Nonlinear Expectation Theory And Application In Risk Measurement

With the development of economy in countries and the increasingly frequent international exchanges,the trading volume and frequency of financial products such as stocks,funds,futures and their derivatives have increased exponentially,which makes the financial relationship more complex between domestic and international.However,due to the uncertainty of financial products,we cannot accurately measure their values at a certain time,which we cannot ignore the risks.Moreover,the impact of financial risks on the economies is continuously expanding and the impact speed is further accelerating in various countries.Therefore,it is greatly significant for the safety and stable growth of economy to construct an appropriate model based on market data and to calculate the corresponding parameters,so that to measure financial risks reasonably and effectively.Early,one of the most famous models for measuring financial risk is markowitz's mean-variance model.Since the mean-variance model can only be measured under strict assumptions,it is not consistent with the real financial market.Subsequently,the VaR model was proposed and applied to risk measurement.VaR model is to use a given degree of confidence to find the quantile of portfolio loss distribution function,which we called it the maximum possible loss.However,the VaR model also has some defects that cannot be ignored.For example,it does not satisfy the sub-additivity,namely,the risk of asset portfolio is not necessarily less than the sum of a single asset,which is contrary to the theory of diversification which is thought diversification can reduce risks in the financial market.Subsequently,Artzner et al.proposed the theory of consistent risk measurement,which effectively solved the problem of measuring diversification in the financial market.Subsequently,a series of risk measurement models,such as CVaR model,ES model,distortion risk measurement model and spectral risk measurement model,is developed rapidly,which satisfying specific market conditions.A more reasonable and effective risk measurement model system was gradually established.However,the financial market is complex and changeable.And the financial risk distribution does not necessarily meet the traditional probabilistic statistical model,which means that the risks we face cannot be measured by the traditional linear expectation theory.Therefore,many scholars actively explore the measurement of financial market's uncertainty under the framework of nonlinear expectation and non-additive measure.Based on the nonlinear expectation theory,this paper studies the risk measurement in financial market.This paper is mainly divided into the following four parts:The first chapter mainly reviews the development of risk measurement theory and nonlinear expectation theory.Firstly,the theory of risk measurement discusses the mean-variance model,VaR model and the variation of VaR model in the framework of linear expectation theory,and then it develops to discuss the properties of risk measurement and its satisfying conditions in the framework of nonlinear expectation.The second chapter mainly introduces the Choquet integral and the premium pricing function under the Choquet integral.This chapter firstly presents the related definitions and properties of capacitors and Choquet integral.Secondly,when the premium pricing function meet the five conditions,it can be proved the existence of reduced distortion function makes the risk premium X can be defined by Choquet integral.Then it is proved when g is a concave distortion function,the risk premium pricing function based on the Choquet integral is a consistency risk measure.The third chapter firstly introduces the generation conditions of g-expectation and condition g-expectation and their related properties.Then we prove that the risk measure constructed by g-expectation is convex risk measure when generator g satisfies convexity.When the generator g satisfies convicity and positive homogeneity,the risk measure constructed by g-expectation is the consistency risk measure.Then it is proved that the use of convex conditions g-expectation can construct dynamically consistent risk measures,which indicates that the consistency risk measures and convex risk measures can be constructed dynamically through g-expectation.So,the importance of g-expectation in risk measures is revealed.In the fourth chapter,the definition of G-expectation and its relative properties are given.It is proved that the G-normal distribution based on G-heat equation is equivalent to the G-normal distribution based on sublinear expectation space.In this paper,the VaR model defined by G-expectation is given,and it is proved to meet the consistency risk measurement under the condition of nonlinear expectationThe fifth chapter mainly summarizes the contents of the front chapters.