Anthony Knockaert went close to opening the scoring in the early stages but his powerful effort from 30 yards failed to force Bogdan into a save.
Bolton's Hungarian goalkeeper was called into action midway through the first half when Martyn Waghorn was gifted space on the edge of the box, but Bogdan's reaction save kept the scores level.
The tight nature of the match continued in the second half, with a rare opportunity being wasted by David Nugent when the striker drilled an effort into the side netting.
Bolton enjoyed a spell of domination towards the end of the second half but Leicester's defence remained stubborn to reduce the home side's chances.
However, it was the visitors who came closest to claiming all three points with less than 10 minutes remaining as the ball fell to Knockaert but once again, Bogadn produced a superb save to earn his side a draw.