First sacred cow: "If the trend started, it will continue" - page 78
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все эти теории глубоко теоертичны))), а практически непригодны.
Because they operate on the notion of the best forecast: the best price forecast for tomorrow is today's price. We can prove that this is the best forecast only if we know a priori what kind of process and distribution we are forecasting, but in practice we don't. How can we prove that a particular forecast/forecast methodology gives the best prediction (in terms of RMS) other than by going through all possible ones, which is unrealistic? And then you don't have to predict the price at a certain point in the future to make money.
I gave two opposite cases where E[x(i+1)]=E[x(i)]. If it's a martingale, that's the best prediction we can make. I.e. no one is saying it's a good prediction. It's a bad prediction, but every other prediction you can make is even worse. That's why this one is the best. And you don't need any over-reverting, everything is already proven. In the case of the mean-reverting process, it's a very good prediction, it's guaranteed money.
"E[x(i+1)]=something" is not a prediction of price at a certain point, it's an estimate of the trend. Even if you have estimated E[x(i+1)] absolutely accurately, it is not a fact that at time i+1 the price will be exactly that. It is a fact that over the long run, on average, the price will show the predicted results.
Всё да, но может быть и шире. E[x(i+1)]=E[x(i)] это не только мартингал.
E[x(i+1)]=E[x(i)] - это флэт, завтра цена будет такая же как сегодня. Это mean-reverting процесс, который так приятно торговать.
Или это random walk, который прибыльно торговать невозможно.
Т.е. рынок можно рассматривать как чередование периодов случайного блуждания с периодами псевдо-стационарности. При этом всегда будет E[x(i+1)]=E[x(i)] и никаких трендов. Такая вот гипотеза.
mean-reverting is an alternation of sub- and super-martingales, local, which have been mistaken for a martingale.
Actually, for me a trend is a sub or super martingale, an opportunity to make a best estimate different from the current value. The martingale itself is also a trend, but it is called a flat. :)
95% of traders in the market ignore and even despise theories. >> 95% of traders in the market lose more than they earn. Can't you see a trend? The GPMorgans and other GoldmanSachs are hoovering up the best mathematicians, see for example Shiryaev's lecture, the link was above in this trid. The trend is still not visible? Well, be happy with that...
I have given two opposite cases where E[x(i+1)]=E[x(i)]. If it is a martingale, it is the best prediction we can make. I.e. no one is saying it is a good prediction. It's a bad prediction, but every other prediction you can make is even worse. That's why this one is the best. And you don't need any over-reverting, everything is already proven. In the case of the mean-reverting process, it's a very good prediction, it's guaranteed money.
"E[x(i+1)]=something" is not a prediction of price at a certain point, it's an estimate of the trend. Even if you have estimated E[x(i+1)] absolutely accurately, it is not a fact that at time i+1 the price will be exactly that. It is a fact that over the long run, on average, the price will show the predicted results.
You're explaining to me the plain truths that I didn't argue with. If we know a priori that e.g. the increments are independent and the distribution is such-and-such, then martingale/sub/supermartingale follows from that. My point is that in practice there is no way to attribute a real process to one of the martingales and/or say that it is a process with independent increments.
Я о том, что на практике нет возможности реальный процесс отнести к одному из мартингалов и/или сказать, что это процесс с независимыми приращениями.
If you just consider the price, then yes, it's like the situation with a slightly wrong coin - you just can't tell it apart from the right one using statistical methods. But if you use additional information, it's better if it's different.
If you just consider the price, then yes, it's like the situation with a slightly wrong coin - you just can't tell it apart from the right one using statistical methods. But if you use additional information, it's better.
Let's take a particular generated random walk. It can be changed systematically by introducing even deterministic dependencies at some points in time. In this case the new series will also be distributed and without knowing the way of adding these dependencies it is practically unrealistic to say from the new series that there are any. The external resemblance to a martingale says nothing about the absence of dependencies in the series.
I would like to clarify a misunderstanding that goes something like this:
"best prediction on E(x[i+1]=E(x[i])".
Why misunderstanding, because the above identity is a special case of
autoregressive equation for a Markovian random process, when the future value of the series
is influenced only by its current state, i.e. the system "remembers",
only today and it does not care about its path to its current state.
This is the so-called Markovian random process.
And in the case of "non-markovian" i.e. when the system "remembers" the path to its
present state and the memory depth is p=(1,2,3,...) that is
the coeff. of autocorrelation AR(i) are not equal to zero at i<=p, and the equation
prediction will be X(i+1)=AR(1)*x(i)+AR(2)*x(i-1)+....+AR(p)*x(i-p+1) ; (1)
and the condition X(i+1)=X(i) will be satisfied as seen from (1), will
be satisfied if p=1 and AR(1)=1;
mean-reverting - это чередование суб- и супер-мартингалов, локальных, которые по ошибке приняли за мартингал.
Собственно, для меня тренд - это и есть суб- или супер-мартингал, возможность делать наилучшую оценку отличную от текущего значения. Сам мартингал - то же тренд, но именуемый флетом. :)
Er, no... Mean-reverting is witty, but wrong. I cheated a bit with mean-reverting, in fact it should be E[x(i)]=constant. Which of course doesn't negate E[x(i+1)]=E[x(i)].
The sub-martingale is clearly a trend. E[x(i+1)]>E[x(i)] its nature may be different, but it's not that important for a general definition. The only question is how often you see sub-martingales in the market. There are papers that claim to have seen this beast and unambiguously identified it. But it's very rare.
это да, но я немного о другом.
возьмем конкретное сгенерированное случайное блуждание. Его можно изменить системно внеся даже детерминированные зависимости в некоторые моменты времени. При этом новый ряд будет так же распределен и не зная способа добавления этих зависимостей практически нереально сказать по новому ряду что таковые имеются. Внешняя схожесть с мартингалом ничего не говорит об отсутствии в ряде зависимостей.
We are talking about the same thing, but in different words.
Э-э-э нет... Про mean-reverting остроумно, но не так. С mean-reverting я чуть-чуть смухлевал, на самом деле должно быть E[x(i)]=константа. Что естественно не отменяет E[x(i+1)]=E[x(i)].
Суб-мартингал это однозначно тренд. E[x(i+1)]>E[x(i)] природа его может быть различна, но это не так важно для общего определения. Вопрос только как часто тебе встречаются суб-мартингалы на рынке. Есть работы, которые утверждают, что видели этого зверя и однозначно его идентифицировали. Но очень редко.
Oh, I see, it's a generalised Orstein-Uhlenbeck process. Well, that's one way of looking at it. Perhaps it even makes sense physically, for the market.