Let's suppose USD/EUR price in USD follows a GBM with $$ dS_t = rS_tdt + \sigma S_tdW_t $$ What process does EUR/USD follow in EUR?


1 Answer 1


This will be the inverse process


Applying Itô's formula the dynamics are then given by

$$d\frac{1}{S_t}=\frac{-1}{S_t^2}dS_t+\frac{1}{S_t^3}dS_tdS_t$$ some simple algebra then leads to $$d\frac{1}{S_t}=\frac{1}{S_t}(\sigma^2 -r)dt+\frac{1}{S_t}\sigma dW_t$$

  • 1
    $\begingroup$ I think there should be -ve sign for the last term. $\endgroup$
    – Riser
    Commented Nov 22, 2014 at 20:05

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