Parameters & Bond Terms
Corporate Bond Terms
Markov Chain Credit Visualizer
8×8 Markov Credit Rating Transition Matrix
1-Year migration probabilities across 7 performing rating classes and absorbing State 8 (Default).
| From \ To | AAA | AA | A | BBB | BB | B | CCC | Default (D) |
|---|
Term Structure of Zero-Coupon Bonds, Default Probabilities & Spreads
| Maturity | Zero Bond ($100 Par) | Physical PD ($\mathbb{P}$) | Risk-Neutral PD ($\mathbb{Q}$) | Zero Yield to Maturity | Credit Spread (bps) |
|---|
Matrix 1: Recovery Rate $\delta$ vs. Rating Grade
5-Year Credit Spread (bps) across bond recovery assumptions and rating tiers.
| Rating \ $\delta$ | 20% | 35% | 50% | 65% | 80% |
|---|
Matrix 2: Risk Premium $\pi$ vs. Maturity $T$
Corporate Bond Price ($) across risk premium multiplier and debt maturity.
| $\pi \backslash T$ | 2Y | 5Y | 7Y | 10Y | 15Y |
|---|
Theoretical Foundations: Jarrow, Lando & Turnbull (1997)
1. Markov Chain Rating Dynamics
In Jarrow, Lando, and Turnbull's (1997) seminal framework, credit risk is governed by a time-homogeneous or inhomogeneous Markov chain on state space $\mathcal{S} = \{1, \dots, K\}$, where state $K$ (Default) is an absorbing state. The $t$-year physical transition matrix is given by matrix power $P(t) = P^t$ or continuous matrix exponential:
$$P(t) = \exp(\Lambda t)$$
where $\Lambda$ is the infinitesimal generator matrix satisfying $\sum_j \Lambda_{i,j} = 0$.
2. The Risk-Neutral Calibration $\mathbb{Q}$
Because physical default probabilities $p_{i, K}$ omit the market price of credit risk, JLT constructs the equivalent martingale measure $\mathbb{Q}$ using state-dependent risk premium scaling functions $\pi_i(t)$:
$$q_{i, K}(t) = \pi_i(t) \cdot p_{i, K}(t)$$
Calibrating $\pi_i(t)$ directly to observed corporate bond yield spreads enables exact no-arbitrage valuation of credit default swaps, collateralized debt obligations, and corporate bond options.