# Probability in Trading and Finance

There are advanced definitions of probability and even versions with negative values but for the purpose of this thread, I’ll concentrate on the basic level.

1. Axiomatic definition.

I. P(α) ≥ 0
II. The probability of a certain event equals 1
III. If two events α and β are mutually exclusive, then P(α+ β) = P(α) + P(β)

This serves as a foundation for developing a theory and proving theorems (along with some theorems from algebra).

2. Classical definition

The probability of an event α equals the ratio of the favorable to the total number of outcomes, provided that they are equally likely:

P(α) = Nf/N

where Nf is the number of favorable outcomes and N is the total number of outcomes.

There are two problems with using the classical definition in trading and investing.

a. Circularity: equal likely means equally probable. This is a known problem with this definition.
b. Outcomes in markets are not equally likely. If we assume that, then it is a fair coin toss

3. Relative frequency definition

If an experiment under consideration is repeated n times and an event α occurs Nα times, then the probability P(α) of an event α is defined as the limit of the relative frequency of the occurrence of α:

P(α) = lim Nα /n, as n → ∞

The frequentist definition is where the game but also the conundrum starts. Apparently, the definition is about the existence of a limit, PROVIDED, n is sufficiently large.

If finance and markets, n is rarely sufficiently large, except in some rare cases.

HFT can provide sufficient samples. In trend-following, samples are never sufficient and averages do not have any significance due to small n. When the returns distributions are leptokurtic, averages converge for very large n. Yet, the frequentist definition is extensively used.

Especially with macro, when the same economic conditions rarely occur, talking about probabilities in a frequentist sense is wrong. For example: “In 5 out of 4 last recessions, the GDP fell by… Therefore, the probability of recession is now 80%.” This is meaningless.

Even if there were a sample of 100 recessions, still the events would not be outcomes of the same experiment but different ones. Even if we consider all recessions to arise from the same experiment, still the sample is small given the complexity of the processes.

4. Probability as a measure of belief

Actually, when people talk about probabilities in markets and finance, they talk about subjective priors, whether they know it or not. The problem with updating priors is p isn’t enough for returns profitability. Expectation decides that.

So, is probability useful in finance, trading, and investing?

It is under specific, limited conditions. What people refer to as probabilities and expectations in most cases, are averages, and hope, knowingly or unknowingly, these averages converge to distribution parameters.

The ramifications for backtesting as serious. Due to low samples, computed expectations, and probabilities in most cases are random variables, and far from convergent. For this reason, backtesting can be used to only reject strategies but never accept, at risk of Type-II error.

For similar reasons, making any inferences about strategy robustness based on calculated averages fails to yield significant p-values. Developers must resort to stochastic modeling and that is even problematic with small samples. We haven’t even touched on data-mining bias here.

All in all, probability as a notion has been relentlessly abused by finance and trading professionals. In most cases, the notion of probability is not even applicable. I include below a link to an article that summarizes the four definitions of probability and a link to a paper.