Short term implied fx rate

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.

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 $$T is the one I'm not completely sure how it would work. Since the fwd points before $$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

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.

• 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. Jun 27 at 9:33