Developing a profitable system(infrastructure) on a (pseudo-)random data

[Equalizer]

Quote from phattails:

I took a look at that paper and I could be wrong in my analysis, but the results seem trivial. The E[X|x>.5]= .75 assuming a Continuous uniform dist. from 0 to 1. So if we are given a value of x, s.t. X>.5 or x<.5. Then our cond. expected values will be .75 and .25 respectively. So how often will any x be less
than .75? of course it's 75% of the time. Does this seem like prediction? If I referenced any value in the sequence that was < or > .5 I could say that 75% of the time I can predict the increment.

Regarding the synthetic data I say start with a simple random walk and start modelling and developing and then work up different stylized facts in order to isolate the relevant features.

The reason why the "best predictor" of the sign of x(i+1) - x(i) must be opposite to x(i) - 0.5 is due to anti-correlation and that x(i) is a UDRV in [0,1].
Equation (3) is wrong, anti-correlation is not -1/4 in that example. Clearly the authors screwed up the calculation of E[x(i)^2] - 3rd term in expansion. I can assure you it is not 1/2 for a UDRV in [0,1].

But the derivation for general distributions looked correct last time I went through this.

 

[Mike805]

Quote from nLepwa:

At each step (or turn):

1. Close your position.
2. If expected move = up, open long position.
3. Else open short position (or stay in cash).

This strategy applied on a process with normal distribution has positive expectancy as long as:

fees < |avgGain|*0.66 - |avgLoss|*0.33.

Or in other words, if your process has a constant step size of s (at each step, the cumulative value is the previous one +/- s), then you have positive expectancy when:

s>fees*3.03

Your equity curve is then directly proportional to the trading frequency, which is linked to s by the sqrt() function and Pi.

Your trading fees determine your equity curve.

Let me guess...

The way around this is ... to discretize price using a step function and to then normalize it with a volatility measure to eliminate fat tails?

One could also use the Arcsine law I mentioned earlier to compute the skew in the distribution.

I would also look at the Monty Hall paradox.

Ninna

Yeah... that's a good starting point. What you're likely to find is that this approach will be often buying at bid and selling at ask; i.e. you'll "profit" from the spread which is, as we all know, a bit more difficult than simple stats

No way on elimination of fat tails though... that's where some clever use of certain distributions will come in handy. Those tails are where all the opportunity is at, never ever try to get rid of them.

Haha, yes this game is an interesting permutation of Monty Hall.

 

[Mike805]

Quote from phattails:

Hi Mike,

Thanks for the thought experiment. When I first glimpsed at this thread I thought you were going somewhere else with the experiment before I had a chance to look at the spreadsheet. Was the motivation for this experiment optimal stopping or was is bayes Monty Hall stuff?

Have you checked out the bandit problems? From what I can remember there's a couple guys that have done a lot of research on that types of stopping problems that have tie in with Jim Simons. Samuelson has an interesting rebuke of the the kelly criterion using the optimal stopping problem and martingales as his motivation.

pt

Optimal Stopping + Monty Hall is a good way of talking about it. The motivation is primarily the application of non-intuitive stats. If one were to try to convince someone of the theory behind monty hall they might have a difficult time... understanding random variables is not an easy task right off the bat.

I remember a former roommate back in college was applying for a job and they asked him to write a computer program that was similar to the monty hall problem. This was back when I had never heard of anything about stats/monty hall, and, for a while, I thought the question was a trick question (mind I was 18 and had never taken a stats class). After seeing it work, it really grabbed my interest. I started really looking into it and then moved on to other things.

There's quite a lot of interesting stuff one can do with tick data and discrete steps. My emphasis has always been on creating a decent distribution from the data, not on so much on the stats that lead to a trading rule... that said, the bandit problems are neat in that they might get one to start thinking in terms of multiple turn based probabilities, which, actually will lead to the discovery of edges IMO. The 3 days down idea is something along those lines.

I'd be interested in anyone posting some decent test results - the 50% result from SP daily closing prices was actually really interesting... maybe in a while I'll post a distribution that may show some better results for that particular time series.

 

[Mike805]

Quote from phattails:

Samuelson has an interesting rebuke of the the kelly criterion using the optimal stopping problem and martingales as his motivation.

pt

Would you mind pointing me to the sources for that? PM me if you'd like.

Thanks,
Mike

 

[Code7]

Quote from Mike805:

The best way to explain this is through an example.

Suppose we have price data that is normally distributed between 100 and 200. In excel this function would be RANDBETWEEN(100,200) or something similar.

One turn = one change in price.

1st turn we have an R = 123 and C = 155. C is now a "high" value.

2nd turn we have an R = 190 and C = 156. C is now a "Low" value.

3rd turn we have an R = 177 and C = 157. C is now a "Low" value.

4th turn etc etc...

Each turn in this case will either have a right or a wrong outcome depending on the value of C(t+1). t=turn.

In the case above

R(1) < C(1) -> C(1) = high. C(1) < C(2) = 156 -> "wrong"

R(2) > C(2) -> C(2) = low. C(2) < C(3) = 157 -> "right"

R(3) > C(3) -> C(3) = low. C(3) < C(4) = 158 -> "right"

etc etc...

If one repeats this process for N - trials, the probability of being "right" = 66%. This is a classic gambling exercise.

There are some interesting characteristics of this process that some here may have seen before. One is that it doesn't matter if C(1),C(2),C(3)... is continuous or not. Second, the random distribution used for comparison is *independent* of C(t).

We now have a way to determine if the next closing price will be higher or lower with a 66% accuracy. So, how do we create a trading rule?

Mike

We can't use this to determine if the next closing price will be higher or lower than the current closing price.

We can just compare price changes. If the next change will be greater than let's say -30, it could still be -15 and price drops again.

Maybe I'm missing something but I can't see how to create a trading rule out of this.