Optimized Risk vs Reward Equation

Once again I must say that I like the conversation going on in this thread. Seems that very reasonable people contribute here. Everyone has experienced both, big drawdowns on their live accounts - maybe even margin calls, and prosperity times where profit went into their favour. With moving back a few steps and taking a look onto both experiences, a person gets a better perspective of what truly was good and what were the main errors that prevented the success.

If this same person learns from its own mistakes, then he has a greater chance of not fail again. At least not on the same mistakes. And if a person learns that a fire is hot, he/she has two possibilities:

a) To completely avoid fire in order to prevent any possible damage.
b) To learn how to deal with fire and try to use it in his/her favour.

In our country we say: 'Donkey goes on ice only once...' and a human supplement would be '...While icehockey players are on it every day'.

It seems that donkey has chosen option a), while icehockey players found a version of the option b). I leave it open for each individual here to decide whether there are similar possibilities in Forex.

In addition, I couldn't agree more with the following statement of twoblink:

People are different, no matter what comparison you make. If differences do not seem obvious from the far distance, they surely exist in a detailed view.

Luckily, there is a math. A logical and fair invention of human civilization that helps understanding things around no matter if your are black or white. We tend to define denominators in order to make things clearer. We learnt apples are not pears and pears are not apples. Going more into details, even apples tend to be different among themselves.

And in Forex things are no different. You can talk about euros, dollars, pips, percentages, drawdowns, etc. But honestly, what is your most essential denominator here?

I would say it is absolutely your profit. Not just today, not just tomorrow. The one on a long term. Denominator's biggest challenge - consistency. Can we handle it?

This could reduce time needed to grasp the complexity of the problem.

Download free Mathematica 7 player (it's like acrobat reader but for math papers, formulas etc...) and lectures, graphs, etc....

This is definitely something I must use

Fractional Derivative


The fractional derivative of http://mathworld.wolfram.com/images/...ve/Inline1.gif of order http://mathworld.wolfram.com/images/...ve/Inline2.gif (if it exists) can be defined in terms of the fractional integral http://mathworld.wolfram.com/images/...ve/Inline3.gifas http://mathworld.wolfram.com/images/...dEquation1.gif(1)

where http://mathworld.wolfram.com/images/...ve/Inline4.gif is an integerhttp://mathworld.wolfram.com/images/...ve/Inline5.gif , where http://mathworld.wolfram.com/images/...ve/Inline6.gif is the ceiling function. The semiderivative corresponds to http://mathworld.wolfram.com/images/...ve/Inline7.gif.

The fractional derivative of the function is given by
http://mathworld.wolfram.com/images/...ve/Inline9.gif=http://mathworld.wolfram.com/images/...e/Inline11.gif(2)


= http://mathworld.wolfram.com/images/...e/Inline14.gif (3)

= http://mathworld.wolfram.com/images/...e/Inline17.gif (4)

= http://mathworld.wolfram.com/images/...e/Inline20.gif (5)

=http://mathworld.wolfram.com/images/...e/Inline23.gif(6)


for http://mathworld.wolfram.com/images/...e/Inline24.gif. The fractional derivative of the constant function http://mathworld.wolfram.com/images/...e/Inline25.gif is then given by

http://mathworld.wolfram.com/images/...e/Inline26.gif=http://mathworld.wolfram.com/images/...e/Inline28.gif(7)

= http://mathworld.wolfram.com/images/...e/Inline31.gif (8)


The fractional derivate of the Et-function is given by

http://mathworld.wolfram.com/images/...dEquation2.gif = http://mathworld.wolfram.com/images/...dEquation2.gif (9)


for http://mathworld.wolfram.com/images/...e/Inline32.gif.

It is always true that, for http://mathworld.wolfram.com/images/...e/Inline33.gif,

http://mathworld.wolfram.com/images/...dEquation3.gif = http://mathworld.wolfram.com/images/...dEquation3.gif(10)


but not always true that
http://mathworld.wolfram.com/images/...dEquation4.gif = http://mathworld.wolfram.com/images/...dEquation4.gif (11)


A fractional integral can also be similarly defined. The study of fractional derivatives and integrals is called fractional calculus.

REFERENCES:

Kilbas, A. A.; Srivastava, H. M.; and Trujiilo, J. J. Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier, 2006.

Love, E. R. "Fractional Derivatives of Imaginary Order." J. London Math. Soc. 3, 241-259, 1971.

Miller, K. S. "Derivatives of Noninteger Order." Math. Mag. 68, 183-192, 1995.

Oldham, K. B. and Spanier, J. The Fractional Calculus: Integrations and Differentiations of Arbitrary Order. New York: Academic Press, 1974.

Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, 1993.



CITE THIS AS:

Weisstein, Eric W. "Fractional Derivative." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/FractionalDerivative.html

This is what we need, at least a variation of it.

Fractional Derivative of a Power

http://demonstrations.wolfram.com/Fr...ativeOfAPower/

(you can download this too and use it in player)

This Demonstration lets you calculate and plot a fractional derivative of http://demonstrations.wolfram.com/Fr...index.en/1.gif. The operator of a fractional derivative is http://demonstrations.wolfram.com/Fr...index.en/2.gif.

