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FXEZ's Quantitative Research In R

  • #41
  • Dec 30, 2016 3:34am Dec 30, 2016 3:34am
Quoting 9047
The following charts demonstrate the for a 45% expectancy a system attain sufficient average reward/pay-off is still profitable (neglecting transnational costs). The equity curves from the simulation: {image} The histogram of the expectancy form 10000 simulations runs of 100 trades each (e.g 1 million simulated trade in total): {image} As you can see 2 (estimated) standard deviations from the (estimated) mean the expectancy is still positive (although not by very much ... especially if transaction costs are thrown into the mix!). 9047.
I'll update this soon with some details on the efficiency frontier .... after all in its simplest form this really is just simple a linear programming exercising.
  • #42
  • Dec 30, 2016 5:41am Dec 30, 2016 5:41am
Quoting FXEZ
A question about your starting formula: It looks like the (1+x) is simply a representation of percent gain and (1-x) would be a percent loss from a starting account of 100%? By multiplying them together (1+x)(1-x) it appears to be the compound interest formula taught in many finance courses. So on the right side of the formula, the >= 1: the 1 represents 100% of starting account?
Sorry that was obvious to me. You start with some money. That's 1 time your account size. If you lose 1% of your account you lose 0.01 time your account. It remains 0.99 time your account. This loss has multiplied your account by a factor 0.99. Same thing if you win 1%, your account gets multiplied by a factor 1.01. (1-x) and (1+x) are the multiplicative factors where x is the fraction of the account you risk. They are independent of the account size. When you win then you lose your account is multiplied by (1+x)(1-x). The sequence win-win-lose-win-lose-lose multiplies the account by (1+x)(1+x)(1-x)(1+x)(1-x)(1-x). BTW because the mulitplication is commutative (a.b=b.a) you can reorder to get (1+x)(1+x)(1+x).(1-x)(1-x)(1-x) or simply (1+x)^W.(1-x)^L where W is the number of winners and L the number of losers. The final account size =startingBalance.(1+x)^W.(1-x)^L depends on the numbers of winners and losers not on their order.

Quote
So on the right side of the formula, the >= 1: the 1 represents 100% of starting account?
The 1 means 1 time your account (100%). You search a MM that increases the account size after a sequence win-loss. The net multiplicative factor (1+x)(1-x) is searched such that it multiplies the account by 1 or more.

Quote
sqrt(-1)
Arrrghhhh. The square root of X is "the positive number that, when multiplied by itself, gives X". Only real numbers can be positive. The imaginary unit is not positive since it isn't a real. sqrt(-1) is just not defined. Especially it is not i!

Quote
Hmm yes I see what you mean about Lord Martin. He gets a bit wild as x approaches 1.0.
Little problem with this curve: unless you borrow money you can't risk more than what you have. x, y and z can't be more than 1 (100% of your money). In fact due to margin requirement they are strictly less than 1.

Quote
so it appears when the account is in 50%+ drawdown you are supposed to risk 100%+ of the account? If it is risk and not bet, then that would translate into leverage much larger than 1.0. It would be like a double or nothing bet at this point right? I'm having problems with the scaling.
Of course! you want a MM such that at the end of the trades the account is multiplied by one or more. If you are in 50% DD you have to at least double the account to achieve this. Since we use RR=1 the gain which is equal to the risk must be greater than the whole account. Therefore you must risk more than what you have!

Quoting FXEZ
When you randomly sample with replacement from that population of trade outcomes you create new trade series that can have different paths and different expectancies because one curve may oversample winning trades in a series while another may oversample losing trades in a series
Amazingly no. It is called bootstrapping in machine learning field. If your sample is big enough this resampling (many many of them) will have equivalent statistics (mean, variance...) than the original unknown distribution.
No greed. No fear. Just maths.
  • #43
  • Dec 30, 2016 8:33am Dec 30, 2016 8:33am
If you flip a fair coin...
Let's say head you win 1 and tail you lose 1. We can also say your gain is either 1 or -1. You can draw this probability distribution:
Attached Image

