随机分析在复杂网络和金融中的应用研究

发布时间：2018-06-23 05:03

  本文选题：鞅 + 测度变换　； 参考：《电子科技大学》2015年博士论文

【摘要】：随机分析是研究金融市场中随机利率下的欧式期权定价问题和复杂动态网络同步问题的重要工具。特别是金融市场中的相关问题,由于其内在的随机性,使得随机分析成为重要的研究工具。本文利用随机分析的理论、方法和技巧研究了随机影响下复杂动态网络的同步和随机利率下的欧式期权定价问题,取得了如下研究成果。一.研究了随机Markov切换的扰动复杂网络系统的函数投影同步问题。通过构造合适的Lyapunov-Krasovskii泛函,有效的采用了不等式分析技巧和Ito公式,设计了控制方案,使随机切换拓扑和扰动情况下的复杂动态网络实现了均方意义下的函数投影同步,进一步给出了驱动网络和响应网络几乎必然同步的白适应控制方案。通过数值仿真,说明了本文获得的结论的可行性与有效性。二.在贴现的零息债券的波动率是一个常数的条件下,给出了随机利率下的灾难期权的显式闭形式公式。尽管价差期权己经被广泛的研究,但是几乎没有论文讨论随机利率下的价差期权定价问题。本文给出随机利率下的两个新颖的价差期权定价模型。这研究假设贴现的债券的波动率是时间t的函数而不是一个常数。本文不仅提出一个好的方法去构造随机利率下的价差期权定价模型,而且提供了新的实验台去理解随机利率下的、各式各样的资产定价模型中的期权价格动态。在一些资产定价模型中,讨论了这模型和测度变换的优点。在一些资产定价模型中,这测度变换是有用的。在随机利率下,本文给出价差期权定价公式,一般化的Black-Scholes-Merton期权定价公式,一个交换期权定价公式以及在跳模型下的一个欧式期权定价公式。最后,给出了价差期权的一些敏感性分析。对于标的股票收益连续条件下的价差期权定价公式,运用了规则网格的计算方法。对于标的股票收益不连续条件下的价差期权定价公式,运用了规则网格和Monte Carlo计算方法。通过数值试验和仿真,演示了随机利率是影响期权价格的重要因素。数值试验表明标的资产价格的波动率显著地影响期权的价值。三.给出了随机利率下的三个新颖的一篮子期权定价公式。对于标的股票收益连续条件下的、低维一篮子期权定价问题,给出了非常有效的计算技巧。进一步,这研究也获得了两个新颖的随机利率下的脆弱期权定价公式。
[Abstract]:Stochastic analysis is an important tool to study the European option pricing problem and the synchronization problem of complex dynamic network under the stochastic interest rate in the financial market. Because of its inherent randomness, stochastic analysis has become an important research tool. In this paper, we use the theory, method and technique of stochastic analysis to study the synchronization of complex dynamic networks under random influence and the pricing of European options at random interest rates. The research results are as follows. I. The problem of function projection synchronization for perturbed complex network systems with stochastic Markov switching is studied. By constructing an appropriate Lyapunov-Krasovskii functional, the inequality analysis technique and Ito formula are used effectively, and the control scheme is designed to synchronize the function projection in the mean-square sense of the complex dynamic network with random switching topology and disturbance. Furthermore, the white adaptive control scheme, which is almost necessarily synchronous between the drive network and the response network, is presented. The feasibility and validity of the conclusions obtained in this paper are illustrated by numerical simulation. II. Under the condition that the volatility of the discounted zero interest bond is a constant, the explicit closed form formula of disaster option at random interest rate is given. Although spread options have been widely studied, there are almost no papers to discuss the pricing of spread options at random interest rates. In this paper, we present two novel pricing models of spread options at random interest rates. This study assumes that the volatility of discounted bonds is a function of time t rather than a constant. This paper not only proposes a good method to construct the pricing model of spread options under stochastic interest rate, but also provides a new experimental platform to understand the option price dynamics in various asset pricing models under stochastic interest rate. In some asset pricing models, the advantages of this model and measure transformation are discussed. In some asset pricing models, this measure transformation is useful. Under random interest rate, the pricing formula of spread option, the generalized Black-Scholes-Merton option pricing formula, an exchange option pricing formula and a European option pricing formula under jump model are given in this paper. Finally, some sensitivity analysis of spread option is given. The regular grid method is used to calculate the pricing formula of the spread option under the continuous return of the underlying stock. In this paper, the regular grid and Monte Carlo method are used to calculate the pricing formula of the price difference option under the condition of discontinuous return of the underlying stock. Through numerical experiments and simulations, it is demonstrated that stochastic interest rate is an important factor affecting option price. Numerical tests show that the volatility of underlying asset prices significantly affects the value of options. III. Three novel basket option pricing formulas under stochastic interest rate are given. For the low dimensional basket option pricing problem under the continuous return of underlying stock, a very effective calculation technique is given. Furthermore, two novel pricing formulas of fragile options under stochastic interest rate are obtained.
【学位授予单位】：电子科技大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：F830.9;F224
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本文编号：2055898