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Financial Institution Prevention Financial Risk Monitoring System Under the Fusion of Partial Differential Equations

Accepté: 23 Jun 2022
Détails du magazine
Format
Magazine
eISSN
2444-8656
Première parution
01 Jan 2016
Périodicité
2 fois par an
Langues
Anglais
Introduction

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 [1]. 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 ρ are important tools for measuring option risk. It plays a vital role in investors' option investment. The leap of risk measurement method should benefit from the introduction of financial risk measurement under partial differential equations [2]. As an important risk management tool, financial risk measurement under partial differential equations has been applied and promoted by various financial institutions. The financial risk measure M under the partial differential equation describes the maximum possible loss of the value of an asset or portfolio of assets in a specific period of time in the future under a given confidence level. VaR = inf(x | p(x) ≥ β). β is the confidence level. x represents the loss of the portfolio at future time Δt. p(x) is the probability distribution function of x.

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 [3]. 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 $[σ_2,σ¯2]$ \left[{{{\underline \sigma}^2},{{\bar \sigma}^2}} \right] . We prove that financial risk measures under partial differential equations are consistent risk measures. On this basis, we give the analytical formula of the financial risk measurement value under the partial differential equation of a single asset under the loss function f(y) = y. We want to measure risk in financial markets from a fresh perspective. This method can propose a more reasonable measurement method for banks, investment companies and securities regulators. This provides a new class of risk measurement techniques and methods for risk management.

Financial risk measurement under the new risk measurement method partial differential equation

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 [4]. 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.

Proposition of financial risk measurement method under partial differential equation

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. f(y) represents the loss faced by a financial asset. y is the market factor that causes the portfolio to lose value. It is a random variable. Let the distribution function of y be denoted by P(y). For any arbitrary α ∈, let $ψ(α)=∫f(y)≤αdP(y)=E[If(y)≤α]$ \psi \left(\alpha \right) = \int_{f\left(y \right) \le \alpha} {dP\left(y \right) = E\left[{{I_{f\left(y \right) \le \alpha}}} \right]}

ψ(α) is the cumulative distribution function. It is a non-subtracting, right-continuous function of α. For any β ∈ (0, 1), the following equation exists: $αβ=inf{α ∈ : ψ(α)≥β}$ {\alpha _\beta} = \inf \left\{{\alpha \, \in \,:\,\psi \left(\alpha \right) \ge \beta} \right\}

Then αβ is the financial risk measurement value under the partial differential equation with the confidence level of β in a certain period of the portfolio, namely $VaR=αβ$ VaR = {\alpha _\beta}

From (1), it is known that the cumulative distribution ψ(α) can be written in the form of E[If(y)≤α]. This is a classic linear expectation. It cannot detect the risk of financial uncertainty. The nonlinear expectation, especially G- expectation, is completely free from the constraints of linear probability and linear expectation. It is a useful tool for revealing the risks of financial uncertainty [5]. We use G- expectation to construct a consistent risk measure to describe the uncertainty of risk, which is a new way of exploring.

The type of loss function f(y) is uncertain in financial markets. Its convexity or concavity depends on the context in which the financial asset exists. The loss function of financial assets in the financial market has the following relationship with the return of financial assets: x = (x1, ⋯, xn)T is n the feasible set of investment weights for investments, and y = (y1, ⋯, yn)T is the loss that caused the portfolio. Then the loss function f(x, y) can be expressed as $f(x,y)=x1y1+⋯+xnyn$ f\left({x,y} \right) = {x_1}{y_1} + \cdots + {x_n}{y_n}

The loss function f(y) = y, f(y) is a linear function of y. Next, construct the risk measure according to the loss function f(y) = y and using G- expectation. We use it to describe uncertainty risk in financial markets.

Lemma 1

Let ρG: H →, define $ρG(y):=E^[−y]$ {\rho ^G}\left(y \right): = \hat E\left[{- y} \right]

Then ρG(y) is the consistent risk measure. where y is the loss of a single asset.

Its research on risk measurement is of high value. It shows that the risk measure constructed by G- expectancy is a consistent risk measure [6]. 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 α.

Definition 2

An acceptable area B = {yY, y is acceptable if it meets the following conditions, then it is a uniformly acceptable area

y1B if y1y2, then y2B;

0 ∈ B but −1 ∉ B;

yB has λ yB to ∀ λ ≥ 0;

y1, y2B has αy1 + (1 − α) y2B to α ∈ [0,1].

Assuming ψ(α) = Ê [αy], we replace the linear expectation Ê(·) in Eq. (1) with the nonlinear expectation Ê(·). In this way, we define the risk measurement method under the theoretical framework of nonlinear expectation [7]. This explains uncertainty risk in financial markets. Therefore, similar to the definition of financial risk measure under partial differential equation, we give the definition of financial risk measure under single asset partial differential equation in financial market.