What is the probability of your gain after playing twice?
Flips are independent. I note H for head and T for tail. There are 4 outcomes.
Inserted Code
result proba gain
HH      0.25   +2
HT      0.25    0
TH      0.25    0
TT      0.25   -2
Attached Image

What if you play more? If you have the distribution for n games you can build the distribution for n+1 games. For each bar of the current histogram you scale the original probability distribution by the value at this bar. You center the original distribution on the bar. You sum all the distributions into one. That's the result. This complex operation is a convolution. It works also with continuous distributions by replacing the sums by integrals.
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What if we do it many times?
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We started with 2 possibilities and we end up with a normal distribution. Let's try with 9047's Sample pay-off histogram:
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Click to Enlarge Name: 9047_payoff.png Size: 10 KB

It works with almost any starting distribution (there are conditions). This is the Central Limit Theorem.
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Click to Enlarge Name: anything.png Size: 10 KB

This is why 9047's plume plot is so similar to my super simple Bernoulli plot (up to his compounding MM): the exact shape of the distribution is not so relevant.

Is it important that the probability is always the same? No.
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No greed. No fear. Just maths.
  • #44
  • Dec 30, 2016 8:44am Dec 30, 2016 8:44am
Cheers Pip. Those visuals really helps in my understanding. In essence all roads lead to the Central Limit Theorem with a large enough sample size. :-)
  • #45
  • Edited 11:17am Dec 30, 2016 9:03am | Edited 11:17am
Quoting PipMeUp
..... Let's try with 9047's Sample pay-off histogram: {image} It works with almost any starting distribution (there are conditions). This is the Central Limit Theorem. {image} This is why 9047's plume plot is so similar to my super simple Bernoulli plot (up to his compounding MM): the exact shape of the distribution is not so relevant. Is it important that the probability is always the same? No. {image}
Absolutely agree the Bernoulli approach is the same thing ..... just a lot simpler .... hell, massively simpler. I do always love seeing central limit theorem in play. It still makes me smile when the bell curve appears. The only "nice" thing about the plume plots is the pretty visualisation over the different scenarios that may happen.

So at the risk of using a cliche .. "there are many ways to skin a cat" (Wiktionary link). All the different approaches and views that arrive at essentially the same answers are what makes threads like this so valuable.

For me the key take away thing is that if you develop a system with an acceptable average (or median) pay off then you can essentially drop your expectation estimate onto the efficiency frontier curve (taking into account operational costs) and you have a guide as how poorly your system can perform before you turn it off (either under certain circumstances or otherwise).

As always - great stuff Pip.

Happy "Gregorian New Year"
  • #46
  • Dec 30, 2016 4:40pm Dec 30, 2016 4:40pm
Nice discussion here.

Playing with random numbers only gets you so far. What about edges or real market data analysis?
  • #47
  • Dec 31, 2016 6:01am Dec 31, 2016 6:01am
Quote
What about edges or real market data analysis?
Or stated differently what are the practical implication of this? Does it give an edge?

Let me answer the second question first: NO (not yet at least...)
Thus far we all implicitly assumed that the edge pre-exists. We saw that without an edge there is no money management that is able to grow an account. A martingale can only pretend to do so until the kiss of death even if you use the smallest possible multiplier explained in post #31. FXEZ shows it graphically on post #35. Way too quickly you should risk more than all you have left to recoup the losses.

What are the practical implication of this?
Compounding MM creates a negative drag on your equity. Only fixed size MM can keep the account safe on this front. A retailer account can't grow fast enough without compounding. Not only you need an edge but it has to be high enough to beat the negative drag created by the money management.

Trading is not easy; it isn't free of charge either. Post #10 shows that the cost of trading is an additional drag to the account. Too often neglected. This cost isn't fix but depends on the leverage. Small leverage is to be used. Retailers can't trade without leverage because of lack of capital. Your edge has to beat both the negative drag of the money management and the costs of trading. Not to make money just to not lose!

alphaomega shows in posts #27 that the variance of trading can easily lure us. A backtest can show a great profit just out of luck. A single realization of the "trades path" is not enough to tell if an edge exists or not. Post #30 Copernicus shows that bootstrapping can help a bit.

