The correct answer to the question is B, which estimates the volatility of the firm's equity at 8%. This is derived using the Merton model, a method for estimating the value of equity and the volatility of a firm's assets. The model is based on the Black-Scholes model, which is used for pricing options. In this case, the model is applied to estimate the volatility of the firm's equity shares.

The Merton model uses the following equation to estimate the value of equity (E):

$E = V \cdot N(d1) - D \cdot N(d2)$

Where:

• $V$ is the firm value.
• $D$ is the debt value.
• $N(d)$ is the cumulative distribution function of the standard normal distribution.
• $d1$ and $d2$ are calculated using the parameters of the firm's value, debt, volatility, and time to maturity.

Given the values from the question:

• Firm value ($V$) = USD 45 million
• Debt value ($D$) = USD 60 million
• $d1$ = 3.217790
• $d2$ = 3.038905

The time to maturity of the debt is 5 years, which is represented as $T - t$ in the model. The volatility ($\alpha$) of the firm value is what we are trying to estimate.

The equations provided in the file are: $d1 = \frac{\ln(V/D) + (r + \sigma^2 / 2)(T - t)}{\sigma \sqrt{T - t}}$ $d2 = d1 - \sigma \sqrt{T - t}$

Where $\sigma$ is the volatility of the firm's value, $r$ is the risk-free rate (not provided in the question), and $T - t$ is the time to debt maturity.

To find the volatility, we only need to use Equation (3) from the file content:

$d2 = (d1 - \sigma \sqrt{T - t})$

Plugging in the values for $d1$ and $d2$, we can solve for $\sigma$:

$3.038905 = 3.217790 - \sigma \sqrt{5}$

Isolating $\sigma$ gives us:

$\sigma = \frac{3.217790 - 3.038905}{\sqrt{5}}$ $\sigma \approx 0.08 \text{ or } 8\%$

This calculation shows that the volatility of the firm's equity is estimated to be 8%, which corresponds to option B. The other options (A, C, and D) are incorrect based on different misinterpretations of the model's equations.