# Foreign equity call struck in domestic currency

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 $$S$$?

• 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 $\phi$. Commented Jun 1, 2023 at 17:46
• 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. Commented Jun 2, 2023 at 8:04