Mathematical induction - Wikipedia, the free encyclopedia I know full well what Induction is. I do not make any recursive calls to my proof...!
Please
read my proof
carefully. If we are only considering two securities, then the maximum my proof allows is charts of one time-step - a chart all securities share.
This is why we must consider numerous securities at once.
An argument that works with N and N+1, must work with any n, if the argument is valid. This is obvious, otherwise it wouldn't be a proof.
And induction isn't recursion, induction is a hypothesis one uses in order to prove that the set of objects that satisfy a property is equal to all the objects in the set.
Let me break down to you exactly what you tried to do.
The existence of valid charting would suggest the following: there is some function P that takes in the chart of a security as a parameter and returns the probability that its price will increase in the next time-step. Charts themselves can be represented by a series of some constant N time-steps. For any chart that would require a smaller time frame, the function could simply give zero weight to the first few time-steps.
Here you're setting up your problem: you want to prove that there is no such function P that returns the probability of a stock increasing at time t+1.
Assume all securities start at an equivalent price at the first time-step; this is a trivial distinction as it designates that a fractional amount of the security was bought. In addition, since charting does not concern itself with any fundamentals, we can assume that charting can be applied equally well to any security or any portfolio of securities.
Assumption #1: All securities start at time T=0 with equal prices.
Assumption #2: Charting works for any number of securities. This is your first induction assumption.
There are so many securities available in the market that we can rather easily assume that there exist at least N+1 (decently) uncorrelated securities.<sup>1</sup>
Now suppose we take N of these securities. Because we are only concerned with N time-steps, we can construct a portfolio of these securities such that the chart of this portfolio (that is, the last N time-steps) is identical to the chart of the leftover security. This is a consequence of basic linear algebra.
Assumption #3. There are N+1 uncorrelated securities. This is your main set.
Logic Step #1. You're taking a subset of your main set, with N securities.
Logic Step #2. Since they're linearly independent vectors, they span Rn. This means that any other vector can be created out of these N independent vectors.
Supposing that charting works, the probability P that the security/portfolio will go up in the next time-step should be identical because the charts are identical. However, the portfolio’s P value is also equal to the weighted sum of the P-values of its constituents, and so, by the transitive property, the security’s P value must equal to the weighted sum of the P-values of the portfolio’s constituents.
Logic Step #3. The probability of the security going up is equal to the weighted probability of the first N securities.
This leaves us with a system of equations, with N unknowns (namely, the weighting of the securities in the portfolio<sup>2</sup>) but N+1 equations! (Specifically: the first N equations dictate that the price of the portfolio is equivalent to the price of the security at each time-step, while the last equation stipulates that the weighted sum of the probabilities of increase equals the security’s probability of increase.) For there to be a valid, unique solution, one of these equations must be redundant.
But by our definition of “(decently) uncorrelated” securities, none of the first N equations (those equations dealing with time-step values) can be redundant.
I'll get back to this below.
Then the last equation must be redundant. But redundant due to which other equation? The answer is obvious when we repeat the above procedure with each security.
This is your induction step in the proof. When you repeat the procedure with every security, you're essentially saying: Take any subset of N securities of these N+1 securities. All of them have this property. This step is necessary so you can show that your proof works for any number of securities. This would be true, if you also proved it to be the case for N=1.
Now getting back to the statement above this one.
The N equations are being multiplied by the solution of the system in order to produce the vector that will be the input of P. Ax=b, A is your matrix, x is the weight vector and b is the N+1 security's price history. Correct?
The prices at the first time step are all equal (as per our definition) and therefore the first equation always states that the sum of the weights is one. If the last equation is redundant due to the first, then clearly the P values of all securities must be equal!
Since you haven't proved that your induction is valid for N=1, you can't use the conclusion from the induction step in your proof.
And because for N=1, your system would have infinite solutions, you can't solve it. Since you can't solve it, your proof is flawed.
