Optimized Risk vs Reward Equation

What I'm saying is, given X lot size at location A, there exists a Y lot size at location B, such that risk is minimized and profit is maximized.

I assume you determine point of entry A and lot size x according to your mm rules and risk appetite



It took me a while to interpret this, English is not my native language; in Serbo-Croatian, some mathematical terms have different names.


OK, so, the derivative of a function is this slope

http://upload.wikimedia.org/wikipedi...ative_line.gif

The second derivative is the derivative of the first derivative of the function. The third derivative is the derivative of the second derivative of the function. I remember this; basically, it's all high school math (at least in Balkans, where you have three types of general high schools(gymnasiums): general one, the ones that give more space to natural sciences (they have lectures involving partial differential equations which other high school types lack) and the third one that favors social sciences).

OK, after that flashback let's move on

A point where the second derivative of a function changes sign is called an inflection point (you mentioned this before, now i know how you got there). At an inflection point, the second derivative may be zero, as in the case of the inflection point x=0 of the function y=x3, or it may fail to exist, as in the case of the inflection point x=0 of the function y=x1/3. At an inflection point, a function switches from being a convex function to being a concave function or vice versa.


http://upload.wikimedia.org/wikipedi...tion_point.gif

Plot of f(x) = sin(2x) from −π/4 to 5π/4; note f’s second derivative is f″(x) = –4sin(2x). Tangent is blue where curve is concave up (above its own tangent), green where concave down (below its tangent), and red at inflection points: 0, π/2 and π



The diagram is very informative, it will give me something to chew on during the weekend.

If these are the random results, i wonder what the results will be when the trend is identified (according to your definition)

Let me summarize this, to see if i got it right.


You scan for a trend, according to definition, and when found, put a lot X (transaction 1, X MUST be larger than Y)at point A. In that moment you do not know if the transaction will be a winner or loser. Using the formula, you determine point B and lot size Y (transaction 2). WHEN and IF the price reaches point B and activates lot Y, in that moment of time and space, you KNOW the lot X will be a winner, regardless of future price moves (the stop is now at the distance B and even if the price retraces and ejects both transactions, 1 in plus and 2 in minus, the overall result will be positive). After the transaction 2 is activated, you hmmm, summarize the points A and B into a point C (located between points A and B) with lot x+y. You then rename C as a new A and calculate new B (with lot that is smaller than x+y, transaction 3). WHEN and IF the new B is reached (transaction 3) you then know the old B (transaction 2) will be profitable.

This is great, I always compound (had the best results with new transaction every 100 pips (but I add transaction 2 at +200 pips) with equal position size, SL and TSL) and only keep the last open transaction below the water, but this approach leaves no room for misinterpretations.

Wouldn't that require that the price movement inside the original order [meaning between original SL and your exit position] to be in some way non-chaotic?

Regardless of market conditions?

Not regardless of market condition; but basically iff (iff = if and only if, not a typo) iff the first position, lot size X at position A is a winner, and reaches point B, then you are guaranteed a few things.

Let me give it in an example everybody is familiar with; school terms.

Suppose you took a class, and it's scored like this:

Midterms are 75%
Homework 10%
Finals are 15%

If you got a 100% on your midterms, you are guaranteed to pass the class with a C. At that point, if your goal is to pass, it's pretty much irrelevant if you did your homework or not. If you get even a 50% on your finals, then you've secured a B no matter what.

That's what I'm saying in FX as well. Given that Almost ALL the weight is on the first position, because given phi-style setup, the lot size X@A will ALWAYS be bigger than lot size Y@B, once you've "scored 100% on your midterm" so to speak, your B and subsequent positions will be profitable.

Furthermore, because A+B = ephemeral C, you can now build upon position C instead of building on position B. Given C = A', then your position size actually increased and your current position is deeper in the money than position B.

bigvlada

You got it! Cool huh???

I originally started with a bunch of "throw a bunch of points and combinations" up in an excel spreadsheet. After I was able to determine that my new equation did indeed cross at a (price/risk) // (price/profit) ratio cross point; then took a derivative of the equation, and give me a way to plug everything I wanted/needed.

There are 4 variables and I can determine the other 3 anytime one is given to me. I can even calculate the phantom 5th variable.

Var 1) Lot size X
Var 2) A distance to B
Var 3) Lot size Y
Var 4) B distance to SL
Phantom Var 5) SL point of A

If I treat A as a B, since I know the SL for B, and because I'm basically taking an integral of B to get A, and if I treat A as B, I will be able to get SL for A by going backwards, I actually get 5 variables total from 1 equation; not only that, fully optimized in risk reward.

