Currency Forward Contract

Summaries

Longing currency forward contract is a commitment to buy foreign currency at a particular rate in the future. So, foreign currency (FC) is just like a kind of asset and you buy it with domestic currency (DC). If you expect the price of FC to rise (more than your contract rate), you long it and you will earn money.

To price the forward contract, think of this: you are buying a unit of FC in the future (T). Now you can borrow 1 unit of FC and exchange it to DC, this is just the spot rate (S0). At T, you will sell DC to buy FC at the contract rate (FR) and pay back the money, so the FR = S0 if there is not interest. But when you borrow 1 unit of FC today, you pay (1+R_FC)^T at time T. However, at the same time, you earn (1+R_DC)^T when you hold the DC during this period. Therefore,

FR = S0/(1+R_FC)^T * (1+R_DC)^T

Similarly, we can derive a more generally form for contract value at time t:

V(t, long) = S(t)/(1+R_FC)^(T-t) – FR/(1+R_DC)^(T-t)

(This means, I borrow 1 unit of FC today and convert to S(t) unit of DC. Then at time T, I have to pay back (1+R_FC)^(T-t) of FC. However, during this period, the S(t) unit of DC also grew by (1+R_DC)^(T-t). So PV of the amount I earn is discounted S(t) – PV of FR due to the presence of interest rates.)

For continuous compounding,

V(t,long) = S(t) exp(-R_FC)^(T-t) – FR exp (-R_DC)^(T-t)

However, is this correct? Should it be more accurate to use:

V(t, long) = S(t)– FR/(1+R_DC)^(T-t) * (1+R_FC)^(T-t)