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 :)


2 Answers 2


This is not an answer but more a comment to the given answer, that I am happy to delete afterwards.

Did you compute this value? It explodes for spot below strike, is roughly 1 for ATMF and declines to 0 if spot is above strike. In the gif below, I set interest $r$ to 3% and no dividends $q$. The x-axes shows different spot values, and the y-axes the respective values for n(d1), 1-N(d1) and Mills (your ratio). For convenience, I also plotted the actual risk neutral probability of the underlying ending up ITM. The blue vertical line shows ATM, the purple horizontal line where the value of Mill's ratio equals 1. As you can see, for your ATM example, maturity and IVOL $\sigma$ do not matter much, as the value will always be close to 1. Insofar, I have doubts it can be interpreted as a proxy for jump in the price of the underlying.

enter image description here

If you change strike it simply moves with ATM. Just looking at the PDF and Complementary CDF seems a lot more meaningful (imho).

enter image description here


Thinking thoroughly about the question, my suggestion would be the following:


  • d1 being standardized price of the underlying
  • cdf(d1) can approximatively be interpreted as the delta of the option or the probability of the option to be in-the-money at expiration of the contract
  • pdf(d1) is the probability distribution of the standardized price of the underlying.

For one stock with two ATM options, i.e., one call and one put, the pdf(d1) would integrate the speed at which the delta could turn in-the-money. Hence, capturing the level convexity of the option and proxy for jump in the price of the underlying.

Hope this helps and makes sense.

Happy to receive your feedback.



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