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CFA Level II Exam · Credit Analysis Models

Structural Models of Credit Risk: The Merton Approach

Updated 7 October 2026 · Fact-checked

A structural model of credit risk, from Merton, treats a firm's equity as a call option on its assets, with strike equal to the debt's face value. Risky debt equals risk-free debt minus a put option on the assets. To solve, find asset value, face value, volatility and time, value the option, then subtract.

Understand Structural Models of Credit Risk

Start with a simple firm. It is financed with equity and one zero-coupon bond with face value K, due at time T. At T, the firm either has enough assets to repay the bond or it does not. Structural models build on this idea: default is linked to the value of the firm's assets compared with its debt.

If asset value V at T is above K, debt holders get K and equity holders keep V − K. If V is below K, debt holders take all the assets, V, and equity holders get zero. So equity at T = max(V − K, 0). That is the payoff of a call option on the firm's assets with strike K. Equity holders have limited liability, so they cannot lose more than they put in.

Debt holders receive min(V, K), which is K − max(K − V, 0). That is a risk-free payoff of K minus the payoff of a put option on the assets with strike K. So today: risky debt = PV of risk-free debt − put on assets. The put is the value of the credit risk. You can also say debt = V − call, because assets = debt + equity.

The put value drives everything. A higher asset volatility, higher leverage (larger K relative to V) or longer maturity raises the put, lowers the risky debt value and widens the credit spread. Equity holders gain from higher volatility because the call is worth more. Debt holders are effectively short the put and lose from it.

Compare this with reduced-form models. Structural models explain why default happens, using the firm's assets and capital structure. Reduced-form models do not model assets. They treat default as a random event that arrives with a hazard rate (intensity), often tied to observable variables such as spreads or macro factors. Structural models have intuition and link equity and debt. Their weaknesses: asset value is not directly observable, the basic model allows default only at maturity, and the capital structure is simplified. Reduced-form models are easier to calibrate to market prices and allow default to come as a surprise.

Key formulas to remember

Equity as a call on assets
E(T) = max(V(T) − K, 0)
V = firm asset value, K = face value of zero-coupon debt. Equity holders own the call.
Risky debt at maturity
D(T) = min(V(T), K) = K − max(K − V(T), 0)
Debt holders hold risk-free debt and are short a put on assets.
Value of risky debt today
D(0) = PV(K) − Put = V(0) − Call
PV(K) is K discounted at the risk-free rate. Put and call both have strike K and maturity T.
Put-call parity on assets
V + Put = Call + PV(K)
Use it to get the put from a given call, or the reverse.
Credit spread
Spread = y − r, where D(0) = K ÷ (1 + y)^T
y is the yield on the risky debt. Use the same compounding as r. With continuous compounding, y = −ln(D(0) ÷ K) ÷ T.
Black-Scholes-Merton inputs
d1 = [ln(V ÷ K) + (r + σ² ÷ 2)T] ÷ (σ√T); d2 = d1 − σ√T; Call = V·N(d1) − K·e^(−rT)·N(d2)
σ is asset volatility, r is continuously compounded. The put uses N(−d1) and N(−d2).
Risk-neutral default probability
P(V(T) < K) = N(−d2) = 1 − N(d2)
This is under risk-neutral probabilities, not real-world ones.

How to solve Structural Models of Credit Risk questions

Use this method for any item set that gives a Merton-style setup, with or without numbers.

  1. 1Identify the firm's asset value V, the face value of debt K (treated as zero-coupon), the time to maturity T, the risk-free rate r and asset volatility σ.
  2. 2Decide who holds what: equity is a call on V with strike K. Risky debt is risk-free debt minus a put on V with strike K.
  3. 3If values are given, apply the identities. Debt = V − Call, or Debt = PV(K) − Put. Use put-call parity to move between the call and the put.
  4. 4If d1 and d2 are needed, compute them and use N(d2) for the probability of no default and N(−d2) for default, both risk-neutral.
  5. 5For the spread, find y from D(0) = K ÷ (1 + y)^T (or the continuous form), then subtract r. Keep the compounding consistent.
  6. 6For direction questions, link each change to the put: higher σ, higher leverage or longer T raises the put, lowers debt value and widens the spread. Equity value rises with σ.
  7. 7For structural vs reduced-form questions, ask whether default is explained by asset value against debt (structural) or modelled as an exogenous hazard-rate event (reduced-form).

Quickest way: Assets = debt + equity, and the put is the gap

When to use it: Use when the vignette gives asset value and option values and asks for debt value, put value or spread.

  1. Write V = D + E. If you have the call, debt = V − call, in seconds.
  2. Find PV(K). The put is PV(K) − D, which is the same as call + PV(K) − V.
  3. Spread: divide K by D to get the gross yield, subtract 1, then subtract r.
  4. For qualitative questions, answer from the put: more risk means a bigger put, lower debt value, wider spread.

Common mistakes in Structural Models of Credit Risk

  • Calling equity a put option and debt a call

    Students mix up the payoff shapes and think of equity as protection against loss.