As far as your 1st,2nd,3rd derivative comment goes; we have slope as 1st derivative, concavity as 2nd; but you will run into the fact that 2nd derivatives "should" be a constant and the constant should be positive.. Otherwise, you are not making a profit when price is > A. ;-) But the fact that I need to have freedom of movement in multiple directions (A movement to B, profitability) and (A movement away from B, a loss with a look for an SL) was a difficult one.

I solved it by look at the properties of "e". e is unique in that it technically "folds" on itself.. You can keep taking derivations in certain situations and it will be .. itself... not unlike a fractal. That means any given position can be considered an A with the purpose of setting a reentry B, or can be considered a B, to set an SL, or multiple entries can be combined to be treated as an ephemeral position and acted upon accordingly.

One more math tidbit, Given X@A, and reaching Y@B as your second entry, the 2nd derivative inflection point of profitability is always ALWAYS more towards B than A with respect to ephemeral position C.. That is the ONLY way you are guaranteed profit. Because the inflection point is picked as the SL, but (while) the ephemeral point is still ITM (In the Money).

BTW.. you can compound more and more lots since ephemeral C gets bigger and bigger; BUT the new additions can NEVER be bigger than the previous positions combined. Otherwise you violate phi, and when you do, then there is no guarantee of profitability because your inflection point will be BEFORE your ephemeral position, so you can actually SL in a OTM (Out The Money) position. So careful!

Twoblink,

That's quite interesting, I have been thinking of doing a bit similar work in the past with decision trees using evolutionary algorithms - my plan never was to have safe version, more kind of optimizing alpha.

I would however tie every decision tree (or those levels you use) to certain kind of entry because (at least I would assume) that different kind of entries require different kinds of adaptation because IMHO the price action inside different entries is not totally chaotic.

I'm really interested to hear if you are able to use this in your actual trading (or are you already?) - spread/slippage is usually a problem (and unknown variable) for these pure mathematical methods.

twoblink, I admire your thread here and am open to read more whenever you feel sharing. I must admit that some time has gone through since I passed all the math exams in highschool and university, so now it hapens that I need to re-read your posts here several times in a row or even help myself with some old notes, to make things clear.

Therefore I am also very thankful to my balkan neighbour bigvlada, who makes illustrative examples with close connections to real forex trading, which makes things even clearer. Puno ti hvala bigvlada!

mikkom,

I use it in my trading right now. Why wouldn't I? Wouldn't you??

I am thinking about putting all this in a book.. but I'm scared the book is going to grow to 300+ pages.

Also, I haven't even done my FX reversal book yet!

I was just wondering if this is just experimental (simulated) stuff or real world, slippage + spread tested material. Great to hear that this is working for you.

I have to cut my post in pieces. Again. The moderators should really think about lifting the number of pictures permitted per post and replace it with total kilobyte size for one post. I would also like to have subscript/superscript option (some superscript is correctly pasted from, word 2007, some isn't).

This weekend I've read The unexpected hanging and other mathematical diversions - a classic collection of puzzles and games from Scientific American by Martin Gardner. The third chapter is most interesting - The transcendental number e. It gives much to think upon, and some clues.

The fundamental nature of e can best be made clear by considering ways in which a quantity can grow. Suppose you put one dollar in a bank that pays simple interest of 4 per cent a year. Each year the bank adds four cents to your dollar. At the end of 25 years your dollar will have grown to two dollars. If, however, the bank pays compound interest, the dollar will grow faster because each interest payment is added to the capital, making the next payment a trifle larger. The more often the interest is compounded, the faster the growth. If a dollar is compounded yearly, in 25 years it will grow to (1 + 1/25) raised to the power of 25, or $2.66+. If it is compounded every six months (the interest is 4 per cent a year so each payment will now be 2 per cent), it will grow in 25 years to (1 + 1/50)raised to the power of 50, or $2.69+.

Banks like to stress in their promotional literature the frequency with which they compound interest. This might lead one to think that if interest were compounded often enough, say a million times a year, in 25 years a dollar might grow into a sizable fortune. Far from it. In 25 years a dollar will grow to (1 + 1/n)raised to the power of n, where n is the number of times interest is paid. As n approaches infinity, the value of this expression approaches a limit that is a mere $2.718. . . , less than three cents more than what it would be if interest were compounded semi- annually. This limit of 2.718. . . is the number e.

If the period is very long, however, even a small interest rate can grow to Gargantuan size. A dollar invested at 4 per cent in the year 1 and compounded annually would in 1960 be worth $1.04 raised to the power of 1960, a number of dollars that runs to about thirty-five figures.

