4.4 Credit Risk & the Merton Model

Basic Prerequisites: Black–Scholes–Merton (the equity-as-call identity is BSM) and Probability & Measure Theory (CDFs, the normal quantile $N^{- 1}$ ). (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

Credit risk is the risk of loss when a borrower does not pay. The pricing problem is subtle: unlike a stock, a bond’s payoff depends on a discrete, terminal, catastrophic event - default - and the firm’s ability to pay is not directly observable. Merton (1974) produced the founding insight in one stroke:

A firm’s equity is a European call option on the firm’s assets, struck at the face value of its debt.

At maturity, equity holders pay off the debt if the firm is worth more than the debt and pocket the residual; otherwise they exercise limited liability, hand the firm to the bondholders, and walk away. Therefore the entire apparatus of Black–Scholes–Merton - the PDE, the risk-neutral measure, $N (d_{1})$ , $N (d_{2})$ - applies directly, with firm value playing the role of the “stock” and debt the role of the “strike”. Risk-neutral default probability is then simply $N (- d_{2})$ : the BSM exercise probability.

This folder is the model topic-folder for the Kwant-Atlas build. It is a hub: it (a) gives the fast formula lookup below (job #1), covering the structural (Merton), reduced-form (intensity), CDS pricing, and portfolio (Vasicek) layers, and (b) routes you to six sub-pages that walk from raw intuition through the structural model, distance-to-default and PD, reduced-form and CDS, the failure modes, and the portfolio extensions.

The one-sentence essence. “Default is an option: equity is a call on firm assets, so a credit spread is the price of a put that the firm’s owners hold against its creditors - and the probability of exercise is a Black–Scholes exercise probability.”

2. Mathematical Ground Truth & Derivations

Quick-Reference Lookup (job #1). All formulas below are transcribed from the verified corpus - Merton (1974) eqs. (10)–(18), Hull (11th ed.) §§24.2–24.9 eqs. (24.2)–(24.10), CreditMetrics (1997) Tables 1.8/6.2, and Vašíček (1987/1991) - and every number in the check column was re-executed and reproduced exactly (§3).

Notation. $V$ firm (asset) value, $E$ equity value, $D$ face value of debt (one zero-coupon bond, maturity $T$ ), $r$ risk-free rate, $σ_{V}$ asset volatility, $σ_{E}$ equity volatility, $R$ recovery rate, $N (\cdot)$ standard-normal CDF, $N^{- 1} (\cdot)$ its quantile, $n (\cdot)$ standard-normal density.

2.1 Structural (Merton) layer. Everything follows from $E = max (V - D, 0)$ , $F = V - E = min (V, D)$ :

Quantity	Formula	Verified check (§3)
Equity (call)	$E = V N (d_{1}) - D e^{- r T} N (d_{2})$	$E_{0} = 100, σ_{E} = 0.40, D = 350, T = 1, r = 5% \Rightarrow V_{0} = 432.9067$
$d_{1}, d_{2}$	$d_{1} = \frac{ln ( V / D ) + ( r + \frac{1}{2} σ _{V}^{2} ) T}{σ _{V} T}, d_{2} = d_{1} - σ_{V} T$	$d_{2} = 2.7900$
Equity-vol identity	$σ_{E} E = N (d_{1}) σ_{V} V$	$\Rightarrow σ_{V} = 0.0926$
Debt (risk-free − put)	$F = V N (- d_{1}) + D e^{- r T} N (d_{2})$	$F = 332.9067$
Risk-neutral PD	$Q (default) = N (- d_{2})$	$26.4$ bp
Risky yield (Merton eq. 14)	$R (t) - r = - \frac{1}{t} ln [Φ (h_{2}) + \frac{1}{d} Φ (h_{1})], d = D e^{- r t} / V$	spread $0.71$ bp
Distance to default	$DD = \frac{ln ( V / D ) + ( μ - \frac{1}{2} σ _{V}^{2} ) T}{σ _{V} T}$	$2.7900$ (RN, $μ = r$ )
PD (real-world)	$PD = N (- DD)$	$μ = 10% \Rightarrow 4.3$ bp
KMV default point	$D^{*} = ST debt + \frac{1}{2} LT debt$ (empirical, Crosbie–Bohn)	-

