📚 Merton Model and Its Applications | 经济考点：默顿模型及其应用解析
In 1974, Robert C. Merton showed that a firm’s equity can be valued exactly like a European call option written on the firm’s assets. The same framework can be used to estimate the probability that the firm defaults before a chosen horizon. This single idea, now known as the Merton Model, forms the backbone of modern credit-risk analysis and is a frequent exam topic in economics and finance syllabuses.
1974年，罗伯特·C·默顿证明了公司股权可以被精确地看作公司资产上的欧式看涨期权。同样的框架还可用于估算公司在给定期限之前发生违约的概率。这一思想如今被称为默顿模型，是现代信用风险分析的基石，也是经济与金融课程中的高频考点。
1. What Is the Merton Model? | 什么是默顿模型？
The Merton Model is a structural credit-risk model: it links a firm’s capital structure to its probability of default. Instead of relying only on credit ratings or accounting ratios, the model uses market data, namely the total asset value V, the promised debt repayment D, asset volatility σ, the risk-free rate r, and the time horizon T.
默顿模型是一种结构化信用风险模型：它将公司的资本结构与违约概率联系起来。该模型不单纯依赖信用评级或会计比率，而是使用市场数据，即总资产价值V、约定债务偿还额D、资产波动率σ、无风险利率r和时间期限T。
The output of the model is a continuous measure of default risk, typically expressed as the probability that the firm’s asset value will fall below its debt obligations at maturity.
模型的输出是违约风险的连续测度，通常表示为公司资产价值在债务到期时低于其债务义务的概率。
2. The Core Idea: Equity as a Call Option | 核心思想：股权作为看涨期权
Consider a firm financed by equity and a single zero-coupon bond maturing at time T, with face value D. If the firm’s asset value V exceeds D, shareholders repay the debt and keep the surplus V − D. If V is less than D, shareholders will rationally choose to default and hand the firm to creditors.
考虑一家仅由股权和一份到期日为T、面值为D的零息债券融资的公司。如果公司资产价值V超过D，股东偿还债务并保留剩余部分V − D；如果V小于D，股东会理性地选择违约并将公司交给债权人。
The shareholders’ final payoff is therefore max(V − D, 0), which is exactly the payoff of a European call option on the firm’s assets, with strike price D and expiry T.
因此，股东的最终收益为 max(V − D, 0)，这恰好是以公司资产为标的、行权价为D、到期日为T的欧式看涨期权的收益结构。
Equity payoff = max(V − D, 0)
This insight allows us to apply Black-Scholes-Merton option pricing mathematics to estimate the value of equity, and from it the market’s view of default risk.
这一洞见使我们能够运用布莱克-舒尔斯-默顿期权定价数学来估算股权价值，并由此推断市场对违约风险的看法。
3. Key Assumptions | 关键假设
- English: The firm’s total asset value V follows a geometric Brownian motion with constant volatility σ.
- 中文：公司总资产价值V遵循几何布朗运动，波动率σ恒定。
- English: The firm has only one class of debt, a zero-coupon bond maturing at T.
- 中文：公司只拥有一类债务，即到期日为T的零息债券。
- English: The risk-free interest rate r is constant over the life of the debt.
- 中文：无风险利率r在债务存续期内保持不变。
- English: Markets are frictionless: no taxes, no transaction costs, and assets are perfectly divisible.
- 中文：市场无摩擦：无税收、无交易成本，资产可以完全分割。
- English: There are no dividends and no other cash payouts before maturity.
- 中文：在到期之前公司不支付股息，也没有其他现金流出。
These assumptions are deliberately simple. They make the mathematics tractable, but as we will see later, they also explain some of the model’s weaknesses.
这些假设被刻意简化，以便于数学处理。但正如后文所述，它们也解释了模型的部分弱点。
4. The Pricing Formula | 定价公式
Under the above assumptions, the market value of equity E is given by the Black-Scholes-Merton formula:
在上述假设下，股权的市场价值E由布莱克-舒尔斯-默顿公式给出：
E = V × N(d₁) − D × e^(−rT) × N(d₂)
d₁ = [ln(V/D) + (r + 0.5σ²)T] / (σ√T)
d₂ = d₁ − σ√T
Here, N(·) is the cumulative standard normal distribution function. The term N(d₂) has a crucial economic interpretation: it is the risk-neutral probability that the firm remains solvent at maturity, i.e. the probability that V exceeds D.
其中，N(·)是标准正态分布的累积分布函数。N(d₂)具有关键的经济含义：它是风险中性下公司在到期时仍然有偿付能力的概率，即V超过D的概率。
The expression D × e^(−rT) is the present value of the promised debt repayment, while V × N(d₁) captures the expected value of the firm’s assets if it remains solvent, weighted by probability.
D × e^(−rT)是约定债务偿还额的现值，而V × N(d₁)则捕捉了公司在有偿付能力情况下的资产预期价值按概率加权后的结果。
5. Distance to Default and Default Probability | 违约距离与违约概率
In practice, examiners rarely ask for the full option price. Instead, they focus on the probability of default. The key quantity is the distance to default (DD):
在实际考试中，考官很少要求完整的期权定价计算，而是重点考察违约概率。关键指标是违约距离（DD）：
DD = [ln(V/D) + (μ − 0.5σ²)T] / (σ√T)
In this version, μ is the expected annual return on the firm’s assets, and ln(V/D) measures how far asset value currently is above the default point in log terms. A larger DD means the firm is further from default.
