Mathematical expectation and trading on the exchange

The average income of a regular casino is comparable in its magnitude only with the profitability of transactions on Wall Street. Smart people have long understood that one can not always count on one's luck and started using statistical methods for the stability of getting their profits.

The casino receives huge sums, because the "probability" or, in other words, the mathematical expectation of the game, is on the side of the gambling house. And regardless of which game to participate, sooner or later the casino wins. The profit of the casino grows even faster if the range of games includes those that end in a relatively fast time - roulette, dice or a few cards.

I think any trader needs to solve three most important tasks for success in his work:

1. To ensure that the number of successful transactions exceeds inevitable errors and miscalculations.

2. Set up your trading system so that the earning potential is as often as possible.

3. To achieve stability of the positive result of their operations.

And here we, working traders, a good help can have a mathematical expectation. This term in probability theory is one of the key. With its help, we can give an average estimate to some random value. The mathematical expectation of a random variable is similar to the center of gravity if one imagines all possible probabilities with points of different mass.

In the case of a trading strategy, the mathematical expectation of profit (or loss) is most often used to evaluate its effectiveness. This parameter is defined as the sum of products of given levels of profit and loss and the probability of their occurrence. For example, the developed trading strategy assumes that 37% of all operations will bring profit, and the remaining part - 63% - will be unprofitable. At the same time, the average income from a successful transaction will be $ 7, and the average loss will be $ 1.4. Let's calculate the mathematical expectation of trade on such a system:

MO = 0.37 x 7 + (0.63 x (-1.4)) = 2.59 - 0.882 = 1.708

What does this number mean? It says that, following the rules of this system, on average we will receive 1.708 dollars from each closed transaction.

Since the resulting efficiency score is greater than zero, such a system can be fully used for real work. If, as a result of the calculation, the mathematical expectation turns out to be negative, then this already indicates an average loss and such a trade will lead to ruin.

The amount of profit per transaction can also be expressed as a relative value in the form of%. For example:

• Percentage of income per 1 transaction - 5%;
• Percentage of successful trading operations - 62%;
• Percentage of loss per transaction - 3%;
• The percentage of failed transactions is 38%;

In this case, the mathematical expectation is (5% x 62% - 3% x 38%) / 100 = (310% - 114%) / 100 = 1.96%. That is, the average transaction will bring 1.96%.

It is possible to develop a system that, despite the prevalence of unprofitable trades, will give a positive result, since its MO> 0.

However, one expectation is not enough. It is difficult to earn if the system gives very little trading signals. In this case, its yield will be comparable to the bank interest. Let each operation give an average of only $ 0.5, but what if the system involves 1000 operations per year? This will be a very serious amount for a relatively short time. From this it follows logically that another shortcoming of holding positions is another distinguishing feature of a good trading system.

If there is a desire to delve deeper into the math of randomness, find out what conditional mathematical expectation, confidence interval and other interesting tools are, we recommend reading the book "Statistics for the trader" (author S. Bulashev). Who knows, perhaps, the chaos of currency movements after reading the book will seem to you just the highest form of order ...