The rise and fall in the value of shares how to describe correctly mathematically?

Question

Mathematics

Guys, the whole brain is broken, please tell me what mathematical formulas can describe the process that occurs on the stock exchange:
A bidder buys 10 shares, because of this, the shares rise in price.
The bidder sells 10 shares, because of this, the shares become cheaper.
In both cases, the exchange took a commission, for example 0.5%.

Here's the problem:
For example, a share costs 100 rubles, a participant bought 10 shares, the price rose to 110 rubles. Exchange commission 10 Shares*100 RUB=1000 RUB*0.5%= 5 RUB
Next, this participant decided to sell these shares at the current price of 110 RUB and we get the following: Exchange commission 10 Shares*110 RUB=1100 RUB*0.5%= 5.5 RUB
In total, after this purchase and sale operation, the participant remained in the black 1100-1000-5-5.5 = 89.5 rubles. those. the exchange participant earned 89.5 rubles from the void. Which shouldn't happen.
What mathematical formula is correct to make so that a participant cannot "earn" by buying shares and immediately selling them, because he is alone on the stock exchange, there are no other participants :)
I.e. it is necessary that, as a result of the purchase and sale procedure, the participant has a balance sheet of zero, or rather -10.5 rubles.

Answer 1

It seems to me that there is rather a misunderstanding of how exchanges work.
1. Alice buys 10 shares of $100 from Bob. 1% goes to broker commission.
Thus, Bob has $1000, and the broker has $10. Alice loses $1010
2. After some time, the shares rise in price and rise in price to $110
(in this case, we understand that there is a buyer who is ready to pay that much)
Vadim buys from Alice her 10 shares at $110. Alice receives $1100, the broker receives another $11 commission.
So what we have:
Alice has $1100. She has a profit of $90
Bob has $1,000
Vadim has 10 shares
From these two trades, the broker made $21
.
Alice could be at a loss if in this case the broker's commission would be higher. Well, or if the commission was charged according to some other principle.

Answer 2

Be careful, there are two radically different places on the market where you can make transactions:
* an exchanger (for example, a bank), they tell you a price that changes periodically and in a sense depends on the volume
* exchange (in the correct version, you work with it through the exchange, but there cuisines that act as an exchanger or even simulators), you can set your own price (transaction limit) or make a transaction at a market price (market) - as in the option above but without a 'hidden' commission (the exchanger is usually an ordinary trader on the stock exchange, calculating its value and adding your commissions).
If we are talking about an exchange and not an exchanger, it is difficult to make any forecasts, because there is a concept - liquidity. It is determined by how many traders like you have placed limit trades counter to you, as well as in dynamics (determined statistically), how quickly the market (traders) fills the list of offers with limit trades (glass). Your transaction may consist of many others, for less, each of which will be at different prices, you can find out only at once by requesting a glass from the exchange (exchanges try to limit their customers in such information, asking for an additional fee for it) and most importantly this information can change very quickly.

so that the participant could not "earn"

There is no simple solution here, the task is comparable to the reverse one - namely, how to make money on speculation, millions of traders around the world are struggling with this, forming a constantly changing trading strategy against each other (in order for one to earn, someone must lose).
Everything changes if you yourself are a broker / exchanger, and a monopolist or a fairly large market participant. You can use information about your customers and their behavior to predict how they will behave in a given situation. For example, the simplest delay in replenishing a deposit can give you guarantees within this time that your clients will be able to make transactions for an amount not exceeding X. You can also see in the list of limit orders which transactions are foreign, which are yours personally and which are your clients (and most importantly, what stops they have), you can always calculate how much you need to make a deal (or how much the market will make deals if you issue the appropriate price tag, like an exchanger) so that the rate shifts. You can see what stops your clients have, i.e. know what volume of trades will be when they start to be executed. Those. further, simple mathematics, how much money you need to spend to move the rate so that your clients' stop trades start to work in order to move the rate further, a stupid example, having calculated all the necessary prices, you put trades for sale at the moment you buy, but with a difference from which you will have an income and your clients will have a loss (since you yourself caused it by moving the market in the right direction). Knowing how the market will behave and manipulating it within the limits of your liquidity, you can shift the expectation of winnings in your favor. from which you will have income and your clients will have a loss (since you yourself have corrected it by moving the market in the right direction). Knowing how the market will behave and manipulating it within the limits of your liquidity, you can shift the expectation of winnings in your favor. from which you will have income and your clients will have a loss (since you yourself have corrected it by moving the market in the right direction). Knowing how the market will behave and manipulating it within the limits of your liquidity, you can shift the expectation of winnings in your favor.
Of course, such manipulators also take risks, because they will receive guarantees only if they are the only or largest on the market, but since this is usually not the case, mathematical statistics come into play.