August 14th, 2013

How ever you want to look at your trading methods, some very basic math apply. For sure... It is not, and can not be: those math things don't apply to me, I've got "my proprietary trading system" of play that circumvents all that. Sure...

Some seem to look at the stock market game as if the same as a casino and play accordingly. They'll defend their methods, use statistical tools and provide explanations as if the game was ruled by "predictable" Gaussian distributions having very "specific odds" for every position they want to take. They will day trade hoping to beat these odds since their "discretionary" trading method has the "ability" to forecast what's coming. Sorry, but there are nuances of importance that can disrupt this kind of casino mentality.

The stock trading game could be considered as a linear or a compounded return game. Its output as an exponential function: (1 + r)^{t} - 1, or linear as in: (1 + rt) – 1, might not sound like much of a difference and some might conclude: who cares if the game is linear or compounded? Some will even say: when I take a position, all I care about is: can I make a profit or not? On a single bet, you get the same profit or loss; there's absolutely no difference:

(1 + r)^{t} - 1 = (1 + rt) - 1 for t = 1. Therefore, ... again, why the distinction?

What is more important is: are your trading methods of the linear type or is your trading strategy compounding? Based on the payoff matrix notation: Σ(**H**.*Δ**P**), which summarizes the outcome of any trading strategy **H** over any trading horizon Δt and any stock selection Δ**P**, this would translate to how do your trading methods behave over the long term? What ever your stock selection Δ**P**, the future value of each day's price differential over the next 20 years is to say the least "unpredictable". The Δ**P** of the stocks in the S&P100 over 20 years would be composed of 500,000 individual price variations.

Nonetheless, there is a stochastic differential equation that can be used to represent stock price series: P(t) = μdt + σdw where μ is the mean average return (the long term drift), while σ is the standard deviation from the mean and dw a Geometric Brownian Motion. The point being that, long term, σdw tends to zero and all that is left is: P(t) = μdt which is just a regression line. If that is all that is left, then, is it not all you can get?

This would justify, in itself, all the academic papers that have for conclusion that buying indexed funds and holding for the long term will have for most expected outcome: μdt. Indirectly meaning: grab the average mean return μ by holding for as long as you can dt. But this is the same as promoting a Buy & Hold solution to the investment problem. Doing this, you win at the game, long term; if you do not get distracted by σdw: the random component of the equation.

But it should be added that in this process there is no alpha generation. There is nothing that you could do that would improve on the expected outcome: E(P(t)) = μdt since whatever you would want to add would tend to zero in value, unless luck was on your side (σdw > 0). No alpha, no over-performance.

Notwithstanding, when looking closely at the problem, you do see some people generating alpha, which would suggest that there are trading methods that can produce more than the expected; trading skills that can generate some alpha, a kind of aberration in itself.

What ever trading method used, we can all agree to some basics. On a single trade, a profit or loss is: Q*(ΔP) = Q*(P(out)-P(in)), which expresses the profit or loss as the price differential between entry and exit scaled by the quantity traded Q. If I made a thousand of such trades in succession, I would be trading linearly. All I could gain would be the sum of price differentials ΔP which are still governed by: P(t) = μdt + σdw. And having done a thousand trades, I do have statistics from which to evaluate some performance metrics. Should I keep the quantity Q constant, it is sufficient to add all the price differentials ΔP from each trade to get: Q* Σ(ΔP), the total generated profits from those thousand trades. The average profit per trade would be: Σ* _{n}*(ΔP)/

Nonetheless, depending on the value of your average profits per trade: Q*Σ* _{n}*(ΔP)/

If I limit myself to: *n**(Q*(ΔP)) hoping that the sum of generated profits exceed a long term objective, this sum should at least exceed a less demanding solution like the Buy & Hold or indexed funds. Not being able to change the price differential matrix Δ**P**, and not being able to predict the random component: σdw of the stochastic differential equation, will have for corollary that I can not predict in which direction or amplitude these price variations will be.

I am left with only 3 variables I can control; and that is *n*, **Q** and Δt. Based on the S&P100 example, Δ**P**, and **Q**, are matrices of 500,000 elements each. A single expression to do 500,000 calculations and provide the sought after solution: the total profit generated by the selected trading strategy **H**. The payoff matrix: Σ(**H**.*Δ**P**) can calculate the generated profits for what ever trading strategy **H** over any Δt of choice.

And here might lie the problem: the Δt of choice. If you need to do 1,000 or 10,000 trades (*n*), it might take some time... For the S&P100 example, you might have many thousands possible trades; 100 EOD (end of day) trading decisions over 20 years.

One could ask which might be better; searching for a combinations of technical indicators and trading methods that can satisfy *n* and Δt, or go the other way around, finding ways to detect and take the *n* ΔPs needed over the trading interval Δt to meet the objective: Σ(**H**.*Δ**P**) > T >> Buy&Hold, which translate to generating a lot more profits than the Buy & Hold.

... to be continued...

Next section will be on the distribution of trends and using compounding trading strategies.

Created... August 1st, 2013 © Guy R. Fleury. All rights reserved.