4.4.4 Reduced-Form (Intensity) Models & CDS Pricing

Basic Prerequisites: 03 · Distance-to-Default & PD and 02 · Structural Model.

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1. Intuition & Practical Objective

The structural model asks why a firm defaults (asset value crosses a barrier). The reduced-form model does not ask why at all - it models default as an exogenous event that arrives with a time-varying intensity (hazard rate) ${\lambda }_t$:

$$\mathbb{P}(\tau \in (t,t+dt]\mid \tau >t)={\lambda }_{t}dt.$$

The default time $\tau$ is the first jump of a counting process. This is the model that credit markets actually use, for three reasons:

1. It calibrates directly to prices. Credit spreads - and especially CDS quotes - are liquid and observable. Reduced-form models invert them into a hazard curve. No unobservable firm value is needed.
2. Default can happen at any instant, not only at maturity - fixing the structural model’s biggest empirical defect (jump-to-default).
3. It prices credit derivatives cleanly. The survival probability $Q(t)=e^{-\lambda t}$ (flat intensity) gives the fair CDS spread in closed form, and the same machinery powers CVA (Gregory, Ch 17).

The practical objective: bootstrap a hazard curve from market spreads and price a CDS to confirm the quoted spread is recovered.

The duality. Structural $=$ cause (asset process, firm fundamentals); reduced-form $=$ symptom (spread-implied hazard). They are linked by Hull’s identity $\lambda (T)=s(T)/(1-R)$ - the spread is the market’s hazard, adjusted for recovery.

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2. Mathematical Ground Truth & Derivations

2.1 Survival probability and the hazard rate

With deterministic intensity $\lambda (t)$, the survival probability to $t$ and the default density are

$$Q(t)=\mathbb{P}(\tau >t)=\exp (-{\int }_0^t\lambda (u)du),\mathbb{P}(t_0<\tau \le t_1)=Q(t_0)-Q(t_1).$$

For flat intensity $\lambda$: $Q(t)=e^{-\lambda t}$ and the cumulative default probability is $1-e^{-\lambda t}$.

2.2 Hazard from a credit spread (Hull eq. 24.2)

A risky bond yields $r+\lambda (1-R)$ in the simplest setup (risk-neutral, expected loss $=$ expected excess return), so

$$\boxed{\lambda (T)=\frac{s(T)}{1-R}}$$

where $s(T)$ is the credit spread to maturity $T$ and $R$ the recovery rate ($\mathrm{LGD}=1-R$). Hull §24.4 refines this by bootstrapping piecewise-constant hazard rates to match observed bond prices exactly (Examples 24.1–24.2); the flat-intensity formula is the first step of that recursion.

Interpretation. $\lambda$ is the risk-neutral default intensity: the market’s expected loss rate per unit time, grossed up by risk aversion. Historical (physical) default frequencies are lower - the same $\mathbb{Q}$-vs-$\mathbb{P}$ gap as in page 03.

2.3 CDS mechanics and valuation (Hull §25.1–25.2)

A credit default swap is insurance: the protection buyer pays a periodic premium (spread $s$ times notional, quarterly in arrears) and receives, on a credit event, a payoff of $\mathrm{LGD}=1-R$. For a unit notional and quarterly payments at $t_k=k\Delta$:

• Premium leg (PV of premiums while alive + accrual on default):

$$A+B=\sum _k\Delta D(t_k)Q(t_k)+\sum _k\frac{\Delta }{2}D(t_k)[Q(t_{k-1})-Q(t_k)],$$

• Protection leg (PV of LGD paid at default):

$$C=\sum _k(1-R)D(t_k)[Q(t_{k-1})-Q(t_k)].$$

The fair (par) spread sets the two legs equal:

$$\boxed{s^{\ast }=\frac{C}{A+B}}$$

(Hull §25.2, “fair spread $s=C/(A+B)$”, Example 25.1). The mark-to-market of an existing CDS is PV(protection) − PV(premiums) at the contractual spread. CDS–bond basis $=$ CDS spread − bond spread; negative when bonds are cheap relative to CDS (funding/liquidity effects; Gregory §14.3.4).

2.4 The continuous approximation

With continuously-paid premium and no accrual, equating legs gives $s\approx \lambda (1-R)=\lambda \mathrm{LGD}$: fair spread $\approx$ hazard $\times$ loss-given-default. This is why $\lambda =s/(1-R)$ in §2.2 is not a coincidence - CDS and spread are two readings of the same intensity.

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3. Computational Implementation - bootstrap hazard, price the CDS, recover the spread

Stdlib only. Start from a quoted $5$-year CDS spread, imply the hazard, build the survival curve, then price the swap from first principles and check that the fair spread equals the quote.

The pricer recovers the input spread to the basis point, closing the loop between the spread-implied hazard and the CDS contract. The $11.75\%$ cumulative 5-year PD is the risk-neutral default probability the market is charging for - visibly higher than any agency’s historical 5-year BB default rate, the $\mathbb{Q}$-vs-$\mathbb{P}$ gap again.

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4. Failure Modes & First-Principles Breakdowns

1. Deterministic intensity ignores spread volatility and jump risk. Real spreads are stochastic and jump on credit events; a single $\lambda$ cannot produce a spread jump. Stochastic-intensity extensions (CIR-type ${\lambda }_t$) and jump-to-default models are the fixes (Gregory §17.6.4).
2. The recovery rate is unobservable and assumed. Everything scales with $(1-R)$; yet recovery is random, correlated with default, and junior/senior-dependent. Hull notes plain-vanilla CDS values are famously insensitive to $R$ when pricing is self-consistent - but getting $R$ wrong biases $\lambda$ and therefore every CVA on that name.
3. Single-name intensity says nothing about correlation. Pricing a portfolio (tranches, $k$th-to-default) needs a dependence structure, usually the Gaussian copula - whose correlation is not a physical quantity and which was at the centre of the 2008 CDO failure (the “correlation smile”). See 06 · Portfolio Credit & Vasicek.
4. Wrong-way risk. Intensity and exposure are not independent in reality (a counterparty’s default probability rises exactly when your exposure to it is largest). Treating them as independent understates CVA - the reason Gregory elevates wrong-way risk to a first-class failure mode.

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5. References

• Hull, Options, Futures, and Other Derivatives
• Jarrow & Turnbull (1995) - Pricing Derivatives on Financial Securities Subject to Credit Risk, Journal of Finance 50(1):53–85
• Gregory, Jon - The xVA Challenge (5th ed., 2025)
• Bielecki & Rutkowski - Credit Risk: Modeling, Valuation and Hedging (2002)

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6. Connected Graph Bridges

• Back: 03 · Distance-to-Default & PD · 02 · Structural Model
• Forward: 05 · Failure Modes · 06 · Portfolio Credit & Vasicek · Index Hub
• Sibling: VaR & Expected Shortfall (the tail of the loss the intensity generates)