Quanto Option Replication Requires Foreign Cash and Currency Conversion

Summary

The document considers whether ordinary delta replication can hedge a quanto option, whose payoff is domestic-currency-denominated despite exposure to a foreign asset. The answer explains that holding delta in the foreign stock and borrowing in domestic cash does not by itself form a self-financing replicating portfolio. A foreign cash account is also needed to support the foreign investment and account for its financing and capital gains.

Because the target payoff is domestic, foreign asset values must be converted through the exchange rate. The answer identifies three components for replication: the exchange-rate-converted foreign stock, the exchange-rate-converted foreign cash account, and domestic cash. It does not quantify hedge error or derive a delta formula, and it offers no market conditions under which a simplified hedge would work. Its core lesson is structural: replication must include the relevant currency and financing accounts, not just the underlying stock and domestic borrowing.

Key ideas

• A delta position in foreign stock financed only with domestic cash is insufficient for self-financing replication.
• A foreign cash account is needed to represent financing and capital gains associated with the foreign asset.
• Foreign asset values must be converted into domestic currency when replicating a domestic payoff.
• The described replicating portfolio includes converted foreign stock, converted foreign cash, and domestic cash.

Tags

# How bad off are we when we use the "regular delta replication" strategy in an FX market on a Quanto?


# How bad off are we when we use the "regular delta replication" strategy in an FX market on a Quanto?












See this question for context:

https://quant.stackexchange.com/questions/32725/dynamic-hedge-of-quanto-options#=

In there, I expressed interest in how well the usual strategy of replicating an underlying stock would work in this case. This strategy would consist of investing Delta (derivative of the option's price wrt stock price) in the foreign stock, and borrowing domestically.

Are there some market conditions under which that strategy actually would work? An idea of one such condition could be that the exchange rate remains constant over the relevant period?

## Answer by Gordon (score 1, accepted)

https://quant.stackexchange.com/a/32802

You can borrow domestically. However, we note the following:

- You can not use your domestic cash to buy foreign stock directly, instead, you also need a foreign cash account.

- Moreover, for the replicating portfolio to be self-financing, the capital gain from the foreign stock and the cash account need to properly adjusted, that is, the foreign cash account is necessary.

- Since you are replicating a domestic payoff, all the foreign asset values need to be converted to domestic currency.

In summary, you need instruments $XS$, $XB^f$, and $B^d$. By missing any of them, you won't be able to have a self-financing replicating portfolio.

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