【摘要】：本學(xué)位論文主要研究期權(quán)定價問題。針對Black-Merton-Scholes模型在假設(shè)上的不足,我們對其做了多方面的拓展。期權(quán)是衍生品的一種,它的一個重要作用就是為投資者的投資組合提供保險功能。近幾十年,期權(quán)市場發(fā)展迅速,其中很主要的一個原因就是我們可以利用模型和方法來對期權(quán)的價格和變動趨勢做出估量。Black和Scholes在構(gòu)造期權(quán)定價公式的同時,引入了標(biāo)的資產(chǎn)服從幾何布朗運(yùn)動,常數(shù)波動率等幾條不符合實(shí)際的假設(shè)。針對這些假設(shè)的不足,本文所做的工作主要集中在兩個方面,其中之一就是關(guān)于常數(shù)波動率假設(shè)的放松。一般而言,決定期權(quán)價格的因素有以下幾點(diǎn)：標(biāo)的資產(chǎn)的價格、利率、到期時間、敲定價格和波動率。在這里,除波動率以外的其它因素基本上是可以在市場中觀察到或者是相對容易估計的。因此關(guān)于波動率建模就成了期權(quán)定價問題的關(guān)鍵。為了解決這一問題,學(xué)者們做了多種嘗試,本文所采用的波動率不確定模型就是其中的一種。它最初的想法很簡單,如果我們無法描繪出波動率的確切變化,那么,至少我們可以確定它的變化范圍,從而給出期權(quán)價格的最優(yōu)區(qū)間。這一定價模型的基本思想是風(fēng)險中性定價,即通過風(fēng)險中性測度計算期權(quán)的價格。但在應(yīng)用過程中,因?yàn)椴▌勇实牟淮_定性,學(xué)者們發(fā)現(xiàn)他們往往需要面對一族相互奇異的概率測度,這為最終價格的確定帶來了困難。且這一困難是在經(jīng)典的概率框架中很難克服的。故而,在本文中我們采用G-期望框架來研究這一問題。G-期望是一種次線性期望,它是由彭實(shí)戈院士在近幾年提出的一個理論框架。這一理論框架是對經(jīng)典概率理論的一個拓展,通過它,我們獲得了一個嶄新的視角來看待經(jīng)典概率論。同時這一理論框架是一個植根于不確定性的理論,這一類不確定性源于人們的未知,是一種奈特所說的不確定性。衍生品定價中的波動率不確定模型就可以歸為這一類,G-期望為我們研究這一模型提供了一個有力的工具。 本文所做的另一方面工作主要是對標(biāo)的資產(chǎn)價格變化行為模式的放松,也即針對標(biāo)的資產(chǎn)服從幾何布朗運(yùn)動這一假設(shè)。為此我們做了兩類拓展,Levy模型和分?jǐn)?shù)布朗運(yùn)動驅(qū)動的模型。Levy模型是一類由Levy過程驅(qū)動的市場模型。Levy過程是對布朗運(yùn)動的一個推廣,它是馬爾科夫的同時還是半鞅,它的分布可以是連續(xù)的亦可以是帶跳的,而且某些Levy過程滿足“厚尾”性質(zhì),這使得Levy模型相對于B-M-S模型在應(yīng)用上具有很大的靈活性。分?jǐn)?shù)布朗運(yùn)動在赫斯特指數(shù)H≠1/2時不是半鞅,從而使得這類市場模型跳出了半鞅的框架。且當(dāng)赫斯特指數(shù)1H1/2時,資產(chǎn)價格之間的增量正相關(guān),并具有長期記憶性,這就使得分?jǐn)?shù)布朗運(yùn)動模型能夠在一定程度上描述市場的分形結(jié)構(gòu)。 不僅如此,對金融變量演進(jìn)過程的探索強(qiáng)烈依賴于過去的信息。受到Arriojas etal(2007)[2]的啟發(fā),我們在如上三類模型中綜合考量了時滯效應(yīng)。 本學(xué)位論文的主要研究成果集中在以下幾方面： 其一,我們構(gòu)造了一類由G-布朗運(yùn)動B驅(qū)動的時滯市場模型,也就是假設(shè)股票價格S滿足下面的隨機(jī)時滯微分方程：其中可以被認(rèn)為是C([-τ,0]；R)值隨機(jī)過程。{B(t),t≥0}是G-布朗運(yùn)動{B(t),t≥0}的二次變差過程。這一模型中的波動率在一定范圍內(nèi)變動,故而這是一類波動率不確定模型。受到Arriojas et al(2007)的啟發(fā),我們同時也考慮了趨勢效應(yīng),即過去的股票價格也許會對現(xiàn)在的價格產(chǎn)生影響。在這一部分之中,我們在Peng(2007), Bai-Lin(2010), Ren(2013,2011)的研究基礎(chǔ)上,討論了這一類隨機(jī)時滯模型的有效性,進(jìn)而將這一模型應(yīng)用于期權(quán)定價之中。 其二,作為與第一部分的比較,我們分別考慮了由Levy過程與分?jǐn)?shù)布朗運(yùn)動驅(qū)動的時滯期權(quán)定價模型,在對Levy過程的跳、分?jǐn)?shù)布朗運(yùn)動的Hurst指數(shù)H以及方程系數(shù)的一些正則性條件限制下,我們分別找到了等價鞅測度與Follmer-Schweizer的最小測度,從而給出相應(yīng)的時滯歐式看漲期權(quán)的定價公式。 其三,作為第一部分與第二部分的例子,我們考慮了由分?jǐn)?shù)布朗運(yùn)動與Levy過程構(gòu)成的無時滯的混合市場模型,在Levy過程的跳是冪跳的情況下,我們證明當(dāng)3/4H1時,該混合市場是完備無套利的并且給出歐式期權(quán)的定價公式的顯示解。 最后,在討論上面這些市場的期權(quán)定價問題時,作為必要的理論基礎(chǔ),我們對G-布朗運(yùn)動進(jìn)行了一些理論探索,拓展了Yamada,Yor以及Yan的研究,獲得了G-布朗運(yùn)動的廣義Ito公式(即Yamada公式)。
[Abstract]:This dissertation mainly deals with the option pricing problem. In view of the deficiency of the Black-Merton-Scholes model, we have done a lot of development. Options are one of the derivatives, an important role of which is to provide insurance for investors' portfolios. In recent decades, the development of the option market is rapid, one of which is that we can use the model and method to measure the price and the change trend of the option. Black and Scholes, at the same time of constructing the option pricing formula, introduced several unrealistic assumptions about the subject's assets, such as the geometric Brownian motion, the constant fluctuation rate, and so on. In view of the deficiency of these assumptions, the work done in this paper is mainly focused on two aspects, one of which is the relaxation of the assumption of constant fluctuation rate. In general, the factors that determine the price of an option are the following: the price of the subject's assets, the interest rate, the expiration time, the finalization of the price and the rate of volatility. In this case, other factors other than the fluctuation rate are basically observable or relatively easy to estimate in the market. Therefore, the model of volatility is the key to the option pricing problem. In order to solve this problem, the scholars have made a variety of attempts, and the fluctuation rate used in this paper is one of them. Its original idea is