With U.S. short-term rates rising and Eurozone rates falling deep into negative territory it's not surprising that the dollar has been gaining against the Euro. Behind these moves is a U.S. economy that appears healthier than the Eurozone economy, and that in turn conditions markets to expect very different monetary policy conditions for the foreseeable future. The dollar's strength to date is salutary and not overdone, so it doesn't present a problem for the U.S. economy. (Question to those who worry about the strong dollar: If a stronger economy is what is driving a stronger dollar, why should a stronger dollar be bad for the economy?)
The chart above compares 2-yr Treasury yields with 2-yr German yields. They have diverging for almost two years now, driven by the growing perception that the Fed would raise rates sooner than the ECB, and more recently, by the ECB's professed willingness to actively pursue easier monetary policy for the foreseeable future.
The chart above makes it clear that interest rate differentials between the U.S. and the Eurozone (blue line) go hand in hand with changes in the Euro/dollar exchange rate (red line). The dollar has strengthened vis a vis the Euro in line with more attractive U.S. interest rates.
The chart above compares the dollar value of the Euro against my calculation of the dollar's Purchasing Power Parity against the Euro (green line), which can be thought of as the exchange rate that would result in prices in the Eurozone being generally comparable to prices in the U.S. By my calculations, the dollar is almost 10% above its PPP value against the Euro, which means that U.S. tourists to the Eurozone should find that things are a bit cheaper there. It's nice to have a strong dollar, and it's not strong enough at current levels to worry about.