Last, but not least, link for Mathematica 7 player download
http://www.wolfram.com/products/player/download.cgi

Now I'm certain that the problem can be assessed and solved easier by combining two or more of these little programs (each one comes with the source code).

These two are the best candidates for now

Automatic Differentiation
http://demonstrations.wolfram.com/Au...fferentiation/

http://img243.imageshack.us/img243/8879/49783758.png
http://img692.imageshack.us/img692/8338/72576731.png
http://img203.imageshack.us/img203/5338/16285563.png
http://img97.imageshack.us/img97/986/11669480.png

Derivatives of Exponential Functions
http://demonstrations.wolfram.com/De...tialFunctions/

http://img692.imageshack.us/img692/4898/22641320.jpg

Comparing twoblink's posts and these programs, all sorts of ideas come to my mind

Saddle Points and Inflection Points
http://demonstrations.wolfram.com/Sa...lectionPoints/

http://img203.imageshack.us/img203/8329/10834852.jpg
http://img215.imageshack.us/img215/4198/popup11.jpg
http://img203.imageshack.us/img203/5060/popup12.jpg
http://img202.imageshack.us/img202/2135/popup13.jpg

The biggest concept I've gained from e, is that e "eats into itself", derivative of ln(e) like state. That is really critical.. Without an equation that behaves similar, you can't continue the pyramiding.

Also, it prevents the "shrinking pyramid" problem that I didn't know how to overcome until now.

Let's say I take a pyramiding scheme via pascal's triangle:
70,56,28,8,1.. if the market continues to climb, I've run out of pyramids to stack!!

This was a big problem, and one that I faced for 2 years. Solution of course was phi.

While not SL {
If A reaches given distance, then enter again, size B.
A = A+B;
Position A = Ephemeral C.
}

This allows for an infinite pyramid size.. and I think is one of those great FX mathematical breakthroughs..

OK, let's get back to math.

Reading this article (http://en.wikipedia.org/wiki/Bolzano...strass_theorem), got me thinking about ways to determine "the procedure to summon the wholy c" (without human/animal sacrifice )

We know that c is somewhere between a and b.


In real analysis, the Bolzano–Weierstrass theorem is a fundamental result about convergence in a finite-dimensional Euclidean space Rn. The theorem states that each bounded sequence in Rn has a convergent subsequence. An equivalent formulation is that a subset of Rn is sequentially compact if and only if it is closed and bounded.


In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is a corresponding point in its domain that the function maps to that value.

The intermediate value theorem states the following: If the function y = f(x) is continuous on the interval [a, b], and u is a number between f(a) and f(b), then there is a c ∈ [a, b] such that f(c) = u.

This captures an intuitive property of continuous functions: given f continuous on [1, 2], if f(1) = 3 and f(2) = 5 then f must take the value 4 somewhere between 1 and 2. It represents the idea that the graph of a continuous function on a closed interval can only be drawn without lifting your pencil from the paper.

The theorem depends on (and is actually equivalent to) the completeness of the real numbers. It is false for the rational numbers Q. For example, the function f(x) = x2 − 2 for x ∈ Q satisfies f(0) = −2 and f(2) = 2. However there is no rational number x such that f(x) = 0, because if so, then √2 would be rational.

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

Darboux's theorem states that all functions that result from the differentiation of some other function on some interval have the intermediate value property (even though they need not be continuous).


Let f : [a,b] → R be a real-valued continuous function on [a,b], which is differentiable on (a,b), differentiable from the right at a, and differentiable from the left at b. Then http://upload.wikimedia.org/math/4/4...113425b813.png satisfies the intermediate value property: for every t between http://upload.wikimedia.org/math/8/5...7c39a2042c.png and http://upload.wikimedia.org/math/6/d...6a1883bd71.png , there is some x in [a,b] such that http://upload.wikimedia.org/math/d/b...e5cc98453.png.

The intermediate value theorem of integration is derived from the mean value theorem and states:

If f is a continuous function on some interval [a, b], then there exists a c with a < c < b such that the signed area under the function on that interval is equal to the length of the interval b − a multiplied by f(c). That is,

http://upload.wikimedia.org/math/f/a...910de87bcf.png


Intermediate value theorem of derivatives

If f is a differentiable real-valued function on R, then the (first order) derivative f′ has the intermediate value property, though f′ might not be continuous.

Sloooooowly geting there.

Pareto efficiency, or Pareto optimality, is an important concept in economics with broad applications in game theory, engineering and the social sciences. The term is named after Vilfredo Pareto, an Italian economist who used the concept in his studies of economic efficiency and income distribution. Informally, Pareto efficient situations are those in which any change to make any person better off is impossible without making someone else worse off.

Given a set of alternative allocations of, say, goods or income for a set of individuals, a change from one allocation to another that can make at least one individual better off without making any other individual worse off is called a Pareto improvement. An allocation is defined as Pareto efficient or Pareto optimal when no further Pareto improvements can be made. Such an allocation is often called a strong Pareto optimum (SPO) by way of setting it apart from mere "weak Pareto optima" as defined below.

