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Can one deem an FX float-to-float swap and a FX forward equivalent on dates immediately after repricing? The reason I am asking, I am hedging something that can be modeled via an FX forward, I was given an FX basis swap to hedge it. I want to safe myself some work and claim that they are equivalent on repricing days (or immediately after) when effectiveness of the hedge would be assessed. I believe, it comes down to how forward points are reflected in the the two instruments. An FX forward is valued as follows:

$$ (F_{t_i,t_M}-F_C) \times N \times \text{df}_{\text{FWD}} $$

I don't know how to do the classic "where" block, so in a sentence, the above means: (FX forward rate at time $i$ until time $M$ (maturity) $F_{t_i,t_M}$ minus FX contracted forward rate $F_C$) times notional $N$ times discount factor $\text{df}_{\text{FWD}}$

A swap's value would be:

$$\sum_{i=k}^{n}{N_{\text{DC}}\times \frac{r_{\text{DC}}}{\text{days}}\times \text{df}_{\text{i,SWP,DC}}} - S_{0} \left(\sum_{i=k}^{n}{N_{\text{FC}}\times \frac{r_{\text{FC}}}{\text{days}}\times \text{df}_{\text{i,SWP,FC}}}\right) $$

Where "DC" = domestic currency, "FC" = foreign ccy, $r$ the applicable interest forward rate and "df" = discount factor. You might have notices I ommitted the final exchange of principal which would add some more complexity and is not important for the question.

The point is that the formulas are obviously different, yet I am trying to claim that the values will be equivalent immediately after repricing. Here's why:

We know that the value of each variable leg will be $100$ in their respective currencies (as any floating bond would be, where forward and discount rates are interconnected. So one could claim that the FX swap's value will be primarily driven by the spot rate $S$, used to translate the foreign currency leg. At this point one would say that the forward will be driven by the difference between the actual forward rate $F$ and the contracted rate so they should NOT be the same. But. Here comes my twisted thinking (or lack thereof). I found out, that in the reality of markets, the forward rate difference in the swap would be hidden in the discount factors as the basis spread, and I am wondering if that, somehow, would make the value equivalent to the forward. Unfortunately don't have access to SWPM to test the "theory". Thoughts?

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Hi PBD, welcome to quant.SE and thanks for posting your question. –  Tal Fishman Jul 13 '12 at 13:31
    
Swaps always get me mixed up, but it might be helpful to begin by considering a swap where only one payment changes hands and show it's equivalence to the forward. A swap with many payments may turn out to equal a series of forward at different times (no idea if that's really the case, but just speculating). –  John Jul 13 '12 at 21:16
    
John - yes, I agree. My only concern here is, that the I'm not entirely sure how to show a "floating rate" forward. In other words, a forward is typically a fixed for fixed exchange of currencies, it's very easy to show, that a fixed-for-fixed swap is a series of such forwards. For floating-for-floating (or basis) swaps this is much trickier it seems. Your comment however made me think, and I am leaning towards the conclusion that a basis swaps is NOT equivalent to a forward. Thanks for this little nudge. :-) –  PBD10017 Jul 14 '12 at 6:04
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The two are not equivalent, because of the cross-currency basis spread (CCBS), which became a risk factor in itself sice 2007, and does depend on term. This practically leeds to a difference in your constantly-assumed notionals (the notional is not constant anymore).

What it happens is that you assume having a constant notional cross-currency swap that exchanges a risk-free overnight rate in one curreny for a risk-free overnight rate in another currency. In reality, one exchanges benchmark rates (say LIBOR), so you should imagine having three swaps: a cross-currency basis spread overnight risk-free rates, a money market basis swap of domestic LIBOR versus domestic overnight risk-free rates and a money market basis swap of foreign LIBOR versus foreign overnight risk-free rates.

This change in notional can actually be seen when doing a mark-to-market cross currency swap instead of your standard one.

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