This is extremely important to our problem:
This type of growth is unique in the following respect: at every instant its rate is proportional to the size of the growing quantity. In other words, the rate of change at any moment is always the same fraction of the quantity's value at that moment. Like a snowball tumbling down a hill, the larger it gets, the faster it expands. This is often called organic growth, because so many organic processes exhibit it.

All these processes are described by formulas based on y = e raised to the power x. This function is so important that it is called the exponential function to distinguish it from other exponential functions, such as y = 2". It is a function that is exactly the same as its own derivative, (this is where I started to giggle ) a fact alone sufficient to explain e's omnipresence in the calculus. Natural logarithms, used almost exclusively in mathematical analysis (in contrast to the 10-based logarithms of the engineer), are based on e.
If the formula (1 + 1/n)" is expanded, one obtains the following well-known infinite series that converges on e:

e = 1+1/1! + 1/2! + 1/3! + 1/4!......

The exclamation mark is the factorial sign. (Factorial 3 is 1*2*3, or 6; factorial 4 is 1*2*3*4, or 24; and so on.) The series converges rapidly, making it as easy as pie, in fact, much easier than pi - to calculate e to any desired number of decimals.

Is there a relation between e and pi, the two most famous transcendentals? Yes, many simple formulas link them together. The best known is the following formula that Euler based on an earlier discovery by Abraham de Moivre:

e raised to the power iп +1=0

The formula unites five basic quantities : 1,0, pi, e and i (the square root of minus one).
This formula struck Benjamin Peirce (a Harvard mathematician and father of the philosopher Charles Sanders Peirce) with the force of a revelation. "Gentlemen," he said one day to his students after chalking the formula on the blackboard, "that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore, we know it must be the truth."

There is a remarkable fraction 355/113 that expresses pi accurately to six decimal places. To express e to six decimals a fraction must have at least four digits above the line and four below (e.g., 2721/1001). It is possible, however, to form integral fractions for e, with no more than three digits above and below the line, that give e to four decimal places.

What integral fraction, with no more than three digits above the line and three below, gives the best possible approximation for the mathematical constant e? The answer is 878/323. In decimal form this is 2.71826. . . , the correct value for e to four decimal places. (Note to numerologists: Both numerator and denominator of the fraction are palindromes, and if the smaller is taken from the larger, the difference is 555.) Removing the last digit of each number leaves 87/32, the best approximation to e with no more than two digits in the numerator and two in the denominator.

What value of n gives the maximum value to the nth root of n? The answer is e.

If real numbers are picked at random from the interval 0 to 1 inclusive, and this continues until the sum of the selected numbers exceeds 1, what is the expected number of numbers that have been chosen? Again the answer is e.

Like pi, e is a transcendental number: it cannot be expressed as the root of any algebraic equation with rational coefficients. Just as there is no method by which a line segment exactly equal to pi (relative to a unit segment) can be constructed with a compass and straightedge, so there is no way to construct a line segment exactly equal to e without violating the classical restraints.

A little lighter approach:
The French entomologist Jean Henri Fabre: "Here we have the abracadabric number e reappearing, inscribed on a spider's thread," he writes in The Life of the Spider. "Let us examine, on a misty morning, the mesh-work that has been constructed during the night. Owing to their hygrometrical nature, the sticky threads are laden with tiny drops, and, bending under the burden, have become so many catenaries, so many chaplets arranged in exquisite order and following the curve of a swing. If the sun pierce the mist, the whole lights up with iridescent fires and beacons a resplendent cluster of diamonds. The number e is in its glory."

As a function of the real variable x, the graph of y=e raised to the power x is always positive (above the x axis) and increasing (viewed left-to-right). It never touches the x axis, although it gets arbitrarily close to it (thus, the x axis is a horizontal asymptote to the graph). Its inverse function, the natural logarithm, ln(x), is defined for all positive x. The exponential function is occasionally referred to as the anti-logarithm. However, this terminology seems to have fallen into disuse in recent times.
http://schools-wikipedia.org/images/190/19096.png
Using the natural logarithm, one can define more general exponential functions. The function
http://schools-wikipedia.org/images/190/19098.png
defined for all a > 0, and all real numbers x, is called the exponential function with base a. Note that this definition of a raised to the power x rests on the previously established existence of the function e raised to the power x , defined for all real numbers.
Exponential functions "translate between addition and multiplication" as is expressed in the first three and the fifth of the following exponential laws:
http://img218.imageshack.us/img218/6...entiallaws.png

These are valid for all positive real numbers a and b and all real numbers x and y.
The importance of exponential functions in mathematics and the sciences stems mainly from properties of their derivatives. In particular,
http://schools-wikipedia.org/images/191/19110.png
That is, e raised to the power x is its own derivative. Functions of the form Ke raised to the power x for constant K are the only functions with that property.

