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?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
This will be the inverse process
$$\frac{1}{S_t}$$
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 $$d\frac{1}{S_t}=\frac{1}{S_t}(\sigma^2 -r)dt+\frac{1}{S_t}\sigma dW_t$$
answered Apr 21, 2014 at 9:03
-
1$\begingroup$ I think there should be -ve sign for the last term. $\endgroup$– RiserNov 22, 2014 at 20:05