# The error term of Hagan's approximation of Black's vol in SABR

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!?

• I think you mean to refer to A-69 and A-66 in the paper, is that correct? – Mats Lind Nov 4 '19 at 15:12
• Thanks! You're right – Sanjay Nov 4 '19 at 15:22
• 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/… – Mats Lind Nov 7 '19 at 9:14