Expected draw down and risk of ruin are explored by 9047 on post #39. He shows the payoff profile is the key indicator. Post #43 I show how very quickly any payoff histogram is equivalent to a simple weighted Bernoulli trial (baised coins with asymetric rewards). A strategy can be summarized by its two key components: winrate and RR. This validates the first posts of the thread by showing that yes they do reflect trading despite their great simplicity.

On post #32 I simplify further by drawing the envelope of all the possible realizations (with 99% confidence) instead of using Monte Carlo. We can visualize the worst expected lowest point of the equity. If we know the winrate and RR of a strategy we can set the risk level to a predefined accepted drawdown/RoR. It shows that we need a high trade frequency and alphaomega shows we must use low trade frequency to limit the cost. Again a tradeoff appears.

We need a strong edge to beat the cost and MM. We also need to know our edge in term of winrate and RR to size the risk. Given the equity curve of a backtest we may try to estimate these two values. The slope of the regression line (in log scale if you use compounding) is an estimator of the expectancy. The dispersion of the equity is an estimator of the variance. From the expectancy and the variance you can know winrate and RR. Unfortunately if you over- or under- estimate the expectancy you will systematically underestimate the variance. With a single backtest you will always over- or under- estimate the expectancy. Post #3, the first graph shows a "backtest" with a great negative slope while the true known edge is exactly 0. Should you estimate the variance with a regression line you could be wrong by a factor 10.

Summary:
Good knowledge of the winrate and RR is required but the risk is always under-estimated.
Fixed size MM is required but percentage of equity is needed.
Small leverage is required but high leverage is needed.
Low trade frequency is required but high frequency is needed.
No greed. No fear. Just maths.
  • #48
  • Dec 31, 2016 6:29am Dec 31, 2016 6:29am
Great summary Pip. So good to get a dose of reality from time to time. :-)
  • #49
  • Dec 31, 2016 8:50am Dec 31, 2016 8:50am
Great post.

A few comments.

Quoting PipMeUp
Compounding MM creates a negative drag on your equity.
The standard name for this is volatility drag.

Quoting PipMeUp
A strategy can be summarized by its two key components: winrate and RR
As with all summaries, some people will be satisfied by this, others will not. For example, if you trade a portfolio of strategies, it's very important to know the correlation between them.

Quoting PipMeUp
Summary:
Good knowledge of the winrate and RR is required but the risk is always under-estimated.
Fixed size MM is required but percentage of equity is needed.
Small leverage is required but high leverage is needed.
Low trade frequency is required but high frequency is needed
Your summary has a lot of implicit assumptions, more importantly the "required" conditions you list assume no apriori edge. So yes, if you have no edge you absolutely must trade at low frequency and with fixed size MM, which will make your trading last longer towards the inevitable 0.

But if you do have an edge, enough to overcome the volatility drag and the other costs, then you move to the "needed" side, and you absolutely should trade at the highest frequency possible.

This is why I asked the original question, because a lot of what was talked about in this thread assumes no edge, which might be reasonable (and true for 99% of traders). But I feel that it kind of misses the point. My opinion anyway, don't want to butt in here.
  • #50
  • Edited 4:27pm Dec 31, 2016 9:19am | Edited 4:27pm
Quoting 000
But if you do have an edge, enough to overcome the volatility drag and the other costs, then you move to the "needed" side, and you absolutely should trade at the highest frequency possible.
Good point 000 and under a diversified portfolio, provided you have a demonstrated edge with individual system contributions, you can then reduce your portfolio volatility by deploying simple yet logical un-correlated systems and then ramp-up trade frequency through timeframe and asset diversification to bring home the returns.

