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Hagans approximation of Black's implied vol in SABR is very! difficult to understand fully. But I want to ask in here if anyone can tell me more about the error term.

Consider the paper: http://web.math.ku.dk/~rolf/SABR.pdf

$\sigma_B$ (the log-normal volatility) can be approximated (see A.69c) through $\mathcal{O}(\epsilon^2)$ where $\epsilon$ is defined as in A.66a-b.

In SABR we set $\epsilon = 1$. Here I arrises my confusion:

Is the error term in Hagan's approximation of $\sigma_B$ really $\mathcal{O}(1)$, and hence independant from all other parameter levels? I find diffucult to believe that $\tau=T-t$ doesn't impact the error term.

Intuitively speaking what does it mean when the error term is $\mathcal{O}(1)$ in this context?** I have actually difficulty understanding the meaning of $\mathcal{O}(x)$ in this context even though I have looked at the definitions a thousand times.

$O(1)$ doesn’t make sense does it!?

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    $\begingroup$ I think you mean to refer to A-69 and A-66 in the paper, is that correct? $\endgroup$
    – Mats Lind
    Nov 4, 2019 at 15:12
  • $\begingroup$ Thanks! You're right $\endgroup$
    – Sanjay
    Nov 4, 2019 at 15:22
  • $\begingroup$ The introduction to app. A in the paper gives me the impression that $\epsilon$ is the parameter in a [singular perturbation problem][1] which the authors uses to get to the model solution. In the original problem $\epsilon$ is by definition 1, and the solution is sought in the "distinguished limit" (in the terminology of singular perturbation problems) where $\epsilon << 1$. I think it could be more useful to look at $\epsilon$ as a parameter this way rather than as an error term. [1]: scholarpedia.org/article/… $\endgroup$
    – Mats Lind
    Nov 7, 2019 at 9:14

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