Risk management system based on probability theory and mathematical statistics

Abstract

The invention relates to the technical field of probability statistical modeling, in particular to a risk management system based on the probability theory and mathematical statistics, and the system comprises a data preprocessing module, a tail distribution estimation module, a tail extreme value fitting module, a dynamic risk assessment module, and a risk early warning and quantification module. According to the method, by extracting the extreme income points in the time sequence data and the market data with remarkable tail characteristics, the accurate capture capability of the tail risk events is improved, distribution analysis of the tail risks is refined by combining quantile interval division and conditional density calculation of the tail data, the quantification capability of risk distribution characteristics is enhanced, and the accuracy of the tail risk events is improved. And furthermore, local extreme value calculation and dynamic analysis of the tail revenue migration rate are combined, so that real-time adjustment of the risk model is realized, the sensitivity and adaptability of risk assessment are improved, and the accuracy and pertinence of risk early warning are enhanced by accurately calculating the interval range of triggering the market risk and the tail risk characteristics.

Description

Technical Field

[0001] The present invention relates to the technical field of probability and statistics modeling, and in particular, to a risk management system based on probability theory and mathematical statistics. Background Art

[0002] The technical field of probability and statistics encompasses the core theories and applications of probability theory and mathematical statistics, mainly covering mathematical foundations such as the distribution of random variables, parameter estimation, and hypothesis testing. It focuses on the quantitative analysis and solution methods for uncertainty problems. This technical field systematically combines means such as mathematical modeling, statistical inference, and data processing, and is widely applied in financial risk control, engineering reliability analysis, and other fields that need to handle complex random phenomena. At the same time, supported by mathematical theories, this technical field is gradually combined with modern computer technologies, and through programming languages and algorithms, it realizes the probability analysis and statistical inference of large-scale data, providing basic tools for scientific research and engineering applications.

[0003] Among them, probability and statistics modeling refers to constructing a mathematical model that reflects the characteristics of random phenomena based on probability theory and statistical theory, and describing and analyzing the behavior and distribution characteristics of random variables. This topic covers the collection and collation of random data, conducts distribution assumptions and parameter estimations through mathematical analysis methods, and realizes the modeling and optimization of probability distributions based on numerical calculation means. Its specific methods include distribution fitting based on mathematical methods, optimizing the solution of model parameters through matrix operations and vector space theory, and combining mathematical inference to complete the hypothesis verification and adjustment of the model, and finally providing a quantitative modeling method to describe the random relationships in complex systems.

[0004] In dealing with extreme return data, the prior art lacks accurate extraction and interval division for tail characteristics, resulting in low analysis accuracy for tail risks. The statistical assumptions of tail distributions usually rely on simple models, which are difficult to comprehensively reflect the complexity of market risk distributions and lack the ability to effectively characterize conditional density features. It has weak dynamic adaptability to market emergencies and cannot adjust the model through real-time data analysis, resulting in lagging risk assessment. In addition, the interval range setting is too rough when the existing system triggers risk warnings, which may miss key risk points or misjudge the risk intensity, affecting the accuracy of market decisions. Summary of the Invention

[0005] The purpose of the present invention is to solve the deficiencies existing in the prior art, and to propose a risk management system based on probability theory and mathematical statistics.

[0006] To achieve the above purpose, the present invention adopts the following technical solution: A risk management system based on probability theory and mathematical statistics includes:

[0007] The data preprocessing module extracts the tail values of returns and their corresponding market time points based on the time series, screens the market data with significant tail characteristics, records the time markers of the tail risk set, calculates the volatility and the dispersion of tail data, marks the extreme return data points that meet the tail characteristics, and generates the tail risk marker values;

[0008] The tail distribution estimation module divides the tail data into multiple quantile intervals based on the tail risk marker values, extracts the return values of each quantile interval, calculates the volatility and distribution density of multiple groups of data, measures the conditional density characteristics of the quantile intervals, and generates the tail conditional density values;

[0009] The tail extreme value fitting module selects the tail data interval with higher volatility based on the tail conditional density values, analyzes the return change trend of the interval data, calculates the local extreme values of the interval data, fits the tail characteristics and the return change data, and generates the return extreme value distribution parameters;

[0010] The dynamic risk assessment module analyzes the tail return changes in the time series based on the return extreme value distribution parameters, calculates the tail return offset rate, adjusts the tail return risk model to reflect market changes, and generates the tail return offset rate;

[0011] The risk warning and quantification module analyzes the risk data of the current tail return interval based on the tail return offset rate, calculates the interval range that triggers the market risk, obtains the market tail risk characteristics, and generates the market risk trigger value.

[0012] The tail risk marker values specifically include the time markers of the tail risk set, the volatility, the dispersion of tail data, and the extreme return data points. The tail conditional density values include the conditional density characteristics, the distribution density, and the volatility of the quantile intervals. The return extreme value distribution parameters specifically refer to the volatility, the local extreme values, and the tail characteristics and return change data of the tail data interval. The tail return offset rate includes the tail return changes in the time series and the tail return offset rate value. The market risk trigger value specifically refers to the interval range that triggers the market risk and the market tail risk characteristics.

[0013] As a further solution of the present invention, the steps for obtaining the tail risk marker values are specifically as follows:

[0014] Extract the tail values and the corresponding market time points according to the return time series, calculate the fluctuation range of the tail return values, screen the data points with significant tail characteristics by matching the correlation between the tail return values and the market time points, and generate the market data set with significant tail characteristics;

[0015] Calculate the volatility of the tail return value based on the market data set with significant tail characteristics, combine the degree of dispersion of the tail return data associated with the time point, evaluate the tail risk through the ratio of volatility to the dispersion amount, record the time mark of the significant tail characteristic data, and establish a tail risk set;

[0016] Judge the extreme return characteristics of the data points that meet the tail characteristics in the tail risk set, calculate through the comprehensive influence of the deviation degree of the tail return value and the volatility, and use the formula:

[0017]

[0018] Generate a tail risk marker value;

[0019] Among them, T r represents the tail risk marker value, x i represents the tail return value, μ represents the mean of the tail return value, γ represents the weight coefficient for adjusting extreme values, σ 2 represents the return volatility, β represents the weight factor of the tail data, and n represents the number of tail data.

