In my research on put options, I come across the ratio: $\frac{(1-\mathcal{N}(d_1))}{\mathcal{N'}(d_1)}$

where $d_1=\frac{\log(S/X)+(r+\sigma^2/2)t}{\sigma \sqrt{t}}$ and $\mathcal{N}(.)$ is the Cumulative Density Function (CDF) while $\mathcal{N'}(.)$ is the Probability Density Function (PDF) for a standard normal distribution.

The fraction $\frac{(1-\mathcal{N}(x))}{\mathcal{N'}(x)}$ is known as the Mills' ratio of $x$, i.e. $\lambda(x)$. While the reciprocal of Mills’ ratio ($1/\lambda(x)$) is known as the hazard (failure) rate, i.e. $h(x)=1/\lambda(x)$. The hazard rate is a function used in credit default securities to answer the question "what is the probability of an event given that the event has not already occured." This function is also described as

\begin{equation} h(x) = \lim_{dx \to 0} \frac{P\left[x \leq X<x+dx | X\geq x\right]}{dx} \end{equation}

However, most applications of the function $h(x)$ are interpreted with respect to time $t$.

In my application, this is different since $d_1$ comes from ATM put options with a maturity of one month. I was thus wondering how I could interpret this function $h(d_1)$ for an ATM put option with a maturity of one month ?

Any suggestions would be greatly appreciated :)


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