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In 1974, Robert Merton proposed a model for assessing the structural credit risk of a company by modeling the company's equity as a call option on its assets. This method allows for the use of the Black-Scholes-Merton option pricing methods. The Merton model is structural because it provides a relationship between the default risk and the asset (capital) structure of the firm.

From an accounting standpoint, the book value of a firm's equity, *E*,
total assets, *A*, and total liabilities, *L*, are
defined by the equation:

These book values for are *E*, *A*,
and *L* are all observable because they are recorded
on a balance sheet. However, the book values are reported infrequently.
On the other hand, only the equity’s market value is observable,
given by the firm’s stock market price times the number of
outstanding shares. The market value of the firm’s assets and
total liabilities are unobservable.

Merton’s model relates the market value of assets, equity,
and liabilities in an option pricing framework. The Merton model assumes
a single liability *L* with maturity *T*,
usually within one year. At time *T*, the firm’s
value to the shareholders equals the difference *A* – *L* when
the assets value *A* is greater than the liabilities *L*.
However, if the debt *L* exceeds the asset value *A*,
then the shareholders get nothing. The value of the equity *E*_{T} at
time *T* is related to the value of the assets and
liabilities by the following formula:

In practice, firms have multiple maturities for their liabilities,
so for a selected maturity *T*, a liability threshold *L* is
chosen based on the whole liability structure of the firm. The liability
threshold is also referred to as the default point. For a typical
time horizon of one year, the liability threshold is commonly set
somewhere between the value of the short-term liabilities and the
value of the total liabilities.

Assuming a lognormal distribution for the asset returns, you
can use the Black-Scholes-Merton equations to relate the observable
market value of equity *E*, and the unobservable
market value of assets *A* at any time prior to the
maturity *T*:

where *r* is the risk-free interest rate, *N* is
the cumulative standard normal distribution, and *d*_{1} and *d*_{2} are
given by

$${d}_{1}=\frac{\mathrm{ln}\left(\frac{A}{L}\right)+(r+0.5{\sigma}_{A}^{2})T}{{\sigma}_{A}\sqrt{T}}$$

$${d}_{2}={d}_{1}-{\sigma}_{A}\sqrt{T}$$

You can solve equation using two approaches:

The

`mertonmodel`

approach, a single point calibration that requires equity, liability, and equity volatility (σ_{E}).This approach solves for (

*A*,σ_{A}) using a 2-by-2 system of nonlinear equations. The first equation is the preceding option pricing formula. The second equation relates the unobservable volatility of assets σ_{A}to the given equity volatility σ_{E}through the equation$${\sigma}_{E}=\frac{A}{E}N({d}_{1}){\sigma}_{A}$$

The

`mertonByTimeSeries`

approach, which requires a time series for the equity and for all other model parameters.If the equity time series has

*n*data points, this approach calibrates a time series of*n*asset values*A*_{1},…,*A*_{n}that solve the following system of equations:$$\begin{array}{l}{E}_{1}={A}_{1}N({d}_{1})-{L}_{1}{e}^{-{r}_{1}{T}_{1}}N({d}_{2})\\ \mathrm{...}\\ {E}_{n}={A}_{n}N({d}_{1})-{L}_{n}{e}^{-{r}_{n}{T}_{n}}N({d}_{2})\end{array}$$

The volatility of assets σ

_{A}is directly computed from the time series*A*_{1},…,*A*_{n}as the annualized standard deviation of the log returns. This value is a single volatility value that captures the volatility of the assets during the time period spanned by the time series.Once the values of

*A*and σ_{A}are known, the*distance to default*(*DD*) is computed as the number of standard deviations between the expected asset value at maturity*T*and the liability threshold:$$DD=\frac{\mathrm{log}A+\left({\mu}_{A}-{\sigma}_{A}^{2}/2\right)T-\mathrm{log}(L)}{{\sigma}_{A}\sqrt{T}}$$

The

*drift*parameter μ_{A}is the expected return for the assets, which can be equal to the risk-free interest rate, or any other value based on expectations for that firm.The

*probability of default*(`PD`

) is defined as the probability that the asset value falls below the liability threshold at the end of the time horizon*T*:$$PD=1-N(DD)$$

`mertonByTimeSeries`

| `mertonmodel`

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