Interpreting the Normal Mills Ratio for At-the-Money Put Options
Summary
The document asks how to interpret the ratio of a standard normal upper-tail probability to its density when evaluated at Black–Scholes d₁ for a put option. It identifies this as the Mills ratio and relates its reciprocal to the normal hazard rate, a concept often used to describe conditional event likelihood after a threshold has been reached. The question is whether that interpretation carries over when the input is an option parameter rather than elapsed time.
One response describes the ratio’s behavior across spot prices: it is near one at the money, grows sharply below the strike, and falls toward zero above it. The response argues that, for an at-the-money option, the value is not very sensitive to maturity or implied volatility and doubts it is a useful proxy for underlying price jumps. It recommends examining the CDF and PDF separately. Another response suggests interpreting the CDF as related to delta or moneyness probability and the PDF as indicating sensitivity, but these are informal suggestions; the discussion provides no derivation or empirical validation.
Key ideas
- The ratio of the normal upper tail to its density is the Mills ratio, and its reciprocal is the hazard rate.
- The document questions whether a time-based hazard interpretation applies to Black–Scholes d₁.
- The ratio varies with spot relative to strike and is near one at the money.
- The discussion doubts that the ratio is a useful proxy for underlying price jumps.
- The suggested interpretations of CDF and PDF are informal and not empirically tested.
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Full text
# How to interpret CDF($d_1$)/PDF($d_1$) from BS model ?
# How to interpret CDF($d_1$)/PDF($d_1$) from BS model ?
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 :)
## Answer by AKdemy (score 1)
https://quant.stackexchange.com/a/73752
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.
If you change strike it simply moves with ATM. Just looking at the PDF and Complementary CDF seems a lot more meaningful (imho).
## Answer by DeltaB (score 0)
https://quant.stackexchange.com/a/73750
Thinking thoroughly about the question, my suggestion would be the following:
Inputs:
- 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.
DeltaBShown 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.