Economic Modelling · Simple models for credit risk

Merton Model for Credit Risk Explained

Q: Why is equity a call option in the Merton model?

Shareholders get the assets left after debt is repaid, but limited liability means they never get less than zero. That payoff is max(V_T − D, 0), the payoff of a European call on the assets with strike D.

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

A structural model ties default to the firm's asset value falling below its debt. A reduced form model treats default as an unexpected event that happens at a random rate, the hazard rate, and does not model the firm's balance sheet. Reduced form models are easier to calibrate to market prices.

Q: What is the default probability in the Merton model?

Under the risk-neutral measure it is N(−d2). Under the real-world measure it is the same expression with the asset drift μ used in place of r. The risk-neutral value is larger than the real-world value when μ > r.

Updated 11 October 2026 · Fact-checked

The Merton model treats a firm's equity as a European call option on its assets, with strike equal to the face value of zero-coupon debt. Debt is assets minus that call. You use Black-Scholes to value equity, get default probability N(−d2), and find the yield and credit spread on the debt.

Understand Structural Models of Credit Risk (Merton Model)

A structural model explains default using the firm's balance sheet. The firm has assets worth V. It owes lenders a fixed amount D at time T. Default happens if assets are worth less than D at T.

Merton's simple set-up: the firm has one zero-coupon bond with face value D maturing at T. Asset value V follows geometric Brownian motion with volatility σ. The risk-free rate r is constant. There is no default before T. Default can only occur at T.

At T, shareholders receive what is left after debt is paid, but never less than zero. So equity at T = max(V_T − D, 0). This is exactly the payoff of a European call on the assets with strike D. Lenders receive min(V_T, D). This equals D minus a put payoff max(D − V_T, 0). So lenders are long a risk-free bond and short a put on the assets. Shareholders hold limited liability, which is the put they effectively own.

Because equity is a call, Black-Scholes gives its value. The probability that V_T < D gives the default probability. Under the risk-neutral measure this is N(−d2). Under the real-world measure you replace r with the real asset drift μ. The value of debt is V_0 minus equity. From the debt value you find the yield, and the credit spread is that yield minus r.

Compare with reduced form models. Those do not link default to asset value. They treat default as a random event with a hazard rate, which is calibrated to market prices. Structural models give economic intuition: higher leverage or higher asset volatility means a larger spread. Their weaknesses are the single debt maturity, default only at T, and spreads that are too low for short maturities.

Key rules to remember

Equity payoff at T: E_T = max(V_T − D, 0); Equity is a call on assets with strike D.
Debt payoff at T: B_T = min(V_T, D) = D − max(D − V_T, 0); Lenders hold a risk-free bond and are short a put on the assets.
Merton equity value: E_0 = V_0 N(d1) − D e^(−rT) N(d2); Black-Scholes call with S = V_0, K = D. No dividends on assets.
d1 and d2: d1 = [ln(V_0 ÷ D) + (r + σ²/2)T] ÷ (σ√T); d2 = d1 − σ√T; σ is the volatility of asset value, not of equity.
Debt value: B_0 = V_0 − E_0 = D e^(−rT) N(d2) + V_0 N(−d1); Also B_0 = D e^(−rT) − put value.
Risk-neutral default probability: P(V_T < D) = N(−d2); For the real-world probability, use μ in place of r in d2.
Yield and credit spread: B_0 = D e^(−yT), so y = −(1 ÷ T) ln(B_0 ÷ D); spread = y − r; Continuous compounding. The spread is always positive when default is possible.

How to solve Structural Models of Credit Risk (Merton Model) questions

Use this order for any Merton-type question. Keep the asset value V and the strike D clearly separate from equity and debt values.

1Write down the inputs: V_0, D (face value of debt), T, r, σ. Check that σ is the asset volatility and that rates are continuously compounded.
2Compute d1 = [ln(V_0 ÷ D) + (r + σ²/2)T] ÷ (σ√T) and d2 = d1 − σ√T.
3Look up N(d1), N(d2) from the tables. Use N(−x) = 1 − N(x) for negative values.
4Value equity: E_0 = V_0 N(d1) − D e^(−rT) N(d2).
5Value debt: B_0 = V_0 − E_0. Check that B_0 is less than D e^(−rT).
6Find default probability: N(−d2) under risk-neutral measure. If asked for the real-world value, put μ in place of r in the d2 formula.
7Find yield y = −(1 ÷ T) ln(B_0 ÷ D) and spread = y − r. State your assumptions and comment on the result.
8 If asked to compare or explain, link effects to leverage (D ÷ V_0), volatility and maturity.

Quickest way: Shortcut: debt = risk-free bond minus put

When to use it: Use when the question gives you equity or put value, or asks only for debt value, yield or spread.

If equity E_0 is given, debt is B_0 = V_0 − E_0 with no need for d1 and d2.
If the put value P_0 is given, debt is B_0 = D e^(−rT) − P_0.
Spread = −(1 ÷ T) ln(B_0 ÷ D) − r. Equivalently, spread = −(1 ÷ T) ln(B_0 ÷ (D e^(−rT))).
For default probability, you only need d2. Compute it and take N(−d2), without calculating d1 separately unless the question needs equity.

