The ATM Limit in Hagan’s SABR Implied Volatility Approximation

Summary

This question asks how the general Hagan SABR implied-volatility expression simplifies at the at-the-money point. It focuses on understanding why the ratio involving the transformed variable and its auxiliary function takes its limiting value, allowing the formula to reduce to the special ATM expression.

The document contains no worked derivation or answer, only a request for help with the algebra and a reference to the original paper. It therefore identifies a useful derivatives-pricing issue but does not establish the limiting argument, provide numerical evidence, or discuss implementation. Readers would need a source that derives the limit to learn the actual steps; this text alone cannot resolve the question.

Key ideas

• The question concerns simplifying Hagan’s SABR implied-volatility approximation at the money.
• It asks why the ratio of the transformed variable to its auxiliary function has a particular limiting value.
• The document provides no derivation or solution to the algebraic question.

Tags

# How to solve special ATM case for Hagan approximation?


# How to solve special ATM case for Hagan approximation?












In Hagan et al's original SABR paper (https://www.next-finance.net/IMG/pdf/pdf_SABR.pdf), how do we reduce (2.17a) to (2.18) for the special ATM case? Could someone help walk me through the algebra? I can't seem to work out the math. It looks like if z/x(z) is equal to 1, then it's rather easy to see, but I can't understand why z/x(z) equals 1. It's been a while since I did actual math, so hopefully this question isn't too silly... Thanks for your help.

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