Rare States and Long-Run Pricing

Abstract

Macroeconomic crises are rare, yet they can have large effects on asset prices. Because the probability of entering a crisis from normal times is small relative to the probability of recovery, the dynamics are nearly reducible: setting the crisis-entry probability to zero makes crisis states unreachable from normal states. We develop a graph-theoretic and spectral analysis of long-run pricing under reducible and nearly reducible dynamics. We show that, at the reducible limit, physical and pricing dynamics induce the same classes of economic states and accessibility relations, but rank their long-run importance differently: by recurrence and by class-specific pricing rates. Moreover, a claim's price depends only on its pricing corridor---the classes of states lying on directed paths from the initial state to states where the payoff is positive---and the highest class-specific pricing rate within this corridor determines the claim's long-run pricing rate. When the reducible-limit corridor excludes the globally pricing-dominant class, we characterize how restoring rare transitions introduces a contribution from that class with a weight that vanishes as the transitions become rarer. This contribution becomes dominant only beyond a crossover maturity, which increases with the rarity of the connecting paths and decreases with the dominant class's pricing-rate advantage. Finally, a consumption-based disaster-recovery application shows how preferences, consumption risk, cash-flow exposure, and recovery timing jointly shape the long-run pricing dominance of disaster states and crossover maturities.