Chapter 21: Credit Risk¶
This chapter develops the mathematical theory of credit risk from two complementary perspectives---structural models grounded in firm value dynamics and reduced-form models driven by default intensities. Starting from the probabilistic foundations of default as a random time, we build pricing frameworks for defaultable bonds and credit derivatives, develop calibration methods for hazard rate curves, address measure-change issues specific to jump-to-default processes, and analyze the model risk lessons of the credit crisis.
Key Concepts¶
Default as a Random Time¶
Default is modeled as a random time \(\tau\) on a probability space, with the default indicator process \(H_t = \mathbf{1}_{\{\tau \leq t\}}\). The mathematical challenge is incorporating default information into the filtration. The background filtration \((\mathcal{F}_t)\) carries market information; the enlarged filtration \(\mathcal{G}_t = \mathcal{F}_t \vee \sigma(\tau \wedge t)\) (progressive enlargement) adds default knowledge as it arrives, while initial enlargement \(\mathcal{G}_t = \mathcal{F}_t \vee \sigma(\tau)\) assumes foreknowledge of \(\tau\). Progressive enlargement preserves semimartingale properties under regularity conditions and is the standard choice for credit modeling, while initial enlargement makes \(\tau\) a \(\mathcal{G}_0\)-measurable random variable and requires the density hypothesis---existence of a regular conditional density \(\mathbb{P}(\tau \in du \mid \mathcal{F}_t) = \alpha_t(u)\,du\)---to ensure tractability. The Azema supermartingale \(G_t := \mathbb{P}(\tau > t \mid \mathcal{F}_t)\) connects the two filtrations and admits a Doob--Meyer decomposition \(G_t = M_t - A_t\) into a martingale and an increasing predictable process. When \(G_t\) is continuous, the hazard process \(\Gamma_t = -\ln G_t\) and intensity \(\lambda_t = \Gamma_t'/G_t\) are well-defined. The immersion property (H-hypothesis)---that every \((\mathcal{F}_t,\mathbb{Q})\)-martingale remains a \((\mathcal{G}_t,\mathbb{Q})\)-martingale---holds automatically under the Cox process construction and enables clean separation of default-free and default-contingent pricing: \(P^d(t,T) = P(t,T) \cdot S(t,T)\). Equivalent characterizations include: for all \(t\), \(\mathcal{F}_t\) and \(\sigma(\tau \wedge s : s \geq t)\) are conditionally independent given \(\mathcal{F}_t\); and \(G_t = \mathbb{P}(\tau > t \mid \mathcal{F}_t) = \mathbb{P}(\tau > t \mid \mathcal{F}_\infty)\). When immersion fails (e.g., in contagion models where one default affects another's intensity), a drift correction \(M_t = \tilde{M}_t + \int_0^{t \wedge \tau} d\langle M, G\rangle_s / G_{s-}\) restores the semimartingale property. The Cox process construction \(\tau = \inf\{t \geq 0 : \Lambda_t \geq E\}\) with \(E \sim \text{Exp}(1)\) and \(\Lambda_t = \int_0^t \lambda_s\,ds\) provides the standard mechanism for generating default times from stochastic intensities, automatically satisfying immersion and yielding \(S(t,T) = \mathbb{E}[\exp(-\int_t^T \lambda_s\,ds) \mid \mathcal{F}_t]\).