[0020] As a further solution of the present invention, the steps for obtaining the tail conditional density value are specifically as follows:

[0021] Based on the tail risk marker value, divide the tail data into multiple quantile intervals, extract the return values of each quantile interval by setting the return value range of the quantile interval, classify the data points inside the quantile interval according to the distribution characteristics of the return values, and combine the statistical characteristics of each interval to generate a set of return values for the quantile interval;

[0022] According to the set of return values of the quantile interval, calculate the volatility of the data in the multi - quantile interval, analyze and statistically calculate its distribution density through the range of return value fluctuations and the degree of data distribution concentration, and calculate and generate the conditional density characteristic value of the quantile interval based on the synergistic relationship between the volatility and the distribution density;

[0023] According to the conditional density characteristic value of the quantile interval, combine the tail characteristics to correct the density weight, and based on the data discreteness and concentration distribution characteristics, use the formula:

[0024]

[0025] Comprehensively calculate and generate the tail conditional density value;

[0026] Among them, C d represents the tail conditional density value, x i represents the return value within the quantile interval, μ i represents the mean of the return values within the quantile interval, γ iThe discrete adjustment factor for the quantile interval, n i represents the number of data points within the quantile interval, σ i is the volatility of the return value within the quantile interval, w j is the weight factor of the distribution density, ρ j is the distribution density of the quantile interval, n is the number of quantile intervals, and m is the calculation dimension of the distribution density.

[0027] As a further solution of the present invention, the steps for obtaining the return extreme value distribution parameters are specifically as follows:

[0028] Based on the tail conditional density values, select the tail data intervals with higher volatility. By calculating the interval volatility and sorting, screen the data intervals with volatility higher than the target threshold. For the screened intervals, extract the return value range by statistically analyzing the characteristics of the interval data points, and establish a set of tail data intervals with higher volatility;

[0029] According to the set of tail data intervals with higher volatility, analyze the return change trend of the interval data. By calculating the return change amplitude and change rate of the interval data, judge the data intervals with significant return change amplitude, and combine the cumulative characteristics of the change trend and the significance of the change rate to screen the data that conforms to the return change trend, and generate the return change trend characteristic value of the interval data;

[0030] According to the return change trend characteristic value of the interval data, combine the local volatility characteristics to calculate the local extreme value of the interval data. Fit the tail characteristics and return change data based on the return change trend characteristic within the interval, and use the formula:

[0031]

[0032] Generate the return extreme value distribution parameters;

[0033] Among them, P e represents the return extreme value distribution parameter, x i represents the return value within the interval, t represents time, represents the change rate of the return value with respect to time, w i is the weight factor of the change rate, β is the local extreme value adjustment factor, μ is the mean value of the return value, γ is the volatility characteristic correction parameter, ρ j is the density parameter of the return change trend, n is the number of interval data points, and m is the calculation dimension of the return change trend.

[0034] As a further solution of the present invention, the steps for obtaining the tail return offset rate are specifically as follows:

[0035] Analyze the tail return changes in the time series based on the profit extreme value distribution parameters. By calculating the continuous differences between the tail return values in the time series point by point, determine the direction of return change by judging the positive or negative nature of the differences, and combine the absolute value of the continuous differences to statistically analyze the change amplitude characteristics to generate a tail return change sequence;

[0036] Calculate the offset amplitude of the tail return according to the tail return change sequence. By analyzing the continuous change rate between return points, combining the statistical characteristics of the offset amplitude, judge the cumulative characteristics of the change rate at different time points, extract the change trend and offset characteristics of the tail return points, and generate tail return change offset parameters;

[0037] Adjust the tail return risk model according to the tail return change offset parameters. By comprehensively considering the change trend characteristics of the tail return points and the market volatility characteristics, use the formula:

[0038]

[0039] Calculate and generate the tail return offset rate;

[0040] where, R s represents the tail return offset rate, x i represents the tail return value, t represents time, is the change rate of the return value with respect to time, w i is the weight parameter of the return point, ρ j is the market volatility characteristic parameter, α is the tail adjustment coefficient, σ j is the volatility of the tail return, n is the total number of return change points, and m is the dimension of the volatility characteristic parameter.

[0041] As a further solution of the present invention, the specific steps for obtaining the market risk trigger value are as follows:

[0042] Analyze the risk data of the current tail return interval based on the tail return offset rate. By detecting the offset amplitude and the time series change trend within the tail return interval, combining the interval return and the market change correlation parameter, judge the risk sensitivity of the tail return and extract its risk characteristics to generate tail return risk data characteristics;

[0043] Calculate the interval range that triggers the market risk according to the tail return risk data characteristics. By statistically analyzing the fluctuation amplitude and distribution density of the tail return risk data, combining the risk trigger parameters and the return offset rate within different intervals, judge the tail interval range with significant risk characteristics to generate a market risk trigger interval;

[0044] Combine the data of the market risk trigger interval to obtain the market tail risk characteristics. By quantitatively analyzing the sensitive parameters of the tail return, use the formula:

[0045]

[0046] Calculate and generate the market risk trigger value;

[0047] Among them, T v represents the market risk trigger value, r i represents the return rate of each tail risk interval, λ is the risk attenuation factor, t i represents the time point in the time series, θ is the adjustment threshold, σ j is the volatility of each trigger interval, n is the total number of risk intervals, and m is the number of calculated trigger intervals.

