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변동성 군집에 관한 수학적 모형에 관한 논고
- Alternative Title
- Essays on mathematical models of volatilities flocking
- Author(s)
- 이상혁
- Alternative Author(s)
- Sang-Hyeok, Lee
- Advisor
- 배형옥
- Department
- 일반대학원 금융공학과
- Publisher
- The Graduate School, Ajou University
- Publication Year
- 2018-02
- Language
- eng
- Keyword
- volatility; flocking; Markov process; Cucker-Smale model; time delay
- Alternative Abstract
- 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.
- Fulltext
- Appears in Collections:
- Graduate School of Ajou University > Department of Financial Engineering > 4. Theses(Ph.D)
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