SABR and ZABR: Interpreting the Local Volatility Function and State Variable

Summary

The document gives a brief explanation of how the SABR model is written in a form that extends to ZABR. It identifies the local scale function as a power law, σ(s) = αs^β, and clarifies that α and β are model parameters governing how volatility varies with the underlying asset level. This addresses why the function is expressed with constants and a power of the state variable.

The question about the financial interpretation of x in an equation from the cited paper remains unresolved in the discussion. The response points readers to the paper’s short-maturity expansion section for context, but provides no further derivation or interpretation. The material is therefore a narrow clarification of the asset process and its parameters, rather than a full account of ZABR calibration, pricing, or empirical performance.

Key ideas

• In the SABR and ZABR setup described, the asset process uses a local scale function proportional to the underlying level raised to a power.
• The constants α and β are parameters that shape how the scale changes with the asset level.
• The discussion does not establish a financial interpretation for the variable x in the cited equation.
• The cited paper’s short-maturity expansion section is suggested as context for understanding x.

Tags

# Understanding the ZABR model (an extension of SABR)


# Understanding the ZABR model (an extension of SABR)












http://janroman.dhis.org/finance/SABR/ZABR%20Andreasen.pdf

In this acticle the SABR model is first presented in another form ( see equation 7 in the article ) and then extended to the so called ZABR model. I have a couple of questions to help me understand the model.



- in figure 1 and 2. Why on earth is $\sigma(s)=c_0s^{c_1}$?

- What is $x$ in equation 7? I mean the financial interpretation. It will make no sense to call it the implied volatility but I am not sure.

Here is an Matlab implementation I found: https://se.mathworks.com/matlabcentral/fileexchange/50328-zabr-stochastic-volatility-smile-modelling

## Answer by Sanjay (score 4, accepted)

https://quant.stackexchange.com/a/46137

I have found the answer to my own question during the last month where the question have been unanswered



- Asset process in SABR and ZABR is $ds(t)=vol*\sigma(s) dW(t)$ where $\sigma(s)=\alpha s^\beta$. These two constants are simply the two parameters (how on earth did i miss that ..... :D )

- I don't have a good answer to the last one, but $x$ is explained in in the second section Short Maturity Expansion

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