I've only seen one 'popular' trader use this divine combo of overlays, (they also cook bacon and eggs together in one pan so maybe it's some sort of fetish or something in their case) but I've dabbled with it for a while as well. The Bollinger Bands and Ichimoku cloud were the first two indicators that I came to use and understand, the fact that they can be used simultaneously without getting too ridiculous is a huge bonus.
Since I do not believe in a wrong way to trade, other than losing a lot; I'll keep my point of view on this combo as straight-up as possible.
Bollinger displays the 'mood' of the price, while the Ichi gives S/R levels as well as providing profit and stop targets a la the 'edge-2-edge' play or the 'c-clamp', giving oversold/bought signals, and bull/bear/indecisive signals via the kumo and the TK. The Ichimoku creates a structure that the Bollinger puts into context by portraying the relative momentum.
Either one of these on their own could arguably do all the things that the other one could in their own way: But, because of this, by hybridizing the two of them the resulting complex can be used like one supercharged indicator.
It's about that simple, and this 4h chart shows a stunning example of the edge-2-edge play mentioned earlier, which is simply when the price enters the kumo/cloud and travels from one edge to the other, either bullishily or bearishly. Here, it was bullish of course.
Adding a Fib to the mix merely enhances the metrics of the Ichi/BB overlay. Any horizontal S/R is going to be 'wedging' with the Ichi/BB on a routine basis, allowing for breakout-trade setups and the usual triangle/wedge patterns. Plus, whatever cosmic Fib voodoo you may happen to worship can also be applied with more context.
This brings me to my final point before this gets too long: The Ichi/BB isn't a substitute for price action analysis, rather it provides a map of the territory. Keeping ones eyes open to price action in whatever form you may already be using it in is a major boon to the Ichi/BB approach.
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