Step 1. Find simple systems that positively respond to particular market conditions, are robust in protecting capital outside those particular market conditions and are applicable to any liquid instrument. Each system needs to display a demonstrated edge over as much data history as you can throw at it. The edge does not need to be great but it needs to contribute weak alpha to the portfolio. The focus should be directed towards robustness as opposed to raw return. Each system contribution must be stress tested to ensure that failure of a single contributing element of the portfolio does not bring the whole portfolio crashing down. Under a very diversified portfolio where each contributing element has a low % contribution to overall equity, the game of finding simple system candidates is far easier than finding a suitable system in isolation as you have the luxury of removing system constraints such as stop losses that can unnecessarily introduce drawdowns into your system.

Step 2. Blend equity curves of as many uncorrelated system contributions as you can find for a portfolio to amplify the signal to noise ratio. This is a continuous process as market conditions are continuously changing. Over time drop those system contributions that are no longer contributing weak alpha to the portfolio. The non-random signals of correlated systems will constructively interfere exacerbating portfolio volatility, whereas the non-random signals of uncorrelated systems will in principle be preserved. Blending uncorrelated systems has a two-fold impact. The random noise of each equity curve is not purely additive through destructive interference principles whereas the valid signals will be retained and more evenly dispersed across the time series through blending of uncorrelated series. The result will be a blended less volatile composite equity curve. Rinse and repeat with uncorrelated additions to the primary portfolio until a desired risk-weighted return outcome is achieved.

Step 3. Diversify, diversify and diversify across timeframes and asset classes to ramp up trade frequency and achieve your desired risk weighted return ensuring that your universe is as evenly weighted and as uncorrelated as possible.

Step 4. Continuously identify new candidates for inclusion in your portfolio that follow the principles of Steps 1-3 above.
  • #51
  • Dec 31, 2016 10:05am Dec 31, 2016 10:05am
Quote
The standard name for this is volatility drag.
I thought the name was "leverage decay".

Quote
For example, if you trade a portfolio of strategies, it's very important to know the correlation between them.
Yes the correlation of a portfolio of strategies is very important to know but the whole portfolio can be regarded as a high level strategy and the conclusion doesn't change. The payoff profile will converge to a winrate/RR equivalent shortly.

Quote
Your summary has a lot of implicit assumptions, more importantly the "required" conditions you list assume no apriori edge.
Quote
if you have no edge you absolutely must trade at low frequency and with fixed size MM
I think you misread me. Under no edge assumption we saw that the optimal fixed size is 0 and the optimal frequency is to never trade. Here I HAVE TO assume a STRONG edge because of the cost and negative drag.

The question is how to estimate correctly the expected return of a strategy (the edge)? Literature says you just can't.
No greed. No fear. Just maths.
  • #52
  • Dec 31, 2016 11:35am Dec 31, 2016 11:35am
Quoting PipMeUp
{quote} The question is how to estimate correctly the expected return of a strategy (the edge)? Literature says you just can't.
That's only a problem for low frequency strategies. You just showed above that a HFT strategy (ie: lots of samples) quickly converges.

To get back to the edge question, anybody here using anything beside level-1 price data?
  • #53
  • Edited 12:55pm Dec 31, 2016 11:59am | Edited 12:55pm
Quoting PipMeUp
{quote}
................I think you misread me. Under no edge assumption we saw that the optimal fixed size is 0 and the optimal frequency is to never trade. Here I HAVE TO assume a STRONG edge because of the cost and negative drag.
The question is how to estimate correctly the expected return of a strategy (the edge)? Literature says you just can't.
Literature is correct! (partially).

We cannot estimate the true edge based on the historical stats of the equity curve.
Blind back testing also does not give true estimate of the edge. Most likely you will fall in the trap which I already described in one of the previous posts (the luck factor).

To estimate the true edge, you need very good understanding(+ statistics) about each specific pattern, and the dynamics (the mechanics) which are the true source of this pattern.

If you do not understand where your edge comes from, most likely you don't have any edge. And what ever results you are getting are just random and temporary.

The bottom line is that trading at its core is all about fundamentals and information.

