I am looking around for a model that kind of makes sense but also gives enough flexibility to fit market prices. If it is of any relevance, my main consideration would be index volatility. The graphs in the file attached use data from options on the eurostoxx 50 from last december.Plotting implied volatilities vs strikes makes it pretty obvious that a simple v- or u-shaped model (pretty much a straight line to the downside and a U shaped curve around the ATM level) does the trick. Those models have vol, slope, call and put curve - they are plain second degree polynomials with linear cut offs that ensure that the wings don't blow up too fast due to the quadratic coefficient. What I don't like about these types of models is that they do not scale nicely in time. That is, plotting vol vs strike for a few maturities makes it very visible that in strike space the smile gets tighter and tighter as expiry time nears. To moderate the trouble one can use some scaling tricks, one of which is delta.So, trying to plot volatilities vs deltas shows that the resulting curves are very similar from one expiry to another (see graphs). It seems to be a more elegant and complete approach than the above. But then I started thinking about fitting a mathematical curve that reproduces that shape and I struggled. Especially, the small deltas effectively "compress" a very large array of strike space. Therefore, while in that far downside strike space the volatilities are quite linear, in the small delta space they shoot up quite hard. I am at a loss trying to find a simple enough model that is suitable for the purpose of representing the volatility in delta space.

These vol smiles vs delta are much better behaved than those I've seen for the US Equity indexes. I gave up on trying to model the smile vs delta for the full range of deltas (roughly, from 0.025 to 0.975) and just went with log moneyness and a used a spline.

Hey Skip,When you say that you gave up modeling for the full range of deltas, do you mean that you managed to model most of the range (roughly, from 0.025 to 0.975)? I find it that this delta space view "behaves" in most index products. Maybe stocks structures are distorted due to the discrete timing of earnings announcements.I do find it very difficult to model the vol curve in delta space. So far I have always used combinations of second degree polynomials and straight lines - those are very easy to control and fit the market quite nicely - but not nicely enough, there are always bits you can't quite fit and instead of trading away nicely parts of it turn into a massive model twisting exercise. It is quite natural to go for a spline approach. How many points would you use? I would say that 5 suffice. Also, do you use an interpolating or a smoothing one? I have messed around with interpolating splines but then raising one point in the curve necessarily lowers others (and that is illogical, since if bidders come somewhere, they are really a little bit everywhere, so you want to make sure you raise everything, albeit not equally) Smoothing splines yield more logical behavior but often you get with control points so far away from the volatility curve that they are difficult to control.

I did this work some while ago, but I vaguely recall being able to somewhat model the implied vols for US equity indexes for deltas in the range 0.025-0.975, but found that they go annoyingly hyperbolic at the edges. The problem arose when I had to extrapolate to estimate implied vols beyond this range. For short term options, the call-adj deltas tend to evolve into binary values of either 0 or 1, so that's where it gets really messy. It just seemed much easier to model IVs vs moneyness, since the relationship is usually linear at the edges of that smile. I recall using a custom smoothing spline that was fairly insensitive to the number of knots used. Not much else I recall about it otherwise.

It is a fair comment that close to expiry there would be major issues - because delta would flip around quite a bit as the underlying moves back and forth through the ATM strike, with a delta-based model your volatility in those strikes will continuously shift loads and you will have to pump your parameters the whole time to stay in line with the market level.Vols in delta space do go hyperbolic at the small deltas. This is due to the fact that even though these small ones may represent, 2.5 percent of the delta range, they represent much more of the absolute strike range. I think that's a problem with the concept of delta itself - consider a product with S=100, atm vol=22, time to expiry = 3 months. You can judge the riskiness of putting on spreads by looking at the spread deltas - 90/110 spread is riskier than the 100/110 which is riskier than the 110/120. But then, looking at spread deltas you could conclude that the 50/40 put spread (massively out of the money) is just as good as the 50/10 (also massively out of the money). The relative values of black and scholes deltas for the far out of the money options are problematic, even if you use volatilities imlpied by the market - possibly this is the reason for the hyperbolas at the edges of the vol vs delta graphs.