Formula from finance: decomposition, overnight index swap

[ducmod]

Homework Statement

Hello!

Please, take a look at the picture attached. I would be grateful for the step by step explanation of the math equation; how it has been derived from f/s (1 + libor eur) - (1 + libor dol) to the one with logs?

Homework Equations

The Attempt at a Solution

Thank you very much!

 

[fzero]

Let's write the left-hand side as
$$ \frac{F}{S} ( 1 + \text{Libor}^{\text{Eur}}) - ( 1 + \text{Libor}^{\text{USD}}) = \left(\frac{F}{S}-1\right) ( 1 + \text{Libor}^{\text{Eur}})+ ( 1 + \text{Libor}^{\text{Eur}}) - ( 1 + \text{Libor}^{\text{USD}}). $$
We now assume that $F/S$ is very close, but not equal, to $1$. This has the consequence that
$$\left(\frac{F}{S}-1\right)\text{Libor}^{\text{Eur}}$$
is small, so the authors drop it. Furthermore, we have a Taylor series for $\ln x$ for $x\approx 1$:
$$ \ln x = (x-1) - \frac{(x-1)^2}{2} + \cdots.$$
Keeping only the first term in the series, we can write
$$\frac{F}{S}-1 \approx \ln (F/S) = \ln F - \ln S.$$
Putting these together, we have (using $1-1=0$)
$$\frac{F}{S} ( 1 + \text{Libor}^{\text{Eur}}) - ( 1 + \text{Libor}^{\text{Eur}}) \approx \ln F - \ln S +\text{Libor}^{\text{Eur}}- \text{Libor}^{\text{USD}}.$$
Finally, we can add
$$ \text{OIS}^\text{USD} - \text{OIS}^\text{Eur} - (\text{OIS}^\text{USD} - \text{OIS}^\text{Eur} )$$
to get the expression in the text.