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For context, I'm working towards constructing a FX implied rate curve based on the fwd points market. As you know, most of the spot rates usually settle in t+2. We can group fwd points in two types, the ones that settle after the spot date $T>T_s$ and the ones that do it before it $T<T_s$.

In the case where $T>T_s$, one could use the usual non-arbitrage argument, if you have 1 USD:

1USD 1USD
Case 1 Case 2
Convert to EUR with the EURUSD spot rate $S_{T_s}$ in $T_s$ Invest in USD rate until $T$
Invest from $T_s$ to $T$ at the EUR rate. Enter into a fwd to convert USD to EUR in $T$
Profit in $T$: $S_{T_s} (1+r_{eur(T_s,T)}\frac{(T-T_s)}{360})$ Profit in $T$: $F_{T_s} (1+r_{usd(T_s,T)}\frac{(T-T_s)}{360})$

Thus creating the usual parity:

$$ F_{T_s,T} = S_{T_s} \frac{(1+r_d[T_s,T](T-T_s))}{(1+ r_f[T_s,T](T-T_s))} $$

In this case if one knows the value of $S_{T_s}$ and asumes a value for $r_f$ one could solve for $r_d$ to get the implied rate.

However the case when $T<T_s$ is the one I'm not completely sure how it would work. Since the fwd points before $T_s$ are the ON and SN points which are actually FX swaps points, in which one could buy/sell the FX but would have to do the opposite operation on maturity date it creates a different dynamic.

For clarity I would describe this point to my understanding, please do tell if I'm making an incorrect assumption at this point:

ON FX points trade: Buy/Sell 1 USD today $t$ at the ON rate (assuming mid values Spot -TN points - ON points), then Sell/Buy 1 USD tomorrow $t+1$ at the TN rate (assuming mid values Spot -TN points).

TN FX points trade: Buy/Sell 1 USD tomorrow $t+1$ at the TN rate, then Sell/Buy 1 USD on the spot date $t+2$ at the spot rate.

With this in mind, I think the investment strategy from $t$ to $T_s$ would be:

For, $t=today<t+1 = tomorrow < t+2 = spot date$

1USD 1USD
Case 1 Case 2
In $t$: Convert to EUR by entering into the ON FX swap selling 1USD at the ON EURUSD rate $S_{ON}$ then invest $S_{ON}$ at the EUR 1 day rate $r_1$ In $t$: Invest in USD rate until $T_s$
In $t$: Enter into an SN FX swap to sell 1 USD at the SN EURUSD rate $S_{SN}$ in $t+1$ In $T_s$ Sell the USD with interest at the spot rate in $T_s$
In $t$: Enter into a spot trade to sell 1 USD at the Spot EURUSD rate $S_{T_s}$ in $T_s$ Profit in $T_s$: $S_{T_s} (1+r_{usd(T_s,T)}\frac{(T-T_s)}{360})$
In $t+1$: Recieve $S_{ON} ( 1+ r_1)$. Since you entered a FX swap the day before, now you have to buy 1 USD at the TN rate $S_{TN}$, assuming $S_{ON}( 1+ r_1)> S_{TN}$. You could use that to buy the 1 USD, and resell it at the $S_{TN}$ with the SN FX swap. Finally, you could reinvest $S_{ON}( 1+ r_1)- S_{TN} + S_{TN}$ at the one day rate from $t_1$ to $T_s$ $r_{1,2}$.
Finally in $T_s$: Recieve $S_{ON}( 1+ r_1)$ (1+$r_{1,2})$. Since you entered a FX swap the day before, now you have to buy 1 USD at the spot rate $S_{T_s}$, assuming $S_{ON}( 1+ r_1)$ (1+$r_{1,2})$> $S_{T_s}$. You could use that to buy the 1 USD, and resell it at the spot rate trade $S_{T_s}$.
Profit in $T_s$: $S_{ON}( 1+ r_1)$ (1+$r_{1,2})$

This would mean that: $$ S_{ON}( 1+ r_1) (1+r_{1,2}) = S_{T_s} (1+r_{usd(T_s,T)}(T-T_s)) $$ If,

$$ r_{1,2} = (\frac{(1 + r_{0,2} t_{0,2})}{(1 + r_{0,1} t_{0,1})}-1) * \frac {1}{t_{1,2}} $$

$$ S_{ON} (1+ r_{t,T_s}(T_s-t)) = S_{T_s} (1+r_{usd(t,T_s)}(T_s-t)) $$

Which would imply in the short term, the spot rate is a forward rate. Does this seem correct?

Thanks to anyone who could provide some feedback.

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  • $\begingroup$ I did not check all details of calculations, but you have the right idea: if you need the foreign currency tomorrow, you need to do a Spot trade and a TN swap (in the backwards direction). If instead you need the currency today you need an additional ON swap (again backwards). This is how maturities < spotdate are handled. And yes, "spot" can be considered the forward rate for T+2 when you start at T or T+1. $\endgroup$
    – nbbo2
    Jun 27 at 9:33

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