Best method to define Take Profit and stop loss in pairs trading
You can set the stop-loss whenever the unrealized PnL drops below a threshold value. For example, if you chose entry signals at 2-sigma, you are expecting that the spread will revert back to mean from this threshold. However, it is possible that the spread continues to fall or rise. Say it reaches 2.5-sigma. To prevent further losses, you can place a stop loss at say 3-sigma.
You can set the take profit whenever the spread reverts back to the mean.
Deciding stoploss is a very important aspect of strategies like pair tradnig which are inherently short gamma (unlike momentum strategies where the signal auto-correct itself). But in most cases they are applied as an after-thought. One common appraoch is a a fixed percentage point move. This is susceptible to whipping (potentially degenarating in to buy-high/ sell-low case) and stoploss hunting. The other method is a relative distance from current profit (trailing stop loss). This is only meaningful when the trade is already in profit.
Ideally, the value of a stoploss depends on the stochastic process of the spread we are trading. Theoritically, if the spread process is a white noise, usually stop losses have negative value-add to the strategy. However if the spread movements are non-random (e.g. serially correlated), they can be useful. If we enter a pair trade at the peak, and our underlying hypothesis is right, we can expect the returns of these strategy to be positively correlated as we converge slowly to the mean from top. This implies stoploss width to be a function of the auto-regressive coeffecient and spread volatility and mean returns. In reality, things are of course more complex. One way to approach is to bake in a stoploss policy within the strategy itself. This can be done in a fashion similar to change point analysis. That is, as part of the strategy, we track the cumulative returns from an entry, and exit the trade if the cumulative returns goes below a certain threshold in terms of the standard deviations of these returns. During the strategy development, we try to establish a stable threshold, given the volatility and time series signature of the underlying prices. For more theoritical details and examples refer to http://alo.mit.edu/wp-content/community/uploads/2015/06/When_Do_Stop-Loss_Rules_Stop_Losses.pdf.