本文選題：可轉(zhuǎn)換債券　切入點：互補問題　出處：《山東大學(xué)》2013年碩士論文

【摘要】：可轉(zhuǎn)換債券是近年來推出的一種重要的金融工具,現(xiàn)今已成為我國市場中一個不可或缺的融資工具。正確評估可轉(zhuǎn)換債券的價值,對應(yīng)其發(fā)行方合理設(shè)計條款,投資方理性投資以及整個可轉(zhuǎn)換債券交易市場的健康發(fā)展都有重要意義。本文建立了兩個可轉(zhuǎn)換債券定價模型,利用有限差分的思想將其寫為互補問題形式,在原對數(shù)光滑函數(shù)的基礎(chǔ)上提出了一種新的求解互補問題的連續(xù)算法。 在第二章中,首先詳細(xì)分析了可轉(zhuǎn)換債券的交易條款,一般情況下,金融市場中可轉(zhuǎn)換債券包含轉(zhuǎn)股,贖回和回售三種交易條款,在數(shù)學(xué)模型上可以表示為可轉(zhuǎn)換債券價值函數(shù)在不同基準(zhǔn)股票價格和利率水平下的的邊界條件。假設(shè)市場投資者是理性的,經(jīng)過適當(dāng)簡化,在Black-Scholse期權(quán)定價理論的基礎(chǔ)上,建立了一個在不同基準(zhǔn)股票價格下反映可轉(zhuǎn)換債券價值的最優(yōu)化模型,而可轉(zhuǎn)換債券包含的三種交易條款則給出了此優(yōu)化問題的可行域。進一步,考慮到市場利率并非一成不變,引入Vas icek模型來描述隨機利率,構(gòu)建了在隨機利率下帶有三種交易條款的可轉(zhuǎn)換債券價值函數(shù)雙因素模型,與單因素一樣,其也是一個帶有等式與不等式約束的最優(yōu)化問題。利用有限差分法的思想,在空間(股票價格與市場利率)水平上對模型進行離散化處理,得到了一個二維平面上的七點差分格式。然后在時間刻度上做類似的處理,將定價模型轉(zhuǎn)化為一個互補問題。 第三章,首先總結(jié)回顧了求解互補問題的光滑算法,給出了光滑函數(shù)定義與經(jīng)典形式的構(gòu)造,介紹了光滑牛頓法和非內(nèi)點連續(xù)法,說明了單調(diào)線和非單調(diào)線搜索算法的執(zhí)行步驟與常用的Armijo與Wolfe條件。然后,在原有對數(shù)光滑函數(shù)基礎(chǔ)上,借鑒互補松弛思想,提出了一種新的光滑函數(shù),分析了此光滑函數(shù)的性質(zhì),證明了全局無限次可微性,對于參數(shù)μ全局嚴(yán)格單調(diào)遞減以及關(guān)于μ的強制性,關(guān)于變量a與b的局部嚴(yán)格單調(diào)遞增性,以及對原互補函數(shù)的收斂性。最后結(jié)合非單調(diào)Armijo線搜索法,構(gòu)造了一個求解互補問題的非內(nèi)點連續(xù)算法。證明了算法的適定性。在算法的性能上,本文中所述算法克服其Lu的算法在求解過程中容易陷入局部最優(yōu)解的局限性,而較之Huang所提出的算法,則在保留其優(yōu)點的同時進一步提高了算法效率,降低了算法的時間復(fù)雜度與空間復(fù)雜度。
[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.
【學(xué)位授予單位】：山東大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2013
【分類號】：O221;F830.91

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本文編號：1676222