Risk measurement is the core and foundation of risk management. Exploring a measurement method that can scientifically and accurately reflect the financial risk characteristics of financial institutions has always been an important topic of common concern to academic circles and financial regulatory authorities. Markowitz first proposed the mean-variance model in 1952. The model uses variance to describe the risk of returns. It has the characteristics of simple operation and strong adaptability . However, it also has the disadvantages of too strict assumptions, not in line with the financial market, and large amount of calculation. For example, it requires the assumption that there is at least a second moment in the profit and loss distribution. But in economics and finance, profit and loss distributions are usually thick-tailed distributions. It does not have a second-order moment. Subsequently, many scholars have improved the mean-variance model and proposed the mean-absolute variance model, the lower half-variance model and so on. These models are all risk measurement methods based on moment information. Although its financial meaning is simple and clear, the calculation is too complicated and difficult to understand.
In addition, the Greek letters Δ, Γ, Φ, ν and
Some scholars put forward the concept of g- expectation and conditional g- expectation through the backward stochastic differential equation BSDE. In this way, we have established the nonlinear mathematical expectation theory under a certain framework. Some scholars have proposed and proved that g- expectation is a necessary and sufficient condition for compatibility risk measurement. Some scholars have defined risk measure and dynamic risk measure by using condition g-expectation in Backward Stochastic Differential Equation (BSDE) theory.
Scientists have found that g- expectation is a useful tool to measure financial risk in financial markets. However, the risk measure induced by g- expectation needs to satisfy a certain set of probability measures. It can well explain the uncertainty of financial asset changes within a certain range, but it cannot reveal the complete uncertainty in the financial environment. Some scholars have introduced the concepts of G- normal distribution and G- expectation under nonlinear expectation . This model has attracted the attention of the international mathematics and financial economic circles. G- expectation is a nonlinear expectation. The risk measure induced by it satisfies the four conditions of the axiomatic system of consistent risk measure. G- expectations need not be built into a given set of probability measures. We use G- expectation theory to describe the uncertainty of financial risk more essentially. The G- normal distribution has a certain relationship with the classical normal distribution. It is a generalization of the latter under sublinear expectations. Its mean is constant. Its value is
The new Basel Accord clearly stipulates that financial institutions use the financial risk measurement under the partial differential equation to measure the only tool for financial risk. At the same time, it is the mainstream method of risk management at present. Its advantages are obvious. It combines the expected future loss with the probability that that loss will occur. It intuitively and simply converts potential losses in the financial market into a number. This helps managers, investors and financial regulators to accurately grasp the actual risks of financial institutions . But our simple measure of financial risk under partial differential equation relying on fixed probability measure is not enough to be used for risk management consistently. Some scholars put forward the axiom of consistency: if a risk measure satisfies the four conditions of monotonicity, subadditivity, positive homogeneity and translation invariance, then the risk measure is a consistent risk measure. Only a risk measure that satisfies the axiom of consistency can be used as a portfolio management tool if it is consistent with the actual size of the risk. Therefore, the consistent risk axiom has become a famous theory for analyzing financial risk. It is found that g-expectation can construct a consistent risk measure within a certain set of probability measures. This paper adopts the definition of financial risk measure under partial differential equation and uses G-expectation to construct a consistent risk measure.
The financial risk measurement under partial differential equation refers to the maximum loss that a financial asset or asset portfolio may suffer in a certain period of time in the future given a certain confidence level.
From (1), it is known that the cumulative distribution
The type of loss function
The loss function
Its research on risk measurement is of high value. It shows that the risk measure constructed by G- expectancy is a consistent risk measure . Therefore, it is reasonable to study risk measurement methods under the framework of G- expectation. A financial risk measure under a partial differential equation is defined as the quantile at a given confidence level
An acceptable area 0 ∈ ∀
Financial risk measures under partial differential equations describe a region
Therefore, the financial risk measurement under the partial differential equation satisfies the monotonicity condition of the consistent risk axiom . The greater the loss, the greater the corresponding risk;
The financial risk measure under partial differential equation satisfies the translation invariance condition of the consistent risk axiom. Losses increase or decrease by a certain amount, and the corresponding risk increases or decreases by the same amount.
The financial risk measure under the partial differential equation satisfies the four conditions of the consistent risk axiom. Then the financial risk measure under the partial differential equation is a consistent risk measure. The greater the loss of the asset, the greater the risk of the asset. The risk of an asset satisfies translation invariance and positive homogeneity. Assets have the nature of risk diversification.
The loss function of a single financial asset in the financial market can be written as
The loss function
The asset has formula (8) in the future when the acceptable area is :
The above conclusion is the financial risk measurement value under the required partial differential equation . It represents the maximum loss value that the asset can suffer in the future under the acceptable region
The loss function
Then the asset has formula (10) under the condition that the acceptable area of the asset is
It is the measure of financial risk under the desired partial differential equation. It represents the maximum loss value that the asset can suffer in the future under the acceptable region 6.
At present, finding a suitable risk measurement method to interpret the uncertainty risk in the financial market is a hot research topic of scholars. Financial risk measurement and g-expectation under partial differential equations are popular risk measurement methods in the world. It has been widely used in risk management. But the calculation relies on the classical probability measure set, which cannot reveal the uncertainty risk in the financial market. G-expectations are not easy to apply and understand. In this paper, the mathematical expression of financial risk measurement under partial differential equations is deeply studied for the uncertainty risk measurement problem in the financial market. We improve the financial risk measurement under partial differential equations on the basis of clarifying that the financial risk measurement under partial differential equations is not a consistent risk measurement. We propose a new class of risk measurement methods. This research model defines
The financial risk measurement method under partial differential equations proposed in this paper fully considers the uncertainty of financial assets. It satisfies the axiomatic system of consistent risk measures and overcomes the shortcomings of existing methods. This paper measures the risk in the financial market with a new idea. It can provide a more reasonable measurement method for banks, investment companies and securities regulators.
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