發(fā)布時(shí)間：2018-06-27 12:30

  本文選題：違約風(fēng)險(xiǎn) + O-U過程��； 參考：《上海師范大學(xué)》2014年碩士論文

【摘要】：在金融數(shù)學(xué)中,用隨機(jī)控制理論研究最優(yōu)投資問題是一個(gè)重要的研究領(lǐng)域。隨著全球經(jīng)濟(jì)的發(fā)展,投資者及投資機(jī)構(gòu)幾乎每天都面臨著投資決策問題,研究各類模型下的最優(yōu)投資問題變得尤為重要.此外,縱觀當(dāng)今國際金融市場,各種金融產(chǎn)品錯(cuò)綜復(fù)雜,金融市場中的違約風(fēng)險(xiǎn)越來越凸顯并被人們所重視.在這樣的大背景下,研究帶違約風(fēng)險(xiǎn)的最優(yōu)投資問題不僅具有重要的理論意義,而且也具有較強(qiáng)的現(xiàn)實(shí)意義.本文選取O-U(OrnsteinΚ Uhlenbeck)過程來描述股票價(jià)格的波動(dòng),原因在于O-U過程比幾何布朗運(yùn)動(dòng)更符合實(shí)際金融市場.利用最優(yōu)控制理論和粘性解理論,研究了O-U過程下帶違約風(fēng)險(xiǎn)的最優(yōu)投資問題、O-U過程下帶違約傳染的最優(yōu)投資問題、O-U過程下可違約債券的效用無差別定價(jià)問題.本文共分為六章： 第一章,闡述了帶違約風(fēng)險(xiǎn)的最優(yōu)投資問題的選題背景、研究意義、國內(nèi)外發(fā)展現(xiàn)狀以及本文的主要研究內(nèi)容. 第二章,簡單介紹了本文所涉及的基礎(chǔ)知識(shí)：O-U過程、違約過程、最優(yōu)控制與粘性解理論、效用無差別定價(jià)等基本定義與定理. 第三章,假設(shè)投資者選擇的投資產(chǎn)品為：銀行存款,O-U過程下的股票以及可違約債券,建立了可違約條件下的最優(yōu)投資問題的模型.以隨機(jī)最優(yōu)控制理論為基礎(chǔ),選取合適的目標(biāo)函數(shù),利用動(dòng)態(tài)規(guī)劃原理,推導(dǎo)出值函數(shù)所滿足的HJB方程,證明了HJB方程粘性解的存在性.最后,利用滿足正系數(shù)條件的有限差分格式進(jìn)行數(shù)值計(jì)算,并對(duì)所得結(jié)果進(jìn)行了分析討論. 第四章,在第三章的基礎(chǔ)上,假設(shè)投資者購買公司A的股票和公司B的債券,兩個(gè)公司均存在違約風(fēng)險(xiǎn),并且兩公司之間具有違約傳染性.通過違約強(qiáng)度的變化來刻畫違約傳染帶給彼此的影響,從投資者的角度出發(fā)建立了該最優(yōu)投資問題的數(shù)學(xué)模型,然后采用有限差分法進(jìn)行近似求解,并對(duì)數(shù)值結(jié)果和參數(shù)進(jìn)行了分析. 第五章,利用效用無差別定價(jià)方法研究了O-U過程下的可違約債券的定價(jià)問題.假設(shè)在可違約債券的有效期內(nèi)投資者可以動(dòng)態(tài)優(yōu)化自己的投資組合,用O-U過程替代傳統(tǒng)的幾何布朗運(yùn)動(dòng)來刻畫股價(jià)的運(yùn)動(dòng)軌跡,分別在投資者購買和不購買可違約債券的兩種情況下,利用動(dòng)態(tài)規(guī)劃原理推導(dǎo)出值函數(shù)所滿足的HJB方程,得到可違約債券的效用無差別價(jià)格,并對(duì)一些參數(shù)進(jìn)行了數(shù)值分析. 第六章,總結(jié)全文并給出了可進(jìn)一步研究的問題.
[Abstract]:In financial mathematics, it is an important research field to study optimal investment problem with stochastic control theory. With the development of global economy, investors and investment institutions are faced with the problem of investment decision almost every day, so it is very important to study the optimal investment problem under various models. In addition, all kinds of financial products are complicated and the risk of default is becoming more and more prominent in the international financial market. Under this background, the study of optimal investment with default risk is not only of great theoretical significance, but also of practical significance. In this paper, the O-U (Ornstein K Uhlenbeck) process is chosen to describe the volatility of stock price. The reason is that the O-U process is more consistent with the actual financial market than the geometric Brownian motion. The optimal investment problem with default risk under O-U process and the optimal investment problem with default contagion under O-U process are studied by using optimal control theory and viscous solution theory. This paper is divided into six chapters: the first chapter describes the background of the optimal investment with default risk, research significance, domestic and foreign development status and the main research content of this paper. In the second chapter, we briefly introduce the basic definitions and theorems of the basic knowledge in this paper, such as: O-U process, default process, optimal control and viscous solution theory, utility nondifferential pricing and so on. In the third chapter, assuming that the investment products chosen by investors are stocks and defaultable bonds in the process of bank deposit, the model of optimal investment problem under the condition of default is established. Based on the stochastic optimal control theory, the HJB equation satisfied by the value function is derived by selecting the appropriate objective function and using the dynamic programming principle, and the existence of the viscous solution of the HJB equation is proved. Finally, the finite difference scheme satisfying the positive coefficient condition is used to carry out the numerical calculation, and the results obtained are analyzed and discussed. In chapter 4, on the basis of chapter 3, suppose that investors buy the stock of company A and the bond of company B. both companies have default risk, and the two companies have the contagion of default. The influence of default contagion on each other is described by the variation of default intensity. The mathematical model of the optimal investment problem is established from the perspective of investors, and then the finite difference method is used to approximate solve the problem. The numerical results and parameters are analyzed. In chapter 5, the pricing problem of defaultable bonds under O-U process is studied by using utility nondifferential pricing method. Assuming that investors can dynamically optimize their portfolios during the period of validity of defaultable bonds, the O-U process is used instead of the traditional geometric Brownian motion to depict the trajectory of stock price. In the case of investors buying and not buying defaultable bonds, the HJB equation satisfied by the value function is derived by using the dynamic programming principle, and the utility of the defaultable bonds is obtained without difference, and some parameters are analyzed numerically. The sixth chapter summarizes the full text and gives the problems that can be further studied.
【學(xué)位授予單位】：上海師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2014
【分類號(hào)】：F830.91;O211.6

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