Thursday, August 20, 2009

Continuous and Discrete

Moral: Wherein we use: Say what? Well, think of the title as a headline aimed at an attention-grabbing.


Of course, any implied reference to the dualities addressed (example) by physics is intended, however tenuous they may be at this point. But we'll get into that later, as are not many Quants from backgrounds in the hard sciences (see Remarks 08/20/2009)?

The idea is to start to address some issues in finance. It'll be slow, as there is a whole lot to cover.

Not that I'm an entire newbie, but it was nice to see that the notion of discrete and continuous has already come up in finance. Note, for instance, this bit in Wikipedia that compares binomial and the Black-Scholes models makes that distinction.

Having mentioned B-S does not mean that we'll get into the Greeks just yet. We have to start much more basically. Consider that the approach is both fundamental and foundational (f&f).

Now, doesn't our intuition tell us that some things are smooth and others are not? And, even if we don't use that particular word, could not any of us make this choice in a situation like sorting some set of objects? Even blindfolded?

But, being technical, we know that smoothness involves differentiability. As we all know, non-smooth, from noise or error, is difficult to handle, in fact causes very bad behavior. The advances in finance that we are to address came from several directions, but two major ones are the means to handle problematic mathematical objects, like the non-smooth (via Ito's and others' work), and to have computational means for this (hey, it would not have worked otherwise - get used to quasi-empirical as a concept, as it'll be used a lot).

As we know, it was a major step to realize that we could have continuous and non-differentiable functions (Bolzano, old KW, et al). Yet, there are other steps forward, like this one out of France (Levy process). Actually, there are too many to cover in brief posts like this.

At any time, various hints at background are all that we can do in order to state some theme. In this case, a whole lot of work has gone into handling randomness and high variability. Some attempts at bringing to fore some intuitive views can be seen, yet they also bear the danger of being problematic (thanks, Hail the Quants!). Actually, the intuitive is no more apt for this downside than is the most polished, theoretically-grounded construct.

The current excuse is that overlays that have people underneath are by definition too difficult, as opposed to physics and its particular entities that are mindless. Yet, consider, please. All that says is that derivatives (see Note 1) placed upon humans, or things that are highly influenced by human activity, have value derivation problems.

Yes, I know, plenty have made money. You know, of course, that the multitude lost out. Call it near-zero, if you must. If you look at overviews, like this one from the WSJ (Reagan's Bull Market), yes, things are a lot different than 1982 (gosh, some of the Quants may not even have been born). But, look at the median income line over that period (if that were a log chart, it would be flat lined).

The intent is not criticism. Rather, let me suggest that there are ways to do this modeling, that is, overlay people with more insightful structures, however, the concept of smoothness will be imperative. How can this be? Ah, that is what we'll be going over through time. But, start to think about boundedness as more than a limiting abstraction.

Hopefully, the discussions will come to code and test sooner than later; actually, that is the goal, though f&f will continue to be of major focus for now.


03/15/2012 -- Okay, might have used incomputability (see post on Alan M. Turing) but stand by the context, the issues, and the need for resolutions. Wake up, quants (you, too, Ben).

01/13/2012 -- A re-look at this. 

10/08/2011 -- The link is more than tenuous. In fact, numeracy (see Reflections on the work of Steve Jobs) ties right into support of the discrete model. Why? For now, we have to transform from continuous space to the not-continuous in order to obtain solutions. One has to wonder if there is some way that 'analog' views might help us grasp better the ways of the continuous (we all know that we're a whole that is much more than the sum of parts -- yes, we do).

02/10/2010 -- We could probably use the auto (and recent events) as a way to characterize the concepts of the blog. Of course, we have the value versus quality mis-think as part of the problem. Business Week reports that Toyota was asking suppliers for a 10% cut. Well, such scrimping would have an effect, even if it was only in looks. However, cutting into the life of a system may appear smart but, actually, relies on the same unstable basis as does a lot of economic thinking.

09/02/2009 -- Let's put undecidability on the table, please.

09/01/2009 -- Oh, one might ask, why do I pick on Quants? A good question that we'll expand upon over time. They're like explorers in a new territory who mainly pillage and exploit when we, the human race, need more those who can map and help us learn. Learn as in real knowledge not just moving monies to certain pockets. Yes, indeed, a map-territory issue here among many others.

08/25/2009 -- Overlays, yes, needing insights from across the pond.

08/20/2009 -- Note 1: 'Derivative(s)' has been used a few times in the posts. The context may imply the usage, hopefully. But, in general, we're talking two types: 1) from finance, where 'derived from' is the proper interpretation (or as one may surmise from posts here and elsewhere, something from nothing), 2) the usual mathematical variety.

Modified: 03/15/2012

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