[0048] Compared with the prior art, the advantages and positive effects of the present invention are as follows:

[0049] In the present invention, by extracting the extreme return points and market data with significant tail characteristics in the time series data, the accurate capture ability of tail risk events is improved. Combining the quantile interval division of tail data and conditional density calculation, the distribution analysis of tail risk is refined, and the quantification ability of risk distribution characteristics is enhanced. Further combining the calculation of local extrema and the dynamic analysis of the tail return offset rate, the real-time adjustment of the risk model is realized, and the sensitivity and adaptability of risk assessment are improved. By accurately calculating the interval range of triggering market risk and tail risk characteristics, the accuracy and pertinence of risk warning are strengthened, providing comprehensive support for risk control in a complex market environment. Brief Description of the Drawings

[0050] Figure 1 is the system flow chart of the present invention;

[0051] Figure 2 is the flow chart of the steps for obtaining the tail risk marked value of the present invention;

[0052] Figure 3 is the flow chart of the steps for obtaining the tail conditional density value of the present invention;

[0053] Figure 4 is the flow chart of the steps for obtaining the distribution parameters of the return extreme value of the present invention;

[0054] Figure 5 is the flow chart of the steps for obtaining the tail return offset rate of the present invention;

[0055] Figure 6 is the flow chart of the steps for obtaining the market risk trigger value of the present invention. Detailed Embodiment

[0056] To make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only for explaining the present invention and are not used to limit the present invention.

[0057] In the description of the present invention, it should be understood that the orientation or positional relationships indicated by the terms "length", "width", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc. are based on the orientation or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore should not be construed as limiting the present invention. In addition, in the description of the present invention, the meaning of "a plurality of" is two or more unless otherwise specifically defined.

[0058] Embodiment 1

[0059] Please refer to Figure 1 , a risk management system based on probability theory and mathematical statistics includes:

[0060] The data preprocessing module extracts the profit tail values and their corresponding market time points based on the time series, screens the market data with significant tail characteristics, records the time markers of the tail risk set, calculates the volatility and the tail data dispersion, marks the extreme profit data points that meet the tail characteristics, and generates the tail risk marker values;

[0061] The tail distribution estimation module divides the tail data into multiple quantile intervals based on the tail risk marker values, extracts the profit values of each quantile interval, calculates the volatility and the distribution density of multiple groups of data, measures the conditional density characteristics of the quantile intervals, and generates the tail conditional density values;

[0062] The tail extreme value fitting module selects the tail data interval with higher volatility based on the tail conditional density values, analyzes the profit change trend of the interval data, calculates the local extreme values of the interval data, fits the tail characteristics and the profit change data, and generates the profit extreme value distribution parameters;

[0063] The dynamic risk assessment module analyzes the tail profit changes in the time series based on the profit extreme value distribution parameters, calculates the tail profit deviation rate, adjusts the tail profit risk model to reflect market changes, and generates the tail profit deviation rate;

[0064] The risk warning and quantification module analyzes the risk data of the current tail profit interval based on the tail profit deviation rate, calculates the interval range for triggering market risks, obtains the market tail risk characteristics, and generates the market risk trigger value.

[0065] The specific values of the tail risk markers are the time markers, volatilities, tail data dispersion amounts, and extreme return data points of the tail risk set. The tail conditional density values include the conditional density characteristics, distribution density, and volatility of the quantile interval. The return extreme value distribution parameters specifically refer to the volatility, local extreme values, tail characteristics, and return change data in the tail data interval. The tail return offset rate includes the tail return changes and tail return offset rate values in the time series. The market risk trigger value specifically refers to the range of the interval that triggers market risk and the market tail risk characteristics.

[0066] Please refer to Figure 2 , and the specific steps for obtaining the tail risk marker values are as follows:

[0067] Extract the tail values and corresponding market time points from the return time series, calculate the fluctuation range of the tail return values, and by matching the correlation between the tail return values and the market time points, screen the data points with significant tail characteristics to generate a market data set with significant tail characteristics;

[0068] First, calculate for all data points in the return time series according to the fluctuation range, and use the formula V i = x i - μ| to calculate the deviation value of each return point, where x i is the return value and μ is the sequence mean. Then, based on the calculated deviation values, screen the data points with deviation values greater than a certain dynamic threshold, associate the time of these data points with the return values, and record them as the preliminary tail data point set. After that, perform a correlation analysis on the time series of the preliminary tail data point set, calculate the time interval distribution between these time points, and extract the data points in the extreme interval to generate a market data set with significant tail characteristics.

[0069] According to the market data set with significant tail characteristics, calculate the volatility of the tail return values, combine the dispersion degree of the tail return data associated with the time points, evaluate the tail risk through the ratio of volatility to dispersion amount, record the time markers of the data with significant tail characteristics, and establish a tail risk set;

[0070] Use the formula to calculate the volatility, where x i is the tail return value, μ is the tail return mean, and n is the number of tail data points. At the same time, calculate the dispersion degree of the tail return values associated with the time points, and use the formula for calculation, where max(x i ) and min(x i ) are the maximum and minimum values in the tail data set respectively. Evaluate the tail risk level through the ratio of volatility to dispersion amount, record the time markers of the data points with significant tail risk levels, and establish a tail risk set.

[0071] Judge the extreme return characteristics of the data points that meet the tail characteristics in the tail risk set, and calculate through the comprehensive influence of the deviation degree of the tail return value and the volatility. Use the formula:

[0072]

[0073] Generate the tail risk marker value;

[0074] Among them, T r represents the tail risk marker value, x i represents the tail return value, μ represents the mean value of the tail return value, γ represents the weight coefficient for adjusting extreme values, σ 2 represents the return volatility, β represents the weight factor of the tail data, and n represents the number of tail data.

