Pricing a Foreign-Currency Equity Call in Domestic Currency

Summary

The document considers a call option on a foreign equity with its strike denominated in the foreign currency, while the payoff is valued in domestic currency. At expiration, the domestic payoff is the exchange rate multiplied by the positive difference between the foreign share price and the strike. The question asks how to price that payoff and whether its terms should be modeled separately.

The answer outlines two risk-neutral pricing routes. One prices the full payoff under the domestic money-market numeraire, which involves a joint, two-dimensional expectation over exchange rates and equity prices. The simpler route values the foreign-currency option under the foreign numeraire and converts its value at the current exchange rate. The document stresses that the relevant dynamics must be consistent with the chosen numeraire. It gives no worked numerical example or assumptions about the underlying market model.

Key ideas

• The domestic payoff depends jointly on the foreign share price and the exchange rate at expiration.
• A domestic-numeraire valuation treats the payoff as a joint expectation over two risk factors.
• A foreign-numeraire approach prices the foreign-currency option first and converts its current value at the spot exchange rate.
• Pricing dynamics must be adjusted consistently when changing numeraires.

Tags

# Foreign equity call struck in foreign currency, expressed in domestic currency


# Foreign equity call struck in foreign currency, expressed in domestic currency












A foreign equity call, struck in foreign currency, i.e. an option to buy one unit of the foreign equity at the strike price of K units of the foreign currency. The value of this claim at the date of expiration is, expressed in the foreign currency, given by $$Z_f = max [S_f (T) - K,0 ]$$

Expressed in terms of the domestic currency the value of the claim at T is $$Z_d = X(T) max [S_f (T) - K,0 ]$$ What is the time no-arbitrage price of the payoff $Z_d$? My first approach is to calculate these two terms $X(T) \cdot S_f (T)$ and $X(T) \cdot K$ separately to derive it's risk neutral dynamics. In this term $X(T) \cdot K$, the strike price $K$ is a constant so what should we do here?

## Answer by Andrea (score 1)

https://quant.stackexchange.com/a/81373

There are 2 ways of doing it.

First, as you suggested (complicated) as a 2-D domestic payoff.

Basically $N_d(0) E^d \left [ \frac{X(T) (S(T) - K)^+}{N_d(T)} \right ]$

which is a (non trivial) 2-D integral.

Alternatively, the easy way is to convert the foreign price at the 0 exchange rate

$X(0) N_f(0) E^f \left [ \frac{(S(T) - K)^+}{N_f(T)} \right ]$

(it goes without saying that the dynamics of everything have to be adjusted to the correct numeraire)

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.