在该式中，μ是公司资产的预期年收益率，ln(V/D)以对数形式衡量资产价值当前高于违约点的程度。DD越大，意味着公司离违约越远。
The default probability is then approximated by:
违约概率则近似为：
P_default = N(−DD)
Because N(·) is increasing, a higher DD gives a lower default probability. This intuitive inverse relationship is the most frequently tested idea in exam questions.
由于N(·)是单调递增函数，DD越高，违约概率越低。这一直观的反向关系是考试题中最常考察的思想。
6. Worked Example | 算例演示
Consider a firm with total asset value V = 100 (million), debt face value D = 80, asset volatility σ = 0.25, risk-free rate r = 0.05, expected asset return μ = 0.08, and horizon T = 1 year.
考虑一家公司：总资产价值V = 100（百万元），债务面值D = 80，资产波动率σ = 0.25，无风险利率r = 0.05，资产预期收益率μ = 0.08，期限T = 1年。
Step 1: Compute d₁ and d₂ | 第一步：计算d₁和d₂
d₁ = [ln(100/80) + (0.05 + 0.5 × 0.25²)] / (0.25 × √1) = [0.2231 + 0.0813] / 0.25 = 1.2176
d₂ = 1.2176 − 0.25 = 0.9676
Step 2: Compute distance to default | 第二步：计算违约距离
DD = [ln(100/80) + (0.08 − 0.5 × 0.25²)] / 0.25 = [0.2231 + 0.0488] / 0.25 = 1.0876
Step 3: Default probability | 第三步：违约概率
P_default = N(−1.0876) ≈ 0.138, or 13.8%
| Quantity | 变量 | Value | 数值 |
| d₁ | 1.2176 |
| d₂ | 0.9676 |
| DD | 1.0876 |
| Default probability | 违约概率 | 13.8% |
A probability of about 14% is high for an investment-grade firm, which tells us that this firm is under significant financial stress.
约14%的违约概率对投资级公司而言偏高，这告诉我们该公司正承受显著的财务压力。
7. Applications in Banking and Finance | 在银行与金融中的应用
- English: Corporate credit assessment. Commercial banks use the Merton Model to set internal credit limits and price corporate loans.
- 中文：企业信用评估。商业银行使用默顿模型设定内部信贷额度，并为公司贷款定价。
- English: Bond pricing. The probability of default feeds directly into the pricing of corporate bonds and credit default swaps.
- 中文：债券定价。违约概率直接进入公司债券与信用违约互换的定价过程。
- English: Bank regulation. Under the Basel framework, banks estimate probability of default (PD) to calculate capital requirements for credit risk.
- 中文：银行监管。在巴塞尔框架下，银行估算违约概率（PD）以计算信用风险资本要求。
- English: Moody’s KMV. The commercialised KMV model, derived from Merton’s work, is widely used to produce expected default frequencies (EDF) for listed firms.
- 中文：穆迪KMV。由默顿研究商业化而来的KMV模型被广泛用于生成上市公司的预期违约频率（EDF）。
The common thread is that the model turns a balance-sheet snapshot into a forward-looking, market-based measure of risk.
这些应用的共同点在于：模型将资产负债表的静态快照转化为具有前瞻性的、基于市场的风险度量。
8. Advantages of the Merton Model | 默顿模型的优势
First, it is forward-looking because it uses the market value of equity, which reflects investors’ expectations about future cash flows, rather than backward-looking accounting data.
第一，模型具有前瞻性，因为它使用股票市场价值，而股价反映了投资者对未来现金流的预期，而非滞后的会计数据。
Second, it is theoretically rigorous. By connecting equity value, firm value, and debt, the model provides a single internally consistent framework for valuation and risk measurement.
第二，模型在理论上严谨。通过连接股权价值、公司价值与债务，它为估值和风险度量提供了内在一致的分析框架。
Third, it gives a continuous default probability, which allows fine distinctions between firms that rating agencies might place in the same category.
第三，模型给出连续的违约概率，能够区分那些在评级机构眼中可能属于同一类别的公司。
9. Limitations and Criticisms | 局限性与批评
- English: The assumption of a single zero-coupon bond is unrealistic for firms with complex capital structures and multiple debt maturities.
- 中文：单一零息债券的假设对拥有复杂资本结构、多重债务期限的公司而言并不现实。
- English: Asset value and volatility are not directly observable; they must be estimated, which introduces measurement error.
- 中文：资产价值与波动率无法直接观测，需要估计，从而引入度量误差。
- English: The normal distribution understates the probability of extreme market moves (fat tails), especially during systemic crises.
- 中文：正态分布低估了极端市场波动的概率（厚尾问题），在系统性危机期间尤为明显。
- English: The model treats default as occurring only at maturity, but in reality firms can default at any time and covenants can trigger early default.
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