本文關(guān)鍵詞： 美式期權(quán) CEV模型 Front-Fixing方法 有限體積法 完全匹配層方法 Newton迭代　出處：《吉林大學(xué)》2014年碩士論文　論文類(lèi)型：學(xué)位論文

【摘要】：本文主要介紹了一種解決美式期權(quán)定價(jià)問(wèn)題的有限體積法,并進(jìn)一步利用該方法解決不變方差彈性(CEV)模型下的美式期權(quán)定價(jià)問(wèn)題。美式期權(quán)問(wèn)題本質(zhì)上是一個(gè)定義在無(wú)界的域上的自由邊界問(wèn)題,因此在應(yīng)用數(shù)值方法解方程時(shí)會(huì)必然有截?cái)嗟倪^(guò)程。本文中,我們?cè)趯⒃瓎?wèn)題整理成線(xiàn)性形式后,先利用Front-Fixing方法將自由邊界變?yōu)橐?guī)則邊界,再利用完全匹配層(PML)方法對(duì)無(wú)界邊取截?cái)�。由于方程中含有未知的自由邊界信�?所以在時(shí)間離散的每一層,我們采用Newton迭代,即在利用有限體積法解方程的同時(shí)通過(guò)Newton迭代得到最優(yōu)的自由邊界值,進(jìn)而得到相應(yīng)的期權(quán)價(jià)格。 對(duì)于CEV模型下的美式期權(quán)問(wèn)題,當(dāng)彈性因子α1時(shí),經(jīng)過(guò)一定變換原問(wèn)題可以轉(zhuǎn)化成美式期權(quán)類(lèi)似的形式,因此可以用相同的處理方法解決。對(duì)于α1的情況由于最終會(huì)轉(zhuǎn)化成有界區(qū)域上的方程問(wèn)題,相對(duì)簡(jiǎn)單,因此本文不做考慮。本文在給出以上兩個(gè)問(wèn)題的基本模型及相應(yīng)算法之后,還將給出兩個(gè)問(wèn)題采用上述方法時(shí)問(wèn)題的正定性證明,對(duì)于每個(gè)算法過(guò)程,其中的迭代步驟均可以總結(jié)為一個(gè)方程組求解問(wèn)題,將其寫(xiě)為矩陣形式后,我們知道,如果右端為正,而左端系數(shù)矩陣滿(mǎn)足M-矩陣定義,則所得的解就能保持正定性。 通過(guò)以上所述步驟,我們可以最終保證我們的方法所得結(jié)果是正確并且合理的。在通過(guò)一系列對(duì)比數(shù)值實(shí)驗(yàn)的驗(yàn)證下,可以觀察到該算法在實(shí)際應(yīng)用中是有效的,而且適用于多種情況。該方法的數(shù)值結(jié)果與差分或有限元方法加細(xì)的結(jié)果吻合,并且在多數(shù)情況下,更光滑,就這一點(diǎn)來(lái)說(shuō)較以往某些方法更接近事實(shí),總體效果是令人滿(mǎn)意的。 隨著時(shí)代的發(fā)展,各類(lèi)期權(quán)的演化,關(guān)于美式期權(quán)及其變形問(wèn)題的研究還將繼續(xù),對(duì)于其他形式的美式期權(quán)是否適用于本文的方法還不得而知,我們也將繼續(xù)在該領(lǐng)域做出努力。
[Abstract]:This paper mainly introduces a finite volume method to solve the problem of American option pricing. Furthermore, this method is used to solve the American option pricing problem under the invariant variance elasticity (CEV) model. The American option problem is essentially a free boundary problem defined in an unbounded domain. Therefore, there must be a truncation process when solving the equation by numerical method. In this paper, we arrange the original problem into a linear form. Firstly, the free boundary is changed into a regular boundary by Front-Fixing method, and then the unbounded edge is truncated by the perfectly matched layer Front-Fixing method, because the equation contains unknown free boundary information. So in every layer of time discretization, we use Newton iteration, that is, we use the finite volume method to solve the equation and get the optimal free boundary value by Newton iteration. Then get the corresponding option price. For the American option problem under the CEV model, when the elastic factor 偽 is 1:00, the original problem can be transformed into a similar form of American option after a certain transformation. Therefore, we can solve the problem with the same method. For the case of 偽 1, it is relatively simple because it will eventually be transformed into the problem of equations in the bounded region. After giving the basic model and the corresponding algorithm of the above two problems, we will also give the positive qualitative proof of the two problems using the above method, for each algorithm process. The iterative steps can be summed up as a problem of solving equations, which is written as a matrix form, we know that if the right end is positive, and the left end coefficient matrix satisfies the definition of M- matrix. Then the obtained solution can keep the positive definiteness. Through the above steps, we can finally ensure that the results obtained by our method are correct and reasonable. It can be observed that the algorithm is effective in practical application and suitable for many cases. The numerical results of the method are in agreement with the finite-element method and are more smooth in most cases. This is closer to reality than some previous methods, and the overall effect is satisfactory. With the development of the times and the evolution of various kinds of options, the study on American options and their deformation will continue. It is not known whether other forms of American options are applicable to this paper. We will also continue to make efforts in this area.
【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2014
【分類(lèi)號(hào)】：F224;F830.9

