Author: Peng, Jing
Title: A study of two behavioral finance models
Advisors: Xu, Zuoquan (AMA)
Degree: Ph.D.
Year: 2022
Subject: Finance -- Decision making
Finance -- Mathematical models
Hong Kong Polytechnic University -- Dissertations
Department: Department of Applied Mathematics
Pages: xvi, 110 pages : color illustrations
Language: English
Abstract: This thesis is concerned with financial models that incorporate components from the burgeoning field of behavioral finance. The goal of behavioral finance is to describe illogical behaviors and anomalies seen in financial markets, as well as to investigate the patterns that emerge as people make decisions. I present two specific behavioral financial models: the first one is a portfolio selection problem in continuous time, and the second one is an optimal insurance design problem. These two models have one feature in common: they both deviate from the traditional paradigm that is built on mathematical assumptions like global convexity (concavity) and linear expectation, resulting in the failure of conventional methods. To tackle them, I reduce them to quantile optimization problems. Using the relaxation and calculus of variation methods, the optimal solutions to them are derived and explicit results are obtained under specific settings.
I begin this thesis by giving a brief historical overview of portfolio selection as well as a summary of the contributions and organization of the thesis. The storytelling of the two models is connected by a detailed illustration of decision-making theory under uncertainty that provides solid grounds and inspiration for modern behavioral economics. Some important prerequisites are presented in the last section of Chapter 1.
In Chapter 2, I present a return-oriented continuous-time portfolio selection model under the cumulative prospect theory. The model is considered in a standard complete and no-arbitrage market, and it also captures the heuristics and biases that occur during the agent's decision-making process. Benchmark and lower bound constraints are introduced to the model to measure performance and control the downside risk. The problem turns out to be a non-classical stochastic control problem, which can be addressed by solving a corresponding quantile optimization problem. The procedure heavily depends on the concept of quantile, which has long been used in nonlinear, nonadditive measures. The problem is converted to a locally concavified optimization problem using the relaxation method, and an optimal solution is derived. The last part of this chapter focuses on deriving the optimal portfolio, which boils down to solving a related partial differential equation (PDE). In particular, explicit expressions are obtained under the Black-Scholes setting.
In Chapter 3, I present an optimal insurance problem where the risk preference of the insured is characterized by the rank-dependent utility theory (RDUT) and the premium principle is based on Wang's class of premium principle. It is required that the insurance policy should not cause an issue of moral hazard, which means both the compensation and retention functions are non-decreasing with respect to the loss. The problem is converted to an equivalent quantile optimization problem. Using the calculus of variation method, the optimal solution is expressed via the solution of an ordinary integral-differential equation (OIDE). A numerical example is provided as well.
This thesis ends up with some concluding remarks and expectations for future work.
Rights: All rights reserved
Access: open access

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