# STIR topics: Implied FX-OIS Basis and FX Forward/Swap Pricing

if someone could provide some clarity on the below:

1. What is meant by 'Implied FX-OIS Basis'? For example: "ON JPY trading at parity, 1W implied OIS basis moved 70BP" and "3M Implied OIS basis moved 25 BP to imply -130 BP (3M LIBOR XCCY is +4 )". My attempt: We have Interest rates implied by FX Swap points compared against what the points would be had we used the term OIS rates for the underlying currencies - the difference being some basis swap (would imply an IBOR v OCR swap if I am correct) against which we can measure, lets call them dislocation, in FX swaps points that strip out the forward looking expectations of OCR rates by the market?

2a. A client wants 5y EURUSD FX Forward Points (mid) - what instrument(s) do we observe to derive the points from spot? I am explicitly asking this way as once we establish the relevant instruments to observe, I will then gain an understanding of what drives FX points beyond the simple 'interest differentials'/CIP most examples use that are sub-1y and just observe actual IBOR data to create a rate curve. My attempt below:

With reference to this topic: Calculating Cross Currency basis swaps, I have access to my own Bloomberg/access to FXFA - it would appear we observe direct or inferred IBOR rates in each currency (short-end: published IBOR rates, such as actual 3m LIBOR, or 3m LIBOR derived from listed futures or FRA, long-end: IRS). BBG then shows us the difference between FX Swap Implied from those curves and Actual FX Swaps points - the difference being attributable to the XCCY Basis Swap. So in effect we need 2 of the following 3 to solve for the third: IBOR curves (whether actual or observed from the market), XCCY Basis Curve and Actual FX Swap Points. Correct?

2b. Let's assume we have two currencies with completely identical interest rate term structures, however the basis is very positive - say IBOR+100bp for the non-USD currency. The points should then be negative by virtue of the interest rate premium implies by the XCCY basis swap on the LHS currency (non-USD/USD)? I am trying to gauge how changes in XCCY Basis impact FX Swap points, although they would appear to be mostly driven by IBOR differentials with some spot component, having graphed FX points vs the difference between term swap yields in various currencies. By spot component, I would also also ask that - in an imaginary world where interest rates are exactly the same between two currencies (both current and implied forward), but the spot rate is 10% different between T and T+1 (nothing else changes) - the swap points would also change?.

2c. Let's assume balance sheet constraints and credit concerns are non-existent and 5y EURUSD points are wildly off market from what we have calculated from point 2a above - how do we arbitrage this in practice? I am asking this, as I believe I have structured this line of questioning so that the answer should be via the instruments observed in 2a. There has to be some upper/lower bound to FX points once arbitrage becomes attractive enough.

Thank you!

Addition: I want to articulate the mechanics of what we are doing - we are borrowing/lending an amount of fixed currency X, against which we are then lending/borrowing a variable amount of currency Y - so we need to know what the effective term deposit/lending rate in each currency is. Given one leg of an FX swap is normally fixed, the difference for the market maker is cleared in the spot market such that the future amount is the fixed notional also, but this adds/subtracts from the notional of the variable amount, which is why the points have some sensitivity to the absolute level of the spot rate.

Implied FX-OIS basis should be pretty simple to "compute", it is the classical "Cross-currency" basis observed in FX Swaps & FX Forwards, that can be backed out when plugging in OIS rates (instead of Libor rates).

## No Arbitrage between FX Spot & FX Forwards:

Taking EUR/USD as an example, we must have for no arbitrage:

$$(1+r_{EUR}) F_{EUR/USD}=S_{EUR/USD}(1+r_{USD}+r_{basis})$$

In words: on the LHS, you take 1 EUR and deposit it into an EUR interest account, then convert it to USD at the end of the interest-baring period using a Forward.

This has to equal in value to the RHS: here, you convert 1 EUR into USD at spot and deposit in into a USD interest account for the same period of time as the Forward maturity on the LHS.

For clarity: $$S_{EUR/USD}$$ is the spot, $$F_{EUR/USD}$$ is the forward.

## Using forwards for funding in foreign currency:

What is the $$r_{basis}$$ term? We can understand that term better if we invert the equation above as follows, to get:

$$(1+r_{USD}+r_{basis})=(1+r_{EUR})\left(\frac{F_{EUR/USD}}{S_{EUR/USD}}\right)$$

Imagine someone has access to EUR currency and wants to use it to obtain USD "funding" for a specific period of time (say 1 year): they know the interest rate $$r_{EUR}$$ at which they can raise the EUR (which is their domestic currency). They also know the FX Spot between EUR and USD (i.e. $$S_{EUR/USD})$$ and the 1-year FX Forward rate between EUR and USD (i.e. $$F_{EUR/USD})$$.

In other words, they can obtain EUR, exchange the EUR for USD at spot and using the 1-y Forward, they can lock into an implied "USD interest rate": the interest rate they will effectively have to "pay" to get access to USD funding for the duration of 1-year.

Someone who has access to USD and wants to obtain funding in EUR for a fixed period of time can the the same in reverse:

$$(1+r_{EUR})=(1+r_{USD}+r_{basis})\left(\frac{S_{EUR/USD}}{F_{EUR/USD}}\right)$$

They can lock into an EUR interest rate, when funding the EUR using their domestic USD currency.

Usually, institutions fund themselves at OIS rates: so the USD institution that has access to USD as their domestic currency could raise this at USD-OIS, whilst the European institution that has access to EUR currency could raise thiis at EUR-OIS.

If the demand for USD via EUR funding is in perfect equilibrium with the demand for EUR via USD funding, the term $$r_{basis}$$ would be zero. But this is hardly the case: the EUR/USD basis changes all the time in line with the prevailing demand for the funding in one currency against the other.

The EUR/USD basis is typically a few BPS (last time I looked it was 17bps added onto the EUR leg, which showed increased demand for EUR funding vs. USD funding).

Btw: te basis term $$r_{basis}$$ described above is directly tradable for maturities longer than 1y via Cross-Currency Swaps. For maturities shorter than 1y (for which Cross-Currency swaps don't trade), it is indirectly reflected in the FX Forwards via the equations above.

## Summary:

$$r_{basis}$$ reflects the prevailing demand for funding in one currency via another currency, for a fixed period of time (term). Each term (i.e. 6m, 1y, 5y, etc) would have it's own $$r_{basis}$$. When OIS rates are used in the equations above, you can back out the Xccy-OIS basis. If Libor rates are used instead, you can back out Libor-Xccy basis.

So in conclusion, taking 6-month tenor as an example: you know the 6m EUR-OIS rate, the 6m USD-OIS rate, the EUR/USD Spot and the 6m EUR/USD Forward: when you plug all of these into the equations above, you can back out the 6m FX-OIS basis for EUR/USD (and you can do this for any other tenor or currency).

FX-OIS basis, depending on the fx pair, basically means, the implied yield vs the OIS basis of the currency pair.

ON JPY trading at parity: USDJPY offered or bid at "0"

1W implied OIS basis moved 70BP: depending if its downward move or upward move, its trading +-70bp vs OIS basis. Upward, therefore, demand for JPY inched up, vice versa demand for USD fell, what that translates to is JPY borrowing cost is now higher vs last point its refereed to. Therefore, now the pricing is USDJPY 1w (previous offered level + 70bp), the opposite will occur if borrowing cost rose for USD.

• "FX-OIS basis, depending on the fx pair, basically means, the implied yield vs the OIS basis of the currency pair." Do you mean for example (3m AUDUSD rate implied from FX swaps) less (3m AUD OIS)? What does the difference/basis represent? – Student Feb 28 at 13:44