August 14, 2013

However you want to look at your trading methods, some very basic math applies. For sure... It is not, and cannot 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 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 a linear or a compounded return game. Its output as an exponential function: (1 + r)^{t} - 1, or linear as in (1 + rt) – 1. It might not sound like much of a difference, and some might conclude: who cares if the game is linear or compounding? 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? Whatever 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 is for the Geometric Brownian Motion. The point is 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 as a conclusion that buying indexed funds and holding for the long term will have the 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 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 expected and trading skills that can generate some alpha, a kind of aberration in itself.

**Trading Linearly**

Whatever trading method is used, we can all agree on 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)/

*n*, where

*n*is the total number of trades. Not all trades resulted in a profit, and this will be seen in the average profit per trade.

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

*n*;

*n*might need to be quite large to make it worthwhile: Q*Σ

_{n}(ΔP). Furthermore, Δt will kick in: it does take time for prices to generate ΔP, it takes more time to generate 1,000 ΔP, and the greater

*n*will be, the more Δt will tend to grow, the more time it will take. You already know your average profit per trade, and you also know how long it took to get there; therefore, depending on your selected trading methods and on your objective, you know how many more trades you will need to make and how long it might take. Even if it is an approximation, you at least have a ballpark figure to work with.

If I limit myself to *n**(Q*(ΔP)), hoping that the sum of generated profits exceeds 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 whatever 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 of possible trades and 100 EOD (end-of-day) trading decisions over those 20 years.

One could ask which might be better; searching for combinations of technical indicators and trading methods that can satisfy *n* and Δt, or going 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...

The next section will be on the distribution of trends and compounding trading strategies.

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