Other ways of saying the same thing include:
•The slope of the graph at any point is the height of the function at that point.
•The rate of increase of the function at x is equal to the value of the function at x.
•The function solves the differential equation y`=y .
•exp is a fixed point of derivative as a functional.
For exponential functions with other bases:
http://schools-wikipedia.org/images/191/19115.png

Thus, any exponential function is a constant multiple of its own derivative.
Furthermore for any differentiable function f(x), we find, by the chain rule:
http://schools-wikipedia.org/images/191/19116.png

The exponential function e raised to the power x can be defined in a variety of equivalent ways, as an infinite series. In particular it may be defined by a power series:
http://schools-wikipedia.org/images/191/19119.png


http://schools-wikipedia.org/images/191/19117.gif
The exponential function (in blue), and the sum of the first n+1 terms of the power series on the left (in red).
Continued fractions for e raised to the power x Via Euler's identity:
http://schools-wikipedia.org/images/191/19135.png

More advanced techniques are necessary to construct the following:
http://schools-wikipedia.org/images/191/19136.png

Setting m = x and n = 2 yields
http://schools-wikipedia.org/images/191/19137.png

Yap, an elegant solution for something that looks like simple problem (but, they said that for Fermat's Last Theorem also. And he said the proof was simple. Maybe he was pulling everyone's leg?). Now I know what I'm looking for, just have to mash it together in a equation.
This smells like the last few pictures, Continued fractions for e raised to the power x and you take turns back and forth doing derivatives and integrals of the equation plus adding the bigger and bigger x+y chunks.


The phantom menace - Forex edition: stop loss of the first position

Let me be perfectly clear, whether I succeed in replicating twoblink's equation or not I know it is not the Holy Grail (sorry folks, try the next thread). It won't change the way I trade. I firmly believe that 50% of the success is in the mental strength of the trader (why is George Soros amongst the best? He survived SS death squads before his 15th birthday - losing money on transaction seems trivial compared to that ); 40% belongs to the money management and only 10% is left for the system.
I am by all standards a greenhorn - a year on demo (it took me six months to double my account - same size as future live one) and I am trading live since march (really picked the appropriate time to learn how to trade - at the start of the financial crisis - the good thing is - If I am able to survive this, everything else will look easier). I threw out all indicators after first two months and spent the rest of the time on perfecting money management, price action and compounding techniques.
Predicting future moves in the market is theoretically possible - BUT... You have to have the ability to be at many places at once in order to hear all the hints, rumors and genuine data, the ability to read people's minds (so you would know what each individual wants to buy or sell and when) and the computer powerful enough to calculate in real time all present and future moves based on aforementioned abilities.
Summa summarum - practically impossible. Using indicators is just speculation - the equivalent would be gutting the fish in order to predict the future. Trading the news is even worse. I've spent a year studying history. The most important subject (that should be taught at every elementary school on the planet)was Introduction to historical studies. Basically, it teaches you how to be objective (being a member/sympathizer of a political party or religion distorts that objectivity to unrepairable state ) and how to value historical sources - how to determine if the photography was tampered with, or if the text was forged (just a basic level). One fine example is Churchill's World War 2 which lacks any mention of computers, enigma machine and is untrue to some extent (the author hides some of it's mistakes). You have to read David Irving's Churchill's war in order to get a better picture (but you have to bear in mind that Irving's interpretations of Nazi Germany have proved highly controversial due to allegations of undue sympathy for the Third Reich and antisemitism, and because of his involvement in the Holocaust denial movement ). Both sources are good for studying the ww2 but you have to know their limitations. The subject teaches you to make hmmm, psychological profile of the author of the historical source and to use the source with that knowledge. While doing press clipping and media analysis, I had the daily task of determining if a particular article is positive, negative or neutral towards the client.
This rant was necessary in order to explain why I think that all news is highly subjective and biased towards certain opinion. (something went up/down or something probably will get up/down) You have to know the psychological profile and motives of the author/institution in order to use that source. For most, this is impossible. I read the news because I find them interesting and entertaining (but I newer dare trade because of them).
Retail segment of Forex trade is perhaps 2% of its total volume (this information is also subjective, but I calculated that no one profits if it distorts it and I take it as a truth, for now ). We have less money, information and thought power (only one brain against the financial institution). The mathematics is the only tool where we can be equal to them.
Using mathematics and logic in trading is not speculating. The predators in animal world do not speculate.They set a trap, an ambush, and wait. IF the prey enters the designated area, they will kill and eat it. What we are trying to do is exactly the same (set the trap/transaction) and wait. IF the price reaches it, we know the proper way to respond in order to get the best possible ending (maximal profit). The formula we are looking for gives that set of rules (we all may have the rules of our own, but they are less effective - the ones based on fish gutting are at the bottom of the barrel). It's just a tool, it won't make you instantly profitable.

