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.