Structural Credit Risk Models¶
Structural models derive default endogenously from firm value dynamics. In the Merton model (1974), the firm's assets follow \(dV_t = (r-q)V_t\,dt + \sigma_V V_t\,dW_t^{\mathbb{Q}}\) and default occurs if \(V_T < D\) at maturity. Equity is a European call on firm value: \(E_0 = V_0 e^{-qT}N(d_1) - De^{-rT}N(d_2)\) with \(d_{1,2} = (\ln(V_0/D) + (r-q \pm \sigma_V^2/2)T)/(\sigma_V\sqrt{T})\), while debt is \(B_0 = V_0 e^{-qT}N(-d_1) + De^{-rT}N(d_2)\). The risk-neutral default probability is \(\mathbb{Q}(\tau \leq T) = N(-d_2)\), and the credit spread is \(s = -\frac{1}{T}\ln(B_0/(P(0,T)\cdot D))\). The Merton model produces characteristic spread term structures: for high-quality firms (large \(V_0/D\)), spreads are hump-shaped, initially increasing then decreasing; for distressed firms, spreads decrease monotonically. First-passage models (Black--Cox, 1976) trigger default at the first time \(V_t\) hits a barrier \(B_t\): \(\tau = \inf\{t \geq 0 : V_t \leq B_t\}\), introducing the possibility of default before maturity. For a constant barrier \(B\), the survival probability has the closed form
where \(d_{\pm} = (\pm\ln(V_0/B) + \mu T)/(\sigma\sqrt{T})\) and \(\mu = r - q - \sigma^2/2\), with the default time density following an inverse Gaussian distribution. The Black--Cox extension allows a time-dependent barrier \(B_t = Ke^{-\gamma(T-t)}\), representing safety covenants that trigger reorganization. The equity-credit connection links equity volatility to asset volatility via leverage: \(\sigma_E = (V/E)\,N(d_1)\,\sigma_V > \sigma_V\), enabling joint calibration of equity and credit markets. This relationship underpins capital structure arbitrage strategies that exploit mispricings between equity options and CDS. The distance to default \(DD = (\ln(V_0/D) + (\mu - \sigma_V^2/2)T)/(\sigma_V\sqrt{T})\) is the foundation of the KMV/Moody's approach, mapping \(DD\) to empirical default frequencies (EDF) via a proprietary database of historical defaults rather than assuming normality. A persistent credit spread puzzle---structural models underpredict investment-grade spreads by 20--70\%---motivates hybrid approaches combining structural intuition with reduced-form tractability.
Reduced-Form Intensity-Based Models¶
Reduced-form models treat default as an exogenous event governed by a stochastic default intensity \(\lambda_t\), defined rigorously through the compensator: \(\mathbb{Q}(\tau \in (t, t+dt] \mid \mathcal{F}_t, \tau > t) \approx \lambda_t\,dt\). The compensated default martingale \(M_t = H_t - \int_0^{t \wedge \tau}\lambda_s\,ds\) is the fundamental building block, ensuring that the compensated default process has zero conditional expectation. Survival probability has the expectation representation \(S(t,T) = \mathbb{E}^{\mathbb{Q}}[\exp(-\int_t^T \lambda_s\,ds) \mid \mathcal{F}_t]\), which reduces to \(S(t,T) = \exp(-\int_t^T \lambda(s)\,ds)\) for deterministic intensity. Common intensity specifications parallel interest rate models: Vasicek \(d\lambda_t = \kappa(\theta - \lambda_t)\,dt + \sigma\,dW_t\) (Gaussian, may go negative); CIR \(d\lambda_t = \kappa(\theta - \lambda_t)\,dt + \sigma\sqrt{\lambda_t}\,dW_t\) (non-negative under Feller condition \(2\kappa\theta \geq \sigma^2\)); jump-diffusion adding \(J\,dN_t\) for contagion effects; piecewise-constant \(\lambda(t) = \lambda_i\) for \(t \in (T_{i-1}, T_i]\) for bootstrapping applications. The credit term structure mirrors the interest rate term structure: survival probability \(S(t,T) \leftrightarrow\) discount factor \(P(t,T)\), hazard rate \(h(t,T) = -\ln S(t,T)/(T-t) \leftrightarrow\) zero rate, forward hazard rate \(\lambda_f(t,T) = -\partial_T \ln S(t,T) \leftrightarrow\) forward rate. Affine intensity models (CIR-type) yield \(S(t,T) = A(\tau)\exp(-B(\tau)\lambda_t)\) with \(B(\tau) = 2(e^{\gamma\tau}-1)/((\gamma+\kappa)(e^{\gamma\tau}-1)+2\gamma)\) and \(\gamma = \sqrt{\kappa^2 + 2\sigma^2}\), preserving analytical tractability while allowing stochastic default risk. The Cox process construction provides the formal link: given an \(\mathcal{F}_t\)-adapted intensity \(\lambda_t \geq 0\), the default time \(\tau = \inf\{t : \int_0^t \lambda_s\,ds \geq E\}\) with \(E \sim \text{Exp}(1)\) independent of \(\mathcal{F}\) automatically satisfies the immersion property.
