Interpreting the Error Order in SABR Implied Volatility Approximations

Summary

The document asks how to interpret the error term in Hagan's approximation of Black implied volatility under the SABR model. It refers to a derivation that gives the approximation through a specified order in a small parameter, then notes that the parameter is set to one in the usual SABR application. The central question is whether this makes the error merely order one and independent of model parameters, especially time to expiry.

The author also asks what big-O notation means in this approximation context and whether an order-one error is meaningful. No answer or derivation is included in the document, so it does not resolve the dependence of the approximation error on maturity or other inputs. It is useful as a statement of a model-validation question, but readers need additional mathematical analysis to quantify the error or determine when the approximation is reliable.

Key ideas

• Hagan's SABR formula approximates Black implied volatility using an expansion in a small parameter.
• The question concerns interpreting the remainder after that parameter is set to one.
• The author wonders whether the error depends on expiry and other model parameters.
• The document raises questions about big-O notation but supplies no explanation or answer.
• Error magnitude and approximation reliability therefore require analysis beyond this discussion.

Tags

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


# 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!?

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