Formally, a (strong/weak) Pareto optimum is a maximal element for the partial order relation of Pareto improvement/strict Pareto improvement: it is an allocation such that no other allocation is "better" in the sense of the order relation.

A so-called weak Pareto optimum (WPO) nominally satisfies the same standard of not being Pareto-inferior to any other allocation, but for the purposes of weak Pareto optimization, an alternative allocation is considered to be a Pareto improvement only if the alternative allocation is strictly preferred by all individuals (i.e., only if all individuals would gain from a transition to the alternative allocation). In other words, when an allocation is WPO there are no possible alternative allocations whose realization would cause every individual to gain.

For our purposes, individual = any transaction in the sequence.



Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set is the set of choices that are Pareto efficient. The Pareto frontier is particularly useful in engineering: by restricting attention to the set of choices that are Pareto-efficient, a designer can make tradeoffs within this set, rather than considering the full range of every parameter.

replace engineering with trading, sounds interesting, does it not?

The Pareto frontier is defined formally as follows.

Consider a design space with n real parameters, and for each design-space point there are m different criteria by which to judge that point. Let http://upload.wikimedia.org/math/3/a...10b3761cb7.pngbe the function which assigns, to each design-space point x, a criteria-space point f(x). This represents the way of valuing the designs. Now, it may be that some designs are infeasible; so let X be a set of feasible designs in http://upload.wikimedia.org/math/e/e...9be9cb21a4.png, which must be a compact set. Then the set which represents the feasible criterion points is f(X), the image of the set X under the action of f. Call this image Y.

Now construct the Pareto frontier as a subset of Y, the feasible criterion points. It can be assumed that the preferable values of each criterion parameter are the lesser ones, thus minimizing each dimension of the criterion vector. Then compare criterion vectors as follows: One criterion vector y strictly dominates (or "is preferred to") a vector y* if each parameter of y is no greater than the corresponding parameter of y* and at least one parameter is strictly less: that is,http://upload.wikimedia.org/math/8/d...737f381e2a.pngfor each i and http://upload.wikimedia.org/math/9/c...437c31ef77.png for some i. This is written as http://upload.wikimedia.org/math/8/2...bfd6d44dd3.png to mean that y strictly dominates y*. Then the Pareto frontier is the set of points from Y that are not strictly dominated by another point in Y.

Formally, this defines a partial order on Y, namely the (opposite of the) product order on http://upload.wikimedia.org/math/d/e...796b7da56a.png (more precisely, the induced order on Y as a subset of http://upload.wikimedia.org/math/d/e...796b7da56a.png), and the Pareto frontier is the set of maximal elements with respect to this order.

Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science. There, this task is known as the maximum vector problem or as skyline query.


http://upload.wikimedia.org/wikipedi...pareto.svg.png

Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point C is not on the Pareto Frontier because it is dominated by both point A and point B. Points A and B are not strictly dominated by any other, and hence do lie on the frontier.

Why the convergence is "good" and divergence is "bad".
(you'll have to figure out what is subscript and what is superscript )

A series ∑an is said to 'converge' or to 'be convergent' when the sequence SN of partial sums has a finite limit. If the limit of SN is infinite or does not exist, the series is said to diverge.
Integral test (a method used to test infinite series of non-negative terms for convergence.): if ƒ(x) is a positive monotone decreasing function defined on the interval [1, ∞) with ƒ(n) = an for all n, then ∑an converges if and only if the integral ∫1∞ ƒ(x) dx is finite.
Using the integral test for convergence, one can show (see below) that, for every natural number k, the series
http://upload.wikimedia.org/math/6/7...e6be2601da.png

still diverges (cf. proof that the sum of the reciprocals of the primes diverges for k = 1) but
http://upload.wikimedia.org/math/3/b...82322f3d99.png
converges for every ε > 0. Here lnk denotes the k-fold composition of the natural logarithm defined recursively by
http://upload.wikimedia.org/math/6/1...8debd67245.png


We don't need no bad boys.

This is most helpful, web integral calculator with help.

http://img692.imageshack.us/img692/3193/integral1.jpg


http://integrals.wolfram.com/index.jsp

Well, it's even easier to just use wolfram alpha search engine
http://www.wolframalpha.com/

Here is the result for query integrate (x^3+x-2)/(x^2+1)

http://img202.imageshack.us/img202/6710/31242897.jpg
http://img217.imageshack.us/img217/843/44098112.jpg
http://img81.imageshack.us/img81/6442/37546893.jpg

And the best thing is the ability to easily expand the series as much as I like.

Additional usefulness is the pdf export thingy, it will allow anyone to supplement this discussion with attached function.

Now I need to figure out how to instruct him to find me a first and second derivative of a function. Hell, I'll bet it can solve the problem, if only I knew how to program it.

The syntax is:

derivate ln(e^x+1)

second derivate ln(e^x+1)

Now I can play with this and tweak a few things.

And here is the example, query for e^x+1