For example if you get your news info 1 minute before 99% of market participants - THIS is TRUE edge. No need for back-testing.......
Of course this is the ideal scenario, and extremely hard to achieve.
The other true edge is if you have information about the positions of the large market participants. So you can front run them.....
Arbitrage and market mispricing due to errors also can provide REAL edge. (but this is almost not applicable in modern markets)

Unfortunately all this information is completely inaccessible to small traders.
-----------------------------------
So we have to rely on other types of info. We have no other choice.
This other type of info have larger degree of randomness build in it. In other words we have a sea of randomness and in that sea of randomness there are some clues swimming around! These clues are pointing to some non random and deterministic models. These clues are the TRACES of the actions made by the people who possess true edges (like the ones which I already described above).

So in essence, we're followers. Always!
We wait for the sharks to attack their prey and to reveal their hand, and then we try to feed ourselves with the leftovers.

Just one example:

If the market is oversold 150 pips (I'm talking, smooth fast trend with accelerating momentum, almost no corrections and without any news) and everyone thinks that the price is too low and it is time to rise, then chances are:

a) that you are in the middle of some major stop hunting event. (planned and executed by the market makers)
b) some big player(s) knows something.......

In both cases your best bet is to either Sell short, or to stay on the sidelines!
And what most people do in a situations like this? They BUY and average down (almost every single time!!!)
This fact by itself gives you a small edge.... at least 50% of the time.
  • #54
  • Edited Jan 1, 2017 8:29am Dec 31, 2016 2:09pm | Edited Jan 1, 2017 8:29am
Guys, I take my hat off to you.
Bravo!

Best
D.P.

P.S.

Quoting PipMeUp
{quote}Summary: Good knowledge of the winrate and RR is required but the risk is always under-estimated. Fixed size MM is required but percentage of equity is needed. Small leverage is required but high leverage is needed. Low trade frequency is required but high frequency is needed.
IMO, retail traders need fixed MM, low leverage (if possible) and high frequency trades...

Quoting alphaomega
{quote}
1/ For example if you get your news info 1 minute before 99% of market participants - THIS is TRUE edge. No need for back-testing.......
2/ The other true edge is if you have information about the positions of the large market participants. So you can front run them.....
3/ Arbitrage and market mispricing due to errors also can provide REAL edge. (but this is almost not applicable in modern markets)
Completely true.

Mostly, retail traders can try to find its edge at 2/ and 3/ since there is no access to insider's source of information.
To realize 2/, retails should have information on the volume cluster formations, for example. Some platforms can give such info.
As to 3/, market mispricing due to errors can be used as well for sure; intraday's structure formation by means of using the market profile technique should be taken into account.

P.P.S. At last there are some researchings about RR; RTS futures market is described - however the results can be extrapolated to Forex since RTS depends on the USD index. Notice that text was written by author in his Russian blog. Source - http://tradetrade.ru/math/2013/06/04...herse-rts.html
Hunting High and Low
  • #55
  • Jan 1, 2017 8:43am Jan 1, 2017 8:43am
Quoting 000
You just showed above that a HFT strategy (ie: lots of samples) quickly converges.
The payoff profile approximates the probability distribution of the strategy. But it's not the true one. If we had the true one we could infer an equivalent distribution solely defined by some winrate and some RR to asset the risk and expectations. Not those of the strategy but a pair of equivalent parameters.

How much the approximate payoff profile represents the true one is unknown. But to give you an idea let's use the coin flip again. After three flips you get Head, Head, Head. Can you accept the coin will always give Head? Of course not. We have to average many of them. FXEZ's Post#3 shows 100000 (a hundred thousands) of flips and the cumulated result is rather far from zero, visually -275 I think. The average is -0.00275. It is supposed to be zero.
At the end of lecture 10 he explains why the more a coin is fair the less you trust it is.
Inserted Video

