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I have American option prices for SPY and need to calculate the equivalent European option price to use in further calculations.

What does it (formally) mean to price the equivalent European option from an American option?

So I have $C_{\text{American}}(K, S, r, T, \delta)$, how do I retrieve $C_{\text{European}}(K, S, r, T, \delta)$?

Edit: are there any methods to price the European put (call) from the market price of the American put (call)?

If the $P_E - P_A \leq 0$ (the American is strictly greater than the European due to exercise premium), will attempts at modeling the time-dependent dividend cash flow provide a model-free approach to price the European counterpart? How do varying interest rates impact these results? Can interest rate variation be safely ignored from some threshold of $T$?

Edit 2: I also can observe the price of the American call. I want to use calls and puts together to strengthen my call or put price curve quotes (or other calculations that are derived from them) knowing that OTM options are much more liquid. So even though I cannot trade the ITM call as well, I can get a better idea of its value for modeling purposes from the corresponding put. This is along the same line of thinking because if I could convert to European options I could utilize strict parity, so a method to price the European counterpart would be pricing the source of disparity between the European and the American.

Edit 3: I have found a resource (from 1987) that makes attempts at analytically valuing the difference of an American exercise vanilla vs European exercise:

REFERENCES

Barone-Adesi, Giovanni & Whaley, Robert E, 1987. " Efficient Analytic Approximation of American Option Values," Journal of Finance, American Finance Association, vol. 42(2), pages 301-320, June.

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    $\begingroup$ You left out the key parameter in this operation: $\sigma$ !! $\endgroup$
    – nbbo2
    Nov 10, 2016 at 17:55
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    $\begingroup$ I agree with @noob2. However, just to be pedantic, you actually left out the model + pricing method parameters. This could be the BS volatility plus some parameters describing an FD scheme for instance (what kind of grid, what kind of scheme), or Heston parameters and parameters describing a Monte Carlo scheme (what discretisation scheme, how many paths). Both matter. Because even if we agree on a BS dynamics for instance we could have different results depending on how pricing is conducted. $\endgroup$
    – Quantuple
    Nov 10, 2016 at 18:48

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First, you need to specify your working modelling assumptions by selecting a (jump)-diffusion framework, or more exactly, by postulating the risk-neutral dynamics of the underlying (e.g. Black-Scholes).

Then, you choose a pricing method, which should allow you to price both American-style and European-style vanillas (e.g. a binomial lattice)

At this point, using the above pricing method, you should be able to calibrate your model parameters so that they allow you to reproduce the American option price.

Eventually, you can use these calibrated parameters (e.g. the vol, forward and discount actor in BS) to price the European-style counterpart of your American contract.

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