A Kind Of Optimal Stopping Under Probability Distortion

In the financial markets, the people’s decision-making behavior such as judgment and choice behavior, always face risks and uncertainties. The behavior is always governed by the psychological, which reflects people’s mental activity. So the psychology is the basis of behavioral finance. Many experimental evidences show that people tend to inflate, intentionally or unintentionally, small probabilities. Yaari’s dual theory of choice uses probability distortion (or weighting), as one of the building blocks of a number of modern behavioral economics theories, has also been extensively investigated in the insurance literature.In the cumulative prospect theory, the individual’s risk attitude is divided into four different types:when the probability of an event is relatively large, the investors in the income state reflect an attitude of risk aversion, the investors in the loss of state performance risk preferences. However, when the probability is very small, the attitude tends to risk appetite for the gain and risk aversion for the loss. Due to the shape of the distorting function also reflects the risk attitude of people, we mainly discuss a kind of optimal stopping under probability distortion in this paper. The paper is mainly divided into the following sections:The first chapter introduces the generation, significance and effect on risk appetite, and the research situation of distortion (weighting) function, and the main content of the paper.The second chapter introduces some related concepts including the distortion function, distortion probability, the formulas and related lemmas of optimal stopping under probability distortion.The third chapter describes how the utility function and distortion function affect individual’s risk attitudes, and related conclusions of optimal stopping strategies when the utility function and distortion function have different shapes.The fourth chapter studies optimal stopping of asset selling. Firstly, we give the model of asset selling, then under the assumptions of asset price following a geometric Brownian motion and so on, we discuss optimal stopping strategy when the investors sell an asset under the condition of probability distortion. And we obtain the optimal stopping to make the investors determine the best selling time and gain the maximum benefit.The last chapter gives a summary of the paper and some prospects.