Risk-Neutral Drift for a Domestic-Currency Foreign Equity Call

Summary

The question considers pricing a call on a foreign equity whose domestic-currency value is defined as the foreign share price multiplied by the exchange rate raised to a fractional power. It asks whether the drift obtained by applying Itô’s lemma under the domestic risk-neutral measure can be used as the asset’s risk-neutral drift, since the usual domestic risk-free drift would ignore that power parameter.

The document presents the modeling issue but gives no answer, derivation, pricing formula, or supporting evidence. It therefore highlights a useful distinction between the standard currency-converted equity and a modified payoff underlying, while leaving unresolved which no-arbitrage assumptions and measure changes apply. Any conclusion would require a fully specified model for the foreign equity and exchange rate, including their volatilities and dependence.

Key ideas

• The underlying is defined as a foreign equity price multiplied by the exchange rate raised to a fractional power.
• The question challenges assigning the domestic risk-free drift to this modified underlying under the domestic risk-neutral measure.
• It proposes deriving the drift with Itô’s lemma but supplies no solution.
• Pricing conclusions depend on the joint model for the equity and exchange rate.

Tags

# Foreign equity call struck in domestic currency


# 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 $S$?

Thanks in advance

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