Deriving FX Swap Points from Cross-Currency Discount Factors

# Calculating FX Swap Points from various interest rate curves (and vice versa)


# Calculating FX Swap Points from various interest rate curves (and vice versa)












I have tried extensively to find the answer to this both on this website and externally. My question is 'how are FX swap points determined?' but would like to lay out my own understanding first (also for the benefit of others).

I have always been under the impression that you can replicate the FX Swap by transacting an Interest Rate Swap (IRS) in both currencies, which we know is an exchange of fixed for floating cash flows, with the floating leg calculated off the LIBOR or foreign LIBOR equivalent.

We now have a fixed v floating obligation in two currencies, and no exchange of principal. To then replicate the FX Swap, we now trade a Cross Currency Basis Swap (CCBS), which exchanges notional at outset (converted at the spot FX rate) as well as floating interest payments in the two currencies. The direction of the CCBS is traded such that our floating obligations via the two IRS traded previously have no been cancelled out, with the actual basis differential on foreign LIBOR remaining (the premium or discount foreign LIBOR is quoted via the CCBS).

My question in all this is - where do OIS fit in with all this, given they are also fixed for floating swaps that reference the equivalent overnight central bank rate (as opposed to LIBOR) and also there is no exchange of principal. If there was a Basis swap that would negate the floating cash flows in both currencies OIS, I could understand that. But as far as I am aware, that doesn't exist.

I guess my ultimate question is how are we solving fixed interest rates in currencies using the OIS, without an equivalent product to cancel out the floating payments (unless with my example above where using IRS + CCBS replicates a fixed loan/deposit i.e. FX swap perfectly)?

FYI for anyone reading this, FX swaps generally have one notional fixed with the differential cleared in the Spot FX market, at which point you can compare the FX swap to a loan/deposit in two currencies.

EDIT: While I know the mechanics of the CCBS aren't as simple as above, I still think the IRS/CCBS combo replicates an FX swap - if it doesn't, what does? What actual instruments can be used to exploit arbitrage?

Thanks

EDIT#2: Hi Attack68 - I know the FX forward/swap is the interest-rate adjusted spot price. To frame my question another way - how would I replicate an FX forward with various underlying instruments (IRS, CCS [fix of Libor] and/or OIS[fix of OCR])?

Some thoughts: - Commonly held that the CCS is back solved from the swap points under 2 years for most G10 pairs, while the inverse is true over 2 years (CCS is used to solve for forward points). This is why I suspect there is a way replicate the FX Swap/Forward with these instruments. - Where do the OIS fit in with all this then? As I questioned above, given CCS fix off Libor but OIS fix off the OCR, I am a bit confused.

I am mindful of the blow up we saw in AUDUSD CCS over the 18/19 holiday period and it's impact on the swap points, which has made mindful of not truly understanding how the points are derived.

If you could lay out a step wise process that would be helpful - I read from above it is only the CCS you need to calculate the discount factors.

Some thoughts: - Commonly held that the CCS is back solved from the swap points under 2 years for most G10 pairs, while the inverse is true over 2 years (CCS is used to solve for forward points). This is why I suspect there is a way replicate the FX Swap/Forward with these instruments. - Where do the OIS fit in with all this then? As I questioned above, given CCS fix off Libor but OIS fix off the OCR, I am a bit confused.

I am mindful of the blow up we saw in AUDUSD CCS over the 18/19 holiday period and it's impact on the swap points, which has made mindful of not truly understanding how the points are derived.

If you could lay out a step wise process that would be helpful - I read from above it is only the CCS you need to calculate the discount factors.

## Answer by Attack68 (score 2)

https://quant.stackexchange.com/a/50960

A single period cross currency swap (XCS) will replictate an FX swap, that is because both instruments involve only 4 cashflows:

2 at the start in 2 different currencies, and 2 at the end in two different currencies. The semantics about how those cashflows are generated might be different but the economics and fair values of the cashflows are obviously the same amounts: otherwise one would have a positive/negative NPV.

In order to calculate the FX Swap points you need to generate the forward FX rates.

$$ f_i = \frac{w_i^*}{v_i} F_0 $$

where $f_i$ is the forward FX rate at date/node $i$ and $F_0$ is the initial/spot FX rate and $v_i^*$ is the foreign discount factor and $w_i^*$ is the domestic discount factor using the foreign currency as a collateral basis, e.g:

$$ \text{(EURUSD)} f_i = \frac{\text{(EUR DF: usd collateral)}\; w^*_i}{\text{(USD DF: usd collateral)}\;v_i} \text{(EURUSD)} F_0 $$

The discount factors are derived from XCSs to satisfy the MTM values of those the cross currency swaps (in this case those which are collateralised in USD).

EDIT

You cannot replicate the mechanics XCS with one or more FX swaps and vice versa. They are distinct products, however, they both rely on the same pricing mechanism. Similar to an FRA and an interest rate future. One cannot replicate the other but they both rely on the same pricing mechanism; another example is interest rate swaps and zero coupon swaps. Furthermore there are non-mtm XCSs and mtm XCSs, you cannot replicate non-mtm XCSs with mtm XCSs. (Replicate means fully reproduce all exposures (delta, gamma, vega, etc) and cashflows without requiring further dynamic hedges)

However, my comment about a single period XCS and FX-Swap being the same is true. Here is an example:

Suppose the following data is available:

USD 3M Libor: 2%, USD 3M OIS: 1.9% EUR 3M Euribor: 1%: EUR 3M OIS: 0.95% EURUSD Fx: 1.10

If you received EUR 100mm EUR/USD 3M XCS @ -10bps you would have the following cashflows in USD (todays FX rate):

```
($-today)       EUR             USD          Date
nominal         -120mm          +120mm       today
ibor            +0.3mm          -0.6mm       3m
basis           -0.03mm         n/a          3m
nominal         +120mm          -120mm       3m
```

Now let's suppose that the CSA (credit support annex) on this trade is USD collateral (this matters because of how will will discount the cashflows) then using the USD OIS rate we can determine 3 of the 4 necessary discount factors:

```
(DFs)           EUR             USD          Date
                1.00000         1.00000      today
                w_i*            0.99527      3m
```

How do we calculate `w_i*`? We use the fact this XCS is mid-market i.e. MTM is zero so the cashflows should sum to zero. If you plug all these values into your excel you will find the solution is that `w_i*` is 0.998.

Now go back up to the top of the page and you can derive $f_i$, the forward exchange rate, which also gives you the 3M FX-swap price of +30 pips:

$$ f_i = \frac{0.998}{0.99527} 1.1 = 1.1030 $$

The prices of FX-swap and XCS are related by these concepts.

(do note that `w_i*=0.998` is not equal to `v_i*=0.99763`, which is the native discount factor evaluated purely with EUR OIS rate)