@vanderpjer

samo opušteno, ovo je samo igra/be cool this is just a game

I believe that just like you can get 10 chefs and give them the same recipe, you'll end up with 10 different dishes; so it is with FX. All 10 dishes can taste great, and all 10 methods for FX might be profitable, so there is no one right way.

But i am weary of so called "experts" who sell potions and lotions without any mathematical proof. I guess you can play blackjack based on the moon's cycle and make money, but when it's all said and done, it's all about the math.

If your expectancy is < 1, you will lose money and over a period of time, you will lose it all. if your expectancy is > 1 then you will make money over the long term. It's really as simple as that, but people seem to love pipe dreams more than reality.

A late and great friend of mine use to say "People are easy to lie to, difficult to teach." so it is with FX'ers.

I regularly spank out 600+ pips a week, but for some reason, "experts" who do 100 pips a month can charge 2K per class and people jam pack them. I've done almost 10000 pips in a month before, but the fact i don't have a morgan stanley name card somehow makes me not an expert. it's strange to me that results don't say squat nowadays, but somehow, a fancy business card and an mba from some ivy league school is suppose to.

I believe, good solid mathematics, expectancy > 1, good understanding and use of MM, and above all use, great psychological fortitude = a winning trader.

I'm moving to another country in 3 days. but after I settle down, writing more books is definitely on the top of my list.

Please let me know what you'd like to read about.. and i'll see if i can oblige.

Have a nice trip twoblink, hope everything goes well with the move.

as for this discussion, I've realized that I'll have to reread all of my high school math, plus a good chunk of theoretical math and partial differential equations. This could take a few weeks (or months). But I'm on a good path, I know that someone did it before.

hmm, book.

Price action?

Fx and math, the forbidden love story? (proving that 100+ most popular indicators are based on faulty presumptions could be a good start, a chapter on averaging down could be useful (in suicide prevention, that is), types of compounding etc... )

Thanks for this thread. I really appreciate the time it takes to put pen to paper, or finger to key, on such matters. If anything you've rekindled my interest in my A level maths (UK) course and it;s something i'm going to be looking back over, deeply.
Man, I struggled with differentiation! I think I may have some questions once i've reread this thread and my notes.

Many thanks

Lakey

First I was going to say that it's really strange that I ran into this thread today. I just began to think of this same idea 2 nights ago. I thought of this same general concept, not with all the complex mathmatical equations tied in with this thread but the basic logic behind it.

First question is do you have a live account or a demo? Demos don't mean anything because as far as I'm concerned regardless of how strong someones psychological will power is, a demo will be traded differently that a live account at some given point.

If you are actually doing 600 pips average a week and have done 10,000 pips in a month with real money you would not have too much trouble getting investor money behind you. If you show investors proof that's all it really takes along with going out of your way to connect with them. I don't know how much you've lost to begin with or how large your down draws are which are huge factors obviously.

This is an exageration but if you walked in to Morgan Stanley HQ with a laptop and a phone call to your broker proving your live account with 20% gains and %1 downdraws monthly and consecutively for 10 years I have a hard time believing you would just be brushed away. There are a lot of bs, backdoor, connections that can be made and are made like that everyday.

Who says you can't just walk into every large or small local investment firm showing the head advisors your proof? Nobody can stop you, and you'll walk out of every one knowing that whoever turned you down looked a gift horse in their mouth.

People talk big, but very rarely prove it to be true.

and if they do attempt to, they accuse everyone of caring too much and refuse to put the effort of "right-clicking" then uploading the file.

The point of this thread is to educate on the math behind the optimized risk vs reward equation. So there is really no point in wondering and arguing over what is another person's DD and live account and what not.

For those who have not noticed and are yakking over live/demo accounts and what not, this is about MM. It is just a part of your system. As mentioned, if your system expectancy is <1, no amount of MM is going to help. So please, lets focus on the math.

Anyway so far these are the equations i have dervied so far and i am stuck on how to carry on. Any ideas?