Strange
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| Total Posts: 1250 |
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Does anyone have a good suggestion for parametrization for equity volatility smile?
I am trying to fit some historical data for analysis purposes. So the key features have to be
(a) parameters can be interpolatable with respect to time
(b) skew should be parametrized with respect to strike, not delta
(c) parameters can be easily fit numerically
(d) parameters are fairly stable for smaller changes
(e) reasonable behavior of the wings
So far I have considered SVI (Gatherals model), SABR (short form, reduced to log-normal form only, but not sure - maybe long form would work) and simple 4th order polynomial.
Polynomial fits simply using weighted OLS (which is why it's the only one i've tried so far), but has truly unpredictable behavior on the wings far OTM. What I like there is that first and second parameter are essentially ATM volatility and simple slope (i.e. sk10).
Before I lounge myself into numerical fitting of the other models (easy enough in R but still a pain), are there any other suggested parametrization? Also, am I missing something important? |
It's buy futures, sell futures, when there is no future! |
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Trev
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| Joined: Mar 2010 |
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Could your wing oscillation be due to Runge's phenomenon? OLS is supposed to deal with this issue, and I noted you said WOLS but I'm not sure what type, but Chebyshev nodes deal with the Runge Phenomenon by oversampling points at the boundary(s) (i.e. wings in your case).
Forgive me if I misinterpreted what you are doing: I deal mostly with time-series and regression tasks but have used polynomial regression recently and ran into similar issues (i.e. wild oscillation at boundary points).
Cheers. |
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How about using orthogonal polynomials to get around this issue
I might do something like:
You have vol at ATMF -> zero'th order polynomial
now vols at ATMF +/- $10 (i.e. a sort of risk reversal and butterfly) -> linear and quadratic coefficient
then vols at ATMF +/- $20 give you the next one or two coefficients
I think once you go beyond 3 parameters (per timeslice) you are going to run into overfitting problems no matter what you do, unless you have a particular problem you are trying to solve (e.g. spread between variance swap and vol swap) in which case you might be better off fitting that "exotic" parameter directly, instead of working from smile data which is likely to be noisy in any case.
edit: inspired by Trev... may be saying the same things in different language |
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Trev
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qxy
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What kind of products are you looking at? And what do you consider far OTM?
For most practical purposes I'd say OLS works perfectly well. The problems with the wings far OTM you mention might just be related to the market being extremely wide in vol points for such options (especially when the tick size is relatively big compared to the option price, e.g. with markets like 0.01-0.03 or 0.05-0.10). This is more of a problem in less liquid products like smaller equities than in indices.
Many market makers set their vol curve using some variation of a quadratic model with some cutoffs at the extremes from where volatility is assumed constant. Then they'll quote around this price with the width if the market being a function of vega, with a minimum width for low vega options. On top of that prices might be skewed in individual strikes based on their position. I'd say it doesn't make sense to try to make your volatility model more complicated than what's used by the people on whose prices you base it. |
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Strange
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| Total Posts: 1250 |
| Joined: Jun 2004 |
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I am looking at index options and single stock options, which is pretty much all you can find in the listed equity space. Anything under 10 delta for my purposes is far OTM and should not be seriously considered in the fitting process (I am ok making a half-a-vol fitting error that far out). Even discounting these, the shape of the skew can be pretty tricky, especially for something like VIX options.
When I look at the quotes, they appear pretty smooth, so it does not appear that the problem with the wings is quote-related. It is more probably somehow related to the fitting process and the phenomena that Trev has described.
So far I am thinking of the two-prong approach - first of all, I need to improve the numerical fitting procedure (maybe it is the weights or maybe it's the fitting function - e.g. i need to make two separate polynomial fits and blend them) and then I need to keep an eye on the changes in polynomial parameters as a function of time. |
It's buy futures, sell futures, when there is no future! |
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Strange
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| Total Posts: 1250 |
| Joined: Jun 2004 |
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I just re-read the approach suggested by silverside and yeah, that might be the right way.