[0075] Formula:

[0076]

[0077] The advantage of the formula is that by adding the absolute value term of the deviation of the tail return value from the mean and the weight parameter, it can more accurately capture the tail risk characteristics, and combined with the weight adjustment of the volatility and the number of data points, it balances the influence of extreme values, and at the same time ensures that the calculation result of the tail risk marker value is more stable and accurate.

[0078] Detailed explanation of the formula and the derivation process of the formula calculation:

[0079] First, calculate the mean μ and volatility σ of the tail return value 2 , use the formula to calculate the mean, use the formula to calculate the volatility. After that, calculate the absolute value of the deviation of each data point from the mean and sum them up, using the formula for calculation, where γ is the adjustment coefficient used to enhance or weaken the influence weight of extreme values. Finally, calculate the tail risk marker value through the formula where β is the weight factor of the tail data and n is the number of tail data points. Assume n = 20, μ = 50, σ 2 = 15, γ = 2, β = 0.5, and substitute them into the formula:

[0080]

[0081] Assume the tail data is x = [70, 60, 45, 30,...], and calculate step by step:

[0082] 1. Calculate Each value is [400, 100, 25, 400,...], and the sum is 4500;

[0083] 2. Calculate the denominator σ2 +β·n = 15 + 0.5·20 = 25;

[0084] 3. Finally, calculate

[0085] The result shows that the tail risk marker value is 180. Combining with the distribution of data in the tail risk set, this value indicates a relatively high extreme volatility of tail returns, which matches the significant characteristics of the tail risk set and can be used for further tail risk decision-making analysis.

[0086] Please refer to Figure 3 , the steps to obtain the tail conditional density value are specifically as follows:

[0087] Based on the tail risk marker value, divide the tail data into multiple quantile intervals, extract the return values of each quantile interval by setting the return value range of the quantile interval, classify the data points within the quantile interval according to the distribution characteristics of the return values, and combine the statistical characteristics of each interval to generate a set of return values for the quantile interval;

[0088] Determine the boundary values of the quantile interval according to the range of the tail risk marker value, count the return value range within the quantile interval, exclude the interference values of non-tail data by setting the upper and lower boundaries of the return value, calculate the distribution of the return values for the data points within the quantile interval, generate a set of return values for each quantile interval, extract the extreme values and average values of the return values of the quantile interval based on the statistical distribution of the return values of the quantile interval as the return value characteristics of this quantile interval, and finally perform normalization processing on the return value characteristics of all quantile intervals to generate a set of return values for the quantile interval.

[0089] According to the set of return values of the quantile interval, calculate the volatility of the data within the multiple quantile intervals, analyze the distribution density by analyzing the volatility range of the return values and the degree of data concentration, and calculate the conditional density eigenvalue of the quantile interval based on the synergistic relationship between the volatility and the distribution density;

[0090] Calculate the volatility for each quantile interval, using the standard deviation formula of the return value Calculate the fluctuation range of the quantile interval, determine the degree of dispersion of the return values of the quantile interval in combination with the magnitude of the volatility, count the frequency distribution of the return values within the quantile interval, use the frequency distribution density as the distribution density characteristic of the quantile interval, analyze the overall characteristics of the return values of the quantile interval through the proportional relationship between the volatility and the distribution density, and calculate and obtain the conditional density eigenvalue of each quantile interval.

[0091] According to the conditional density eigenvalue of the quantile interval, perform density weight correction in combination with the tail characteristics. Based on the distribution characteristics of data discreteness and concentration, use the formula:

[0092]

[0093] Generate the tail conditional density value through comprehensive calculation;

[0094] Among them, C d represents the tail conditional density value, x i represents the return value within the quantile interval, μ i represents the mean of the return values within the quantile interval, γ i is the discrete adjustment factor of the quantile interval, n i represents the number of data points within the quantile interval, σ i is the volatility of the return value within the quantile interval, w j is the weight factor of the distribution density, ρ j is the distribution density of the quantile interval, n is the number of quantile intervals, and m is the calculation dimension of the distribution density.

[0095] Formula:

[0096]

[0097] The benefit of the formula is that by introducing multiple parameters such as the return value deviation, volatility, and distribution density within the quantile interval into the calculation model, it comprehensively considers the discreteness, central tendency of the return value, and its distribution characteristics in the tail data, thereby more accurately measuring the conditional density of the tail data and improving the accuracy and granularity of tail data analysis.

[0098] Detailed explanation of the formula and the derivation process of formula calculation:

[0099] Substitute the following parameter values for calculation:

[0100] x i : The return values within the quantile interval are [12, 15, 20, 10, 18], corresponding to 5 data points respectively;

[0101] μ i : The mean of the return values within the quantile interval is μ i = 15, obtained through the formula obtained;

[0102] γ i : The discrete adjustment factor is set to 2, varying in direct proportion to the discreteness of the return value;

[0103] n i : The number of data points within the quantile interval is 5;

[0104] σ i : The volatility of the return values within the quantile interval is Calculated to get σ i = 3.74;

[0105] wj : The weight factor of the distribution density is set to 1 and varies in direct proportion to the weight of the tail characteristic data;

[0106] ρ j : The distribution density within the quantile interval is 0.2, which is obtained through frequency distribution calculation;

[0107] n: The number of quantile intervals is 1;

[0108] m: The dimension of distribution density calculation is 1;

[0109] Calculation steps:

[0110] 1. Calculate the numerator:

[0111]

[0112] Calculate the denominator:

[0113]

[0114] Calculate C d :

[0115]

[0116] This result indicates that the tail conditional density value C d = 12.05 means that within the current tail data quantile interval, the discreteness of the return value is relatively large, and the synergistic effect of volatility and concentrated distribution is relatively low. Through this value, the importance and conditional probability characteristics of the tail data in the overall data distribution can be further analyzed, providing a quantitative basis for the risk assessment of the tail data.

