Interpolating Vanilla Option Volatility Smiles Across Expiries
Summary
The document considers how to construct an implied volatility surface from listed call and put options when the intended use is pricing other vanilla options. The proposed starting point is to interpolate each expiry’s smile with cubic splines across strike or delta, then interpolate total variance across expiries for at-the-money-forward volatility. It identifies a dependency in that workflow: without a zero curve or forward, the forward may need to be inferred from call-put parity before locating the at-the-money point and estimating its volatility.
The central unresolved problem is how to carry smile shape between expiries; the author asks whether a model such as local volatility is appropriate. The response emphasizes that option style and available inputs matter, and that deriving implied volatility from prices requires assumptions about rates and dividends. It notes that splines are straightforward but may introduce arbitrage or approximation error. No interpolation method is selected or validated, and the discussion does not provide arbitrage tests or a completed surface construction.
Key ideas
- A proposed workflow interpolates smiles within expiries and total variance across expiries for at-the-money-forward volatility.
- Forward levels and an approximate zero curve and dividend yield may be needed to infer implied volatility from prices.
- Spline interpolation is simple to apply but may introduce arbitrage or approximation error.
- The document leaves cross-expiry smile interpolation unresolved.
- European and American options may require different treatment.
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Full text
# simple volatility surface interpolation # simple volatility surface interpolation I'm trying to build an implied vol surface from some listed options. In particular I have data for calls and puts for different strikes and expiries. I'm not looking to price on the interpolated vols any exotic payoffs, just other vanilla options. To keep things very simple, I'm thinking to use the following approach: - Use cubic splines for smile interpolation on either delta or strike, for a given expiry - Use total variance interpolation across expiries, for ATM vol Where ATM is ATM forward (i.e. ATM strike = fwd) While the first point is probably fine, I think the second has two main issues. First of all, I don't have a zero curve or forward. So I would need to first get the fwd from the call-put parity, for each expiry. Then use the interpolation of 1. to get the ATM implied vol, and then use that vol for the total variance curve (so building the ATM curve on some interpolated values). Second, 2. only works for the ATM vol curve, and the key issue becomes, how do I interpolate smiles across expiries? So my question is, what's the best way to interpolate smiles between expiries? Also would an another approach e.g. local vol, be more appropriate in this situation? (given that I'm not looking to price exotics) ## Answer by yungpadewon (score -1) https://quant.stackexchange.com/a/72009 First, are you building the surface for european style options or american style, different complexities altogether for american style.. that being said, what data do you have to begin with? just price data? if so, then yes you need to find a way to calculate the zero curve & the dividend yield to get an IV. Once you have price data, and some approximation of zero curve & dividend yield, you can convert from price space to vol space quite easily. Spline would certainly be the easiest to do - but also would contain the most arb // margin of error. I'd be more than happy to give further thoughts once I know what data / inputs you are working with and what is your arb tolerance level
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