Math Trading. Expected value. Probability and reality (martingale, averaging, order grids)

Mathematics is one of the most important sciences of our beautiful world. We begin to study it from school, then move on to university and work. Mathematics plays a particularly important role in trading. Since the market is moving according to mathematical and physical principles. Knowledge of mathematics, even to a minimal extent, is very important for any trader, no matter what securities he trades. The question is, how much knowledge should a trader have in trading.

In the process, the trader often broadens his horizons by reading various articles on websites and communicating on forums. Some books assure the trader of the minimal use of mathematics, while others require a clear, disciplined approach to trading using mathematical methods.

Mathematics plays a very important role in the work of the trader, as the market today is very fast and constantly changing. To understand the essence of trading from the inside, you need to understand the mathematics of trading. As Peter Lynch said: “Everyone has brains to monitor the market. And if you know math, at least at the elementary school level, you can handle it. ”

The math of trading is as important as discipline. Imagine that you play poker, and you need to calculate all the possible moves and chances of your opponents in advance. In poker and trading, as in any other business, an unspoken rule applies; the higher the risk, the greater the profit. So if you ever decide to trade other securities besides currency pairs, don't forget that you will need math skills.

Expected value
Expectation is one of the most important indicators of a trader’s performance in the market. This indicator is calculated as the sum of the products of each possible gain and loss and the probability of making a profit or loss.

For example, take a trader with a strategy and the ability to get 40% of positive deals for \$ 6, and lose 60% of deals for \$ 2. The expected value will be calculated by the following formula:

MO = (0.4 x 6) + (0.6 x (-2)) = 2.4 + (-1.2) = 1.2;

We get the number 1.2, which means that our mathematical expectation for each transaction will be equal to 1 dollar and 20 cents.

As a result, we obtain that mathematical expectation is one of the most effective indicators for identifying the profitability of a trading system. Mat expectation can be negative and positive. Everything is simple, if the expectation is positive, then traders earn money, if the negative means lose. Moreover, the higher this indicator, the faster the trading account will grow - this is logical.

If the expectation is negative, you will lose money, and if this continues for a long time, you will merge. Therefore, when you see that you are starting to lose money, you need to change your money management system and strategy.

Probability theory
Probability Theory is a branch of mathematics that studies the laws of random phenomena. Random events may be incompatible and joint. Incompatible this is when the appearance of one object excludes the appearance of another, the rest of the case will be called joint. Probably, many traders will not agree with this, but I think that one of the most important keys to success in trading is the theory of probability. A trading strategy cannot be profitable if the bias of probability is not in your favor.

Example:

The box has 10 red and 10 green apples. At random, one apple is taken. The event apple turned red, and the event turned green are incompatible.

Probability is a quantitative assessment of the possibility of an event. The probability indicator is calculated in the range from 0 to 1.

An event equal to "0" is impossible;

An event with an indicator of "1" is reliable.

And if these two probabilities intersect and do not change the probabilities of the other, then such an event is called independent.

The probability of a random event “A” is the ratio of the number “n” of elementary events making up event A to the number of all events “N”.

P (A) = n / N

A well-known example is the loss of an eagle when tossing a coin equals A = ½ = 0.5

Martingale
Martingale (from French Martingale) is a risk management system. The system was originally developed for playing in a casino. But gradually, this system began to be adopted by financiers who play on the stock exchange.

System structure:

The game starts with an already pre-approved bet or deal;
Further, after each loss, the player / trader must increase the bid / deal serially. For example, classically 1,2,4,6, etc .;
If the trader / player makes a profit, according to the rules of the system, the player must return to the minimum approved rate / deal.
When a player / trader makes a final profit, after a series of unsuccessful transactions, he wins back the previously merged capital and earns. But not many people know that the profit in this party is only the initial rate. Experienced traders know this, and understand that if suddenly a force majeure happens on the market, then they may lose the entire deposit. Therefore, the verdict on this system is this: in general, a player almost never loses money, and if he loses it, he only loses money and often earns, but not much.

An example of a system working in trading:

The trader opens a deal with a total volume of 100 US dollars. The deal is winning, the trader earns. The second transaction of the trader, also concluded at \$ 100, is unsuccessful. And then the Martingale system turns on. In the transaction you lost \$ 100, then the next deal needs to be opened with a volume of \$ 200. The transaction is triggered, and the player earns \$ 200, but since he lost \$ 100 in the previous transaction, in the end he gets only \$ 100 of net profit.

Pros and cons of the system:

The system is very easy to use and very popular. Many traders think that it is without a loss and make mistakes. There is 1 drawback in this system and this mathematical expectation is equal to “0” in this case. That is, by concluding transactions, you are simply acting out. This is one of the main disadvantages of the system. Since to work on this strategy you will need a rather large deposit, and only a few can afford it. And yes, all this seems easy on paper in theory, but in real trading everything can happen the other way around.

One of the main advantages of this system is the sense of market. For example, if you work on this strategy for 2-3 months, you will quickly learn to understand the market and even further changing the strategy will make it easier for you to learn how to make money.

Averaging (from the English. Average) this is one of the varieties of Martigale methods used in financial markets. Averaging refers to reopening a transaction in the opposite direction of the market. This tactic is used by both professionals and beginners. Beginners, for example, use this system in case of accidental, unreasoned entry into the market. They open the opposite transaction, so that when the price is profitable to close both transactions at "0" or in a small plus. But professionals, they work according to a clear system, since they already understand the market well, they boldly open positions and take good profits.

Conclusion
This article describes only the basics of mathematics in trading. There are many speculators in the world; each has his own strategy and his own trading style. You can come up with many more criteria for evaluating trade. If you evaluate each characteristic separately, this will not give you a complete picture, but their combination will give.

Each trading strategy has both positive and negative sides. There is no perfect strategy. The effectiveness of the strategy is determined by the mathematical expectation; the larger it is, the greater your earnings.