You can also watch the three next on the same topic.
Inserted Video

Inserted Video


Quoting alphaomega
Just one example:
If the market is oversold 150 pips (I'm talking, smooth fast trend with accelerating momentum, almost no corrections and without any news) and everyone thinks that the price is too low and it is time to rise, then chances are:
a) that you are in the middle of some major stop hunting event. (planned and executed by the market makers)
b) some big player(s) knows something.......
In both cases your best bet is to either Sell short, or to stay on the sidelines!
And what most people do in a situations like this? They BUY and average...
Data confirm this. If price is oversold it keeps going down for some time.
http://www.forexfactory.com/showthre...94#post9336994
Happy new year to all
No greed. No fear. Just maths.
  • #56
  • Edited 9:38am Jan 1, 2017 9:25am | Edited 9:38am
Quoting PipMeUp
If you flip a fair coin... Let's say head you win 1 and tail you lose 1. We can also say your gain is either 1 or -1. You can draw this probability distribution: {image} What is the probability of your gain after playing twice? Flips are independent. I note H for head and T for tail. There are 4 outcomes. result proba gain HH 0.25 +2 HT 0.25 0 TH 0.25 0 TT 0.25 -2 {image} What if you play more? If you have the distribution for n games you can build the distribution for n+1 games. For each bar of the current histogram you scale the original probability...
Thoughts about a stop-loss

Standart Deviation is used as a parameter for volatility description. The higher the volatility, the larger the Sigma is. So the Sigma is taken as a measure of risk, as a measure of a custom stop loss.
Attached Image
Hunting High and Low
  • #57
  • Jan 1, 2017 9:30am Jan 1, 2017 9:30am
Quoting Dr.Phoenix
To realize 2/, retails should have information on the volume cluster formations, for example. Some platforms can give such info.
Indeed, this is one of the inputs I use in my conv nets, for the minute-by-minute sources. I'm now trying to integrate sparser sources (hour-by-hour or longer).

Also, related to the above discussion, the loss function can be a bit tricky to define. The same for figuring out exactly what you want to predict.
  • #58
  • Edited 10:09am Jan 1, 2017 9:58am | Edited 10:09am
Quoting 000
{quote} Indeed, this is one of the inputs I use in my conv nets, for the minute-by-minute sources. I'm now trying to integrate sparser sources (hour-by-hour or longer). Also, related to the above discussion, the loss function can be a bit tricky to define. The same for figuring out exactly what you want to predict.
Problem however is that Forex (to be unregulated space) doesn't give you such data, or we can not take it without permission. So I am interested to look at more endeavors of yours (if you have).

About stop-loss. I just meant that Sigma to be calculated in pips could be used as a custom stop-loss, for example.
Actually, Market Statistics v4.2 indicator (published at FF) can calculate Standart Deviation to be Sigma.

Best
D.P.
Hunting High and Low
  • #59
  • Jan 1, 2017 10:08am Jan 1, 2017 10:08am
w/o spread, probability calculations makes no sense

Am on the same page w/ @alphaomega

Quoting alphaomega
Although I have zero experience with R (or things like matlab) and relatively modest experience with other languages (mostly mql, c and some web, html..) over the years I've managed to make quite a few experiments regarding algo trading (and trading prob.stats. as a whole). So, I am able to share some of my experiences and insights. I have no intention to post my code here, only my theoretical conclusions and philosophical insights based on my research. If the OP considers the following for off topic - feel free to delete my post! ------------------------------------------------------------------------------------------------...
WaBaS Where are Buyers and Sellers
  • #60
  • Jan 1, 2017 10:57am Jan 1, 2017 10:57am
Quoting Dr.Phoenix
Problem however is that Forex (to be unregulated space) doesn't give you such data, or we can not take it without permission. So I am interested to look at more endeavors of yours (if you have).
Indeed, Forex as a whole doesn't. But individual Forex brokers do publish such information.

Quoting Dr.Phoenix
About stop-loss. I just meant that Sigma to be calculated in pips could be used as a custom stop-loss, for example.
When I said loss function I didn't mean stop-loss. A loss function is a formula used to compute how "good" your machine learning prediction is.

In regards to stop-loss, they are an integral part of the system. For example, a mean reversion system will typically not use a stop loss, as opposed to a trend following one. Of course, you would have a "master" stop loss, but that one should only be hit once, and you should reevaluate the system afterwards.
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