[0117] Please refer to Figure 4 , and the specific steps for obtaining the parameters of the extreme return distribution are as follows:

[0118] Based on the tail conditional density value, select the tail data interval with higher volatility. By calculating the interval volatility and sorting, filter out the data intervals with volatility higher than the target threshold. For the filtered intervals, extract the return value range by statistically analyzing the characteristics of the interval data points, and establish a set of tail data intervals with higher volatility;

[0119] When screening for intervals with high volatility by analyzing the tail conditional density values, the volatility values of all tail intervals are sorted in ascending order. By using the statistical characteristics of volatility, determine which intervals have volatility values exceeding the 80th percentile of the entire interval. These intervals are regarded as intervals with relatively high volatility. Subsequently, analyze the distribution of return values within these intervals. Combining the time stamps of the return values, statistically calculate the extreme value ranges of the return values within each interval. Mark the part where the return value exceeds twice the standard deviation of the interval mean as the significantly volatile return interval. Combine these marks to generate a preliminary range of intervals with relatively high volatility; further correct these interval ranges. Introduce the local volatility change characteristics of the time series, divide the return values within the interval into windows, recalculate the volatility within each window, and screen for windows where the local volatility exceeds the median. Through the above screening and correction, establish a set of tail data intervals with relatively high volatility.

[0120] Based on the set of tail data intervals with relatively high volatility, analyze the return change trends of the interval data. By calculating the return change amplitude and change rate of the interval data, determine the data intervals with significant return change amplitudes. Combine the cumulative characteristics of the change trend and the significance of the change rate to screen for data that conforms to the return change trend, and generate the return change trend characteristic values of the interval data;

[0121] By calculating the cumulative change amplitude of the return change for each interval, use the formula for the change rate of the return value over time to calculate the change rate for each time period. The formula for the change rate is where Δx i represents the return difference value between adjacent time periods, and Δt represents the time interval between adjacent time periods. Based on the calculation results, screen for time periods with significant change rates, accumulate the return values within these time periods, and analyze the upward and downward trends of the return value changes through the directional changes of the cumulative change trend. Classify the intervals with clear return change trend directions as significantly trending intervals. Combine the continuous changes in the return direction to fit the return change trend, and further combine the time periodic change trend of the return data within the interval. Analyze the local periodic change trend of the return through the method of cycle division, and generate the return change trend characteristic values of the interval data.

[0122] According to the return change trend characteristic values of the interval data, combine the local volatility characteristics to calculate the local extreme values of the interval data. Fit the tail characteristics and return change data based on the return change trend characteristics within the interval, using the formula:

[0123]

[0124] Generate the return extreme value distribution parameters;

[0125] where, P e represents the return extreme value distribution parameter, and x iRepresents the revenue value within the interval, and t represents time. Represents the rate of change of the revenue value with respect to time, w i Is the weight factor of the rate of change, β is the local extreme value adjustment factor, μ is the mean value of the revenue value, γ is the volatility characteristic correction parameter, ρ j Is the density parameter of the revenue change trend, n is the number of interval data points, and m is the calculation dimension of the revenue change trend.

[0126] Formula:

[0127]

[0128] The advantage of the formula is that it dynamically corrects the local extreme values by introducing the rate of change of revenue and the volatility adjustment factor, and at the same time optimizes the calculation of local characteristics by combining the density weights of the revenue distribution.

[0129] Detailed explanation of the formula and the derivation process of the formula calculation:

[0130] Assume that there are 5 data points in a certain interval, and their revenue values are 10, 12, 8, 15, and 9 respectively, and the time interval is 1 unit of time. First, calculate the rate of change of revenue The calculation results are 2, -4, 7, -6 respectively. Then, calculate the weight factor of the rate of change according to the formula Set the weight factor w i Allocated according to the proportion of the magnitude of the rate of change, which are 0.2, 0.15, 0.3, 0.2, and 0.15 respectively. The rate of change after weight adjustment is calculated to be 0.4, -0.6, 2.1, -1.2. Subsequently, combined with the local adjustment factor β = 1.5 and the local mean μ = 10.8, calculate the weighted sum of the revenue deviation values, calculate the volatility characteristic correction parameter γ = 2, and the density weight parameter ρ j Sampled from the periodic distribution density result and normalized to 0.25, the result is obtained:

[0131]

[0132] This result shows that the revenue extreme value distribution parameter P e Is optimized by combining the local rate of change and volatility characteristics. The result value reflects the dynamic distribution characteristics of revenue extreme values, and has a strong correlation with the change trend and volatility adjustment of interval revenue. Further through P e Can be used to analyze the extreme value characteristics and distribution status of revenue.

[0133] Please refer to Figure 5 , and the specific steps for obtaining the tail revenue offset rate are as follows:

[0134] Analyze the tail return changes in the time series based on the parameters of the extreme value distribution of returns. By calculating the continuous differences between the tail return values in the time series point by point, determine the direction of return change by judging the positive or negative nature of the differences, and combine the statistical characteristics of the change amplitude of the absolute values of the continuous differences to generate a tail return change sequence;

[0135] Extract every two adjacent data points of the tail return time series, calculate their differences and record the signs of the differences to judge the direction of return change. Determine the change amplitude characteristics by the absolute values of the differences. Then, screen out the significantly changing return points by setting an amplitude threshold, so as to divide the tail return points into positive change points, negative change points and no change points. Combine the cumulative characteristics of the time intervals and change amplitudes of the return points to form the tail return change trend. By cumulatively summing the directionality and change values of the significantly changing points, generate the change amplitude sequence of the tail return points; To clarify the overall distribution characteristics of the tail return changes, calculate the mean and standard deviation in the change amplitude sequence, further classify the significantly changing points, respectively count the proportions of each type of points, and record the characteristics of their corresponding time points. Form a time series of the tail return change distribution through the classification results to generate a tail return change sequence.