Pricing with Default Risk¶
The fundamental credit pricing formula for a claim paying \(C(T)\) at maturity if no default and \(R(\tau)\) at default is
where intensity acts as an additional "credit discount rate." The first term values the survival-contingent payoff, the second the recovery at default. Recovery conventions critically affect pricing: Recovery of Face Value (RFV) pays \(R \cdot F\) at default, giving \(P^d(0,T) = F\,\mathbb{E}[e^{-\int_0^T(r+\lambda)\,ds}] + RF\int_0^T \mathbb{E}[e^{-\int_0^u(r+\lambda)\,ds}\lambda_u]\,du\); Recovery of Treasury (RT) pays \(R \cdot F \cdot P(\tau,T)\) at default, giving \(P^d(0,T) = F\,P(0,T)[R + (1-R)S(0,T)]\) when rates and default are independent; Recovery of Market Value (RMV, Duffie--Singleton) pays \(R \cdot P^d(\tau^-,T)\), yielding the elegant formula \(P^d(0,T) = F\,\mathbb{E}^{\mathbb{Q}}[\exp(-\int_0^T(r_s + (1-R)\lambda_s)\,ds)]\) with the loss-adjusted intensity \((1-R)\lambda\) replacing the raw intensity. The credit spread in the intensity framework satisfies the approximation \(s \approx (1-R)\bar{\lambda}\), linking spreads to hazard rates via the loss-given-default. The term structure of credit spreads \(s(T) = -\frac{1}{T}\ln\frac{P^d(0,T)}{P(0,T)}\) can be upward-sloping (increasing default risk over time), downward-sloping (near-term distress), or humped, depending on the intensity dynamics. Typical recovery rates range from 50--70\% (senior secured) to 0--10\% (junior/equity), with 40\% as the standard CDS assumption.
Credit Derivatives¶
A credit default swap (CDS) exchanges periodic premium payments for default protection on a reference entity. The protection leg has present value \(\text{PV}_{\text{prot}} = (1-R)\,N\int_0^T D(0,t)\lambda_t S(0,t)\,dt\) and the premium leg \(\text{PV}_{\text{prem}} = s \cdot N \cdot \text{RPV01}\) where the risky PV01 is \(\text{RPV01} = \sum_{i=1}^n \delta_i D(0,t_i)S(0,t_i)\) plus an accrued premium adjustment. Setting legs equal gives the par CDS spread
which simplifies to \(s_{\text{par}} = (1-R)\lambda\) for constant intensity and rates. Mark-to-market is \(\text{MTM} \approx -(s_{\text{new}} - s_{\text{old}}) \times N \times \text{RPV01}\). Post-2009 conventions use standardized coupons (100 or 500 bp) with upfront payments \((s_{\text{par}} - s_{\text{fixed}}) \times \text{RPV01} \times N\). Valuation adjustments (XVA)---CVA for counterparty default risk \(\text{CVA} = \mathbb{E}[\text{Exposure} \times \text{LGD}]\), DVA for own default, FVA for funding costs, MVA for margin---extend the classical pricing framework to account for bilateral credit risk and real-market frictions. Total return swaps (TRS) transfer total economic exposure of a reference asset (price appreciation plus coupons) in exchange for a floating rate plus spread, providing synthetic exposure without direct ownership. CDS indices (CDX, iTraxx) pool single-name CDS into standardized baskets, and tranched products (equity, mezzanine, senior) allocate portfolio losses in priority order, with pricing depending critically on default correlation.
Change of Measure and Credit Risk¶
Default intensities differ across measures: \(\lambda_t^{\mathbb{Q}} = \lambda_t^{\mathbb{P}} + \text{credit risk premium}\), with risk-neutral intensities (from CDS) systematically exceeding physical default frequencies (from historical data) due to investors' demand for compensation for bearing default risk. The gap widens substantially during stress periods. Girsanov's theorem for jump processes extends measure changes to models with jump-to-default: the compensated default process \(M_t = H_t - \int_0^{t \wedge \tau}\lambda_s\,ds\) remains a martingale under measure change, with the intensity transforming as \(\lambda_t^{\mathbb{Q}} = \lambda_t^{\mathbb{P}} + \theta_t\) where \(\theta_t\) reflects the market price of default risk. Girsanov applies jointly to diffusive components (drift change) and jump components (intensity change) of the model, with the Radon--Nikodym derivative accounting for both: \(dQ/dP|_{\mathcal{G}_t}\) contains an exponential martingale for the continuous part and a multiplicative correction for the jump part. The consistency conditions link compensators, measure changes, and recovery assumptions, ensuring that discounted asset prices remain martingales under the pricing measure. The survival measure conditions on no default (\(\tau > T\)), simplifying pre-default calculations by removing the default indicator from expectations and effectively working on the background filtration \((\mathcal{F}_t)\) alone.
