The term "derivative" usually refers to agreements by someone to pay various amounts based upon what happens with assets which are owned by someone else. To use an analogy (which should actually make clear the problem), suppose a building owner buys a fire insurance policy on his building. The policy represents an agreement by the insurance company to pay whatever would be necessary to repair the building in case of a fire. In some sense, the owner of the building now owns two things: the building, and the insurance policy. The theory behind derivatives is that the building and the policy should be regarded as separate assets, so that the holder of the building would own something whose value might be greatly reduced in case of a fire (since he would no longer receive payments to cover his losses in such cases) while the holder of the policy would have something that would only have value if the building caught fire.
If the idea of someone holding an insurance policy on a building in which they have no property interest doesn't smell right, that's because it shouldn't. While regarding the building and the policy as separate assets might make seem superficially to make sense, it's fundamentally broken. An insurance policy on a building doesn't just represent an unconditional cash payout if a fire occurs. It gives the policy holder the right to demand cash in exchange for transferring to the insurance company the right to recover the loss, by salvaging damaged property, suing a negligent third party who caused the fire, etc.. Only the holder of the property would have the right to attempt such recovery, and thus only the holder of the property would be able to trade such right for a cash payout.
While one could set up an "insurance policy" which didn't require any attached rights of recovery, that would still leave another problem. If someone starts of a company and sells 1,000 shares, and subsequently the company goes bust and has to be liquidated, the value of those shares will be 1/000 of the amount by which the monies brought in by liquidation exceed its debts. Whether liquidation brings in money far in excess of the debts, or whether it brings in nothing, there will be no question of whether the company's assets will be sufficient to pay its shareholders what they are owed, since what the shareholders are owed is defined in terms of the company's assets.
The problem with derivatives is that since they aren't attached to any type of asset, it can often be difficult, and in some cases impossible, to determine whether the issuer will actually be able to pay them. Essentially, what happens is that the issuers of derivative accepts various wagers; if they lose, they may make good on the wagers or--if they lose more bets than they can pay--be forced to welsh on them. A company that accepts wagers on which it would not be able to make good on effectively cheats the other person even when the company legitimately wins and doesn't have to pay out, but there's no general way by which the other person could know that. Given that, it's not difficult for someone to formulate a combination of wagers such that they have a 99.99% chance of winning something significant and a 0.01% chance of losing a thousand times more than they could possibly pay. Such a combination would have a 99.99% chance of yielding enough "mostly-free" money to allow a person to set up an off-shore haven to which they can flee in the 1-in-10,000 case where their bet goes bad. The possibility of being caught short isn't an accident--it's a deliberate strategy.
I've seen some nice houses lately and mortgage interest rates are low. I've joked to my wife that I should apply for a 100 year mortgage. (She sees no humor in this.) The idea of a mortgage that exceeds one’s expected remaining lifetime somewhat resembles the situation you describe in your last paragraph.