# Extrapolation of the volatility smile

Are there any market practices to extrapolate the volatility smile for equities? I already have an arbitrage free interpolated call prices data and I'm looking for a method to extrapolate beyond the last available data. I tried the method described here : S. Benaim, M. Dodgson, and D. Kainth. An arbitrage-free method for smile extrapolation. Technical report, Royal Bank of Scotland, 2008. 1, 2, but I have arbitrageable prices (Plus there is no indication that beyond the vicinity of the last point, we obtain arbitrage free prices.

• It depends what you want them for. Do you just want something that looks nice? What do you want to price with it?
– will
Oct 13, 2017 at 9:18
• It's to price some exotic payoffs with a local vol model. Oct 13, 2017 at 9:51
• If they're the kind of exotics you can actualyl price properly with LV, then you can probably just get away with flat extrapolation.
– will
Oct 13, 2017 at 9:54

I'm a fan of fitting a distribution, and then implying vols from that in the wings.

You'll get an arbitrage free surface, that makes sense.

A very simple example is to use gaussians to build a pdf, and then numerically price the options from it. Here's an example fitting FTSE options using just 2 gaussians to create the implied pdf and resulting smile:

Where the fit to the options is good:

You're of course free to use other distributions as your basis functions to create your pdf, as well as mixtures of different functions - what i liike about doing this is that you get a very nice smooth distribution. No spline nodes causing weird behaviour in the derivatives, and it integrates to 1.

You just price the options by numerically integrating your pdf.

I welcome anyone to point me at a particularly nasty smile from the past and i will happily fit it and put the results here.

• Agreed this is probably the best option at the smile level. But won't you have a hard time imposing absence of calendar arbitrage with mixtures? With SVI it's a bit the opposite, easy to impose absence of cal arb, but too rigid to account for exotic smile shapes (+ not really arbitrage free). SSVI is clearly too rigid even though arbitrage free. Also there is a question of how do you extrapolate over time (i.e. for maturities that are not listed in the market). Oct 13, 2017 at 9:42
• @Quantuple so, something you can do for interpolation is to start the fit at the next tenor with the same params as the previous tenor (such that they hopefulyl do not deviate too far), and then you can interpolate the parameters. Unlike the various SVI models, interpolating the parameters here still gives you nice smooth surfaces.
– will
Oct 13, 2017 at 9:53
• One thing that surprised me when looking at your implied vol. plot: I would have expected the smile to flatten out at both ends when using a pure normal mixture for the log-returns? Oct 13, 2017 at 10:02
• @LocalVolatility I disagree. For paths which go down, we're goign to see the log normal vol going up, since the proportional vol will increase. if i use a single normal variable, the smile is pretty much exactly $\frac{\sigma}{\sqrt{K/F}}$. I haven't bothered to work out why.
– will
Oct 13, 2017 at 10:14
• OK - I guess the misunderstanding ist that I thought you model log-returns while you seem to model prices as normal mixtures. Correct? Oct 13, 2017 at 10:17