixture collapse is not closed in the affine surrogate family. The current affine-local collapse (Method 0) is the production baseline.
- Candidate methods: M0 affine-local, M1 affine-regional/moment-matched, M2 quadratic-local, M3 quadratic-regional, M4 small-mixture preservation, M5 continuation-value recursion, M6 T-step projected continuation, Tail shortcut (coefficient stabilization / geometric extrapolation).
- Truth Level A: exact one-step log-sum-exp at a grid of states. Truth Level B: MC regime-path-conditional pricing.
- Stages A-F: one-step diagnostics → multi-step truth proxy → strip comparison → continuation architecture → T-step solver → tail shortcut.
Procedure
- Stage A: one-step operator diagnostics on multiple state design sets (local box, wider box, Gaussian cloud, stress points).
- Stage B: conditional MC over regime paths as multi-step benchmark.
- Stage C: strip recursion comparison at horizons up to T_bar=1400, with error decomposition by horizon bucket.
- Stage D: continuation-value architecture comparison.
- Stage F (out of order): tail shortcut — geometric strip extrapolation after a finite burn-in.
- Bansal-style log-linear approximation tested as a special case against exact grid benchmarks.
Results
- Geometric tail extrapolation at h_burn=100 delivers 14x speedup with decay state-independent (CV < 0.032%), validated across representative parameter draws (Exps 0-5, 7R).
- Bansal log-linear approximation: FAIL — 67-72% Jensen gap at 50-year horizon, confirmed by exact grid benchmarks. The log-linear closure is structurally biased under regime switching.
- M0 (affine-local collapse) has a 13-17% structural bias at the production parameterization that remains across horizon buckets.
- T_bar=1400 validated as production by T_bar=5000 sweep (delta_LL=0.0 across 46 theta vectors, with monitor_tbar=true safety).
- Riccati convergence: A → -infinity is correct (exp(A)→0); Apartment is the slow asset (needs T_bar~50K at the current MAP).
Analysis
The tail shortcut is the clear operational win from this program — a 14x speedup with negligible accuracy loss, because strip decay is empirically state-independent and the geometric series is an exact envelope. The M0 affine-local collapse remains the production method, since the 13-17% structural bias it introduces is acceptable for likelihood ranking (as validated by the MC pricing program, program-mc-finite-horizon-pricing). The Bansal-style log-linear approximation is a dead end for this model class. Continuation-value and T-step methods remain largely exploratory.
Claim updates
- principal-eigenvalue-determines-long-run-asset: tested_by, strength moderate. The geometric tail extrapolation is empirically justified by the principal-eigenvalue theory — strip decay rate converges to a single regime-averaged value governed by the spectral radius of the pricing operator, consistent with the Hansen-Scheinkman prediction.
- sdf-multiplicative-factorization-decomposes-long-run: tested_by, strength weak. The tail behavior (state-independent asymptotic decay) is empirically consistent with the factorization where the long-run component dominates at high horizons, but the program does not directly verify the factorization itself.
Follow-up
- M0 structural bias (13-17%) motivates the MC pricing program (program-mc-finite-horizon-pricing) as the production pricing layer.
- Geometric tail shortcut integrated into
SimMdlPrices/src/mc_tail.jl. - See also: constrained-estimation-branch-summary, basin-finder-complete-program-summary.