For european options, we know the delta of a synthetic forward position (long call short put with the same strikes and maturities) because of the put-call parity. However the P-C parity does not apply to american options. Is there a nice formula for the delta of a synthetic forward of american options?

  • $\begingroup$ There is no closed formula for the Delta of American puts, this interesting post shows how they differ qualitatively from Europeans: the delta is lower for the American on the left side of the graph. quant.stackexchange.com/questions/35163/… $\endgroup$ – noob2 Apr 13 '20 at 16:15
  • $\begingroup$ noob2: Thanks for providing the link! It is even true that there is no formula for delta of european option in some sense (it is model dependent), however for european synthetic forward the delta is 1 (with a discount factor). So I am just wondering if there is a formula for the delta of american synthetic forward even if we dont have the formula for one leg. $\endgroup$ – CABLE Apr 13 '20 at 16:28
  • $\begingroup$ I believe qualitatively Delta is going to be slightly less than one if it is built from American options (which BTW means that is not a perfect Synthetic Forward, though probably close enough for most uses, unless the Strike is very low) but I don't know any quantitative answer. $\endgroup$ – noob2 Apr 13 '20 at 16:54
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    $\begingroup$ I think it is probably the opposite (greater than 1) according to your link. For non-dividend paying underlying, american call delta is the same as european call delta however amercian put delta is more negative than european put delta. $\endgroup$ – CABLE Apr 13 '20 at 17:02

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