Merton Model: How Asset Volatility Determines d1 at 50% Default Probability
Summary
The document asks how to interpret the Merton structural credit model when the probability of default is 50%. Under the stated setup, this corresponds to d2 being zero, but it does not imply that d1 is also zero. The relationship between the two quantities includes asset-value volatility and the square root of the time to maturity, so d1 is higher than d2 when those inputs are positive.
The response explains that estimating d1 requires the volatility of the firm’s asset value process. It notes that this volatility is typically inferred from equity prices. The exchange provides no worked estimation procedure or numerical example, and the result depends on the model inputs and assumptions. The key lesson is that a 50% default probability fixes d2, while d1 additionally reflects volatility and horizon.
Key ideas
- A 50% default probability in the stated Merton setup corresponds to d2 equal to zero.
- d1 differs from d2 by asset-value volatility multiplied by the square root of time to maturity.
- Estimating d1 therefore requires an estimate of asset-value volatility.
- The response says this volatility is typically estimated from equity prices.
Tags
Full text
# Merton model d1 and probability of default
# Merton model d1 and probability of default
What is the value of $d_1$ when the probability of default is 50%
I know that: $$ \begin{aligned} d_2 &= 0 \\ \mathcal{N}(d_2) &= 50\%\\ 1- \mathcal{N}(d_2) &= \mathcal{N}(-d_2) = 50\% \end{aligned} $$ But I don´t know if $d_1 = 0$ or different.
## Answer by siou0107 (score 2)
https://quant.stackexchange.com/a/49980
Since $d_1 = d_2 + \sigma\sqrt{\tau}$, you need to know the volatility of your asset value process. You typically estimate it from equity prices (see e.g. Hull's book).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.