Suppose that the asset value of a firm evolves according to a lognormal diffusion, as in Merton (1974). The current value of the firm's assets is $100 million, and its volatility is 24.24%. Suppose too that the firm has only one issue of debt outstanding: zero-coupon debt with a maturity of three years, and a face value of $70 million. Finally, suppose that the risk-free rate of interest is 4% (continuously-compounded terms) for all maturities. What is the risk-neutral probability of the firm defaulting at maturity of the debt?
A) 17.12%
B) 19.23%
C) 20.02%
D) 21.22%

Question

Quizplus · Accepted Answer

Using Merton's model, the probability of default can be calculated as follows:
$$
p_{default}=N\left(-\frac{\ln \left(\frac{D}{V} ight)+(r+\frac{\sigma^2}{2})T}{\sigma\sqrt{T}}ight)
$$
where $N(\cdot)$ is the cumulative distribution function of the standard normal distribution, $D$ is the face value of debt, $V$ is the current value of assets, $r$ is the risk-free rate of interest, $\sigma$ is the volatility of assets, and $T$ is the time to maturity of debt.

Substituting the given values, we get:
$$
p_{default}=N\left(-\frac{\ln \left(\frac{70}{100} ight)+(0.04+\frac{0.2424^2}{2})	imes 3}{0.2424\sqrt{3}}ight)=0.1923=19.23\%
$$

Therefore, the risk-neutral probability of default at maturity of the debt is 19.23% (option B).

Suppose That the Asset Value of a Firm Evolves According