Choosing Implied-Volatility and PDE Methods for Local Volatility Calibration

# Local volatility calibration methods (PDE grid vs using directly the IV smoothed surface)


# Local volatility calibration methods (PDE grid vs using directly the IV smoothed surface)












When we calibrate our local volatility we need to have create a smoothed volatility arbitrage free implied vol surface(for example SSVI or Sabr).

Let's imagine the smoothed implied vol surface as a black box were we give a strike and expiry and we get back our implied vol.

Now there is a well known formula which can give the local vols as function of implied vols if we assume that we have available in continuous grid of implied vols:

$\sigma_{\mathrm{Dup}}(T,K)^2 = \frac{ \frac{\partial w}{\partial T} }{1 - \frac{y}{w} \frac{\partial w}{\partial y}+ \frac{1}{4}\left( - \frac{1}{4} - \frac{1}{w} + \frac{y^2}{w^2} \right) \left(\frac{\partial w}{\partial y}\right)^2 + \frac{1}{2}\frac{\partial^2 w}{\partial y^2} }$

Given the smoothed and arbitrage free implied vol surface we can compute the above derivatives numerically and we can extract any local vol we like.

But I still see that one of the very popular approaches (something similar presented in the paper "Volatility Interpolation" from Jesper Andreasen and Brian Huge) where they attempt to solve a ill posed minimization problem with some smoothness constraint in a PDE grid combined with the forward Dupire PDE for instance. They also assume some structure of the local vol (piecewise constant in time liner in the strike dimension) in order to be able to apply this FDM approach.

This seems more complicate and also modelling-wise the piecewise constant in time dimensions feel very restrictive. and much less precise (e.g. the linear interpolation in strike dimension

I feel I am missing something. Maybe the results from the first method are not smooth enough? Maybe there are other drawbacks that I am missing?

## Answer by Jesper Tidblom (score 1)

https://quant.stackexchange.com/a/80481

The choice of volatility model very much depends on the application and the requirements coming from this.

The SSVI model or SABR model (and other similar models) surely produces smooth arbitrage free surfaces. The problem is that those models have very few parameters. You will typically never be able to fit the market data to the model except approximately. You might have hundreds of option prices/market volatilities, but only a few free parameters to use for the fitting procedure (four for the SABR model). The surface is then arbitrage free, but just a very rough approximation of the market implied surface.

This might be enough for your application if you for example want to use it for some, not so exact risk measurement calculations. If you need the surface to get a good estimate of the price of some exotic derivative this choice of model will not be so good however.

The Huge/Andreasen model has a flexible number of parameters depending on how many option prices are given and the fit is generally excellent. The constant interpolation in their model can be replaced by linear interpolation to get an even better fit and more stable calibration procedure. Even splines can be used, but I have read that this does not really improve things much from the linear case. I also want to say that they only do this rough interpolation for their so called local volatility "proxies" which is then used to calculate option prices for all times and strikes which is then used to calculate implied and local volatility. This does not mean that they use piecewise constant volatilities in the final surface.

I also want to mention that some models are excellent for producing a volatility smile for one maturity, but are less suitable for producing an entire surface. Sometimes you are only interested in the volatility for a specific maturity. For example you might want to price some exotic volatility dependent derivative with a specific maturity as good as possible compared to the given market data. Then you typically want a smooth implied density of the underlying implied by the market data (by which you can price derivatives with the same maturity, or get any volatility for that maturity). The density is the second derivative of the call price with respect to the strike. So a three times differentiable call price interpolation would be required to produce a smooth density. Also the interpolation should be arbitrage free.

The above is a rather hard problem to solve in general. I just implemented an article from last year by Le Floc'h about exactly this. https://arxiv.org/abs/2305.13791 This is from the Arxiv so it is not as polished as an article in a journal, but the idea is simple to understand and it produces an excellent fit to the data even though it is a bit of a numerical challenge to implement.

Also note that there is not just one correct answer here when it comes to creating a smile or surface. There are infinitely many ways of creating arbitrage free surfaces matching the market data.So there will never be a model that produce some kind of "correct" result. The result will always vary a bit depending on the model, but typically we want it to be as consistent with the market data as possible.