Inferring Domestic FX Rates from Forward Points

# Implied forward rate with forward points


# Implied forward rate with forward points












I've been studying the application and derivation of a domestic implicit rate in a FX contract when your input are the forward points and the foreing rate. Let's establish some with some ideas first.

The FX market revolves around the spot rate and it's forwards around forward points. Which are added/substracted to the spot rate to get the forward rate that corresponds each settlement date.

Let's use for example the EURUSD forward points (just to be clear, price of 1 EUR in USD), and using mid values for simplicity. Consider a spot rate of 1.1049 EUR per USD. With the usual spot (t+2) settlement, in this case march 22.

And the next forward points

| T | Dates | Pts |
| ON | march 21 | 0.733 |
| TN | march 22 | 0.273 |
| SN | march 23 | 0.278 |
| 1W | march 29 | 2.09 |
| 2W | april 25 | 4.93 |

Now, from this one could calculate the forward rate to those settlements, for example for the 1Week forward would be: 1.105109. And by equating this to the usual no-arbitrage forward pricing formula get:

$$ fwd_{t_0,1W} = S_0 \frac{(1+r_d(1W-t_0))}{(1+ r_f (1W-t_0))}= 1.105109 $$

However, since we are working with the spot rate which would be settled in t+2, it seems to me that the correct equation would be:

$$ fwd_{t_0,1W} = S_0 \frac{(1+r_d[t_2,1W](1W-t_2))}{(1+ r_f[t_2,1W](1W-t_2))}= 1.105109 $$

In which $ r_d[t_2,1W]$ would be the forward domestic (usd) rate from $t_2 $to 1W settlement, or in this case march 22 until march 29.

From here let's assume you already have a term structure as the foreing euro rate, from which you could determine the $ r_f[t_2,1W]$. In which case you could solve the equation for $ r_d[t_2,1W]$ to know the implicit domestic rate. And more generally you could solve for all the settlements and through an interpolation method find any $ r_d[t_2,T]$ to create a spot zero curve. However you are lacking the information of $r_d$ from $t_0$ to $t_2$.

So now, if you want to value another forward on the spot rate in $t_0$ and with a given forward price K, which expire in T, I would do this:

$$ \frac{(fwd_(t_2,T) - K )}{(1+ r_d[t_0,T](T-t_0) )} = \frac{(fwd_(t_2,T) - K )}{(1+ r_d[t_0,t_2](t_2-t_0) )(1+ r_d[t_2,T](T-t_2)} $$

So how does one know $r_d[t_0,t_2]$ ? I'n guessing through the ON and TN forward points, but I'm not sure the same forward equation holds, since those points are before the spot date.

Also any insight on how those points are used in the market would be appreciated.

Much help appreciated.