Introduction
Mathematics offers a wealth of tools and resources for applications in a variety of fields. One significant use of mathematics is in finance since strategic planning and forecasting require accurate models and quantitative analysis. Trading, in particular, reflects trading strategies in which strategic reasoning is necessary to maximize profits. This paper will evaluate which of the mathematical tools traders routinely use to study the organized market and multiply their own profits.
Functional Analysis
The dynamics of trading metrics over time is, in fact, a time series that can be analyzed with functional analysis. For example, the change of a share price during a limited time can reflect the continuity of time series, that is, the situation when there are low fluctuations during short periods of time. In this case, a trader sees a continuous growth or decrease of the measured parameter, which allows for making guesses about how the function will behave in the future. In other words, the absence of sharp fluctuations while maintaining the continuity of the function allows one to judge the market trends and make predictions about the future of this parameter. On the contrary, if the time series turns out to be noisy, that is, if it constantly goes up and down, it may lead a trader to believe that it is necessary to invest in the share price when it is falling and sell when it is rising, which means that in this case short-term investing is most likely.
Meanwhile, functional analysis implies the use of differentiating functions to determine their rate of change. Strictly speaking, to get the derivative for a function, such as a time series, one needs to study the limit of the increment of that function to the increment of time, as shown in the equation below. In simple words, this means studying the rate of change of the indicator understudy in a segment of time. Differentials of financial functions are thus used to determine the rate of change in the price of a stock. Too fast change can indicate high volatility of the studied parameter, which increases investment risks. On the contrary, too slow a change may mean that a trader should use long-term strategies to manage finances in this market.
It should also be emphasized that the behavior of a financial indicator in the market is often random and difficult to predict, which means that the ideas of Brownian motion are applicable to it. That is why stochastic calculus is recommended in financial mathematics because, in practice, time series are rarely smooth. The stochastic function calculus implies the normality of the distribution of a variable on a broad scale, which allows us to treat nonsmooth functions as some general change in the stock price combined with random variables, as shown in the equation below. In other words, normal algebraic calculus does not work for nonsmooth functions in trading, so stochastic methods allow us to cover this gap.
Linear Regression
In addition, keep in mind that any time series is ultimately a visual representation of a correspondence table of the measured parameter as a function of the change in the argument. For such correspondences, it is appropriate to use linear regression models to determine the overall upward trend. The automatic calculation of this model allows determining not only the equation of a particular function with a certain accuracy, expressed by the coefficient of determination, but also the behavior of the linear function: it can fall or rise. The form of this function allows judging what happens to the studied financial parameter over time, which is valuable knowledge for a trader. Another advantage of using linear regression is the possibility of mathematical forecasting. Since a linear function is infinite, it is possible to know the value of a stock in the future with its extrapolation if the market does not undergo serious fluctuations. It is clear that the value of such knowledge is not absolute because forecasting is never exact, but it allows for correcting the trader’s strategic behavior and understanding the general pattern the market follows.
Conclusion
To summarize, financial mathematics uses a large number of tools, among which functional analysis and linear regression are important resources for studying the behavior of a financial indicator over time. Effective trading cannot exist without a mathematical application because only it allows for managing finances in a meaningful and planned way — otherwise, the investor risks losing all the profit of trading. The paper showed that the tools of functional analysis and forecasting with the help of linear regression could be used to study the market.