Learning how to hold the bow can be complicated. Each part of the hand/fingers serves a different role. In addition, the bow hold must be flexible and not tense/rigid. We do many exercises to learn how to relax our fingers, to shape our hands, and how to place the bow on to our instrument. Good bowing technique starts with taking the time to set up a correct bow hold.
Phrases that I use during this process include . . . thumb bumps, macaroni pinkies, relaxed fingers, and floppy fingers. We also learn that the tip of the thumb and pinkie is close to the fingernail, not on the side of the fingerpad. It is important that the performer keep the hand rounded after placing the thumb. The thumb does not hold the bow on it's side. When the thumb is placed, there should be a circular opening in the hand, about the size of a ping pong ball.
The bow hold must be completely relaxed. I tell the students that the hand doens't really hold the bow, it guides the bow. When I see excessive tension in the hand, I have them take the bow away, shake out the hand, and reset the bow hold. Shaking out the hand relieves tension. Kids are often so earnest when trying to hold the bow that they squeeze all of the muscles in their hands. Tension interferes with being able to engage the elbow and wrist properly.
Here are some pictures that demonstrate parts of the bow hold. Most parts of the bow hold are similar for violin/viola/cello/bass. The main exception is the placement of the pinkie. The pinkie is placed on top of the stick for violin/viola students. Cello/bass students curve the pinkie down over the stick. The second photo with the blue background shows more easily the complete bow hold than the first photo.
Notice that we NEVER hold the bow with our fist!