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Formula from finance: decomposition, overnight index swap

  1. Jul 29, 2015 #1
    1. The problem statement, all variables and given/known data
    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?

    2. Relevant equations


    3. The attempt at a solution
    Thank you very much!
     

    Attached Files:

  2. jcsd
  3. Jul 29, 2015 #2

    fzero

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    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.
     
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