Comparing Approximation Formulas for Shifted-SABR Normal Volatility

Summary

The document raises a question about two approximations for normal implied volatilities under shifted SABR. One formula is attributed to a 2016 paper, while a later 2020 paper gives a different formula for the same quantity. The author wonders whether the expressions agree to the same order of approximation and asks for a proof.

No formulas, derivation, answer, or numerical comparison are included. The material introduces a narrow issue in SABR approximation theory rather than teaching a pricing method or resolving equivalence. Readers would need to consult the cited papers and perform the algebraic analysis themselves; the document does not establish that the approximations are equivalent.

Key ideas

• The question concerns normal implied volatility in the shifted-SABR model.
• Two papers present different approximations for that quantity.
• The author asks whether the formulas agree at the same approximation order.
• No proof or comparison is supplied.

Tags

# SABR Model, Hagan et al. approximated formulas


# SABR Model, Hagan et al. approximated formulas












In P. Hagan, D. Kumar, A. Lesniewski, D. Woodward, “Universal Smiles”, Wilmott, issue 84, Jul. 2016 there is an approximated formula for shifted-SABR normal implied volatlities (par. 1.2).

In a following paper, P. Hagan, A. Lesniewski, G.E. Skoufis, D. Woodward, “Digital option valuation with the SABR model”, Wilmott, issue 107, May 2020 there is a different formula for the same shifted-SABR normal implied volatlities (eqs. 3.11 and 3.12).

I suppose they are equivalent, at least at the same order of approximation. If so, does anyone know a proof?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.