Optimized Risk vs Reward Equation

I have found this optimized equation thread by twoblink incredibly fascinating. All of the contributors have been doing a great job with their attempts to engineer the equation. I dont have anywhere near the advanced math skills to further the development of this thread but was wondering if this subject has now died or if there remains any interest?

Lets keep this thread going as Im sure twoblink will continue to provide quidance, as he already has, provided we keep the enthusiam up!

Cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

The interest is there, but the problem is hard, or should I say: The spirit is willing but the flesh is weak.

I've mentioned the resemblance to the Fermat's Last Theorem before, the problem looks easy on the surface, but dig a little deeper and you'll need a lot of branches of mathematics in order to solve it. It is a slow, trial and error process.

I'm looking at the problem like Will Smith in I robot, I won't get an answer until I ask the right question, because that will be the proof that I understood all aspects of the problem. Answer without the question is meaningless (42 anyone? ).

As for fear of mathematics, here's a great on line tool to brush up on math skills, with equations, graphs, quizzes, questions, examples and accompanying excel files.

http://www.zweigmedia.com/RealWorld/tccalcp.html

This one is particularly interesting to us, it deals with exponential functions and models

http://www.zweigmedia.com/RealWorld/...frames2_2.html

with a good explanation of our old friend, number e

http://img134.imageshack.us/img134/9366/26876760.jpg

I'll attach appropriate excel sheet, so you can play with e for yourself.


My earlier assumption, using limits to determine sum was wrong, I think twoblink used line integrals to do it, for some segment of the line of the function. I'm still not sure if it is perhaps the line between two saddle points of the function (that would require function with multiple saddle points) or between one saddle point and infinity (I know, the divergence shows its ugly head).

In mathematics, a line integral (sometimes called a path integral or curve integral; not to be confused with calculating arc length using integration) is an integral where the function to be integrated is evaluated along a curve.

The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve).

In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given field along a given curve. More specifically, the line integral over a scalar field can be interpreted as the area under the field carved out by a particular curve. This can be visualized as the surface created by z = f(x,y) and a curve C in the x-y plane. The line integral of f would be the area of the "curtain" created when the points of the surface that are directly over C are carved out.

more here: http://en.wikipedia.org/wiki/Line_integral

here's some more excel sheets about derivatives, slopes, graphs, etc... to visualize our discussion.

And in the end, second derivative grapher, function grapher and Riemann sum grapher

But, start with the first chapter of this on-line course and work your way to derivatives, integrals etc... Then read the thread again and give some fresh ideas about the possible solution to the problem.

I (with wife in tow) just moved 6000+ miles.. i am sitting in my newly rented house, with no furniture, a really small netbook, and some OJ in the frig. That's it. So given the circumstances, I apologize for not being more involved in the thread.

As far as it being like Fermat's.. I seriously doubt that.. but I have to say that either this is a problem that nobody has documented tackling, or nobody before me has succeeded.. Then again, before I put out my first book, I don't think anybody made an attempt to scientifically document what a "trend" is by scientific definition..

This is presented more like the "Godel Escher Bach" problem presented towards the beginning of the book, you can attack it by testing everything, or breaking it at the equation level.

If I have a position A lot(s) @ X location; there is an infinite number of points I can pick for B lot(s) @ Y location; The concept is quite simple, but you'll find that when you move one, all 3 move; so again, the only way is to move them all together, hence the continuing reference to a mini-max.

My friend who's math teacher, told me that I have been doing it all wrong these last 2 years; (just talked to him last week) and that I should have used a matrix set.. Since I got an F in my matrix class.. I'm just going to throw it out there, and let those who know matrix and would like to go that route, to deal with it.

http://en.wikipedia.org/wiki/Saddle_point

I assume you're referring to the "MU problem" where by jumping out of the system one realizes that there is no way to derive MU from the given axioms.

That sounds like my experience with respect to this conjecture (proof). I seem to just keep coming up with simple algebraic proofs that there is NO maximally effective pairing of distance and lot size for an add-on entry. But that may simply mean that I'm defining the "effectiveness" of the add-on entry differently than you did.

