Understanding the ZABR model (an extension of SABR)

Question

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

Answer 1

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