2.2 Reduced-form (intensity) layer (Jarrow–Turnbull 1995; Hull §24.4):

Quantity	Formula	Verified check
Hazard from spread	$λ (T) = \frac{s ( T )}{1 - R}$ (Hull 24.2)	$s = 150$ bp, $R = 40% \Rightarrow λ = 250$ bp/yr
Survival / default	$Q (t) = e^{- λ t}, PD (t) = 1 - e^{- λ t}$	$Q (5) = 0.8825$ , $PD = 11.75%$
CDS fair spread	$s^{*} = \frac{C}{A + B}$ (PV protection ÷ PV premium+accrual)	recovers $150.00$ bp from $λ = 2.5%$
Continuous approx	$s \approx λ (1 - R)$	$= 150$ bp

2.3 Portfolio credit layer (Vašíček one-factor / Gaussian copula, Hull 24.10; Bluhm §1.2.3):

Quantity	Formula	Verified check
Conditional PD	$p (x) = N (\frac{N ^{- 1} ( p ) - ρ x}{1 - ρ})$	-
Asymptotic loss CDF	$F (θ) = N (\frac{1 - ρ N ^{- 1} ( θ ) - N ^{- 1} ( p )}{ρ})$	$p = 2%, ρ = 15%$ : $F (0.1763) = 0.999$
Loss quantile (Basel “formula”)	$θ_{q} = N (\frac{N ^{- 1} ( p ) + ρ N ^{- 1} ( q )}{1 - ρ})$	$θ_{0.999} = 0.17633$ (17.63% of EAD)
Portfolio UL (2 loans)	$UL^{2} = \sum_{i} p_{i} (1 - p_{i}) + 2 ρ p_{1} (1 - p_{1}) p_{2} (1 - p_{2})$	Bluhm (1.13)

Critical scaling caveat. In the structural model the observable is equity, not firm value; $(σ_{V}, V_{0})$ must be solved from the two-equation system, and that inversion is the single largest source of error (§5). In the reduced-form model the spread is given and PD is implied - the two layers calibrate to the same data but answer different questions (real-world vs risk-neutral).

3. Computational Implementation - the hub engine

This runs on the standard library only (math.erf gives the exact normal CDF, so there is no numpy/scipy dependency). It solves the Merton two-equation system by fixed-point iteration, then reports distance-to-default, PD, and the model credit spread.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

Firm value is unobservable - the model must invert equity into $(V_{0}, σ_{V})$ , an ill-conditioned problem whose error is amplified into the PD (a 1% error in $σ_{E}$ moves PD by $\approx 9%$ here).
Default only at maturity / continuous asset paths - real firms jump to default between coupons; the diffusion cannot produce a surprise (the “credit-spread puzzle”: the model gives $\sim 0.7$ bp where markets quote $100 +$ bp).
Independence is false - defaults cluster; asset correlation $ρ$ drives the entire tail of a portfolio loss distribution, and getting $ρ$ wrong mis-prices every CDO tranche.

5. References

Merton, Robert C. - On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, Journal of Finance 29(2):449–470 (1
Hull, John C. - Options, Futures, and Other Derivatives (11th ed.)
Bluhm, Overbeck & Wagner - Introduction to Credit Risk Modeling, 2nd ed. (2010)
Gupton, Finger & Bhatia (J.P. Morgan) - *CreditMetrics™
Vašíček, Oldřich - Probability of Loss on Loan Portfolio (KMV, 1987) and Limiting Loan Loss Probability Distribution (1991)
Gregory, Jon - The xVA Challenge (5th ed., 2025)

6. Connected Graph Bridges

Foundational base: Black–Scholes–Merton · Probability & Measure Theory
Sibling topic: Value at Risk & Expected Shortfall (the portfolio-loss machinery this folder feeds) · Extreme Value Theory & Fat Tails (the tail of the loss distribution)
Sub-pages (in-folder): 01 From Zero · 02 Structural Model · 03 Distance-to-Default & PD · 04 Reduced-Form & CDS · 05 Failure Modes · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05

Kwant Atlas

Explorer

4.4 Credit Risk & the Merton Model

1. Intuition & Practical Objective

2. Mathematical Ground Truth & Derivations

3. Computational Implementation - the hub engine

4. Failure Modes & First-Principles Breakdowns

5. References

6. Connected Graph Bridges

4.4.1 Credit Risk from Zero

4.4.2 The Merton Structural Model

4.4.3 Distance to Default & Default Probability

4.4.4 Reduced-Form (Intensity) Models & CDS Pricing

4.4.5 Failure Modes & Real-World Practice

4.4.6 Advanced Extensions