Orange Juice, by the way, is quite yummy.

It's like a formula for circumference or area of a circle, it is always 2*r*pi and r^2*pi regardless of the size of the circle. I've been looking at this from a wrong perspective. The positions are related to one another like sphere in a sphere, or the monoliths on the moon and Europa or even like Russian babushka dolls, their proportions are identical.

The x and SL of a in first position are defined by our MM rules, all subsequent positions are their representations with same proportions.The x tells us how big the monolith is and Sl determines how close the monoliths are to one another. One saddle point and one inflection point is enough. That is why the time frame is irelevant, on 5min chart you'll have small stop loss and small distance between positions, on weekly, they will be proportionally larger.

I did try with the hessian matrix but found it too complicated. I am more familiar with the function approach.

Definition of graph of function of two variables:
Suppose that f is a function of x and y. The graph of f consists of all points in space of the form (x,y,f(x,y)) where (x,y) is in the domain of f.

A function f of two variables is a relation that assigns to every ordered pair
of input values (x,y) in a set called the domain a unique output value denoted by f(x,y). The set of output values is called the range.

The graph of a function of two variables is a surface in three dimensions.
A contour line of a surface is a graph in three dimensions where the value z = f (x, y) is held constant.
A cross section of a surface is a graph in three dimensions where the value of
either the x or y variables is held constant.


Using this?

In mathematics, the trapezoidal rule (also known as the trapezoid rule, or the trapezium rule in British English) is a way to approximately calculate the definite integral

http://upload.wikimedia.org/math/6/0...c6462db380.png

The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoid and calculating its area. It follows that

http://upload.wikimedia.org/math/d/3...8ab3699c52.png


http://upload.wikimedia.org/wikipedi...lustration.png

The error of the composite trapezoidal rule is the difference between the value of the integral and the numerical result:
http://upload.wikimedia.org/math/5/b...8d9a109ad6.png

It follows that if the integrand is concave up (and thus has a positive second derivative), then the error is negative and the trapezoidal rule overestimates the true value. This can also been seen from the geometric picture: the trapezoids include all of the area under the curve and extend over it. Similarly, a concave-down function yields an underestimate because area is unaccounted for under the curve, but none is counted above. If the interval of the integral being approximated includes an inflection point, then the error is harder to identify.

more here:
youtube example
http://numericalmethods.eng.usf.edu/...t_example.html

powerpoint example
http://numericalmethods.eng.usf.edu/...continuous.ppt


this example is most interesting, it is situation when we do not know a or b
http://numericalmethods.eng.usf.edu/...erivation.html

everything
http://numericalmethods.eng.usf.edu/...idal_rule.html

I also added my old excel sheet in attachment where you can adjust length between positions, SL and lot size. You really can extract 4600 or 18200 pips out of 1000pip move. Problem is, it is not optimized.

ps. xlsx (excel 2007 file format) can't be uploaded.

capitalist88.. you are missing a critical element; you are where I was before..

Khinchin's Constant

In number theory, Aleksandr Yakovlevich Khinchin proved that for almost all real numbers x, the infinitely many denominators ai of the continued fraction expansion of x have an astonishing property: their geometric mean is a constant, known as Khinchin's constant, which is independent of the value of x.

http://mathworld.wolfram.com/KhinchinsConstant.html



Most commonly, it is used to describe a convergent sequence or convergent series, where it essentially means that the respective series or sequence approaches some limit.

The rational number obtained by keeping only a limited number of terms in a continued fraction is also called a convergent.

http://mathworld.wolfram.com/Convergent.html

Functions of multiple variables can be differentiated with respect to either of their variables, the other variable being understood to be held constant during the differentiation. The partial derivatives are written as ∂f/∂x, ∂f/∂y, and so on. The symbol "∂" is used (instead of "d") to remind us that there is more than one variable, and that we are holding the other variables fixed.

Interpretation

∂f/∂xis the rate at which f changes as x changes, for a fixed (constant) y.

