A famous example of the efficacy (and also problems inherent in) mathematical modeling is the concept of Dynamic Hedging ← a transcript of an excellent documentary on the subject. The model was created by Black and Scholes, for which they were awarded the Nobel prize in Economics. Under certain reasonable assumptions, their model is guaranteed to reduce risk in proportion to the amount of investment one has made (which seems a bit counter-intuitive at first). The best analogy up with which I can come is that Dynamic Hedging is a lot like opening a casino. The model ensures that the odds will always be in the house's favor, so in the long run you will, in expectation, make a profit. The problem is that every once in a while someone will likely "win big", and your casino will have to make a payout to them. That's fine, as long as you have enough liquid money to make that payout; you'll eventually recoup the loss in the long run. The problem is that, unlike a casino, the instantaneous risks of the market are unbounded. For example, there is unbounded risk associated with shorting a stock. Therefore, as long as you have an unlimited supply of liquid cash (or credit) to deliver those payouts along the way, you are guaranteed to at least break even. Black and Scholes' model was implemented in the form of a number of hedge funds, most notably Long-Term Capital Management, which enjoyed spectacularly consistent returns in the 40% range throughout the 1980s and most of the 1990s. The problem (I'm greatly simplifying things here) was that eventually the fund got so big that it was no longer able to borrow/acquire enough liquid capital to sustain itself.
So what's happening these days? The financial concepts of arbitrage and hedging are theoretically guaranteed to be risk-free. The idea is quite simple:
- A widget is being sold by party X in London for $5.
- At the exact same point in time, due to market fluctuations, the same type of widget is being bought by party Y in New York for $6.
- If we can find out about #1 and #2 quicker than Y can find that same information, then we can buy the widget from X and sell it to Y at a $1 profit.
That's why we currently see the "big banks" investing millions (and some speculate billions) of dollars into high-tech computer centers very close to the big stock exchanges. (Sorry, that video is in Nederlands, but a lot of it is just dubbed English, and many Dutch words are mutually intelligible with English.) Since almost all trading is done electronically these days, the quicker one can get information from and make trades on the markets, the greater chance one has at exploiting arbitrage. In practice, though, stochasticity of the market does induce risk, which has led to widespread use of statistical arbitrage.
I feel I should elaborate on the "science" vs. "art" component of this question, and why I brought up the question of their differentiation in the original set of comments. In my mind, the classifications of science and art are not mutually exclusive. The primary defining characteristic of science is a method by which predictions about a previously mysterious process can be made. Art is the display, application, or expression of a method, almost always with the intention of affecting others' emotions or intellect. By these definitions, science and art are not mutually exclusive, e.g., I believe one could make an argument that cooking is both a science and an art.
Anyway, without getting sidetracked too much, I also want to mention models. I think Newtonian mechanics and Newton's law of universal gravitation are great examples. Newton's models for the way the universe works are extremely good at "predicting the future", at least on the scale of ordinary interactions here on Earth. Unfortunately, Newton's models break down if we change the scale. For example, if we look at a planetary scale, they don't accurately predict the perihelion of Mercury; it wasn't until the turn of the 20th century that models were created that could both accurately predict mechanics on the scale of humans and also mechanics on the scale of planets (most notably Einstein's general relativity). Given that we know there are phenomena in the universe that Newton's model is incapable of predicting, does that mean Newton's model isn't science or is "delusional in [its] science"? Of course not. In fact, we still use Newton's model for many tasks (e.g., engineering) down on the human scale, because it's a lot easier to use than the more robust models.
(I apologize for all of the Wikipedia references; I just think they're a good way to get a high-level understanding of some of these concepts.)