Why Practitioners Parameterize Implied Volatility Surfaces

# Why parameterize the Black Scholes implied volatility surface?


# 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?

## Answer by AFK (score 20, accepted)

https://quant.stackexchange.com/a/11584

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

## Answer by FinanceGuyThatCantCode (score 3)

https://quant.stackexchange.com/a/33811

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!