∂f/∂yis the rate at which f changes as y changes, for a fixed (constant) x.

more here: http://people.hofstra.edu/Stefan_Wan...Calcsumm8.html

Graph of a Function of Two Variables The graph of the function f of two variables is the set of all points (x, y, f(x, y)) in three-dimensional space, where we restrict the values of (x, y) to lie in the domain of f. In other words, the graph is the set of all the points (x, y, z) with z = f(x, y).

http://people.hofstra.edu/Stefan_Wan...ypic/cs8_4.gif

Geometric Interpretation of Partial Derivatives

If f is a function of x and y, taking the partial derivative ∂f/∂x and evaluating it at (a, b) amounts to holding y constant at y = b and finding the rate of change of f at x = a. Thus, the partial derivative is the slope of the tangent line to this curve at the point where x = a and y = b, along the plane y = b.

http://people.hofstra.edu/Stefan_Wan...ypic/cs8_6.gif

∂z I
∂y I(a, b) is the slope of the tangent line at the point P(a, b, f(a, b)) along the slice through y = b

If f is a function of x and y, then f has a local maximum at (a, b) if f(a, b) „ f(x, y) for every (x, y) in a small neighborhood of (a, b). A local minimum is defined in a similar way. f has a saddle point at (a, b) if f has a local minimum there along one slice and a local maximum along another slice.

The function illustrated below has has a relative minimum at (0, 0), a relative maximum at (1, 1), and saddle points at (1, 0) and (0, 1).

http://people.hofstra.edu/Stefan_Wan...ypic/cs8_7.gif

Attached a two variable function grapher, in excel.

And that missing element involves the probability of price reaching the NEXT entry point prior to reaching the current stop?

No, it involves the framework as a context-er.. I know that's not actually an English word, but don't know how else to describe it.

Capitalist.. quick question, have you figured out what you are using for your Z-axis?

The z-axis being the criteria by which to determine the "effectiveness" of a given new entry point and amount? No. I was using total profit/loss if the stop gets hit, but that just gave me divergence.

However, I realized a couple of days ago that there are actually three independent variables that we can manipulate:

x: the proportion of the new entry size with respect to the initial position size

y: the proportion of the distance from the current average entry price (AEP) to the new entry price with respect to the distance from the AEP to the current stop (commonly called "R")

z: the proportion of the distance from the old stop to the new stop with respect to the distance from the old AEP to the new entry point.

To use a concrete example, suppose I set:

x = 1/2
y= 2
z = 1/2

If my current position is 1000 units with a stop 50 pips away, then these values mean that I would add 500 units (x=1/2) if price moves in my direction by 100 pips (y=2), and I would move the stop forward by 50 pips (z = 1/2).

The goal is determining what three values of x, y & z are the "best," but first the trick is to determine what "the best" actually means and how it's measured. This is something that seems to flit in and out of the edges of my brain. As I look at various values of the three parameters, I can see why I prefer some values over others, so I must determine a way to quantify those preferences.

"context-er" meaning the builder or provider of context?

By the way, from where did you move 6,000 miles, and where did you end up?

hmm, the optimal function must have boundaries, logarithmic one as lower and exponential one as higher.

In computer science, an algorithm is said to be asymptotically optimal if, roughly speaking, for large inputs it performs at worst a constant factor (independent of the input size) worse than the best possible algorithm.

More formally, an algorithm is asymptotically optimal with respect to a particular resource if the problem has been proven to require Ω(f(n)) of that resource, and the algorithm has been proven to use only O(f(n)).

In numerical analysis (a branch of mathematics), the speed at which a convergent sequence approaches its limit is called the rate of convergence. Although strictly speaking, a limit does not give information about any finite first part of the sequence, this concept is of practical importance if we deal with a sequence of successive approximations for an iterative method, as then typically fewer iterations are needed to yield a useful approximation if the rate of convergence is higher. This may even make the difference between needing ten or a million iterations.

I'm trying to figure out the optimal rate of convergence for a two variable function.