Common mistakes in Structural Models of Credit Risk (Merton Model)

Using N(d1) or N(−d1) as the default probability.
d1 and d2 look alike and both appear in the equity formula.
Fix: Default probability is N(−d2). N(d1) is the delta of the equity call, not a probability of default.
Using equity volatility instead of asset volatility in d1 and d2.
Equity volatility is what you can observe in the market, so it feels natural to use.
Fix: The model assumes assets follow GBM. σ is asset volatility. Equity volatility is higher than asset volatility because equity is a leveraged claim.
Mixing real-world and risk-neutral measures.
Both a default probability and a price are asked for in one question.
Fix: Use r for pricing equity and debt and for the risk-neutral default probability. Use μ only when the question asks for the real-world probability of default.
Treating D as the present value of debt, or forgetting to discount D.
D is the amount owed at T, which is easy to confuse with today's value.
Fix: D is the face value due at T. Its present value at the risk-free rate is D e^(−rT). Risky debt is worth less than that.
Computing the spread as ln(D ÷ B_0) without dividing by T, or without subtracting r.
The yield formula is remembered halfway.
Fix: y = (1 ÷ T) ln(D ÷ B_0). Spread = y − r. Check that the answer is a small positive number.
Saying structural and reduced form models are the same, or that Merton allows default at any time.
Both give default probabilities and spreads.
Fix: Merton links default to asset value and allows default only at T. Reduced form models use a hazard rate and do not model the firm's assets.

Worked examples

Example 1

A firm has asset value V_0 = ₹100 crore with asset volatility σ = 20% a year. It has one zero-coupon bond of face value D = ₹80 crore due in T = 1 year. The risk-free rate is 5% a year continuously compounded. Using the Merton model, find (a) the equity value, (b) the risk-neutral default probability, (c) the debt value, and (d) the yield and credit spread.

Show the solution

ln(100 ÷ 80) = ln 1.25 = 0.22314. r + σ²/2 = 0.05 + 0.02 = 0.07.
d1 = (0.22314 + 0.07 × 1) ÷ (0.2 × 1) = 0.29314 ÷ 0.2 = 1.4657. d2 = 1.4657 − 0.2 = 1.2657.
From tables, N(d1) ≈ 0.9286 and N(d2) ≈ 0.8972.
D e^(−rT) = 80 × e^(−0.05) = 80 × 0.95123 = 76.098.
(a) E_0 = 100 × 0.9286 − 76.098 × 0.8972 = 92.86 − 68.28 = about ₹24.59 crore.
(b) Default probability = N(−d2) = 1 − 0.8972 = about 0.103, or 10.3%.
(c) B_0 = V_0 − E_0 = 100 − 24.59 = about ₹75.41 crore. This is below 76.098, as it should be.
(d) y = ln(80 ÷ 75.41) ÷ 1 = ln(1.0609) = about 0.0591, so 5.91%. Spread = 5.91% − 5% = about 0.91%, or roughly 91 basis points.

Answer: Equity ≈ ₹24.59 crore; risk-neutral default probability ≈ 10.3%; debt ≈ ₹75.41 crore; yield ≈ 5.91%; spread ≈ 0.91% (small differences arise from table rounding).

Example 2

A company has assets worth ₹200 crore and zero-coupon debt with face value ₹150 crore due in 2 years. The risk-free rate is 6% a year continuously compounded. In the Merton model the equity is valued at ₹70 crore. Find the value of the debt, the value of the implied put option on the assets, and the credit spread.

Show the solution

Debt = assets − equity = 200 − 70 = ₹130 crore.
Risk-free value of the debt: 150 × e^(−0.06 × 2) = 150 × e^(−0.12) = 150 × 0.88692 = ₹133.04 crore.
Put value = risk-free debt value − risky debt value = 133.04 − 130 = about ₹3.04 crore.
Yield: y = (1 ÷ 2) ln(150 ÷ 130) = 0.5 × ln(1.15385) = 0.5 × 0.14310 = 0.07155, which is 7.155%.
Spread = 7.155% − 6% = 1.155%.

Answer: Debt = ₹130 crore; implied put = about ₹3.04 crore; credit spread = about 1.155% a year.

Exam tips

Write 'equity = call on assets, strike D' at the start. Examiners give marks for setting up the option analogy, even if the arithmetic goes wrong.
Show d1, d2, N(d1) and N(d2) clearly. Most marks are for the method, so state the formula in standard notation before substituting.
If the question gives equity or a put price, use the shortcut B_0 = V_0 − E_0. Do not recompute Black-Scholes.
For discussion parts, link a higher D ÷ V_0, higher σ, or longer T to a higher default probability and wider spread, and state the model's limits: default only at T, one debt issue, constant r and σ.
In MCQs, check whether the question asks for a risk-neutral or real-world probability, and whether it asks for yield or spread.

Practice questions from Simple models for credit risk

Structural Models of Credit Risk (Merton Model) 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 (Merton Model): frequently asked questions

Why is equity a call option in the Merton model?

Shareholders get the assets left after debt is repaid, but limited liability means they never get less than zero. That payoff is max(V_T − D, 0), the payoff of a European call on the assets with strike D.

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

A structural model ties default to the firm's asset value falling below its debt. A reduced form model treats default as an unexpected event that happens at a random rate, the hazard rate, and does not model the firm's balance sheet. Reduced form models are easier to calibrate to market prices.

What is the default probability in the Merton model?

Under the risk-neutral measure it is N(−d2). Under the real-world measure it is the same expression with the asset drift μ used in place of r. The risk-neutral value is larger than the real-world value when μ > r.

How do I get the credit spread from the Merton model?

Find debt value B_0 = V_0 − E_0. Then the yield is y = −(1 ÷ T) ln(B_0 ÷ D), and the spread is y − r. Higher leverage and higher asset volatility both widen the spread.