Nope, the US Constitution,20th Amendment, says that the term of the President end at noon and their successors start then. The swearing in is required before he executes any official duties.
For the rest it is just easier to quote the Constitution:
"If, at the time fixed for the beginning of the term of the President, the President elect shall have died, the Vice President elect shall become President. If a President shall not have been chosen before the time fixed for the beginning of his term, or if the President elect shall have failed to qualify, then the Vice President elect shall act as President until a President shall have qualified; and the Congress may by law provide for the case wherein neither a President elect nor a Vice President elect shall have qualified, declaring who shall then act as President, or the manner in which one who is to act shall be selected, and such person shall act accordingly until a President or Vice President shall have qualified."
If both were to die then it the House would select an eligible person for President and the Senate the VP. Obama would not be eligible since he already served two terms.