If we have a set of volatilities for different maturities and different strikes and we wish to interpolate for a period/strike not on the grid what is the market standard way of doing this please? By way of example we have 2 and 3 year 4 and 5 % strike levels but want 2.5 year 4.5% strike. I am assuming vols. are quoted as Black vols.

Could try by turning the vols into prices, doing a spline surface interpolation and moving back into vols again by inverting the blacks formula? Vols are lumpier than prices so clearly interpolation over them would be a little crankier - but if you only have 4 coordinates on your surface (2yr,4%), (3yr,4%), (2yr,5%) and (3yr,5%) any interpolation scheme's going to be a little best-guess.

Can you recommend a paper that details how one might go about doing this please?

errr, not sure of a paper on its own but here's the basic set up.1) take a cap of maturity T and strike K with vol V. convert this vol into a price using black scholes [it's in Hull]2) do this for your entire vol surface until you have a price surface3) in numerical recipes (you can find it online) there are details of different interpolation routines; linear, cubic spline and i think it talks about a 2d-spline interpolation (this is what you need for a smooth surface interpolation)4) 2d spline interpolate your price surface to get the price for the unknown point (2.5yr,4.5%)5) take this cap price and calculate the flat vol that it implies (can be done by simple bisection).hope this is clearer, j

Hi Firstly you should use VARIANCE interpolation.as variances are additive (statistical property) all interpolation (or extrapolation) should be done with the variance and then take the square root. For example, when calculating forward vol, we use the relationship:v1^2(t1-t0) + v12^2(t2-t1) = v2^2(t2-t0) ; ie we can add variances. ==> forward vol = v12 = SQRT([v2^2(t2-t0) - v1^2(t1-t0)]/(t2-t1))with regard to your problem, there are many forms of interpolation (ie linear .......cubic spline) check out QuantLIB. It has a wide range of methods.

Yes thank you very much.

Mutley, Doublebarrier,If I understand well, there are two proposals :* interpolation on cap price along a strike* interpolation on caplet variance (forward-forward volatility) along a strikeI have several questions :* do you first clean your inputs from the broker market ? how is it done ?* do you interpolate in one direction (time) and then in the other direction ?* do you recommand particular interpolation technic ?RegardsCalculator

My relatively simple solution is this, does it sound somewhat reasonable? I will take vol surface from broker market and linearly interpolate across time and skew on the cap/floor vols. I will then compute fwd/fwd vols on a deal specific basis if I need to, but otherwise will just use the interpolated vol. So for example if we have 2 & 3 year 3% and 4% caps and they are as follows 2y3% 10, 2y4% 12, 3y3% 14, 3y4% 16. My 2 1/2 y 3.5% would then be 13%. From a risk perspective "vol" exposure would be allocated across the 4 reference caps. Any comments gratefully appreciated.

Anybody any comments on the method suggested please?

I would caution against an interpolation based scheme. One can never be sure if the interpolated vols are self-consistent and arbitrage-free. This is critical for hedging especially for a desk with huge exposure over a large number of strikes and tenors. That's why mathematically-consistent stochastic vol models such as SABR are typically used and calibrated against market data for pricing within a smile environment.

shame on you, let people do the interpolation and start trading. They should soon find out the error of their ways.

Alvinkam and RikhadI would like to share your thoughts about a few questions:Can't the caplet vol bootstrapping (from the flat vol) safely (from a theoretical pt of view) be made before any proper calibration phase?And isn't the hedging improvement motivations more important than simple "arbitrage avoiding" logics for one who chooses a more complex model like SVMs? (sometimes, the smoothing of the input curve is done or should be done even in this later case)regards

Yes, some smoothing/interpolation can be 'safely' done before calibrating a stochastic vol model. What I am cautioning against is the usage of vol interpolation for actual pricing. This is because one may unknowingly run into circular pricing inconsistencies leading to potentially disastrous losses from simple long-short arbitrages from counterparties. A stochastic model, being mathematically consistent, at least protects you against this (at the price of some model risk).