【共引文獻(xiàn)】

相關(guān)期刊論文 前4條

1 陳耀輝;孫春燕;;美式期權(quán)定價(jià)的一個(gè)非線(xiàn)性偏微分方程[J];長(zhǎng)江大學(xué)學(xué)報(bào)(自然科學(xué)版)理工卷;2010年01期

2 徐海文;張黔川;楊成;雷開(kāi)洪;;半正定單調(diào)變分不等式的CPC算法[J];四川師范大學(xué)學(xué)報(bào)(自然科學(xué)版);2009年04期

3 劉棠,張盤(pán)銘;美式期權(quán)定價(jià)中非局部問(wèn)題的有限元方法[J];數(shù)學(xué)的實(shí)踐與認(rèn)識(shí);2003年11期

4 岑仲迪;徐愛(ài)民;樂(lè)安波;;固定利率抵押貸款模型的基于自適應(yīng)網(wǎng)格的有限差分法[J];系統(tǒng)科學(xué)與數(shù)學(xué);2014年01期

相關(guān)博士學(xué)位論文 前9條

1 任筱鈺;關(guān)于委托—代理問(wèn)題的研究方法和模型[D];浙江大學(xué);2010年

2 王國(guó)治;期權(quán)定價(jià)的路徑積分方法研究[D];華南理工大學(xué);2011年

3 劉韶躍;數(shù)學(xué)金融的分?jǐn)?shù)次Black-Scholes模型及應(yīng)用[D];湖南師范大學(xué);2004年

4 鄧國(guó)和;市場(chǎng)結(jié)構(gòu)風(fēng)險(xiǎn)下雙指數(shù)跳擴(kuò)散模型期權(quán)定價(jià)與最優(yōu)投資消費(fèi)[D];湖南師范大學(xué);2006年

5 陳啟宏;[D];復(fù)旦大學(xué);1999年

6 蔣賢鋒;跳躍擴(kuò)散過(guò)程下的實(shí)物期權(quán)及在電力投資中的應(yīng)用[D];東北財(cái)經(jīng)大學(xué);2007年

7 成鈞;同倫分析法在非線(xiàn)性力學(xué)和金融學(xué)中的應(yīng)用[D];上海交通大學(xué);2008年

8 宋海明;常數(shù)波動(dòng)率和隨機(jī)波動(dòng)率下美式期權(quán)定價(jià)問(wèn)題的數(shù)值解法[D];吉林大學(xué);2014年

9 宋麗平;整體和非限制實(shí)施下永久經(jīng)理期權(quán)的最優(yōu)實(shí)施策略[D];蘇州大學(xué);2013年

，

本文編號(hào)：1442224