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The Merton model: company equity as an option, and where a credit spread comes from

A shareholder with limited liability holds something that pays off only if the company is worth more than its debts, which is exactly a call option. Taking that seriously gives a price for the equity, a price for the debt, a default probability and a credit spread, all from one formula.

CM2 · Financial Engineering & Loss Reserving · 12 min read

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Value a company equity and debt with the Merton model, find the risk-neutral default probability and the credit spread, and state the limitations

The observation the model is built on

A company is worth V. It owes D, repayable at time T. At that date one of two things happens:

  • if VT > D, the debt is repaid and the shareholders keep the rest, VT − D;
  • if VT < D, the company defaults, the lenders take what there is, and the shareholders get nothing — not a negative amount, because their liability is limited.

So the shareholders' payoff is max(VT − D, 0). That is a European call option, written on the company's own assets, with the face value of the debt as its strike. The insight is that limited liability is an option, and it has been there in the company's legal structure all along.

Once that is seen, everything follows from the option pricing already in the syllabus. Nothing new has to be derived: the whole model is Black-Scholes with the labels changed.

The three valuations

Treat the asset value as lognormal with volatility σV, and let r be the continuously compounded risk-free rate. Then:

E0 = V0Φ(d1) − De−rTΦ(d2)

d1 = [ln(V0/D) + (r + ½σV2)T] / (σV√T)
d2 = d1 − σV√T

In words: the equity is worth what a call option on the firm's assets is worth, struck at the nominal value of its debt.

The debt then follows two ways, and the fact that they agree is the most useful check in the topic.

From the balance sheet. Debt plus equity is the whole company, so B0 = V0 − E0. Nothing is assumed here beyond the accounting.

From the lenders' position. Lending to a company with limited liability is the same as lending risk-free and simultaneously writing a put on the assets, struck at D — because if the assets fall short, the lender absorbs the shortfall:

B0 = De−rT − put(V0, D, T)

Those two expressions are identical, and the identity is put-call parity written in corporate-finance language. Computing both and comparing them catches any arithmetic error in the normal distribution values, and the worked example below shows them agreeing to nine decimal places.

The second form is also the more informative one, because it names the thing the lender is short of. The value of the put is the value of the default option, and it is the whole of what makes risky debt cheaper than safe debt.

The default probability

risk-neutral P(default) = P(VT < D) = Φ(−d2)

In words: the chance the assets end below the debt, which is the chance the option finishes out of the money.

The word risk-neutral has to stay in that sentence. This is not the probability of default a credit analyst would forecast, because it is computed under the measure in which every asset earns the risk-free rate rather than its real expected return. A company's assets are expected to grow faster than the risk-free rate, so the real-world default probability is lower — usually substantially. The risk-neutral figure is the right one for pricing and the wrong one for a capital model that is supposed to describe what will happen.

The credit spread

The lender pays B0 today and is promised D at time T. The continuously compounded return on that, if everything is repaid, is the debt's yield:

yield = −(1/T) ln(B0/D)
credit spread = yield − r

In words: work out the rate that turns what the lender paid into what it is promised, and subtract the risk-free rate. What is left is the compensation for the default risk.

Four things move the spread, and all four come straight out of the formula. It widens with more leverage (D closer to V), with higher asset volatility, and — less obviously — it can widen or narrow with term, because a longer term gives more time to default and also more time for the assets to grow. It does not depend on the assets' expected return, for exactly the same reason an option price does not.

What the model says about conflicts of interest

This is the part of the model that earns its place in a professional syllabus, and it is a one-line consequence of the option view.

The shareholders hold a call option. An option's value rises with the volatility of the thing it is written on. So shareholders gain from increasing the riskiness of the company's assets, even when that does nothing whatever to the total value of the firm. Since the total is fixed, whatever they gain the lenders lose.

The worked example puts a number on it: raising asset volatility from 30% to 40%, with the asset value unchanged, moves about 16 crore from the lenders to the shareholders. Nothing was created; the claim on the existing assets was redistributed.

That is the mechanism behind a great deal of real financial practice. It is why loan covenants restrict what a borrower may invest in, why regulators cap the risk a bank may take with depositors' money, and why the incentive to gamble rises as a company approaches distress — when the option is nearly worthless, volatility is the only thing that can revive it. Being able to state that chain is usually worth more marks than the arithmetic.

The limitations, which are serious

  • Neither V0 nor σV is observable. A company's traded equity is observable; the market value and volatility of its underlying assets are not. In practice they are inferred from the equity price and equity volatility by solving the model backwards, which means the model is used to produce its own inputs.
  • Real debt is not one zero-coupon bond. Companies have bank loans, bonds of several maturities, overdrafts and leases, with different seniorities. Collapsing all of it to a single D and a single T is a large simplification.
  • Default is assumed to happen only at T. In reality a company can fail at any time, including while technically solvent, because it runs out of cash. Models that allow default the first time the asset value touches a barrier are the standard repair.
  • Recovery is assumed to be the whole of the remaining assets. Bankruptcy has costs, and the lenders get less than VT.
  • Short-dated spreads come out too small. This is the best-known empirical failure. A lognormal process can hardly fall far in a month, so the model says a healthy company's one-month debt is almost risk-free, and observed spreads are wider than that. Jumps in the asset value are what the model is missing.

The fair summary is that the Merton model is a structural explanation rather than a pricing engine: it says why a spread exists and what moves it, and it is routinely out by a factor on the level.

Questions students ask

If the company is worth more than its debt, why is default possible at all?

Because what matters is the asset value at time T, not today. The example below has 500 crore of assets against 300 crore of debt — comfortable cover — and still a 19% risk-neutral chance of default, because five years at 30% volatility is plenty of time for 500 to become less than 300. Cover today and cover at maturity are different questions, and the gap between them is what the volatility and the term are measuring.

