In this paper by Jackel (2006), on page 2, he writes:

The normalised option price $b$ is a positively monotic function in $\sigma \in[0, \infty)$ with the limits $$ h(\theta x) \cdot \theta \cdot\left(\mathrm{e}^{x / 2}-\mathrm{e}^{-x / 2}\right) \leq b<\mathrm{e}^{\theta x / 2} \quad \quad (2.5) $$ wherein $h(\cdot)$ is the Heaviside function. In order to understand the asymptotic behaviour of (2.2) from a purely technical point of view, let us recall [AS84, (26.2.12)] $$ \Phi(z)=h(z)-\frac{\varphi(z)}{z}\left[1-\frac{1}{z^2}+\mathcal{O}\left(\frac{1}{z^4}\right)\right] \text { for }|z| \rightarrow \infty \quad \quad (2.6) $$ with $\varphi(z)=\mathrm{e}^{-z^2 / 2} / \sqrt{2 \pi}$. Equation (2.6) highlights a common practical issue with the cumulative normal distribution function: when its argument $z$ is significantly positive, as is the case here for deeply in the money options ${ }^2$, $\Phi(z)$ becomes indistinguishable from 1 , or has only very few digits in its numerical representation that separates it from 1. The best way to overcome this problem is to use an implementation of $\Phi(z)$ that is highly accurate for negative $z$, and to only ever use out-of-the-money options when implying Black volatility ${ }^3$.

In the footnote for [3], he writes:

"This is the reason why put-call parity should never be used in applications: it is a nice theoretical result but useless when you rely on it in your option pricing analytics."

What does he mean by this? Why is put-call parity useless in practice when many papers use it "in-practice"?



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