    Fix: Equity gets V − K only when assets exceed debt, so it is a call. Debt holders are short the put.

  • Using face value K instead of PV(K) in the debt formula

    Both K and PV(K) appear in the same setup, so the discounting step is forgotten.

    Fix: Risky debt today = PV(K) − put. Discount K at the risk-free rate over T first.

  • Treating N(−d2) as a real-world default probability

    It looks like a plain probability.

    Fix: It is a risk-neutral probability, computed with r as the drift. Say so in your answer.

  • Thinking higher asset volatility helps debt holders

    Students assume more volatility means more upside for everyone.

    Fix: Debt upside is capped at K, while downside remains. Volatility raises the put, lowers debt value and widens the spread. Only equity benefits.

  • Mismatching compounding in the spread calculation

    The risk-free rate is given as continuous in one place and annual in another.

    Fix: Compute y with the same convention as r before subtracting. Check the vignette wording.

  • Saying reduced-form models use asset value and capital structure

    Both families are called credit models, so the descriptions blur.

    Fix: Structural: asset value against debt. Reduced-form: an exogenous default process with a hazard rate, calibrated to market or observable data.

Worked examples

Example 1

A firm has asset value of 120 (USD millions). It has a single zero-coupon bond with face value 100 due in one year. The risk-free rate is 5% (annual compounding). A European call on the assets with strike 100 and one-year maturity is valued at 26.5. Questions: (1) What is the value of the risky debt? (2) What is the value of the put on the assets? (3) What is the approximate credit spread? Options for (3): A 1.0%, B 1.95%, C 6.95%.

Show the solution
  1. Debt = V − Call = 120 − 26.5 = 93.5.
  2. PV(K) = 100 ÷ 1.05 = 95.2381.
  3. Put = PV(K) − Debt = 95.2381 − 93.5 = 1.7381, about 1.74. Check with parity: 26.5 + 95.2381 − 120 = 1.7381.
  4. Yield on the debt: 100 ÷ 93.5 = 1.06952, so y = 6.95%.
  5. Spread = 6.95% − 5% = 1.95%.

Answer: (1) Risky debt = 93.5 million. (2) Put = about 1.74 million. (3) Credit spread is about 1.95%, which is option B. Option C is the yield, not the spread.

Example 2

An analyst uses a Merton model for a firm with asset value 200 and zero-coupon debt of face value 150 due in three years. The model gives N(d2) = 0.82. Questions: (1) What is the risk-neutral probability of default? (2) If asset volatility rises and everything else is unchanged, what happens to the equity value and the credit spread? (3) Which feature of the basic model leads to understated short-term spreads for a healthy firm? Options for (1): A 18%, B 82%, C 64%.

Show the solution
  1. Default occurs if V(T) < K. The risk-neutral probability is N(−d2) = 1 − N(d2) = 1 − 0.82 = 0.18.
  2. Higher volatility raises the value of both the call and the put on assets.
  3. Equity is the call, so its value rises. Debt = PV(K) − put, so debt value falls and the yield rises, widening the spread.
  4. In the basic model, default can occur only at maturity, and asset value moves continuously. For a healthy firm with a short time left, the put is nearly worthless, so the model implies very low spreads. Surprise defaults are not captured, unlike in reduced-form models.

Answer: (1) 18%, option A. Option B is the probability of no default. (2) Equity value rises and the credit spread widens. (3) Default only at maturity with continuous asset paths, so short-term default risk is understated.

Exam tips

  • Write V = D + E and the payoffs at the top of your rough work before reading the questions. Most answers follow from them.
  • When a vignette gives the call value, use debt = V − call. Do not recompute the model.
  • For direction questions, go through the put: volatility, leverage and maturity up means put up, debt down, spread up. Equity moves the other way on volatility.
  • Read whether d2 or N(d2) is given. N(d2) is the no-default probability. N(−d2) is the default probability.
  • On structural vs reduced-form comparisons, name what each model uses: assets and capital structure, or a hazard rate. Then name one strength or weakness.

Structural Models of Credit Risk in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Structural Models of Credit Risk: frequently asked questions

Why is equity a call option on firm assets?

Equity holders get what is left after debt is repaid, so their payoff is max(V − K, 0). That is the payoff of a call with strike K. Limited liability stops equity going below zero.

How is risky debt valued as a put?

Risky debt equals risk-free debt minus a put on firm assets with strike equal to the face value. The put is the value of the credit risk the debt holders bear. You can also get the same debt value as asset value minus the call.

What is the difference between structural and reduced-form credit models?

Structural models link default to firm assets falling below debt, so they use asset value, volatility and capital structure. Reduced-form models treat default as an unexpected event driven by a hazard rate and are calibrated to observable data. Reduced-form models handle surprise defaults better.

What does N(−d2) mean in the Merton model?

It is the risk-neutral probability that asset value ends below the debt's face value at maturity, which means default. It is not a real-world probability. Its complement, N(d2), is the probability of no default.