# 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 $c_2$?

• You should check the contract specs, for instance read this quick note on nikkei index futures , cmegroup.com/trading/equity-index/files/… Apr 21 '15 at 4:37
• @pyCthon, The contract specs does not answer the question of how to calculate the fair value, nor does the link you provided.
– RRG
Apr 21 '15 at 6:09

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 $X(t,T) = X_t\frac{P_2(t,T)}{P_1(t,T)}$.

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.

• Can you elaborate. Will the fair value depend on the interest rates of the two currencies and the forward exchange rates? What will the exact form be?
– RRG
Apr 18 '15 at 10:05
• Mmm....For risk neutrality pricing to work (replacing mu with risk ree rate r)you need compelete markets.The forward price of a traded quantity isn't hence the market's expected value in the future, as there is an arbitrage between the spot price and forward price via interest rate linking. Model currency is just modlling/base currency, is not magical. Because of the cross currency basis spreads which are forecasted by finger in the wind. Apr 20 '15 at 4:54