변동성 군집에 관한 수학적 모형에 관한 논고
Essays on mathematical models of volatilities flocking
- 주제(키워드) volatility , flocking , Markov process , Cucker-Smale model , time delay
- 발행기관 아주대학교
- 지도교수 배형옥
- 발행년도 2018
- 학위수여년월 2018. 2
- 학위명 박사
- 학과 및 전공 일반대학원 금융공학과
- 실제URI http://www.dcollection.net/handler/ajou/000000027572
- 본문언어 영어
- 저작권 아주대학교 논문은 저작권에 의해 보호받습니다.
초록/요약
We introduce a mathematical model for the movement of asset’s volatility in the financial market when a shock is arrived in the market. It consists of three parts: to develop a volatility-flocking model, to include time delay of the communication in the model, and to provide a series scheme for approximating the solutions of the model with time delay. First, we propose a mathematical model of asset return's volatility flocking, which is often observed during recessionary periods or after adverse economic shocks. Our model is based on a multiplicative noise based on Wiener process. In order to understand complex properties of volatilities we apply Cucker-Smale (C-S) flocking mechanism, and regime switching modulated by hidden Markov chain. We analyze the model mathematically, and provide numerical solutions based on several experiments. In obtaining the solutions, we analyze volatilities with its ensemble average and the fluctuation around the average. Secondly, we provide the model with time delay. In this model, time delay affects the ensemble average, not the fluctuation. By the results, we infer that the effect of time delay changes the market's average without changing the individual asset prices. This implication is consistent with the empirical tendency after a financial crisis. When a shock arrives in the market, we analyze the behavior of the ensemble averages, and the relationship between the asymptotic behavior of the averages and the initial data with time delay. Finally, we present a series scheme for time delay dynamics to obtain approximate solutions. Using the asymptotic behavior of the model with time delay, we build the scheme. The scheme provides two advantages with analyzing the solutions: the two dimensional problem is transformed into one dimensional problem; the first term in the series scheme approximates the solution when time delay is sufficiently small.
more목차
1 Introduction 1
1.1 Overview of volatility 1
1.1.1 Stylized facts of volatility 1
1.1.2 Stylized facts of volatility after nancial crisis 3
1.2 Preliminaries 3
1.2.1 The simple Cucker-Smale model 3
1.2.2 The hidden Markov chain 4
2 Volatility flocking with hidden Markov chain 6
2.1 Introduction 6
2.2 Dynamics for volatility flocking: Case without noise 8
2.2.1 Dynamics for volatility flocking 8
2.2.2 The special case: Market having two stock 10
2.3 Dynamics for flocking volatility: Noisy case 13
2.3.1 Ensemble average 16
2.3.2 Fluctuations 21
2.4 Numerical simulation 24
3 A flocking volatility model with time delay and Markov chain 28
3.1 Introduction 28
3.2 A model 29
3.2.1 A model for volatility flocking with time delay 29
3.2.2 The behavior of the ensemble average 30
3.3 Properties of delay di erential equation 31
3.4 Estimation of the ensemble average 45
3.4.1 The case for p00p11 -p01p10 = 0 46
3.4.2 The case for p00p11 - p01p10≠0 47
4 Small delay approximation 50
4.1 Introduction 50
4.1.1 Motivation 50
4.1.2 Main result 51
4.2 Series solution with time delay 52
4.2.1 How to build a new series solution 52
5 Conclusion 72
References 74
