## Default Probability by Using the Merton Model for Structural Credit Risk

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. The Merton model uses the Black-Scholes-Merton option pricing methods and is structural because it provides a relationship between the default risk and the asset (capital) structure of the firm.

A company balance sheet records book values—the value of a firm's equity E, its total assets A, and its total liabilities L. The relationship between these values is defined by the equation

`$A=E+L$`

These book values for E, A, and L are all observable because they are recorded on a firm's balance sheet. However, the book values are reported infrequently. Alternatively, only the equity’s market value is observable, and is 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.

The Merton model relates the market values of equity, assets, and liabilities in an option pricing framework. The Merton model assumes a single liability L with maturity T, usually a period of one year or less. At time T, the firm’s value to the shareholders equals the difference AL when the asset value A is greater than the liabilities L. However, if the liabilities L exceed the asset value A, then the shareholders get nothing. The value of the equity ET at time T is related to the value of the assets and liabilities by the following formula:

`${E}_{T}=\mathrm{max}\left({A}_{T}-L,0\right)$`

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 to a value 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:

`$E=AN\left({d}_{1}\right)-L{e}^{-rT}N\left({d}_{2}\right)$`

In this equation, r is the risk-free interest rate, N is the cumulative standard normal distribution, and d1 and d2 are given by

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

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

You can solve this equation using one of two approaches:

• The `mertonmodel` approach uses single-point calibration and requires values for the equity, liability, and equity volatility (σE).

This approach solves for (AA) using a 2-by-2 system of nonlinear equations. The first equation is the aforementioned option pricing formula. The second equation relates the unobservable volatility of assets σA to the given equity volatility σE:

`${\sigma }_{E}=\frac{A}{E}N\left({d}_{1}\right){\sigma }_{A}$`

• The `mertonByTimeSeries` approach requires 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 A1,…,An that solve the following system of equations:

`$\begin{array}{l}{E}_{1}={A}_{1}N\left({d}_{1}\right)-{L}_{1}{e}^{-{r}_{1}{T}_{1}}N\left({d}_{2}\right)\\ ...\\ {E}_{n}={A}_{n}N\left({d}_{1}\right)-{L}_{n}{e}^{-{r}_{n}{T}_{n}}N\left({d}_{2}\right)\end{array}$`

The function directly computes the volatility of assets σA from the time series A1,…,An 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.

After computing the values of A and σA, the function computes 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:

`$DD=\frac{\mathrm{log}A+\left({\mu }_{A}-{\sigma }_{A}^{2}/2\right)T-\mathrm{log}\left(L\right)}{{\sigma }_{A}\sqrt{T}}$`

The drift parameter μA 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 of the asset value falling below the liability threshold at the end of the time horizon T:

`$PD=1-N\left(DD\right)$`