Endogenous Default Boundaries
The classic Merton model assumes default can ONLY happen at maturity ($T$). In reality, a firm can default at any time if it violates covenants or runs out of cash. This leads to First Passage Time models (like Black-Cox, 1976).
The Concept
Instead of checking solvency only at the end, we define a Barrier (usually the Debt level).
- If the Asset Value ($V_t$) hits this Barrier at any point before maturity, Default is triggered immediately.
- This is mathematically similar to a Barrier Option (Down-and-Out Call).
Endogenous vs Exogenous
- Exogenous Boundary: The default point is fixed by the debt contract (e.g., "Default if assets < 100").
- Endogenous Boundary: The default point is decided by the shareholders. Modeling when shareholders will choose to default optimally.
- Shareholders might keep a zombie company alive by injecting cash, hoping for a recovery.
- They will only "throw in the towel" (default) when the option value of waiting drops to zero.
Benefits of Barrier Models
- Reflects Reality: Companies default throughout the year, not just on bond maturity dates.
- Higher Default Probabilities: Since the asset path can hit the barrier due to volatility (even if it recovers later), these models typically predict higher (and more accurate) default risks for high-volatility firms.
Note
Analogy:
- Merton: You drive a car blindfolded. You only check if you are on the road at the finish line.
- Black-Cox: You default the moment your car touches the guardrail (Barrier) at any point during the race.
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