Clear nail polish won't wash out.

Put on an OveGlove and burn the end. Use your fingers to squeeze the hot end into a nice smoot round shape.

We do this with nylon ski ropes. Burn the end, squeeze into a point and reinsert the end into the rope. Ski ropes work like those silly Chinese handcuffs kids used to play with and can't be pulled apart then.

Your cord looks more twisted so you won't be able to slide it into itsself but you can burn it and smooth the end if it is nylon.

I recommend melting/burning the ends, if the cord is nylon. If it is something other than nylon, glue it with carpenters glue. Carpenters glue is similar to Elmer's White glue and dries clear, but it's also waterproof and should hold up to washing. You could also just tie a single knot near the end and let it unravel to the knot for a more tassle like look.