Dec. 20, 2018

No one seems to want to consider how much randomness there is in stock prices. The question should be, why? It might be the most important question of all. Even if not, it would still be more than worthwhile to investigate how much there is. Depending on the answer, it could simply force us to rethink, remodel, or at least transform our trading strategies and the way we play the game.

The problem is: how should we define randomness in stock prices in the first place?

Do we go the martingale route, even if it does not apply, or define it as a Gaussian distribution with extras? We already know that stock price distributions are not normal or I.I.D., but more like some Paretian distribution thing with fat tails. Regardless, declaring it something or whatever will not change the nature of that randomness or make it more predictable.

In stock trading, it is what you do at or from the right edge of a price chart that matters. If you are not in the market, all you can express are lost opportunities. It is following your decision to enter into a trade at a chart's right edge that your profit or loss starts to accumulate.

You might find and define stock patterns over past data, but when it comes to playing the game, you will have to declare those patterns before they occur, not after, where you cannot profit from them since you will again be at the right edge of the price chart.

One thing that is almost for sure, the stock price will not remain idle over extended periods of time. You will not get: p(t) = p(0); therefore, forget about the martingale. You might cross p(0) many times, but most probably, as time increases, you will get further and further away from p(0), either on the positive or negative side. Stock prices can go up for decades just as they can drop to nothingness in a hurry. In either of those scenarios, did they mean reverse?

It goes like this: if you do not address the notion of randomness in your trading strategy, does it go away? Can it be ignored!!!

**The Normal Distribution**

Should we see this trading game as a 50/50 I.I.D. proposition? Then, most likely, we will have to consider that what we are looking at is basically gambling with no assurance of an immediate or long-term payoff.

Gambling away does not necessarily mean you lose the game. It only says that you are gambling. And, therefore, might still win the game by chance. Not as an inherently bad proposition, but more from what will be a multitude of 50/50 bets with their mathematical, statistical built-in expectations and consequences.

Any bell-shaped distribution might appear as close to a 50/50 proposition, even a Paretian distribution with fat tails. This is what we are faced with when trading; our next decision will be as if at the right edge of that price chart. As if asking the question: will it be heads or tails?

What is your risk on such a proposition? Not much. Your long-term expectancy is zero. Therefore, no win, no loss. In all logic, in a no-win and no-loss scenario, whatever risk there might be should be considered minimal as long as you play for a long time and your running stake far exceeds the variance boundaries of the game. This will require making small bets in order to survive any losing streaks of any length.

The all-in scenario on every bet will nonetheless assure you to lose the game. You could invest for the long term that way but should not trade in that fashion. Full market exposure, for a trader, should be defined as using many small bets. Whatever your betting system, it is like a series of returns and could be expressed as: F(0) ∙ r* _{j}*(t) = F(0) ∙ ∏ (1 + r

*) for i = 1, ..., n, and where if ∃r*

_{i}*= -1, it will result in: F(0) ∙ r*

_{i}*(t) = 0.*

_{j}If there is a fee to play the game, then, in all probabilities, you have a negative proposition where you are bound to lose the longer you play. What you might get out of such a game is simple too: some form of entertainment you will have paid for. It is just like a little hamster on its running wheel, just running to nowhere at whatever the speed.

That you model price movements to stochastic equations does not change the nature of the problem, the randomness does not go away.

In the last notebook, a Python one-liner: **return_vec = np.random.randn(n_assets, n_obs)** generated normally distributed return series with mean zero and standard deviation of one.

The optimizer used in that notebook was able to extract even low drift values. To start moving the scale, it was sufficient to add a drift with a magnitude of *0.0001σdt*. That is 0.01% of the considered standard deviation (σ) of one. Doing so moved the distribution's average expected mean from zero to *0.0001σdt*. To gain a sense of its magnitude, it is like adding one cent per day to a $100 dollar stock that had an average $1.00 standard deviation. With the vast majority of its price moves (99.7%) within +/- $3.00 (+/- 3σ).

A 2-cent upward drift per day on the same $100 dollar stock looks insignificant. And yet, it was sufficient to explain the long-term upward drift on randomly generated stock prices. The added drift had for impact: r* _{j}*(t) = r

*(0) + (µ*

_{j}*+ 0.0002σ*

_{j}*)dt + σ*

_{j}*dW*

_{j}*.*

_{j}The mean (µ) was set to zero, and sigma (σ) was set to one by construction. While the initial rate of return at the intercept was zero at time *t* = 0: r* _{j}*(0) = 0. Thereby, leaving: r

*(t) = (µ*

_{j}*+ 0.0002σ*

_{j}*)dt + σ*

_{j}*dW*

_{j}*. which explains the stochastic equation in terms of sigma and its Wiener process. Making the*

_{j}*0.0002σdt*the most significant part since the impact over the long term from the Wiener process would tend to zero.

My latest book (**Beyond the Efficient Frontier**) stresses the fact that if you could extract more than what the market has to offer, it would require that you increase the *0.0002σdt* part of the equation or add something to it. It would be this excess that would translate into alpha. Already, the *0.0002σdt* upward drift appears as if built-in. Even if you could factor it out, it would remain as if the average secular upward trend we see, on average, in stock prices.

If we raised the average long-term rate of return to *0.0004σdt*, then we would declare the added *0.0002σdt* as alpha. These are not big moves. A *0.0001σdt* is about 2.5% per year on a $100 dollar stock. But then again, the game is compounding over time. And one thing the book does demonstrate is that a little alpha can go a long way.

Created... December 20, 2018, © Guy R. Fleury. All rights reserved.