Bogdan primarily frustrated the Foxes in the first half but made a superb second-half save to thwart Anthony Knockaert.
Schmeichel was busier after the break but perhaps his most notable moment was a double save from Lee Chung-yong just before half-time.
After getting off to a winning start against then-league leaders Cardiff on Saturday in his first game in charge, Bolton boss Dougie Freedman named an unchanged starting line-up.
Leicester were frustrated in the early stages by the in-form Bogdan as the Hungarian goalkeeper produced parrying saves to thwart David Nugent, Danny Drinkwater and Lloyd Dyer.
Bogdan also made a superb save to keep out Martyn Waghorn’s powerful strike.
Bolton continued to ride their luck as De Laet’s shot clipped the top of the crossbar but some late pressure almost saw the hosts take the lead just before half-time.
Chung-Yong Lee’s long-range shot caused Schmeichel problems, with the Leicester keeper pushing the ball back into the South Korean’s path. Lee took another shot but Schmeichel pushed it wide. From the resulting corner, Darren Pratley volleyed over.
Schmeichel was called into action again shortly after the break, this time to keep out Kevin Davies’ header.
The Dane denied Bolton once more when he turned a Mark Davies effort around the post for a corner.
Leicester threatened again midway through the second half but Nugent shot into the side-netting following Knockaert’s lovely lofted pass.
Bogdan then pushed away a fiercely-struck shot from Waghorn before making a superb save at his near post from Knockaert after Bolton were caught with a short corner.
Schmeichel spilled a shot from Chris Eagles late on but Morgan was on hand to clear away the loose ball and an entertaining clash finished goalless.