# 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?

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

• 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) 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
• 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
Jun 10 '14 at 1:50

I like parameterizations for many reasons. Let's say you have an SPX smile with 125 strikes that can reasonably be traded. Many parameterizations reduce these 125 strikes down to 5 parameters. Also, the parameterizations can smooth out what can often be very noisy data on the wings. In addition, some of these parameterizations can have parameters that can quickly give a trader an immediate intuition as to what the smile looks like. Now imagine having a time series of volatility surfaces - these parameterizations do a nice job of condensing a massive amount data to a merely large amount of data.

For example, I tend to use the SVI parameterization (although I have found that it is very tough to fit inside of the very tight bid ask for a lot of the short dated expiries in the last couple years for SPX). The usual parameters for SVI are not intuitive, but I can easily translate those 5 parameters into intuitive parameters such as ATM, Skew (first derivative of the smile ATM), Kurtosis (second derivative of the smile ATM), and the left hand/right hand side asymptotic slopes (SVI is linearly asymptotic in implied variance).

Another cool feature of the SVI using these intuitive parameters is that I can almost surely (in the probabilistic sense) uniquely invert them back to the raw parameters used to calculate vols. I like this because I can shock the intuitive parameters - most likely shock ATM or skew and then invert back to new raw parameters. Shocking the vol surface helps me to understand stress risks. The only downside with this technique is that after too big a shock, we might not be able to invert - in other words the mapping from raw svi parameters to the intuitive parameters is not surjective - a square root of a negative number will alert you to this!

• Are your intuitive params basically the svijw parameters? Or your own set?
– will
Apr 21 '17 at 20:46
• Similar. At my old shop, we had ATM, Skew, Kurtosis along with some wing params. That form that was used was flawed in some ways, but I liked having intuitive parameters. I did that with SVI - then Gatheral published the SVIJW parameters. Mine are slightly different, but similar. He uses min vol - I used second derivative ATM of the smile instead which gives some sense of butterflies. If one prefers minvol, they should use the minvol offset from ATM (i.e. the min is 2 vols less than ATM) - that way if you do an ATM shock, you don't have to shock the minvol as well to maintain the shape. Apr 21 '17 at 20:55
• Makes sense. I like a parameterization of the firs tfew derivatives at the atm and then some stuff for the wings - things where you can actually guess them from a smile rather than a lot of the stuff you see, but makes arbitrage harder to check, if you care about that.
– will
Apr 21 '17 at 20:58
• In order to check for arbs, I go back to the raw params and try to solve for the strike with the lowest price digital option over an unreasonably big range of strikes - if the lowest price is negative, there is an arb - if not, we are almost certainly arb free as far as exploding vols go (which then implies butterfly arbs don't exist for SVI) Also, trying to make a smooth-ish term structure of intuitive parameters makes more realistic surfaces in the face of sparse and/or bad data - trying to avoid calendar arbs, helps to make the term structure of intuitive params smooth. Apr 21 '17 at 21:08