Kioysohi Ito has died.

His work is the foundation of the calculus of random motion, used in physics and misused in finance. Why it is important is that share prices, indeed all prices, have a random component. This random component is best modeled using a mathematics which can show the range of possible areas.

The key to his work is Ito's Lemma, which is like the chain rule for ordinary calculus. The chain rule is one of the fundamental parts of calculus, and it is really the scaled up version of what everyone learned in geometry, that two things that are equal to a third thing, are equal to each other. Well the chain rules says that if you know the rate of change of two things, that are related to a third thing, then you can relate the change of the two things. Or think of it as three children holding hands and running around. The child on the right is holding hands with the child in the center, the child on the left is holding hands with the child in the center, so you can relate the child on the left to the child on the right, by the movement of the child in the center. The chain rule is for differentiation, that is finding out the rate of change of something at a point. It produces the substitution rule.

An example of a local martingale is a craps table with a betting limit. Up to the betting limit, you can get your money back by doubling your last losing bet. However, when you reach the limit, that's the end of it.

So Ito, by giving a replacement for the chain rule in Ito's Lemma, allowed building up a calculus of random motion that mirrored the calculus of deterministic motion.

With this, the power of the calculus could be applied to stochastic processes, things with random parts.

All from a simple insight about equilibrium.

Now here is where scenario fulfillment gets in to play. Ito's lemma requires that the process be adapted - that is dependent only on past information, and that it be a local martingale. Many of the large derivative vehicles that are now tumbling to the ground were designed with Ito calculus. However, Garbage In -> Garbage Out. The first assumption was that we would never reach the point where an issuer couldn't double up the bet and get the money back. The second is that prices always only reflected past information plus the random unknowable component.

So to go back to the craps table, all you need to know is your last bet, and the outcome is based on the throw of the dice which could go either way for you or against you.

However, what if the dice are loaded, and you don't know that the casino has a betting limit?

Well then you can lose big.

But wait, there's more. What if, knowing that you could always win the original bet by doubling up and doubling up and doubling up, that you made a bet for a great deal more more money with someone else, before going in. So for example, let's say you make a bet that if you win, the side bet pays off 10 times what you win. But if you lose, you pay 10 times what you lose. Then you go in, and lose big, and just as you reach for the dice on the double up, you are told "Sorry the casino is closing, so you have to come back tomorrow." Well there's no tomorrow for you, because waiting at the door is a guy that you owe 10 times as much as you are down. Even if you think you can come back in the morning, and win it back, it doesn't matter.

That's what is happening with the financial system, and all because an obscure and brilliant mathematician came up with a mathematics which works very well for molecules bumping around and diffusing outwards, and not so well for share prices where the molecules really do have minds of their own.