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Suppose there is an underlying XYZ trading at \$100. Suppose I also have the price of exchange traded XYZ options for \$60, \$80, \$90, \$100, \$110, \$120, \$140 with 60 dte, 90 dte, and 120 dte.

Are there methods to extrapolate this data to the price of options with different strikes and dte? Continuing with the example, could I robustly estimate the pricing of a $87.5 put expiring in 30 days? What about 14 or even 7 days?

In the real world, at least for very liquid underlyings and options, very roughly, how accurate is this? If there is a method, I can of course evaluate how good it is by comparing its output for options that are also exchange traded but are not included in the model.

To be clear, my end goal is estimating the price of options, more specifically the price at which they actually are actually traded, without having access to the price of those specific options, but having access to a bunch of prices from the rest of the option chain.

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There is a difference between extrapolation and interpolation, and strike-wise and time-wise components. I will consider all 4 cases.

If you are interpolating for a different strike between strikes of existing options, for single expiration, then any parametric or non-parametric methods mentioned by @raptor22 will work fine, and you can trivially check for no-arb bounds.

Extrapolation on the same expiration is little trickier - Gatheral's models are arbitrage free in the wings, but others may not be. This is where you have to check resulting vols against Roger Lee's skew bounds.

Interpolating in time is trickier. Usually you can assume smooth evolution of vol surface ( or a part of it ) if there is no deterministic event between expirations ( like earnings announcement or FED meeting ) and use your favorite vol model to interpolate ( probably better in LMM than in strikes ) Checking for no calendar arbitrage is discussed further here

Extrapolating in time is the most complex of 4 cases. In practical settings either you have to define volatility dynamics for very short-term high gamma options, or for long term high vega options, with possible macro factors in mind. Once I was asked to price 10+ year out SPX puts, and after all the modelling did not have any strong confidence in the number.

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  • $\begingroup$ I'm accepting this answer because it's more complete and I have a better picture of the problem now, and more things to research. $\endgroup$ – Eugenio Bruno May 6 at 17:11
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You would usually use the following steps:

  1. Express your quotes in the Black-Scholes (BS) parametrization: by inverting BS formula, you now have implied volatilities for each strike and tenor that you observe.
  2. Then, you can interpolate/extrapolate the whole implied volatility surface using a methodology that guarantees absence of arbitrage in the resulting quotes.
  3. Once you have built your implied volatility surface, you can get your quote from this surface by applying BS.

I have assumed European exercise style options for simplicity but you could do the same for American options by replacing BS by e.g. a binomial tree. Also, there are many technicalities involved with building such a surface, such as: using the right funding rate (risk-free rate in BS), dividends, and mainly guarantee the absence of arbitrage in the resulting quotes. Some models:

  • Gatheral's SVI and SSVI parametric models.
  • Other methods based on non/semi-parametric models (splines etc.), e.g. this article

If you would like a quick and dirty approach that could still give reasonable results in your exemple, first try to just do a linear/splines interpolation between the implied volatilities in the strike direction.

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