Our mathematics points to the universe being a singularity at the time of the Big Bang. That doesn't necessarily mean that it is a black hole. In fact, there are really no accepted theories out there that describe what transpired at the beginnning of the Big Bang. We only know what happened starting 10^-36 or so seconds after it began. As far as exploding black holes, there is a device called Hawking radiation that allows black holes to lose mass. This is achieved by "virtual particles" being created by quantum mechanical fluctuations around the event horizon of the hole. Virtual particles can be created in all empty space, because it requires no energy loss (an example of virtual particles is the spontaneous creation of electrons and positrons which then annihilate eachother). So imagine an electron and positron near a black hole being created spontaneously. Then there is a probability that one would get sucked into the event horizon, while the other could escape. Therefore the virtual particles do not annihilate eachother, and so the black hole can lose some radiation. At a certain point, say, around 1 billion grams, the black hole radiates so quickly that it becomes undone. That is the only way for a black hole to explode without an external influence. Of course we have to assume here that Hawking radiation actually exists, and black holes can get that small. There is no physical proof as of yet.