Some currency pairs often move in correlation to each other, either in the same direction or in opposite directions. Correlated pairs include EUR/USD and GBP/USD, USD/CHF and USD/JPY, AUD/USD and GBP/USD, EUR/USD and NZD/USD. Opposite pairs are GBP/USD and USD/JPY, GBP/USD and USD/CHF, EUR/USD and USD/CHF, AUD/USD and USD/CAD, AUD/USD and USD/JPY.
Downward pressure on EUR/USD and GBP/USD indicates Dollar strengthening against the two (you can always take a quick look at the US Dollar index for confirmation).
Dollar strengthening, directly, has little effect on EUR/GBP. EUR/GBP is a measure of the relative strength of the EUR against the GBP. However, if you believed that USD was strengthening against the EUR more than GBP, you might deduce that the GBP should be strengthening against the EUR... (i.e. from an 'equilibrium' starting point, the relative drop in Dollars you can get for your Euro is more than the relative drop in Dollars for your Sterling -> the EUR has weakened more than GBP, so EUR/GBP should be bearish too). There are opportunities to trade this through triangular arbitrage, but this is the arena of institutional black boxes.
...the tricky thing is that USD strength doesn't necessarily mean EUR or GBP weakness, and vice versa, ad infinitum. There is a subtle difference between taking a view on the USD as a single currency, and taking a view on EURUSD and GBPUSD - the strength or weakness of the USD is fundamentally driven by macroeconomics, and on shorter timeframes, the demand for it through repatriation of funds, etc... in particular risk aversion supports the dollar, while risk appetite rallies the Dow Jones Index - a view on the Dollar alone might involve a portfolio in currencies, commodities, and Bonds, and it follows that a much more sophisticated approach is needed to build such a portfolio profitably (with specific views on the term structure of interest rates, inflation expectaions, but I'm guessing here). You could trade currencies in this way, but you'd need 'real money' (i.e. actually take delivery of the currency) and a fundamental view on all the currencies involved. More Soros than a spread bet!
So, for currency pairs trading, on short time-frames (by this I refer to hours and days), inter-currency pair analysis is usually more effort than it is worth. It is also important to remind yourself of how you arrive at a bullish or bearish view in the first place - for one I know one can identify opportunities from price action - so this is what you should concern yourself with first and foremost. If you believe that a level of resistance in GBP/USD will not be breached, you are in fact saying that the demand will exhaust and there will be excess supply in the market; you are not specifying whether the change in the exchange rate is driven by USD strength or GBP weakness. The effects that the strengthening of any particular foreign exchange has on exchange rates can be seen across the markets, but these are driven by fundamentals, and so are beyond the capabilities of most traders.
Having said all that, inter-currency pair analysis should play some role in your trading, in the process of good risk management. There are established relationships between some currency pairs (and some commodities, too). Traditonally, for instance there is a correlation between the Canadian dollar and oil since Canada is one of the major suppliers of oil, in particular to the USA market. This means that as oil prices rise the value of the Canadian dollar is likely to strengthen as a higher oil price leads to a healthier trade surplus. Likewise, should coal or other commodity prices falter, the value of the Australian dollar is prone to weaken while should commodity prices strengthen, the dollar will usually weaken. Where correlations especially help is to help you manage your exposure - a simple example is EUR/USD and USD/CHF - in these pairs for instance having a long and short position respectively might expose you to excess risk. Like them or loathe them, correlations are useful in this regard. If you are holding multiple positions, a worthy addition to a trading system is some model of correlation between the major pairs, and your expectation of the 'realised' correlation over the lifespan of your trades.
Are there any currencies that are uncorrelated with the dollar? Not really. Even weird crosses like NZDJPY, though not explicitly correlated, get affected by USD spikes. The whole currency world is connected in a weird way.
Another point (misnomer) is the role of hedging in the currency markets. The way that currencies are traded means that if you are long GBP/USD, and decide to hedge your position with EUR/USD, you actually end up with a GBP/EUR position - the long and the short USD trades cancel each other out completely. So, beware if you think you are hedging a volatile pair, say EUR/JPY, because in fact you will end up exposed to a position that you didn't buy or sell in it's own right.
The exception to the above is inter-commodity hedges, and securing a price. If I am going to get some winter sun over Christmas in the USA, I might decide that I will hedge the uncertainty of the exchange rate, and buy my Dollars now, then I can do the sums before I go to know how much I have to spend on the shopping.
The other exceptions would be to hedge something other than a currency trade with a currency trade - for example, I might decide that I am Bearish on Oil, but it's too volatile, so I hedge my position by going long EUR/USD - the two have been very highly correlated recently, so as long as Oil goes down more that EUR/USD goes down, I am OK. Be wary of trades like this though, as these relationships do not last forever; if the correlation between EUR / USD and Oil drops, my hedge becomes less efficient, and I am left exposed to additional, rather than reduced, volatility.
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