Whenever any speculative opportunity is presented before you, there is a built-in null hypothesis regarding that opportunity. That null hypothesis is, **the opportunity is most likely “fair” or “efficient”, meaning that over time, you should not expect to make or lose money if you continued to take this opportunity indefinitely.**

Your goal as a trader is to determine which opportunities allow you to reject this null hypothesis based on 2 areas, the distribution of the outcomes, which will be referred to as * x* throughout the course, and the payoff structure of the opportunity, which will be referred to as

*.*

**f(x)**If either of these areas individually, or combined provides any advantage one way or the other, you can reject the null hypothesis and speculate accordingly.

But just because you can reject the null hypothesis, doesn’t mean the opportunity should be taken. Take for example casino games.

Casino games are not fair because the house builds in their edge in either *x* (example: green spaces on a roulette wheel) or in *f(x)* (example: payoff odds they offer for landing on a green space).

Unfortunately for us though, we can only take one side of the bet so even though we can reject our null hypothesis when it comes to betting on roulette, we can not benefit from it because we are restricted to only being able to take one side of the bet.

Thankfully in the market you are able to take either side of most opportunities you will encounter. What follows is everything you need to know in order to determine which side of the opportunity you want to take.

**When it comes to x, what we are looking for in order to reject our null hypothesis, is deviations from what would be considered “normally distributed” or “normal”.**

**When it comes to f(x), what we are looking for to reject our null hypothesis is payoffs based on x that are not “fair”, relative to the probability of that payoff occuring.**

Before we dive deeper into these 2 areas, it’s important to understand that an edge can not exist alone in one or the other. If there is an edge in one area, it can ONLY be realized from the other.

A coin toss can be a speculative opportunity worth taking even if the coin (* x*) is “normal”, meaning heads and tails are equally likely to occur, as long as the payoff (

*) is not “fair”, based on 50% probability of each event occuring. In practical terms this would mean, on heads you make more than you lose on tails, or vice versa.*

**f(x)**On the flip side of that, a coin toss can also be an opportunity worth ignoring even if heads is 25% more likely to occur than tails, because the payoff is “fair”, meaning whenever you get heads, you only make $1 and when you get tails, you lose $3.

So * x* and

*are intrinsically linked in that, an opportunity in one, can only be realized or captured by the other.*

**f(x)**## Understanding *x*

In trading, x typically is the distribution of the asset you are trading’s price but it can also be a specific component of the assets price, such as volatility. In our null hypothesis, it is assumed that this distribution of price is normal, and if you plotted it out, it would form a bell curve.

This means the distribution itself would be perfectly random at all times. This is typically what economists are referring to when they say that markets are “perfectly efficient”.

In a perfectly normal distribution, the next move is not only always just as like to be up as it is down, but it’s also just as likely to be as far down as it is up. In this “normal” world, there is no oppurtunity, anywhere in the distribution itself, that can be taken advantage of in order to generate a consistent return, one way or the other.

It’s always “fair”, or “efficient”. The only way you could make a return would be if ** f(x)** was not “fair”.

Some typical properties of a normal distribution are:

- The distribution’s skew is 0.
- The volatility of the distribution is constant.
- The kurtosis of the distribution is 3.
- The median, and mean of the distribution are the exact same.

There are many other properties or characteristics of a normal distribution you can measure, but these are the easiest to use in order to quickly understand whats happening.

Now in order to reject the null hypothesis that the distribution of the asset’s price is “fair” or “efficient”, we need to consistently see within the distribution, characteristics that can not exist if the distribution were truly normal.

Things like excess kurtosis, consistent skew, etc. Once you can confidently reject the null hypothesis, you can start to build a strategy around this unnormal characteristic.

## Understanding *f(x)*

The next area that can allow you to reject the null hypothesis is the payoff structure of the opportunity and whether or not it’s fair.

Something is considered fair if the ratio between the potential risk and reward of the opportunity is exactly equal to the probability of that particular outcome.

Let’s look again at a coin toss. Assuming the coin is normal and has truly 50/50 odds, the payoff is considered fair if the ratio between the risk, and the risk plus the reward is .5.

So if the payoff is risking $1 to make $1, the ratio would then be ($1 / ($1 + $1) = 0.5), the exact probability of the outcome being heads or tails.

Let’s take a more complicated distribution like a six-sided dice. Each side is equally likely to occur with (⅙) odds of occurring or roughly 16.67%.

As long as the reward for each side is the same, the payoff is “fair” because ($1 / ($1 + $1 + $1 + $1 + $1 + $1) = .1667 or 16.67%.

Now assume one side of the dice is more likely to show up than all the other sides, the payoff is no longer fair because not all sides have a 16.67% chance of showing up.

Or imagine one side had a higher payoff than the others, but they all were likely to occur with the same probability. The payoff would also not be “fair” for that side as it’s ratio would be ($2 / ($1 + $1 + $1 + $1 + $1 + $1) = .3334 or 33.34%, which is twice the probability of that side showing up on a dice roll.

This means whenever that side does show up, you get paid twice what would be considered “fair”.

Putting it all together…

The most important thing to understand here is, that any discrepancy between the probability of an event occurring, and the ratio of the payoff from that event, is your advantage or disadvantage depending on which side of the bet you take.

Having a consistent deviation from what is either “normal” or “fair” is the only way you can successfully speculate on anything. No amount of discipline or psychology can save you if there is no opportunity at a fundamental level.