Pricing a Futures Contract Across Two Currencies
Summary
The document addresses fair value when a futures contract is settled in one currency while its underlying spot price and dividends are quoted in another. Its central principle is to express the cash flows in a consistent currency and use the appropriate interest-rate discounting and foreign-exchange conversion. The accepted answer derives the relationship through a no-arbitrage replication: fund the foreign-currency stock purchase and dividends, then account for the cost of that financing in the domestic currency. The forward exchange rate links the two currency prices at maturity.
A second answer emphasizes that pricing uses a chosen numeraire and that one cannot assume risk neutrality under two numeraires simultaneously. The discussion gives a conceptual derivation, but the displayed steps use simplified discount factors and do not spell out all conventions, such as continuous compounding or dividend timing. Applying the relationship therefore requires consistent rate, FX, dividend, and settlement assumptions for the specific contract.
Key ideas
- Cross-currency futures fair value requires converting cash flows into a consistent currency.
- A no-arbitrage replication compares financing a foreign stock purchase with domestic investment and later purchase.
- The forward exchange rate connects the foreign-currency contract value to its domestic-currency value at maturity.
- Pricing is tied to a selected numeraire, so risk-neutral assumptions must be applied consistently.
- Rate conventions, dividend timing, and contract settlement details affect practical application.
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Full text
# Futures fair value with spot in different currency
# Futures fair value with spot in different currency
The fair value, $F$, for a futures contract is
$ F = S(1+rt) - D,$
where $S$ is the underlying spot price, $r$ is the interest rate, $t$ is the time to maturity, and $D$ is the dividends.
What is the corresponding fair value if the futures contract pays in currency $c_1$ and the spot price and dividends are in currency $c_2$?
## Answer by AFK (score 2, accepted)
https://quant.stackexchange.com/a/17504
You can either
- borrow cash now convert it and enter a forward contract for the stock in ccy2 and repay your loan at maturity
- invest your cash at the domestic risk free rate and buy the stock at maturity.
If there is no arbitrage between domestic and foreign markets, the two strategy lead to you receiving the stock 100% of the time so their cost should be the same.
In the first strategy, if $D$ is the total value at time $T$ of the dividends received by a stock holder between $t$ and $T$ then you need to pay $$ \frac{S_t}{P_2(t,T)} - D \qquad P_2(t,T) = (1+r_2)^{-1} $$ at time $T$ in ccy2 so you need to invest $$ P_2(t,T)(\frac{S_t}{P_2(t,T)} - D) $$ at the risk free rate $r_2$ to get this amount at time $T$. So you need to convert $$ X_t (S_t - P_2(t,T)D) $$ at time $t$ in domestic ccy 1 to fund the strategy. So at time $T$ you have to repay $$ \frac{X_t}{P_1(t,T)} (S_t - P_2(t,T)D) = X_t\frac{P_2(t,T)}{P_1(t,T)} (\frac{S_t}{P_2(t,T)} - D) $$ in domestic ccy 1. $X(t,T)$ is the forward FX rate. So the price of your quanto forward contract at time $t$ for maturity $T$ in ccy 1 is the price of the foreign contract times the forward FX rate $X(t,T) = X_t\frac{P_2(t,T)}{P_1(t,T)}$.
## Answer by Drew (score 2)
https://quant.stackexchange.com/a/17424
You need to express everything in the same currency, by converting it appropriately. You cant be risk neutral with respect to two numeraires at the same time, so the price you get will be in the numeraire with which you are risk-neutral to. This is called Seigel's Paradox.
So either convert the S, D or convert the F. It will likely be the S and D.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.