# 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.

• Main question: What is the exact formula that generates the graphs in Figure 3. $$IV(k)=...?$$
• 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

## 1 Answer

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

• The main question: Look at page 9 from "the ODE can be rearranged to $$f'(y)=....=F(y,f)$$". For given $$\alpha, \beta, rho, \epsilon$$ and $$\gamma$$ we have all the relevant information to solve this ODE, hence finding $$f(y(t))$$ which is denoted as $$f(y)$$ in the paper. Then we can compute the implied normal (Bachelier) volatility by $$x = z^{1-\gamma}f(y) \text{ (defined at page 8)}$$ $$IV_{bacheier}(K) = \frac{spot-K}{x}$$

• 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