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I'm trying to get a solution for the foreign equity call struck in domestic currency, where the foreign equity in domestic currency is defined as $S=S^fX^\phi$ with $0<\phi<1$, instead of the normal $S=S^fX$ (See Bjork 2020 for the standard setting).

Here it would be incorrect to assume that $S$ has a drift of $r_d$ (domestic rf) under $\mathbb{Q}^d$, as we would totally disregard the $\phi$ parameter. Is it ok to assume that the $\mu_s$ resulting from an ito's lemma of $S=S^fX^\phi$ under $\mathbb{Q}^d$ is the risk-neutral drift of $S$?

Thanks in advance

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  • $\begingroup$ We have zero incentive to see Bjork 2020 for the standard setting. If Bjork solves that pricing problem with $\phi=1$ you should show how that's done and highlight where you get stuck at other $\phi$. $\endgroup$
    – Kurt G.
    Commented Jun 1, 2023 at 17:46
  • $\begingroup$ Well if you have the risk-neutral dynamics of both $S^f$ and $X$ then yes just apply Ito's lemma to $S^f X^\phi$. And whatever drift you get, that will be the risk neutral drift. As an aside, if $S = S^f X$ then I'd definitely choose another symbol for the product $S^f X^\phi$ to avoid confusion. $\endgroup$
    – Frido
    Commented Jun 2, 2023 at 8:04

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