[0136] Calculate the offset amplitude of the tail return according to the tail return change sequence. By analyzing the continuous change rate between the return points, combine the statistical characteristics of the offset amplitude, judge the cumulative characteristics of the change rate at different time points, extract the change trend and offset characteristics of the tail return points, and generate the tail return change offset parameter;

[0137] For each return change point in the time series, extract its change amplitude and change time interval, calculate the point-to-point change rate, identify abnormal change points through the analysis of the continuity of the rate, mark the points with a rate change rate exceeding one standard deviation of the mean as significantly offset points, and at the same time count the rate change characteristics of all significantly offset points. By extracting the points with significant rate change amplitudes and their offset trends in the time periods before and after, combine the mean and standard deviation of the cumulative change amplitude to screen out the return points with prominent offset trends, calculate the cumulative offset value of the tail return through the time-weighted mean, and quantify the overall offset amplitude in combination with the statistical characteristics of the significantly offset points to generate the tail return change offset parameter.

[0138] Adjust the tail return risk model according to the tail return change offset parameter. By comprehensively considering the change trend characteristics of the tail return points and the market volatility characteristics, use the formula:

[0139]

[0140] Calculate and generate the tail return offset rate;

[0141] Among them, R s represents the tail return offset rate, x irepresents the tail revenue value, t represents time, is the rate of change of the revenue value with respect to time, w i is the weight parameter of the revenue point, ρ j is the market volatility characteristic parameter, α is the tail adjustment coefficient, σ j is the volatility of the tail revenue, n is the total number of revenue change points, and m is the dimension of the volatility characteristic parameter.

[0142] Formula:

[0143]

[0144] The benefit of the formula is that by integrating the revenue change rate and the tail revenue volatility characteristics in the time series, the change rate of the tail revenue is calculated in a weighted form, combined with the market volatility characteristics and the tail adjustment coefficient, effectively reflecting the overall change trend and risk characteristics of the tail revenue.

[0145] Detailed explanation of the formula and the derivation process of the formula calculation:

[0146] 1. Parameter acquisition:

[0147] x i represents the i-th tail revenue value in the time series, obtained by extracting each point of the tail revenue time series;

[0148] t represents the time point, and the time stamp corresponding to each revenue point in the time series is extracted;

[0149] represents the rate of change of the revenue value with respect to time, through the formula Calculated;

[0150] w i is the weight parameter of the revenue point, set according to the significance of the revenue point (for example, points with a change amplitude exceeding the mean standard deviation are given higher weights), and the specific weight value is calculated by statistically significant point ratios;

[0151] ρ j is the market volatility characteristic parameter, obtained by calculating the volatility of the tail revenue in each time period (formula: Obtained;

[0152] α is the tail adjustment coefficient, calculated based on the volatility degree of the tail revenue and the distribution of significant change points;

[0153] σ j is the volatility of the tail revenue, obtained by the standard deviation of the market revenue data;

[0154] n is the total number of revenue change points in the time series, obtained by the length of the tail revenue sequence;

[0155] m is the calculation dimension of the volatility characteristic parameter, and the total number of volatility characteristic values for each period is obtained through time period partitioning.

[0156] Parameter assignment:

[0157] x i : Set the tail profit values in the time series as [3.5, 4.2, 3.8, 5.1, 4.7];

[0158] t: Set the corresponding time points as [1, 2, 3, 4, 5];

[0159] Calculation

[0160]

[0161] Calculate the weight w i : Set through the distribution of significant points. Assume w 1 = 1.5, w 2 = 1, w 3 = 2, w 4 = 1.2;

[0162] ρ j : Calculate through the market volatility. Assume the segmented volatility is [0.8, 1.0, 1.2];

[0163] α: The tail adjustment coefficient is assumed to be set to 0.5 according to the proportion of the distribution of significant points;

[0164] σ j : Assume the tail volatility is [0.4, 0.5, 0.6];

[0165] Formula calculation:

[0166] Calculate the numerator part:

[0167]

[0168] Calculate the denominator part:

[0169]

[0170] Calculation formula:

[0171]

[0172] This result shows that the offset rate of the tail profit is 1.208, which is related to the market volatility characteristics and the weight after tail adjustment, indicating that the current change rate of the tail profit is at a relatively high level in the overall market movement. This result is directly used for parameter correction of the tail risk adjustment model.

[0173] Please refer to Figure 6, the steps for obtaining the market risk trigger value are specifically as follows:

[0174] Analyze the risk data of the current tail income interval based on the tail income offset rate. By detecting the offset amplitude and the change trend of the time series within the tail income interval, combining the interval income and the market change correlation parameter, judge the risk sensitivity of the tail income and extract its risk characteristics to generate the tail income risk data characteristics;

[0175] By extracting the income change value of each time period in the tail income offset rate, detect the income offset situation of adjacent intervals in the time series. Specifically, it includes calculating the absolute value of the income change of each interval to obtain the offset amplitude, combining the change rate of the tail income and the trend characteristics of the time series, analyzing whether the high-offset amplitude area has risk characteristics, judging the sensitive interval by calculating the correlation between the change amplitude of the income offset rate and the tail income volatility, further extracting the sensitivity of the tail income to market changes, combining the market environment parameters of the tail interval, quantifying the risk sensitivity of the tail income, and generating the tail income risk data characteristics through the above quantitative analysis.