Definition 3

Financial risk measures under partial differential equations describe a region B that is consistent and acceptable given an uncertain financial market [8]. The mathematical expression for the maximum possible loss in the value of a financial asset over a specified period of time in the future is as follows: $GVaR(y)=inf{α ∈ : E^[α−y]∈B}$ GVaR\left(y \right) = \inf \left\{{\alpha \, \in \,:\,\hat E\left[{\alpha - y} \right] \in B} \right\}

Among them, y is the asset loss caused by a certain financial asset in a period of time. If a certain risk measurement method is a consistent risk measurement method, it is more in line with the risk measurement of financial markets. In fact, the financial risk measurement under the partial differential equation defined in Definition 3 is a consistent risk measurement method [9]. The following theorem answers this conclusion. Financial risk measures under partial differential equations are consistent risk measures.

y1, y2H. If y1y2, then −y1 < − y2. There is αy1α y2 for any α ∈. Then $GVaR(y1)=inf{α ∈ : E^[α−y1]∈B}≥inf{α ∈ : E^[α−y2]∈B}=GVaR(y2)$ \matrix{{GVaR\left({{y_1}} \right) = \inf \left\{{\alpha \, \in \,:\,\hat E\left[{\alpha - {y_1}} \right] \in B} \right\}} \hfill \cr {\ge \inf \left\{{\alpha \, \in \,:\,\hat E\left[{\alpha - {y_2}} \right] \in B} \right\} = GVaR\left({{y_2}} \right)} \hfill \cr} .

Therefore, the financial risk measurement under the partial differential equation satisfies the monotonicity condition of the consistent risk axiom [10]. The greater the loss, the greater the corresponding risk;

y1, y2H, ∀c ∈. From the normality of nonlinear expectations, it can be known that $GVaR(y1+c)=inf{α ∈ : E^[α−(y1+c)]∈B}=inf{α ∈ : E^[α−y1]−c∈B}=GVaR(y1)+c$ \matrix{{GVaR\left({{y_1} + c} \right) = \inf \left\{{\alpha \, \in \,:\,\hat E\left[{\alpha - \left({{y_1} + c} \right)} \right] \in B} \right\}} \hfill \cr {= \inf \left\{{\alpha \, \in \,:\,\hat E\left[{\alpha - {y_1}} \right] - c \in B} \right\} = GVaR\left({{y_1}} \right) + c} \hfill \cr}

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.

λ > 0. From the normality and positive homogeneity of nonlinear expectations, it can be known that $GVaR(λy1)=inf{α ∈ : E^[α−λy1]∈B}=λinf{γ ∈ : E^[γ−y1]∈B}=λGVaR(y1)$ \matrix{{GVaR\left({\lambda {y_1}} \right) = \inf \left\{{\alpha \, \in \,:\,\hat E\left[{\alpha - \lambda {y_1}} \right] \in B} \right\}} \hfill \cr {= \lambda \inf \left\{{\gamma \, \in \,:\,\hat E\left[{\gamma - {y_1}} \right] \in B} \right\} = \lambda GVaR\left({{y_1}} \right)} \hfill \cr}

Where $γ=αλ>0$ \gamma = {\alpha \over \lambda} > 0 . Therefore, the financial risk measurement under the partial differential equation satisfies the positive homogeneity condition of the consistent risk axiom [11]. It reflects the benefits of not diversifying risk; $∀y1,y2∈HGVaR(y1+y2)=inf{α ∈ : E^[α−(y1+y2)]∈B}=inf{α+γ ∈ : E^[α−y1]+E^[γ−y2]∈B}≤inf{α ∈ : E^[α−y1]∈B}+inf{γ ∈ R: E^[γ−y2]∈B}=GVaR(y1)+GVaR(y2)$ \matrix{{\forall {y_1},{y_2} \in H} \hfill \cr {GVaR\left({{y_1} + {y_2}} \right) = \inf \left\{{\alpha \, \in \,:\,\hat E\left[{\alpha - \left({{y_1} + {y_2}} \right)} \right] \in B} \right\}} \hfill \cr {= \inf \left\{{\alpha + \gamma \, \in \,:\,\hat E\left[{\alpha - {y_1}} \right] + \hat E\left[{\gamma - {y_2}} \right] \in B} \right\}} \hfill \cr {\le \inf \left\{{\alpha \, \in \,:\,\hat E\left[{\alpha - {y_1}} \right] \in B} \right\} + \inf \left\{{\gamma \, \in \,\,R:\,\hat E\left[{\gamma - {y_2}} \right] \in B} \right\}} \hfill \cr {= GVaR\left({{y_1}} \right) + GVaR\left({{y_2}} \right)} \hfill \cr}

Among them γ ∈. Then the financial risk measurement under the partial differential equation satisfies the positive homogeneity condition of the consistent risk axiom [12]. That is, the investment portfolio has the property of risk diversification.

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.

