Topic: Wall Observer BTC/USD - Bitcoin price movement tracking & discussion - page 27991. (Read 26710175 times)

OK, it seems that I was using a definition of "log Brownian" that is not the standard one used in finance.

Indeed I was assuming that the increments D(i) = Z(i+1)-Z(i)  = log(P(i+1)/P(i)) were normal variables with zero mean, so that Z(i) would be a Brownian variable as it is usually defined in other areas - with no trend.  (Note that I am a prof of computer science, not economics!)

However, as you point out, by that definition the expected price E(P(i0+n)) would grow exponentially with n; which does not make sense in the trading context, where the "efficient market hypothesis" demands E(P(i0+n)) = P(i0).  (Or does it? See below.)

We agree at least that in a log-Brownian model the increments D(i) = Z(i+1)-Z(i)  = log(P(i+1)/P(i)) should be assumed to be independent random variables, yes?

The standard way to achieve E(P(i0+n)) = P(i0), in finance, seems to be: assume that the increments D(i) are Gaussian variables with slightly negative mean, mu = -sigma^2/2.  That is, one assumes a slight negative trend in the log-price Z(i) so that the broadening of the log-normal distribution of P(i0+n) as n increases preserves the mean P(i0).   Is that correct?

That assumption satisfies the "efficient market hypothesis", but implies (as you pointed out) that the price is slightly more likely to go down than to go up at each step.  Then Prob(P(i0+n) < P(i0)) increases with with the stride n.  Which seems weird too.

We can get rid of this weidness by assuming a probability distribution for D(i) such that E(D(i)) = 0, E(exp(D(i))) = 1.  It seems that these two conditions cannot be obtained with a Gaussian distribution, except in the limit when sigma → 0.  However, they can be achieved with other distributions that are symmetric about zero, especially if they have fatter tails than the Gaussian.  And, indeed, the most obvious deficiency of the log-Brownian model seems to be that, in real data, the distribution of the increments is not Gaussian.

If my math is correct, the distribution of the n-step increments too would satisfy both conditions: E(Z(i0+n)) = Z(i0), and E(P(i0+n)) = P(i0).  Moreover the distribution of Z(i0+n), being the convolution of n symmetric distributions, would be symmetric about Z(i0), implying that Prob(Z(i0+n) < Z(i0)) = 1/2, and hence Prob(P(i0+n) < P(i0)) = 1/2.

With these assumptions, even though the distribution of P(i0+n)/P(i0) approaches a log-normal distribution as n increases (by the central limit theorem), it remains sufficiently "log-abnormal" to satisfy those conditions (which a true log-normal distribution cannot achieve, it seems).

Perhaps you can tell me what would be a convenient "fat-tailed" symmetric distribution to assume for the increments D(i) that would satisfy both conditions.  (Perhaps a mixture of Gaussians with zero mean whose variance has log-normal distribution?  I would have a justification for that choice...)

Finally, about the "efficient market hypothesis": shouldn't it say that E(P(i0 + n)) = P(i0)*Q^n, where Q > 1 is the typical ROI factor of a generic investment per time step?  That is, if E(P(i0+n)) = X, then P(i0) should be less than X, otherwise other investments would be more profitable.

With that modification to the "efficient market hypothesis", the distribution of D(i) must satisfy E(exp(D(i)) = Q, not 1; and one can achieve that even with a zero-mean Gaussian if desired.  In that case one would have a legitimate log-Browninan model (with Gaussian increments) such that that E(Z(i0+n)) = Z(i0), E(P(i0+n)) = P(i0)*Q^n, and Prob(P(i0+n) < P(i0)) = 1/2.  Does this make sense?

EDIT: not sure whether the distribution must/may have fatter tails than a Gaussian.  Too sleepy to think now...

Until a better system than fiat actually takes hold I'm putting my faith in the logic that any better system will prevail (it was an act of faith when I first started mining Bitcoin). Until then I the exponential price trend to continue, it can only be stalled  by responsible fiat creation, and that is unlikely even if TPTB succeed in unification of all central banks.

Now you are childish.

Poker players raise if they have 52/48 but not if it is 48/52.

I am calculating 90% odds for the certain outcome. If you think the odds are 95%, you should bet with me for +EV.

50% is obviously not same as 90%.

I am tired of writing to people who don't know about betting enough to justify opening their mouth but nevertheless do it.

So no bets here. The other thread will perhaps develop into a probability machine where you will always be able to check if your feeling corresponds with the majority, and bet with the ones who think the opposite.

Some basics if you want to use the exponential trendline, or contend against its use. Two things must be observed:

[same old drivel as always]

Every exponential price trend that grows faster than population, inflation, and total economic output must perforce cease to hold at some point.  Unchecked, a 10x increase per year would mean that 1 BTC would buy the whole Earth within a few decades.

So, the question is not IF, but WHEN will Bitcoin's price stop following the historic exponential trend?

(Do I need to post again the Worldcom price chart? It followed an exponential trend much more closely and for a longer time than Bitcoin so far...)

Price trends should not be trusted without undersanding the fundamentals behind them.  Some bulls here have their explanation for Bitcoins historical exponential trend.  I have mine,  The conclusions seem to be very different...

I think we can agree on the following:

- If Bitcoin never makes it big and does not grow over its current 0.02% adoption of the target, it fails, and price per unit may collapse even from the present level.
- If it does make big and its adoption grows to even 500 million people, the price per unit is multiples of what it now is.

Every exponential price trend that grows faster than population, inflation, and total economic output must perforce cease to hold at some point.  Unchecked, a 10x increase per year would mean that 1 BTC would buy the whole Earth within a few decades.

So, the question is not IF, but WHEN will Bitcoin's price stop following the historic exponential trend?

(Do I need to post again the Worldcom price chart? It followed an exponential trend much more closely and for a longer time than Bitcoin so far...)

Price trends should not be trusted without undersanding the fundamentals behind them.  Some bulls here have their explanation for Bitcoins historical exponential trend.  I have mine,  The conclusions seem to be very different...

Every exponential price trend that grows faster than population, inflation, and total economic output must perforce cease to hold at some point.  Unchecked, a 10x increase per year would mean that 1 BTC would buy the whole Earth within a few decades.

So, the question is not IF, but WHEN will Bitcoin's price stop following the historic exponential trend?

(Do I need to post again the Worldcom price chart? It followed an exponential trend much more closely and for a longer time than Bitcoin so far...)

Price trends should not be trusted without undersanding the fundamentals behind them.  Some bulls here have their explanation for Bitcoins historical exponential trend.  I have mine,  The conclusions seem to be very different...

In the meanwhile, masterluc suggests red candles in the 'analysis never ends' thread ...

Still waiting for bitcoin  death:) Just a bit more, few months maybe
I know i am just a bit impatient:)

Nice poll question Adam most interesting in a while, the results so far are surprising.