一种新的求解互补问题的光滑化算法及在可转债定价中的应用

发布时间：2018-03-28 11:56

本文选题：可转换债券　切入点：互补问题　出处：《山东大学》2013年硕士论文

【摘要】：可转换债券是近年来推出的一种重要的金融工具,现今已成为我国市场中一个不可或缺的融资工具。正确评估可转换债券的价值,对应其发行方合理设计条款,投资方理性投资以及整个可转换债券交易市场的健康发展都有重要意义。本文建立了两个可转换债券定价模型,利用有限差分的思想将其写为互补问题形式,在原对数光滑函数的基础上提出了一种新的求解互补问题的连续算法。在第二章中,首先详细分析了可转换债券的交易条款,一般情况下,金融市场中可转换债券包含转股,赎回和回售三种交易条款,在数学模型上可以表示为可转换债券价值函数在不同基准股票价格和利率水平下的的边界条件。假设市场投资者是理性的,经过适当简化,在Black-Scholse期权定价理论的基础上,建立了一个在不同基准股票价格下反映可转换债券价值的最优化模型,而可转换债券包含的三种交易条款则给出了此优化问题的可行域。进一步,考虑到市场利率并非一成不变,引入Vas icek模型来描述随机利率,构建了在随机利率下带有三种交易条款的可转换债券价值函数双因素模型,与单因素一样,其也是一个带有等式与不等式约束的最优化问题。利用有限差分法的思想,在空间(股票价格与市场利率)水平上对模型进行离散化处理,得到了一个二维平面上的七点差分格式。然后在时间刻度上做类似的处理,将定价模型转化为一个互补问题。第三章,首先总结回顾了求解互补问题的光滑算法,给出了光滑函数定义与经典形式的构造,介绍了光滑牛顿法和非内点连续法,说明了单调线和非单调线搜索算法的执行步骤与常用的Armijo与Wolfe条件。然后,在原有对数光滑函数基础上,借鉴互补松弛思想,提出了一种新的光滑函数,分析了此光滑函数的性质,证明了全局无限次可微性,对于参数μ全局严格单调递减以及关于μ的强制性,关于变量a与b的局部严格单调递增性,以及对原互补函数的收敛性。最后结合非单调Armijo线搜索法,构造了一个求解互补问题的非内点连续算法。证明了算法的适定性。在算法的性能上,本文中所述算法克服其Lu的算法在求解过程中容易陷入局部最优解的局限性,而较之Huang所提出的算法,则在保留其优点的同时进一步提高了算法效率,降低了算法的时间复杂度与空间复杂度。
[Abstract]:Convertible bond is an important financial instrument introduced in recent years, and now it has become an indispensable financing tool in the market of our country. The rational investment of investors and the healthy development of the whole convertible bond market are of great significance. In this paper, two pricing models of convertible bonds are established, which are written as complementary problems by using the idea of finite difference. Based on the original logarithmic smooth function, a new continuous algorithm for solving complementarity problems is proposed. In the second chapter, the trading terms of convertible bonds are analyzed in detail. In general, convertible bonds in the financial market include three kinds of trading terms: equity conversion, redemption and resale. The mathematical model can be expressed as the boundary conditions of the value function of convertible bonds at different benchmark stock prices and interest rates. Assuming that the market investors are rational and simplified properly, on the basis of Black-Scholse option pricing theory, An optimization model for reflecting the value of convertible bonds at different benchmark stock prices is established, and the feasible region of this optimization problem is given by the three trading terms contained in convertible bonds. Considering that the market interest rate is not fixed, the Vas icek model is introduced to describe the stochastic interest rate, and a two-factor model of convertible bond value function with three trading terms under the stochastic interest rate is constructed, which is the same as the single factor model. It is also an optimization problem with equality and inequality constraints. Using the idea of finite difference method, the model is discretized at the spatial level (stock price and market interest rate). In this paper, we obtain a seven point difference scheme on a two dimensional plane, and then do a similar treatment on the time scale, and transform the pricing model into a complementary problem. In the third chapter, the smooth algorithms for solving complementarity problems are summarized and reviewed, the definition of smooth function and the construction of classical form are given, and the smooth Newton method and non-interior point continuous method are introduced. The executive steps of monotone line and non-monotone line search algorithms and the common Armijo and Wolfe conditions are explained. Then, a new smooth function is proposed based on the original logarithmic smooth function and the idea of complementary relaxation. The properties of this smooth function are analyzed, and the global infinity degree differentiability is proved. For the parameter 渭, the global strictly monotone decline and the mandatory for 渭, the locally strictly monotone increment of variables a and b is proved. And the convergence of the original complementary function. Finally, a non-interior point continuous algorithm for solving the complementarity problem is constructed by combining the non-monotone Armijo line search method. The fitness of the algorithm is proved, and the performance of the algorithm is proved. The algorithm in this paper overcomes the limitation that Lu's algorithm is easy to fall into the local optimal solution in the process of solving the problem. Compared with the algorithm proposed by Huang, the algorithm preserves its advantages and further improves the efficiency of the algorithm. The time complexity and space complexity of the algorithm are reduced.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2013
【分类号】：O221;F830.91

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