Here are directions on how to make a loop on the end of your fly line. This same process can be used to splice running line to a shooting head if you make one your self. I use to use it in the 50's to make a double taper silk line into two weight forward fly lines. This will work with lines up to 6wt or so. Any bigger and you need to lengthen the stripped area.
Strip off the outer cover on the tip of the fly line for 5/8". You can soak the tip with Finger Nail Polish or Acetone and it makes the coating softer. I use the blade of my office scissors and don't use the Acetone. Once you have broken through the finish on one side of the line the rest of the finish will peal off. You can see there is a small tag left on the line that was not removed.
Take your fly tying scissors and cut off the tag that was not cleanly removed.
You now have 5/8" of braided core exposed. Take a bodkin or needle and start to un-ravel the braided core. You need to work right next to the un-raveled braid so the bodkin will pull through the braid un-raveling more core. If you insert the bodkin too far from the un-raveled end you can't pull the bodkin through the braid. You will see what I mean when it happens. Don't force the bodkin. Just take it slow and easy.
Continue to unravel the core until all of the exposed core is un-braided.
Now bend over the line onto its self to form the size of loop you want. Mark the fly line at the spot on the main line that the un-braided core is laying next to. I mark the space with a black marking pen. The space should be 1/2' instead of 5/8" like the un-braided end of the line. When you lay the line straight again it will look like in this picture.
Now you want to remove the outer cover on the main fly line like you did on the tip. It should look like this photo and be 1/2" wide.
Now here is the tip of all tips. Make your loop again and press the tag end and main line together together with the two bare parts of the line together. The butt of the un-raveled end should be right over the front of the bare spot on the main line. Take a piece of Scotch Tape and fasten the two lines together. Now you don't have to worry about the tag end and the main line from moving out of position.
This photo shows that the Un-branded tip of the line is longer than the gap on the main line.
Take your fly tying scissors and trim the tip of the line so the un-raveled is just shorter than the gap on the main line.
Now the fun begins. Take a bobbin loaded with the thread you want to use. I usually use 6/0 Uni-thread. You are going to use your bobbin to wrap this connection to make your loop. First make sure the end of the line and the gap on the line is lined up. Use your fingers to roll the line and tip together to compact the line and tip. Use your left hand to grip the main line and the tip just about in the middle of the unraveled tip. You also need to grip the 6/0 thread with this hand and fingers. So you are holding the thread, the unraveled tip and the main line in between your thumb and fore finger of you left hand. Your thumb nail should be facing you. Now roll up any slack thread with your right hand onto the bobbin. You want the bobbin hanging about 3" below the splice point. Now take you right thumb and fore finger and grasp the other side of the splice with the two thumbs faceing each other.
So you are standing with the splice held in the center with your two thumbs and fore fingers, with the thumbs facing each other and the thumb nails are almost touching. The bobbing is hanging down from the junction of your thumbs with about 3" of thread. OK, here we go. Start to swing the bobbing back and forth until the momentum of the swing lets you swing the bobbin around the splice in a circle. If you have the bobbing adjusted properly, and you won't, line will pull off of the bobbin and you will begin to wrap the splice. Wrap toward the left hand first so you can capture the end of the thread. After several wraps you can cut off the end of the thread just like you were starting the thread on a hook. You may have to adjust the bobbin several times until the thread feeds onto the splice but stays about the same distance from the splice. Using your two thumb nails you can direct where the line wraps onto the splice. Wrap beyond the splice on both ends a very short distance onto the main line and the looped line. With a little practices you can get a really decent wrap.
When you have finished wrapping the splice to your satisfaction you take a 6" piece of thread and fold it in half forming a loop. Lay the loop on the splice and hold in place. Wrap over the thread a few turns and then cut the thread from the bobbin leaving a 2" piece on the splice. Place the end of the thread into the loop and pull the end of the thread under the last few wraps. Trim up the thread end and you are done except for sealing the splice.
A comment about adjusting the bobbin so it will swing around the splice. You can practice this using a nail or some other small diameter item before you start your loop.
You need to seal the splice so moisture won't get into the core of the line. I like to use Plio-Bond. It is thin and penetrates the splice forming a good seal but still has some flexibility. You can use Super Glue but it drys hard. I believe Shoe Goop would also work well.
The Finished Loop.
Here is a picture of a finished loop except it needs a coat or two more of Plio-Bond.