[0176] Calculate the interval range that triggers the market risk according to the tail income risk data characteristics. By statistically analyzing the fluctuation amplitude and distribution density of the tail income risk data, combining the risk trigger parameters and income offset rate within the differential interval, judge the tail interval range with significant risk characteristics to generate the market risk trigger interval;

[0177] By statistically analyzing the intervals with large fluctuation amplitudes in the tail income risk data, screen the risk data areas with significant fluctuation characteristics. Specifically, it includes using the fluctuation amplitude values in the tail income risk data to calculate the standard deviation of the income fluctuation amplitude of each interval, combining the income volatility and the risk sensitive parameter, analyzing the fluctuation significance of each interval, delimiting the characteristic range of the high-risk interval by setting the quantitative standard of the risk sensitive parameter, further combining the setting of the risk trigger condition, analyzing the correlation between the tail income fluctuation and the market fluctuation, and generating the market risk trigger interval by integrating the above data range.

[0178] Combine the data of the market risk trigger interval to obtain the market tail risk characteristics. By quantitatively analyzing the sensitive parameters of the tail income, using the formula:

[0179]

[0180] Calculate and generate the market risk trigger value;

[0181] Among them, T v represents the market risk trigger value, r i represents the rate of return of each tail risk interval, λ is the risk attenuation factor, t irepresents the time points in the time series, θ is the adjustment threshold, and σ j is the volatility of each trigger interval, n is the total number of risk intervals, and m is the number of calculated trigger intervals.

[0182] Formula:

[0183]

[0184] The advantage of the formula is that by introducing the risk decay factor λ, it dynamically reflects the time decay characteristics of the yield within the risk interval, and combines with the volatility parameter σ j to quantify the impact of market volatility on risk, and improve the calculation accuracy and time relevance of the risk trigger value.

[0185] Detailed explanation of the formula and the derivation process of the formula calculation:

[0186] r i is directly obtained through the yield monitoring value in the tail income risk data. Assume that the interval yields are 0.02, 0.03, 0.05, 0.04, 0.01 respectively; t i is the time interval of each time point in the time series, set to 1, 2, 3, 4, 5 days respectively, and is directly extracted through the time mark; λ is the risk decay factor, obtained by fitting the historical market risk data, set to 0.1; σ j is the volatility parameter, calculated through the formula for calculating the income volatility within the interval where x k is the yield sequence, is the average yield. The calculated volatilities of each interval are 0.005, 0.004, 0.006, 0.003 respectively; θ is the adjustment threshold, obtained by fitting the tail interval offset amplitude and volatility, set to 0.01;

[0187] Substitute into the formula:

[0188]

[0189] Calculate step by step:

[0190]

[0191] The result shows that the market risk trigger value is 4.02, which represents the risk trigger level of the current tail income interval. When the risk data exceeds this value, it may trigger market fluctuations.

[0192] The above are only the preferred embodiments of the present invention, and do not limit the present invention in other forms. Any person skilled in the relevant art may use the technical content disclosed above to make changes or modifications into equivalent embodiments with equivalent changes and apply them to other fields. However, as long as it does not depart from the technical solution content of the present invention, any simple modification, equivalent change and modification made to the above embodiments based on the technical essence of the present invention still fall within the protection scope of the technical solution of the present invention.

Claims

1. A risk management system based on probability theory and mathematical statistics, characterized in that: The system comprises: The data preprocessing module extracts the return tail value and its corresponding market time point based on the time series, screens the market data with significant tail characteristics, records the time mark of the tail risk set, calculates the volatility and the discrete amount of the tail data, marks the extreme return data points that meet the tail characteristics, and generates the tail risk mark value; The tail distribution estimation module divides the tail data into multiple quantile intervals based on the tail risk marker value, extracts the return value of each quantile interval, calculates the volatility and distribution density of multiple groups of data, measures the conditional density characteristics of the quantile interval, and generates the tail conditional density value; The tail extreme value fitting module selects the tail data interval with higher volatility based on the tail conditional density value, analyzes the return change trend of the interval data, calculates the local extreme value of the interval data, fits the tail characteristics and the return change data, and generates the return extreme value distribution parameters; The dynamic risk assessment module analyzes the change of tail returns in the time series based on the extreme value distribution parameters of the returns, calculates the tail return deviation rate, adjusts the tail return risk model to reflect market changes, and generates the tail return deviation rate; The risk warning and quantification module analyzes the risk data of the current tail return range based on the tail return deviation rate, calculates the range that triggers market risk, obtains the market tail risk characteristics, and generates a market risk trigger value.

2. The risk management system based on probability theory and mathematical statistics according to claim 1 is characterized in that: The tail risk marking value specifically includes the time mark, volatility, tail data discreteness, and extreme return data points of the tail risk set; the tail conditional density value includes the conditional density characteristics, distribution density, and volatility of the quantile interval; the return extreme value distribution parameters specifically refer to the volatility, local extreme values, tail characteristics and return change data of the tail data interval; the tail return deviation rate includes the tail return changes and tail return deviation rate in the time series; the market risk trigger value specifically refers to the interval range that triggers market risk and the market tail risk characteristics.

3. The risk management system based on probability theory and mathematical statistics according to claim 2 is characterized in that: The steps for obtaining the tail risk mark value are specifically as follows: Extract the tail value and the corresponding market time point according to the return time series, calculate the fluctuation range of the tail return value, and screen the data points with significant tail characteristics by matching the correlation between the tail return value and the market time point to generate a market data set with significant tail characteristics; According to the market data set with significant tail characteristics, the volatility of the tail return value is calculated, and the tail risk is evaluated by the ratio of volatility to discreteness in combination with the discreteness of the tail return data associated with the time point, and the time mark of the significant tail characteristic data is recorded to establish a tail risk set; The extreme return characteristics of the data points in the tail risk set that meet the tail characteristics are judged, and the calculation is performed through the combined influence of the deviation degree of the tail return value and the volatility, using the formula: generating a tail risk marker value; Among them, T r represents the tail risk marker value, x i represents the tail return value, μ represents the mean of the tail return value, γ represents the weight coefficient for adjusting extreme values, σ 2 represents the volatility of returns, β represents the weight factor of tail data, and n represents the number of tail data.