Credit Risk Calibration¶
CDS spreads are the primary calibration instruments because they isolate default risk and are relatively liquid. Bootstrapping hazard rates extracts a piecewise-constant intensity curve \(\lambda(t) = \lambda_i\) for \(t \in (T_{i-1}, T_i]\) by sequentially solving \(s_i \times \text{RPV01}_i = (1-R) \times \text{PV}_{\text{prot},i}\) for each maturity, with survival probabilities \(S(0,T_i) = \prod_{j=1}^i e^{-\lambda_j \Delta T_j}\)---directly analogous to yield curve bootstrapping. The protection leg with piecewise-constant intensity is \(\text{PV}_{\text{prot}}(T_i) = (1-R)\sum_{j=1}^i \lambda_j \int_{T_{j-1}}^{T_j} D(0,t)S(0,T_{j-1})e^{-\lambda_j(t-T_{j-1})}\,dt\). The initial guess \(\lambda_i^{(0)} \approx s_i/(1-R)\) seeds Newton--Raphson or Brent iteration to convergence \(|f(\lambda_i)| < \epsilon \sim 10^{-8}\). Practical considerations include: standard liquid CDS maturities (6M, 1Y, 2Y, 3Y, 5Y, 7Y, 10Y); sensitivity to recovery \(\lambda \propto s/(1-R)\) so that changing \(R\) from 40\% to 30\% increases implied hazard rates by ~14\%; possible negative hazard rates from inverted spread curves; and extrapolation beyond the last maturity. Identifiability is a fundamental challenge: higher hazard rate with higher recovery produces the same CDS spread as lower hazard rate with lower recovery, so recovery is typically fixed at 40\% to break the degeneracy. Joint calibration to CDS and bonds rarely achieves perfect consistency due to liquidity differences, funding premia, and basis effects. Historical vs risk-neutral calibration distinguishes physical-measure parameters (from default databases) from risk-neutral parameters (from market prices), with the risk premium bridging the two. Stability---smooth day-to-day parameter evolution, robustness to small quote perturbations---often outweighs theoretical precision, motivating regularization and smoothing of the hazard rate curve.
Model Risk in Credit Markets¶
The 2008 crisis exposed catastrophic model risk in credit markets. The Gaussian copula model \(X_i = \sqrt{\rho}\,Z + \sqrt{1-\rho}\,\epsilon_i\) for portfolio default---where \(Z\) is a systematic factor, \(\epsilon_i\) are idiosyncratic, and default occurs when \(X_i < \Phi^{-1}(p_i)\)---assumed constant pairwise correlation \(\rho\) and Gaussian dependence. While this enabled rapid CDO tranche pricing through a single correlation parameter, the assumptions failed spectacularly as correlations spiked during the crisis and a single \(\rho\) could not fit equity, mezzanine, and senior CDO tranches simultaneously (the "correlation smile"), indicating fundamental model misspecification. Senior tranches rated AAA experienced unprecedented defaults, with actual losses far exceeding model predictions. Wrong-way risk---positive correlation between exposure and counterparty default probability---amplifies losses beyond what independence-based CVA calculations predict, as exposure increases precisely when the counterparty is most likely to default. Standard intensity models assuming independence between exposure and default dangerously underestimate tail losses. Correlation and systemic risk from macroeconomic shocks, sectoral contagion, and cascading failures causes diversification benefits to vanish precisely when they are most needed; single-name models fundamentally cannot capture these portfolio-level dynamics. Stress and crisis behavior reveals that models calibrated in normal markets fail qualitatively during regime shifts: default intensities jump, correlations spike, and liquidity evaporates, invalidating extrapolation from calm-period calibration. Key lessons: tail dependence (Student-\(t\), Clayton, or vine copulas) matters more than average correlation; model-dependent pricing in illiquid markets creates dangerous circularity (models price instruments whose market prices then calibrate the models); stress testing with regime-switching and extreme scenarios is indispensable; and model humility---recognizing structural limitations rather than optimizing within a misspecified framework---is the most reliable risk management tool.
Role in the Book
Credit risk modeling integrates several threads from earlier chapters: the Black--Scholes option pricing framework (Chapter 6) underpins the Merton model, first-passage times connect to Brownian motion and the reflection principle (Chapter 2), intensity-based models parallel the short-rate framework (Chapter 18), and measure-change techniques extend Girsanov's theorem (Chapter 4) to jump processes. The calibration and model risk themes echo Chapter 17, while the copula and correlation methods relate to the multivariate dependence structures implicit in portfolio modeling. The credit term structure analogy---survival probability as discount factor, hazard rate as zero rate---provides a unifying perspective connecting interest rate and credit markets.