Relating SABR Parameters to Implied Volatility Level, Skew, and Convexity

Summary

The document asks how to recover the SABR parameters for the lognormal case, where beta equals one, from the local Taylor expansion of implied volatility around the forward. It labels the constant term as at-the-forward volatility, the linear coefficient as skew, and the quadratic coefficient as convexity, then presents formulas relating those quantities to initial volatility, vol-of-vol, and correlation.

The supplied answer points to a more general derivation based on expanding the Hagan SABR implied-volatility approximation, including cases with beta other than one. It does not show the derivation itself, so the parameter formulas are stated rather than developed or checked in the document. The material is therefore a useful pointer for SABR calibration theory, but readers need the referenced derivation to follow the steps and understand approximation limits.

Key ideas

• A Taylor expansion around the forward summarizes implied volatility through its level, skew, and curvature.
• For beta equal to one, the document gives mappings from those expansion coefficients to SABR parameters.
• The cited approach derives the relationship by expanding the Hagan SABR implied-volatility approximation.
• The response points to a broader treatment that also covers beta values other than one.

Tags

# Relation between SABR parameters and Taylor expansion parameters


# Relation between SABR parameters and Taylor expansion parameters












Suppose a SABR model framework (with $\beta=1$)

$$dF_t=\sigma_t S_t dW^{S}_{t}$$

$$d\sigma_t=\alpha \sigma_t dW^{\sigma}_{t}$$

$$dW^{S}_{t}dW^{\sigma}_{t}=\rho dt$$

I know that the Implied Volatility induced by SABR model can be expressed in a very readable way, by the following equation:

$$\sigma^{SABR}(x)= a +bx +\frac{1}{2}cx^{2}+O(x^{3})$$

$a$ is at-the-forward implied volatility level, $b$ is implied volatility skew, $c$ is implied volatility convexity.

Then, the relation between model parameters and the $a,b,c$ parameters is:

$$\sigma_{0}=a$$

$$\alpha= \sqrt{6b^{2}+3ac}$$

$$\rho=\frac{2b}{\sqrt{6b^{2}+3ac}}$$

Could you please provide a detailed derivation of the three equations above relating model parameters ($\alpha, \sigma_{0}, \rho$) with ($a, b, c$) parameters?

Thank you

## Answer by jherek (score 2)

https://quant.stackexchange.com/a/82099

The more general case $\beta = 1$ or $\beta \neq 1$ is derived in Explicit SABR Calibration Through Simple Expansions by Le Floc'h and Kennedy.

It is based on a Taylor expansion of the Hagan SABR approximation.

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.