# Why parameterize the Black Scholes implied volatility surface?

I know that SVI volatility surfaces are very popular among financial practitioners. I understand that this is not really a model for some underlying asset (such as Black Scholes, Heston etc.) but merely a parametrization of the Black Scholes implied volatility surface.

Another example is the Malz FX Volaility parametrization.

My question is: Why do practitioners prefer these parametrization to the plain Black Scholes implied surface?

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There is no "plain Black Scholes implied surface" because implied volatilities come from options market prices (calls and put). If you had a whole continuum of call prices $C : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$, $(T,K) \mapsto C(T,K)$ you would get a implied volatility function $\sigma_I : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ describing your implied volatiliy surface by inverting the Black Scholes formula for each expiry and strike: $$C(T,K) = Call_{BS}(T,K,\sigma_I(T,K)).$$

But there is only a finite number of strikes and maturities available on any market so you only get a finite number of implied volatilies $\sigma_I(T_i,K_j)$. Instead of a whole surface, you just have a cloud of points. There is an infinite number of surfaces passing through these points and each of them corresponds to a different family of marginal distributions for your price process $(S_T)$ (at least if the surface satisfies no arbitrage conditions).

So in order to get an actual surface you need to interpolate/extrapolate between points while making sure the surface you get is arbitrage free. This is not easy because the buttefly condition $\partial^2_{KK} C(T,K) \geq 0$ (convexity of the call payoff = positivity of a butterfly) translate to a second order differential inequality for implied volatility. This imposes strict and non explicit restrictions on your interpolation procedure. This is why pratictioners prefer to start from a parametrization which is arbitrage free by design and then try to fit it to the cloud of implied volatility points.

For details, see "Arbitrage Free Implied Volatility Surfaces" by M. Roper http://www.maths.usyd.edu.au/u/pubs/publist/preprints/2010/roper-9.pdf

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Thanks for your answer. I understand that you can perhaps not interpolate linearly or even piecewise polynomially because you will then get problems with calculating local volatility. However, my question is still why different interpolations or parametrization are used. Why not just fit a polynomial of of degree n to n+1 points? (considering just the smile here, i.e. regarding the maturity dimension of the surface) – DoubleTrouble Jun 8 '14 at 11:00
Maybe I didn't emphasize this point enough but you really want your implied volatility surface to be arbitrage free. Interpolating between points to get a twice differentiable surface is very simple. Making sure it is arbitrage free is not because one of the conditions translates into $\mathcal{L} \sigma_I(T,K) \geq 0$ where $\mathcal{L}$ is a second order differential operator in $K$. See Durrleman’s Condition in Thm 2.9 of this article maths.usyd.edu.au/u/pubs/publist/preprints/2010/roper-9.pdf. This makes the parametrization approach much more appealing. – AFK Jun 8 '14 at 14:11
Thanks a lot for your answer, and for the reference! – DoubleTrouble Jun 8 '14 at 14:45
You are welcome. I edited my response based on your comment. PS: Durrleman's dissertation is also very enligthening. – AFK Jun 8 '14 at 15:47
Very nice answer, and I liked the paper you linked to, thanks. (+1) – Matt Wolf Jun 10 '14 at 1:50