Given I have all other parameters lined up with the market (borrow rate, dividends...), if I imply volatility using CRR tree from american call and put with the same strike and expiry, will I always get the same vol parameter?

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    $\begingroup$ Generally not, unless market prices are also generated from a CRR model. American options are path-dependent and models with different dynamics leads to different prices and thus different CRR implied volatilities. $\endgroup$ – LocalVolatility Jun 8 '18 at 7:41
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    $\begingroup$ Agreed. Also even with a fixed model these instruments embed different volatility information in general. Consider for instance 2Y American listed options, with a huge dividend to fall in say 6M. The American call price will essentially be a function of the volatility up to 6M due to high early exercise probability. On the other hand, the American put price will depend on the volatility up to 2Y. So you see that calibrating to each individual instrument does not give the same information. You could try to price the European counterparts and make sure that their implied vols match. $\endgroup$ – Quantuple Jun 8 '18 at 8:45

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