A concrete representation of financial risk measurement under partial differential equations

The loss function of a single financial asset in the financial market can be written as f(y) = y. ∀ α ∈, αf(y) = αy. It is both a concave function of y and a convex function of y.

The loss function f(y) is a concave function of y. g(y) = αf(y) is a convex function of y. Then Ê[αy] has the following expression $E^[α−y]=12πσ¯2∫−∞+∞(α−y)exp(−y22σ¯2)dy12πσ¯2∫−∞+∞αexp(−y22σ¯2)dy+12πσ¯2∫−∞+∞(−y)exp(−y22σ¯2)dy$ \matrix{{\hat E\left[{\alpha - y} \right] = {1 \over {\sqrt {2\pi {{\bar \sigma}^2}}}}\int_{- \infty}^{+ \infty} {\left({\alpha - y} \right)\exp \left({- {{{y^2}} \over {2{{\bar \sigma}^2}}}} \right)dy}} \hfill \cr {{1 \over {\sqrt {2\pi {{\bar \sigma}^2}}}}\int_{- \infty}^{+ \infty} {\alpha \exp \left({- {{{y^2}} \over {2{{\bar \sigma}^2}}}} \right)dy +} {1 \over {\sqrt {2\pi {{\bar \sigma}^2}}}}\int_{- \infty}^{+ \infty} {\left({- y} \right)\exp \left({- {{{y^2}} \over {2{{\bar \sigma}^2}}}} \right)dy}} \hfill \cr}

The asset has formula (8) in the future when the acceptable area is : $inf{α ∈ : 12πσ¯2∫−∞+∞αexp(−y22σ¯2)dy+12πσ¯2∫−∞+∞(−y)exp(−y22σ¯2)dy∈B }$ \inf \left\{{\alpha \, \in \,:\,{1 \over {\sqrt {2\pi {{\bar \sigma}^2}}}}\int_{- \infty}^{+ \infty} {\alpha \exp \left({- {{{y^2}} \over {2{{\bar \sigma}^2}}}} \right)dy +} {1 \over {\sqrt {2\pi {{\bar \sigma}^2}}}}\int_{- \infty}^{+ \infty} {\left({- y} \right)\exp \left({- {{{y^2}} \over {2{{\bar \sigma}^2}}}} \right)dy \in B} \,} \right\}

The above conclusion is the financial risk measurement value under the required partial differential equation [13]. It represents the maximum loss value that the asset can suffer in the future under the acceptable region B.

The loss function f(y) is a convex function of y. g(y) = αf(y) is the concave function of y. Ê[αy] has the following expression $E^[α−y]=12πσ_2∫−∞+∞(α−y)exp(−y22σ_2)dy=12πσ_2∫−∞+∞αexp(−y22σ_2)dy+12πσ_2∫−∞+∞(−y)exp(−y22σ_2)dy$ \matrix{{\hat E\left[{\alpha - y} \right] = {1 \over {\sqrt {2\pi {{\underline \sigma}^2}}}}\int_{- \infty}^{+ \infty} {\left({\alpha - y} \right)\exp \left({- {{{y^2}} \over {2{{\underline \sigma}^2}}}} \right)dy}} \hfill \cr {= {1 \over {\sqrt {2\pi {{\underline \sigma}^2}}}}\int_{- \infty}^{+ \infty} {\alpha \exp \left({- {{{y^2}} \over {2{{\underline \sigma}^2}}}} \right)dy}} \hfill \cr {+ {1 \over {\sqrt {2\pi {{\underline \sigma}^2}}}}\int_{- \infty}^{+ \infty} {\left({- y} \right)\exp \left({- {{{y^2}} \over {2{{\underline \sigma}^2}}}} \right)dy}} \hfill \cr}

Then the asset has formula (10) under the condition that the acceptable area of the asset is B for a period of time in the future. $inf{α ∈ : 12πσ_2∫−∞+∞αexp(−y22σ_2)dy+12πσ_2∫−∞+∞(−y)exp(−y22σ¯2)dy∈B }$ \inf \left\{{\alpha \, \in \,:\,{1 \over {\sqrt {2\pi {{\underline \sigma}^2}}}}\int_{- \infty}^{+ \infty} {\alpha \exp \left({- {{{y^2}} \over {2{{\underline \sigma}^2}}}} \right)dy +} {1 \over {\sqrt {2\pi {{\underline \sigma}^2}}}}\int_{- \infty}^{+ \infty} {\left({- y} \right)\exp \left({- {{{y^2}} \over {2{{\bar \sigma}^2}}}} \right)dy \in B} \,} \right\}

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.

Conclusion

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 GVaR(y) = inf{αÊ[αy] ∈ B} to measure uncertainty risk when the loss function of a financial asset is f(y) = y. We prove by theorem that GVaR(y) is a consistent risk measure. Based on the expression of the one-dimensional G-normal distribution, this paper gives the specific form of financial risk measurement under the partial differential equation of a single asset.

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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