In the index space, with respect to log(K/F)/sqrt(t), there is a slightly negatively convex skew which continues pretty smoothly out to really low strikes, but reaches a minimum at some high strike (probably 15ish delta calls) and from there continues nearly out to really high strikes. Sort of looks like a check-mark. |
It's buy futures, sell futures, when there is no future! |
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Strange
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| Total Posts: 1250 |
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| actually, upon some tinkering, i see that using 3rd order polynomial produces better fit then 4th order. also, i think if i add flat extrapolation for the local maximum on the put wing side, i can get more or less predictable behavior for the wings |
It's buy futures, sell futures, when there is no future! |
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mib
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| Total Posts: 335 |
| Joined: Aug 2004 |
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| What you are observing is that OLS fitted polynomial fits are non-local (i.e. a small input change at one strike may lead to large output changes at rather distant strikes). You need something with less rigidity, e.g. Hermite splining + boundary conditions outside of the liquid range. |
Head of Mortality Management, Capital Structure Demolition LLC |
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granchio
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| Total Posts: 1416 |
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I would just add a couple of cents.
The fitting method you choose might depend on the parameterisation. But there is a lot of work available on fitting and optimising, so at the end of the day, that is just a technical issue, though you might be reinventing the wheel by recoding it all.
IMHO, the parameterisation is where most of the interesting work is . In my experience, none of the ones you mention work solidly across many equity underlyings and maturities, though of course it depends on what margins of error you need. It also depends whether you are happy to fit different parameters at different maturities or if you want something global.
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"Deserve got nothing to do with it" - Clint |
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Strange
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| Total Posts: 1250 |
| Joined: Jun 2004 |
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| Merci! I don't really need an ideal fit, as long as it's within bid/offer for close-to-the-money strikes. For far OTM I am ok with being a vol off or so. |
It's buy futures, sell futures, when there is no future! |
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granchio
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| Total Posts: 1416 |
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I forgot one thing (though of couse you know it): having the right zeros and forwards is critical of course - and it does add a pre-processing layer to the work which is not trivial at all.
I believe your requirements (within bid/ask atm, only a bit out far otm), are ~doable on liquid equities with a global parameterisation, but there are a lot of caveats of course. Feel free to shoot me an email offline if you want to discuss more. |
"Deserve got nothing to do with it" - Clint |
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Strange
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| Total Posts: 1250 |
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| Yup, which is a separate problem. Given that longest maturity I'll deal with is 2 years, zeros are not as crucial. Right now, I am using CMT treasury rates from H15 data, though I might switch to fed fund futures or something along these lines |
It's buy futures, sell futures, when there is no future! |
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granchio
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| Total Posts: 1416 |
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yes rates maybe not so crucial, but it can make the difference between an OK fit and a very good one. it's cost/benefit, time you spend on it vs improvement in volsurface.
i haven't been fitting for a few days, but using a snapshot of SPX prices from 2 aug (source: delayed bloomberg) one can fit a global parameterisation from sep12 to dec14within bid ask nearly every where ( a few far otm a tad out). one can probably do even better, but i spent only 5 mins on it.
I attach a couple of pics
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"Deserve got nothing to do with it" - Clint |
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Strange
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| Total Posts: 1250 |
| Joined: Jun 2004 |
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I agree with you, in an ideal world you want to do everything properly, first get a good yield curve, then get a good forward curve (preferably, separating borrow from div yield), then vols and then the fit. However, from what I see, errors/inconsistencies in snapped bid/offer prices is a much bigger source of problems.
Really, a "really good" fit is not worth my time at the moment - as long as vols within +/- 10 delta are are ok, i am happy with it. This is not for market-making, this is for historical analysis and understanding the dynamics. Also, I will need to run this on some 2k stocks/indices and resolve errors/problems, so I'd not want to make it too complex.
However, once it's up and running, I will be trying to get incremental improvements. |
It's buy futures, sell futures, when there is no future! |
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nice...
