I have market quotes on vol only for 10D, 25D, 37D & ATMF for 1 month to 12 months.I am trying to interpolate vol to get a vol surface.Lets say I want to get the vol for 15D and 45 days.Here goes my problem: either using bi-lineal interpolation or bi-cubic splines interpolation I can first interpolate for tenor and then for delta level or vice-versa. The result is different.Is there any rationale for starting with tenors or otherwise for starting with delta levels?thanks,Osvaldo

A vol surface is just an arbitrary deterministic function for volatility in terms of time and strike. There's no real justification for assuming any particular form for this function, other than you need to prevent arbitrage. In practice this means ensuring that:1. dC/dK <0 2. d2C/dK2 >0 3. dC/dt >0Where C is the price of a call option.

Osvaldo,May i read the D as delta of Calls? Then for each time _point_ you have not enough data even for usefull single smiles regarding 1. and 2., at least you need data _around_ ATM. Estimating against time developement is serious (rule 3 as theta would not be enough in any model) - you really want it for delta = 0.1 and 1.5 month? Generally i dislike splines for vol fitting in time: either you go through a SV model (giving you trouble for short times, the smile and far OTM) or others, somewhat not theoretical found. As soon as you see it depends on how you proceed, you might have questions whether it might be a reasonable approach at all.The practical aspect might be for example: you have an OTC trade with expiry = 30 Dec 2002 (or in June) ... Here you can see that vol <-> price may be critical if you think on interpolating time.

I forgot to say that I also have 10D (10 delta), 25D, 37D & ATMF for puts (in the case of ATMF is the same one), so I have three market observed points around the ATMF on each side.I tried cubic splines and looked nice but sometimes (depending on the shape of the surface) some anomalies appear like 15D being higher than 10D due to the cubic function but having little to do with real market pricing. On the other hand I tried linear interpolation to avoid the aforementioned problem but I am subject to several break-points in the surface.And in both cases I have a slightly different result if I interpolate first by tenor & then by strike or viceversa.

Osvaldo,That sounds better. But (besides that fitting is not the very way without having a desk to check it and not the favourite one here) let me try to explain why you should not work with splines.The 'simpliest' things is to fit a smile - for every expiry. If you do that you first might accept that you do not have prices, but may be assumed prices (or settlement) and i assume your prices are correct (which is by no means clear if you have quotes).Then if you are fitting the smiles you are already neccessarily ignoring that positions are often adjusted against another through time (for example: a bottom is assumed and now in the long run one wants to play the correction of an inverse vol structure).This already are situations where you need a desk to judge the market (and one source for your troubles) - it means, that you do not fit 'local w r t time', but only 'punctual'. But let us stay to 1 expiry. If you now use splines you have a tendency to fit against prices to exact - which is also only 'to a point', not L^2 (which you need since BS-Vol here only gives you a v for any exercise K for fixed spot which changes in sequel). Thus it is neither clear that it depends C^2 on the spot nor that the risk neutral density gives a 'reseasonable' pdf - even if you respect some convexity on the price: there is also some convexity on the vol (but not a strict one). Or otherwise stated: within BS the price is an analytic function and thus its inverse vol is as well and splines do not fit in that setting.If you on the other side start with time axis you also would not merely fit a curve against data, but would have a model in mind which describes somewhat over time. Might it be Heston. Then you will get out a _theoretical_ behaviour over the expiries and you are fitting to determine the parameters - which is not done through splines (they do not obey the global nature of parameters). But if you take for example Heston the fit for the parameters is already given _except_ the short expiries and cross-checking the front and back month against that often fails (short time behaviour may merely determined from the long run). Again: even if you have a model backing out an ATM vol functional (and not explicitly analytic like Heston according to Gatheral (?) ) you will hardly catch its 'nature' through splines.If you ignore all model and desks then it is really arbitrary giving you no advantage do estimate anything ...So at least: do not use splines.

Thanks for your comments Avt.What should I read about the Heston model, since I dont know it?

Osvaldo,May be you look at Lewis book (at the bookshop). For splines find attached an article (N Jackson et al) which might help you.

I tried cubic splines and looked nice but sometimes (depending on the shape of the surface) some anomalies appear within BS the price is an analytic function and thus its inverse vol is as well and splines do not fit in that setting.you will hardly catch its 'nature' through splines.which is not done through splines (they do not obey the global nature of parameters). Are you referring to B-splines? And, the use of splines as "global approximant" as opposed to "local approximant" ?

WaaghBakri, a local approximant does not make sense, that's what i want to say.