Calculating the fair value of an exchange rate

Question

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 $V_{T}$

Answer 1

"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.