4.4.1 Credit Risk from Zero

Basic Prerequisites: Probability & Measure Theory (CDFs, expectation) and Black–Scholes–Merton (a call $=\max (S-K,0)$ can be priced from no-arbitrage alone).

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1. Intuition & Practical Objective

This page builds the why of credit-risk modelling with no prior credit knowledge needed. The objective is one idea, and it is the founding idea of the entire field:

Limited liability means the shareholders of a levered firm hold a call option on the firm’s assets. The strike is the debt’s face value; the “underlying” is the value of the firm; the expiration is the debt’s maturity.

Start with the dumbest question: why does a bond ever default? A firm’s assets are worth something stochastic, $V$. It owes its creditors a fixed amount $D$ at maturity. If $V_T>D$, the owners pay the debt and keep $V_T-D$. If $V_T\le D$, the owners are not obligated to contribute the shortfall - limited liability lets them walk away and hand the firm to the bondholders. So the shareholders’ payoff is

$$E_T=\max (V_T-D,0),$$

which is exactly the payoff of a European call option with underlying $V$, strike $D$, maturing at $T$. The debt’s payoff is its complement, $\min (V_T,D)$ - the residual claim.

Three “aha”s:

1. Default is a rational exercise decision, not an accident. The shareholders default precisely when exercising would be a bad trade: when the call is out of the money. Default probability is therefore an exercise probability - and BSM tells us the risk-neutral exercise probability is $N(-d_2)$.

2. Everything about a firm’s credit quality lives in two unobservable numbers: asset value $V$ and asset volatility ${\sigma }_V$. Equity’s price and volatility are observable and are functions of $(V,{\sigma }_V)$, so in principle we can invert them. That inversion is the whole empirical content - and the whole fragility - of the model.

3. A credit spread is put premium. Since $V=E+F$ and $F=\min (V,D)=D-\max (D-V,0)$, a risky bond is a riskless bond minus a put on the firm’s assets (put–call parity). The bondholders are effectively short a put written to the shareholders. The credit spread is what it costs to be short that put.

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2. Mathematical Ground Truth & Derivations

The payoff split. At maturity the total firm value divides without loss,

$$V_T=E_T+F_T,E_T=\max (V_T-D,0),F_T=\min (V_T,D).$$

Equity is a call; debt is riskless-minus-put. Using put–call parity for a call with underlying $V$ and strike $D$,

$$E=VN(d_1)-D{e}^{-rT}N(d_2),$$

so the debt value is

$$F=V-E=\underset{\text{risk-free bond}}{\underbrace{D{e}^{-rT}}}-\underset{\text{put written to shareholders}}{\underbrace{[D{e}^{-rT}N(-d_2)-VN(-d_1)]}}.$$

The risk-neutral default probability. In the risk-neutral world $V_T$ is lognormal with drift $r$ and volatility ${\sigma }_V$, so

$$\mathbb{Q}(V_T<D)=\mathbb{Q}\left(\ln \frac{V_T}{V}<\ln \frac{D}{V}\right)=N(-d_2),d_2=\frac{\ln (V/D)+(r-\frac{1}{2}{\sigma }_V^2)T}{{\sigma }_V\sqrt{T}}.$$

This is the exact statement: the risk-neutral probability of default is the Black–Scholes risk-neutral exercise probability. (Section 03 separates this from the real-world PD, which uses the physical drift $\mu$.)

Where the model lives. Merton (1974) derives this directly from his eq. (10) - the PDE satisfied by $f(V,t)$, the equity value - and notes (his words) that it is “identical to the equations for a European call option on a non-dividend-paying common stock where firm value corresponds to stock price and $B$ corresponds to the exercise price.”

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3. Computational Implementation - the payoff split and the kinked claim

The simplest possible demonstration: enumerate terminal firm values and show that equity is a call payoff and debt is the risk-free payoff minus a put. Stdlib only.

Read the table: below $V_T=100$ the equity is flat at zero (the call is out of the money - shareholders walk away) while the debt absorbs the entire shortfall. Above $V_T=100$ the equity rises one-for-one and the debt is capped. Equity is a convex, kinked claim - exactly a call. The flat-at-zero region is default.

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4. Failure Modes & First-Principles Breakdowns

1. The “it defaults when it runs out of cash” trap. Merton’s default is a maturity-event driven by the asset value crossing the debt face - not a liquidity event. A firm can default with cash on hand (strategic default) or survive with negative equity (rolling debt). The model captures only the terminal claim structure.
2. Asset value is a fictional, unobservable variable. Nobody observes "$V$". The model’s inputs are equity price and equity vol; the firm’s asset value and vol are inferred, and the inference is ill-conditioned (see 05 · Failure Modes).
3. One zero-coupon bond is a caricature. Real firms have layered, coupon-paying, callable, covenant-laden debt. Merton’s own paper extends to coupon and callable bonds (§VI), but the clean “default only at $T$” result is the vanilla case.

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5. References

• Merton, Robert C. - On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, Journal of Finance 29(2):449–470 (1974).
• Bluhm, Overbeck & Wagner - Introduction to Credit Risk Modeling, 2nd ed. (2010)
• Hull - Options, Futures, and Other Derivatives

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6. Connected Graph Bridges

• Base: Black–Scholes–Merton · Probability & Measure Theory
• Continue: 02 · The Merton Structural Model · Index Hub
• Forward: VaR & Expected Shortfall (what to do with the loss distribution this option payoff generates)