I suppose it would be cheeky to ask what "magic formula" can fit out to 2y with a single parametrization
I also keep a time series of exchange traded index option quotes but currently I'm not doing much with it apart from checking some marked vol surfaces, so for me a simple method works well, generally there are enough quotes that splining on K/S is enough. |
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granchio
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thanks - certainly we can't discuss it on the forum, it's quite recent work and some details are still ongoing.
we are aiming at longer maturities as well, of course it is a give and take - if you want to fit from 2 months to 10 years with a single global parameterisation, you will have to accept that you will not market-make 6 months vanillas around it with a 2 bps bid/ask (you will need some adjustments to do that).
On the other hand, i think if you want to focus on the shorter end, say up to 2 years as strange is looking at, you can do quite a lot with a single global, though
scaling up to 2000 underlyings over hundreds or thousands of different past days will require some serious infrastructure work.
If you are not trading or calculating risk, scenarios etc, but just checking marks, I agree splining is fine... why overkill (though the forward fitting still has to be tackled). |
"Deserve got nothing to do with it" - Clint |
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I suppose this is sort of related to the current discussion. The image below is the IV surface of SPX call options,with applying certain filters. For example, a moneyness below 0.9 and above 1.1 were rejected and a time to maturity of less than 7 days was rejected. The surface doesn't seem right to me, at least from an intuitive point of view. Does anyone have any comments?
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| Did you mean to define Moneyness as K/S? Also, how did you generate the surface and why does it not intuitively look right to you? The answer to this question should suggest what is wrong with the fitting or interpolating procedure. |
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Strange
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| Total Posts: 1250 |
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Out of curiosity, are there any semi-accepted models for full surface parametrizations? E.g. something that would fit both the term structure and the skew? I remember reading something about exponential models along the lines of Nelson-Siegel-Swensen, but I can't recall where it was by now.
PS. what's a quick and dirty way to constrain the polynomial to be non-negative at any value? Obviously, some sort of non-negative function of y, but which one? |
It's buy futures, sell futures, when there is no future! |
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granchio
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| Total Posts: 1416 |
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>>Out of curiosity, are there any semi-accepted models for full surface >>parametrizations? E.g. something that would fit both the term structure and the >>skew?
Semi-accepted? i.e. both published and successfully used in practice (i.e. decent, robust fit) ? I am not aware of any (equity at least).
ITO33 claims to work well, but as far as I know (and I could easily be wrong) it is not an analytical parameterization of the volsurface, rather a pricing model combining stoch vol, jumps (and maybe other effects) that needs to be solved numerically - which is very good for certain uses but less for others.
I don't think we'll publish ours widely- but if after some more testing we really like it then we will make it available as "beta" on a small scale. |
"Deserve got nothing to do with it" - Clint |
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deeds
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Don't know if this helps, but, doesn't Dupire have an entropy based approach that is available in Bloomberg? (I think there's a white paper describing it, too)
D |
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radikal
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I think that both SVI and SABR are not going to fit well for you across multiple tenors. Personally, I've never found much success with SABR in the equity space, particularly the 10-20 delta regions where I feel like the market has much more curvature than SABR reflects.
I generally have a more MM perspective on this stuff -- being consistently off more than 1/10th of a vol in anything > 10 delta is somewhat a problem. (And why vega weighting your fit isn't a bad idea) |
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MacJack
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Dear Strange and Grangio,
We recently came up with a (maturity-dependent) arbitrage-free version of SVI: http://arxiv.org/abs/1204.0646
It is basically an extension of SVI (we call it `Surface SVI').
You might be interested.
If you have any questions or remarks, I will of course be happy to discuss them.
Best, |
"Beer is a proof that God loves us and wants us to be happy". Benjamin Franklin |
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granchio
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| Total Posts: 1416 |
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Very recent revision I see, good! Will look and revert if comments, thanks
BTW, there is no email in your profile, and the link does not seem to work
(EDITed for completeness) |
"Deserve got nothing to do with it" - Clint |
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