1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Formula from finance: decomposition, overnight index swap

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

    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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted