In the Merton Model, what is the formula for d1?

In the Merton Model, the formula for d1 represents the standardized value that is key to determining the probabilities related to a firm's default risk when applying a structural approach. The correct formulation for d1 is particularly important for calculating the value of equity as a call option on the firm's assets.

The correct equation for d1 incorporates the natural logarithm of the ratio of the firm's asset value (V0) to the debt (D), the risk-free rate (r), and the volatility of the firm's assets (sigma v). The term (r + sigma v^2/2) reflects the drift of the process describing asset value over time, which affects the expected returns on the firm's assets. This drift is essential because while assessing the risk of default, it’s important to account for the expected growth rate of a firm's assets, which is approximated by the risk-free rate adjusted for volatility.

As a result, the proper formulation for d1 captures these dynamics effectively, ensuring that both the risk-free rate and the risk-adjusted factors related to asset value changes are integrated into the calculation. This allows for a more accurate assessment of the firm's default probability and the valuation of its equity in the context of a call option on assets. The logarithmic term