Skip to content
All library documents

SABR Implied Volatility: Heat-Kernel Derivation and Approximation

Article Quant Q&A · Author: Dabshffabjvs
Read the original

Summary

The document asks where the familiar SABR implied Black–Scholes volatility expression comes from. The replies point to two related derivation routes: a heat-kernel expansion connected with differential geometry, and the derivation in Hagan and coauthors’ work on managing smile risk.

A key clarification is that the commonly cited SABR expression is not an exact closed-form solution. It is an approximation obtained through expansions. The response identifies relevant sections of the cited treatments but does not reproduce the derivation, state the formula, or discuss its assumptions and accuracy across parameter regimes. The material is therefore a pointer to the mathematical foundations rather than a complete explanation. Readers should distinguish an analytical approximation for implied volatility from an exact closed-form solution when using the SABR model for volatility-smile analysis or pricing.

Key ideas

  • The SABR implied-volatility expression is derived using expansion methods.
  • One cited approach relates the derivation to heat kernels and differential geometry.
  • Hagan and coauthors’ smile-risk treatment is identified as another source for the derivation.
  • The resulting expression is an approximation rather than an exact closed-form solution.
  • The document points to references but does not provide assumptions or accuracy guidance.

Tags

Derivatives pricingVolatilityOptionsStatistics
Full text
# SABR Model Closed Form Solution


# SABR Model Closed Form Solution












I've been researching the SABR model and one of the main benefits it seems is that you can obtain a closed for solution of the implied BS volatility in certain cases.

In all the papers I've read, I have not found any proofs/reasoning as to where this solution actually comes from.

Does anyone know/can link me to a derivation of it?

This is the formula I am referring to.

Thanks

## Answer by M. Jeunesse (score 5)

https://quant.stackexchange.com/a/26379

It comes from Heat Kernel expansion and differential geometry.

See Theorem 6 and Section 8 of http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1717676&download=yes

## Answer by Kiwiakos (score 1)

https://quant.stackexchange.com/a/26381

The derivation is in "Managing Smile Risk" by Pat Hagan et al. A copy is here:

http://www.math.ku.dk/~rolf/SABR.pdf

It is not closed form, but rather an approximation based on expansions.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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