Mathematical modeling of the probability that a currency exchange rate will reach a certain level over a future time interval is considered as an extrapolation problem, where the use of mathematical tools for describing models and algorithms is associated with the advantages of a mathematical approach to multi-stage information processing, searching for methods of their solution and transformation into programs.
Stages and Formulation of the Mathematical Modeling Problem
Mathematical modeling in general is not a straight path to the goal, but a repeated return to already passed stages, their repetition with modified data — a sequential approach to a satisfactory variant. On the first stage, there is an assessment of the real situation from the position of the existing a priori model and the goal, and as a result, on the second stage, a substantive model is formed, reflecting the problem statement. This model is formed in the ‘native’ language of the problem: mechanics, physics, economics, biology, sociology, etc. The third stage: the structure of the model (that is, the most suitable mathematical apparatus) is chosen, the type and number of equations, the type of functions. On the fourth stage, if necessary, details of the model are specified (necessary approximations are made, equation coefficients are adjusted). Checking the quality of the resulting construction using criteria, the choice of which is dictated by the goal of the modeling, is carried out on the final, fifth stage. If the quality of the model is unsatisfactory, the procedure is repeated from the beginning or from an intermediate stage — the next approximation is made [1].
Despite the infinite number of situations, objects, and goals introducing their own specifics into the process, it is possible to identify the main stages of modeling. The work starts with considering the available information about the object (experimental data on it itself or similar objects; theories developed for describing the researched class of objects; intuitive representations, etc.) from the position of the research goal, obtaining and preliminary analysis of the series of observed quantities, and ends with the use of the obtained model to solve a specific task. But this process is usually iterative and is accompanied by multiple repetitions, returns to the initial and intermediate points of the scheme, successive approaches to a ‘good’ model.
Depending on the origin of the time series, two qualitatively different situations arise when setting the modeling problem. First: when observations represent a realization of some mathematical model (a system of equations) obtained by numerical methods. For this case, the terms ‘reconstruction’ or ‘recovery’ of equations are fully appropriate. It is much simpler to check the quality of the model here, since there is a ‘true’ original equation — it can be compared with the results of the modeling and the properties of its solutions. In addition, theoretical conditions for the effectiveness of modeling methods for various classes of systems can be formulated. In the second, qualitatively different case, when the time series was obtained as a result of measurements of a real process, there is no single correct model and the success of modeling cannot be guaranteed.
Extrapolation as the Initial Stage of Building Final Forecasts
One can only marvel at the ‘unreasonable effectiveness of mathematics’ if a ‘good’ model is obtained [2,3]. For example, in the mid-1990s, a team of American students visited the most powerful casino in Las Vegas and returned home with several million dollars. Smart students from a technical university played blackjack every weekend for a month and won large sums. These guys were from the world-famous Massachusetts Institute of Technology. Their millions in casino wins revived the stagnant war between players and casinos, started 40 years ago by Professor Edward Thorp. He realized that among other games in the casino, blackjack is the most calculable, when evaluated mathematically. In most money-based games — roulette, dice games, lotteries — past events do not determine future events. But this does not apply to such a game as blackjack.
The fundamental problem in games is finding opportunities for bets with positive expectation. A similar problem in investing is finding opportunities for investing with ‘excess’, taking into account risk adjustments, profitability. As soon as such favorable opportunities are identified, the player or investor must decide what part of their capital to put at stake (invest). Interest in ‘excess’ profitability has existed, at least since the eighteenth century, with discussions of Daniel Bernoulli’s St. Petersburg Paradox. But players also need to know how to manage money. On stock markets (including the securities market), the problem is similar, but more complex. A player who is now an investor seeks ‘large profit with a manageable level of risk’. In both these cases, Edward Thorp used the Kelly criterion, which maximizes the expected value of the logarithm of income (‘maximizes the expected logarithmic utility’) [4].
The general problem of extrapolation is to find the values of a function describing the change of a parameter over time at a point lying outside the observation interval of the given function, which allows the use of extrapolation for forecasting purposes. So far, it has found wide application as an approach to forecasting simple predictive models. Extrapolation defines the trends of future development of the studied phenomenon under the condition that the regularities of this phenomenon, which have formed in the past, will exist in the future. These regularities define the most stable features of the forecasted process — its trend, assuming that it can be described by any function.
Fig. 1. Exchange market indicators as indicators of future trends on Forex.
In applied mathematics, it is often necessary to solve the following extrapolation problem. Let it be known that for the sequence {z} the first N terms are known. Can this information be used:
1) to establish that this sequence converges to a limit;
2) to find this limit;
3) to estimate the error with which this limit was found;
4) using some criterion, to estimate the reliability of the error estimation.
At first glance, this task may seem unsolvable — too little information. Indeed, in general, N first terms completely do not define the sequence, and starting from the N-th number, the sequence can behave unpredictably. Such tasks are called ill-posed in mathematics. However, such tasks occur very frequently in practice. Predicting the position of a moving body, forecasting the conditions of operation of various systems, many design and control problems can be classified as tasks of this type. Examples show that the initial elements of the sequence contain much more information about its limit than we assume, and the question is how to extract it.
One way to estimate the error is to compare the calculated value with the extrapolated one. Extrapolation is also used to accelerate the convergence of sequences. Known convergence acceleration methods are based on the fact that from the original sequence {z} a new sequence is sought, which tends to the same limit, but faster. In some cases, by accelerating the convergence, results that could not otherwise be obtained within acceptable time are achieved. Statistical methods occupy an important place in the system of forecasting methods. The application of forecasting implies that the pattern of development, acting in the past (within the dynamic series), will remain in the forecasted future, i.e., the forecast is based on extrapolation. The accuracy of the forecast depends on how well-founded the assumptions about preserving the actions of those factors that formed the main components of the base dynamic series will be. Therefore, any forecasting in the form of extrapolation of a series should be preceded by careful study of long dynamic series, which would allow determining the trend of change. Since the trend of development can also change, data obtained by extrapolating the series should be considered probabilistic, as a kind of estimates.
FAQ
What is mathematical modeling in the context of currency exchange rate prediction?
Mathematical modeling involves creating a structured representation of real-world processes, such as currency exchange rates, using mathematical tools to analyze and predict future values based on historical data.
How does extrapolation work in forecasting?
Extrapolation estimates future values of a parameter by extending observed trends beyond the current data range, assuming that past patterns will continue into the future.
Why is the Kelly criterion relevant to financial decision-making?
The Kelly criterion helps determine the optimal amount of capital to invest in a bet or trade, aiming to maximize long-term growth while managing risk effectively.
