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Let $u$ be the continuously-compound borrowing rate in the UK and $r$ that in continental Europe. An investor wants to fix a exchange rate $V_{T}$ in a forward contract in which $V_{T}$ euros are exchanged for one pound at a time $T$.
I now want to know what the fair value of $V_{T}$ is if the current rate of the pound is $C_{0}$ euros. I know that the growth factor for continuously-compound with borrowing rate of the UK is equal to $\cfrac{F}{e^{ut}}$ where $F$ is the face value at time $t$ so for example if we are in time $T$ the growth rate would be $\cfrac{V_{T }}{e^{uT}}$, this is about all I know but that doesn't help me with anything. Can someone explain to me what I'm supposed to do here to calculate the fair value for $V_{T}$

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    $\begingroup$ Hint: the idea here is that there should be no arbitrage. Thus you should get the same amount, in sterling, if you just invest your pounds for the period at the UK rate or if you convert your pounds to euros, invest at the EU rate, and then convert back to pounds. $\endgroup$
    – lulu
    May 5, 2017 at 14:15
  • $\begingroup$ So I should have that $C_{0}e^{rT}=V_{T}e^{uT}$? $\endgroup$ May 5, 2017 at 14:23
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    $\begingroup$ That's all it is (trusting that your conversion factors all go the same way, of course). $\endgroup$
    – lulu
    May 5, 2017 at 14:27

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"Fair" means no opportunity for arbitrage (taking advantage of price differences to make a profit). In other words, we should have the same amount of wealth at time $t$ regardless of what action we take.

If we know $1$ pound = $C_0$ euro at the present time, then at time $t$ we'll have $e^{ut}$ pounds and $C_0e^{rt}$ euro. Divide both sides by $e^{ut}$ and we see that at time $t$, the exchange rate should be $1$ pound = $C_0e^{(r-u)t}$ euro, so this should be the agreed-upon exchange rate in the forward contract.

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