There is an explanation of Risk Neutral Default Probability using a Firm's Equity price here - https://www.mathworks.com/help/risk/default-probability-using-the-merton-model-for-structural-credit-risk.html.

However, there is an equation which states that

$\sigma_E = \frac{A}{E} N \left(d_1\right) \sigma_A$

How is this formula derived?


As you see in the third equation on that Mathworks page, the Merton model postulates that the value of equity equals the value on a residual claim on a company's assets after the creditor has been repaid. Economically speaking, equity is a call option on the asset value $A$ with strike price equal to the liability $L$, the formula for which is

$$ E=AN(d_1)-Le^{-rT}N(d_2) $$

We further note that the variance of the asset process is (with a bit of handwaverianism)

$$\sigma^2\left(\frac{dA_t}{A_t}\right)\equiv \sigma_a^2dt$$

Finally, we know for a call option that $ \frac{\partial E}{\partial A}=N(d_1)$ which is also colloquially called Delta. Thus

\begin{align} E&=AN(d_1)-Le^{-rT}N(d_2) \\ \Rightarrow dE&=N(d_1)dA\\ \Rightarrow \frac{dE}{E}&=\frac{1}{E}N(d_1)dA\\ \Rightarrow \frac{dE}{E}&=\frac{A}{E}N(d_1)\frac{dA}{A} \end{align}

and ultimately

$$ \sigma_E\equiv \sigma\left(\frac{dE}{E}\right)=\frac{A}{E}N(d_1)\sigma\left(\frac{dA}{A}\right)=\frac{A}{E}N(d_1)\sigma_A $$


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