4. The risk management system based on probability theory and mathematical statistics according to claim 3 is characterized in that: The steps for obtaining the tail conditional density value are specifically as follows: Based on the tail risk marker value, the tail data is divided into multiple quantile intervals, the return value of each quantile interval is extracted by setting the return value range of the quantile interval, the data points within the quantile interval are classified according to the return value distribution characteristics, and the return value set of the quantile interval is generated in combination with the statistical characteristics of each interval; According to the set of return values ​​in the quantile interval, the volatility of the data in the multiple quantile intervals is calculated, and the distribution density is statistically analyzed by analyzing the fluctuation range of the return value and the concentration degree of data distribution. According to the synergistic relationship between the volatility and the distribution density, the conditional density characteristic value of the generated quantile interval is calculated; According to the conditional density characteristic value of the quantile interval, the density weight is corrected in combination with the tail characteristics. According to the discreteness and concentration distribution characteristics of the data, the formula is adopted: The tail conditional density value is generated by comprehensive calculation; Among them, C d represents the tail conditional density value, x i Represents the return value within the quantile interval, μ i Represents the mean of the return value within the quantile interval, γ i is the discrete adjustment factor of the quantile interval, n i represents the number of data points in the quantile interval, σ i is the volatility of the return value within the quantile interval, w j is the weight factor of distribution density, ρ j is the distribution density of the quantile interval, n is the number of quantile intervals, and m is the calculation dimension of the distribution density.

5. The risk management system based on probability theory and mathematical statistics according to claim 4 is characterized in that: The steps for obtaining the extreme value distribution parameters of the return are specifically as follows: Based on the tail conditional density value, a tail data interval with a higher volatility is selected, and data intervals with a volatility higher than a target threshold are screened by calculating and sorting the interval volatility. The return value range of the screened intervals is extracted by statistically analyzing the characteristics of the interval data points, and a set of tail data intervals with a higher volatility is established; According to the set of tail data intervals with higher volatility, the return change trend of the interval data is analyzed, and the data interval with significant return change amplitude is determined by calculating the return change amplitude and change rate of the interval data. The data that meets the return change trend is screened by combining the cumulative characteristics of the change trend and the significance of the change rate, and the return change trend characteristic value of the interval data is generated; According to the income change trend characteristic value of the interval data, the local extreme value of the interval data is calculated in combination with the local volatility characteristics, and the tail characteristics and income change data are fitted according to the income change trend characteristics within the interval, using the formula: Generate extreme value distribution parameters of returns; Among them, P e represents the extreme value distribution parameter of returns, x i represents the return value in the interval, t represents time, represents the rate of change of the return value over time, w i is the weight factor of the rate of change, β is the local extreme value adjustment factor, μ is the mean of the return value, γ is the volatility correction parameter, ρ j is the density parameter of the return trend, n is the number of interval data points, and m is the calculation dimension of the return trend.

6. The risk management system based on probability theory and mathematical statistics according to claim 5 is characterized in that: The steps for obtaining the tail return deviation rate are specifically as follows: Based on the extreme value distribution parameters of the returns, the tail return changes in the time series are analyzed, the continuous differences between the tail return values ​​in the time series are calculated point by point, the positive or negative difference is judged to determine the direction of the return change, and the absolute value of the continuous difference is combined with the statistical change amplitude characteristics to generate the tail return change sequence; Calculate the offset amplitude of the tail return according to the tail return change sequence, analyze the continuous change rate between the return points, combine the statistical characteristics of the offset amplitude, judge the cumulative characteristics of the change rate at the differentiated time points, extract the change trend and offset characteristics of the tail return points, and generate the tail return change offset parameter; The tail return risk model is adjusted according to the tail return change offset parameter, and the formula is adopted by combining the change trend characteristics of the tail return point and the market volatility characteristics: Calculate and generate the tail return deviation rate; Among them, R s represents the tail return deviation rate, x i represents the tail return value, t represents time, is the rate of change of the return value over time, w i is the weight parameter of the profit point, ρ j is the market volatility characteristic parameter, α is the tail adjustment coefficient, σ j is the volatility of tail returns, n is the total number of return change points, and m is the dimension of the volatility characteristic parameter.

7. The risk management system based on probability theory and mathematical statistics according to claim 6 is characterized in that: The steps for obtaining the market risk trigger value are specifically as follows: Based on the tail return deviation rate, the risk data of the current tail return interval is analyzed. By detecting the deviation amplitude and time series change trend in the tail return interval, combined with the interval return and market change correlation parameters, the risk sensitivity of the tail return is determined and its risk characteristics are extracted. Generate tail return risk data characteristics; Calculate the interval range that triggers market risk based on the tail return risk data characteristics, and determine the tail interval range with significant risk characteristics by statistically analyzing the fluctuation range and distribution density of the tail return risk data, combined with the risk trigger parameters and return deviation rate in the differentiated interval, to generate a market risk trigger interval; Combined with the data of the market risk trigger interval, the market tail risk characteristics are obtained, and the sensitive parameters of the tail return are quantitatively analyzed, using the formula: Calculate and generate market risk trigger values; Among them, T v represents the market risk trigger value, r i represents the rate of return of each tail risk interval, λ is the risk attenuation factor, t i represents the time point in the time series, θ is the adjustment threshold, σ j is the volatility of each trigger interval, n is the total number of risk intervals, and m is the number of calculated trigger intervals.

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