Where does the asset volatility come from?

From the equity, by inverting the model. Because equity is a call on the assets, its volatility is related to the asset volatility through the option's delta — roughly, equity volatility is asset volatility multiplied by the delta and by the ratio of asset value to equity value. Observing the equity price and its volatility gives two equations in the two unknowns V0 and σV, which are solved numerically. This is how the model is used commercially.

Is this how rating agencies work?

Not directly. Agency ratings are largely judgemental, built from financial ratios, industry assessment and management review. Structural models of this kind are used alongside them, and the fact that a model-implied default probability and a rating sometimes diverge sharply is itself treated as a signal worth investigating.

How does this relate to the two-state credit models in the syllabus?

They answer the same question from the other end. A structural model such as this one explains default as a consequence of the asset value falling, and derives the default probability. A reduced-form or intensity model takes the default rate as the primitive and calibrates it to observed spreads, without asking why. Structural models are better at explaining, reduced-form models are better at fitting, and that trade-off is exactly what a comparison question is after.

A worked example

A company's total assets are worth ₹500 crore. It has a single issue of zero-coupon debt with a nominal value of ₹300 crore repayable in 5 years. The volatility of the asset value is 30% a year and the continuously compounded risk-free rate is 6%.

The company is invented for this example. It is not any real business.

  1. d1 and d2

    ln(500/300) = ln 1.6666667 = 0.5108256
    (r + ½σ2)T = (0.06 + 0.045) × 5 = 0.5250000
    σ√T = 0.30 × 2.2360680 = 0.6708204
    d1 = (0.5108256 + 0.5250000) / 0.6708204 = 1.5441177
    d2 = 1.5441177 − 0.6708204 = 0.8732973
    Φ(d1) = 0.9387201,   Φ(d2) = 0.8087495
    Φ(−d1) = 0.0612799,   Φ(−d2) = 0.1912505

  2. The equity

    De−rT = 300 × e−0.30 = 300 × 0.7408182 = 222.245466
    E0 = 500 × 0.9387201 − 222.245466 × 0.8087495
       = 469.360036 − 179.740912 = ₹289.62 crore

  3. The debt, two ways

    From the balance sheet:

    B0 = 500 − 289.619123 = 210.380877

    From the lenders' position — risk-free debt less the default put:

    put = 222.245466 × 0.1912505 − 500 × 0.0612799
        = 42.514566 − 30.649977 = 11.864589
    B0 = 222.245466 − 11.864589 = 210.380877

    The two routes agree to nine decimal places. They share only the four normal values, so the agreement confirms both the arithmetic and that the parity relationship has been applied the right way round.

    And the default option is now priced: ₹11.86 crore. That is what the lenders have given away by lending to a company with limited liability rather than to the government, and it is 5.34% of the ₹222.25 crore the debt would have been worth if it were safe.

  4. The default probability

    risk-neutral P(default) = Φ(−d2) = 0.1913

    About a 19% chance, on assets that today cover the debt one and two-thirds times over. The figure is driven by the five-year term and the 30% volatility, not by the current cover. Remember the qualifier: this is under the risk-neutral measure. If the assets are really expected to grow at, say, 10% a year rather than the risk-free 6%, the real-world probability is appreciably lower.

  5. The credit spread

    B0/D = 210.380877 / 300 = 0.7012696
    yield = −(1/5) ln 0.7012696 = −0.2 × (−0.3548628) = 0.0709726
    spread = 0.0709726 − 0.06 = 0.0109726, that is 110 basis points

    So the market is pricing this debt at a yield of 7.10% against a risk-free 6.00%. Note how modest the spread is relative to the 19% default probability — because default does not mean total loss. The lenders still receive the assets, which is why the expected loss, 5.34% of the risk-free value, is far below the default probability.

  6. Who gains if the company takes more risk?

    Now raise the asset volatility from 30% to 40%, leaving the asset value, the debt and the term exactly as they were. Nothing about the company's worth has changed; only the spread of possible outcomes has.

    d1 = (0.5108256 + (0.06 + 0.08) × 5) / (0.40 × 2.2360680) = 1.2108256 / 0.8944272 = 1.3537442
    d2 = 1.3537442 − 0.8944272 = 0.4593170
    σV = 30%σV = 40%Change
    Assets500.00500.000.00
    Equity289.62305.59+15.97
    Debt210.38194.41−15.97
    Risk-neutral P(default)0.19130.3230+0.1318
    Credit spread110 bp268 bp+158 bp

    The shareholders are ₹15.97 crore better off and the lenders are ₹15.97 crore worse off, to the paisa, because the two must still add to 500. The default probability has risen from 19% to 32% and the spread has more than doubled.

    This is the result to be able to explain in words, because it is what the model is for. The shareholders hold an option, an option is worth more when the underlying is more volatile, and the company's total value is fixed — so the gain can only have come out of the lenders' pocket. Existing lenders cannot reprice; they simply hold something worth less than it was yesterday. That is why debt contracts restrict what the borrower may do, and why the incentive to take the gamble is strongest when the company is already in trouble and the option is close to worthless.

  7. A sanity check on the whole example

    Equity 289.62 plus debt 210.38 is 500.00, the asset value. The debt's market value, 210.38, lies below both its nominal 300 and the 222.25 it would be worth if risk-free, as it must. The default put, 11.86, is positive and small relative to the equity. And the spread, 110 basis points, is in the range real markets charge a moderately leveraged company. If any one of those had failed, the arithmetic would need rechecking before a word of commentary was written.

Where this fits in CM2

This page explains one idea from CM2 Financial Engineering & Loss Reserving. The full CM2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

Other CM2 topics explained

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