simple, and if we can't paint the exact change in the rate of volatility, at least we can determine its range of changes, giving the optimal range of the option price. The basic idea of this pricing model is the risk-neutral pricing, i.e., the price of the option is calculated through the risk neutral measure. But in the application process, because of the uncertainty of the fluctuation rate, the scholars have found that they often need to face a family of mutually strange probability measures, which brings difficulties to the determination of the final price. And this difficulty is difficult to overcome in a classical probability framework. Therefore, in this paper, we use the G-expectation framework to study the problem. G-expectation is a kind of sublinear expectation, which is a theoretical frame made by the academician of Pengtango in recent years. This theoretical framework is an extension of the classical probability theory, through which we have obtained a new perspective to view the classical probability theory. At the same time, the theoretical framework is a theory rooted in uncertainty, which comes from the uncertainty of the people, and is a kind of uncertainty that Nate has said. The uncertainty model of the volatility in the pricing of derivatives can be classified as this category, and G-expectation provides a powerful tool for us to study the model. On the other hand, the work on the other hand is the relaxation of the model of the change of the price of the subject's assets, that is, the object's assets are subject to the pseudo-geometric Brownian motion. Let's do two kinds of expansion, Levy model and fractional Brownian motion drive mode. Model. The Levy model is a class of market models driven by the Levy process The Levy process is a generalization of the Brownian motion. It is Markov and semi-linear. The distribution can be continuous or hop-free, and some Levy processes satisfy the "thick end" properties, which makes the Levy model have a great flexibility in application with respect to the B-M-S model. The fractional Brownian motion is not a half-time at the Hurst index, H-1/2, so that this kind of market model is out of the box and when the Hurst index is 1 H1/2, the increment between the asset prices is positively correlated and has long-term memory property, so that the fractional Brownian motion model can describe the fractal structure of the market to a certain extent Furthermore, the exploration of the process of the evolution of financial variables is strongly dependent on the past The information is inspired by the Ariojas et al (2007)[2], which, when considered in three types of models, The main research results of this dissertation are focused on In one of the following aspects, we construct a class of time-delay market models driven by G-Brownian motion B, that is, it is assumed that the stock price S satisfies the following stochastic delay differential equation: to be C ([--,0]; R) Value random process. {B (t), t {0} is G-Brownian motion {B (t), t {0} The rate of fluctuation in this model is varied within a certain range, so this is a class of waves The dynamic rate uncertainty model. Inspired by the Arriojas et al (2007), we also take into account the trend effect, that is, the stock price in the past may be right now On the basis of the research of Peng (2007), Bai-Lin (2010) and Ren (2013,2011), the validity of this class of stochastic time-delay models is discussed, and this model is applied in this part. Second, as a comparison with the first part, we consider the time-delay option pricing model driven by the Levy process and the fractional Brownian motion, and the Hurst index H of the fractional Brownian motion and one of the equation coefficients in the jump and fractional Brownian motion of the Levy process. Under the constraints of some regulative conditions, we respectively find the minimum measure of the equivalent confidence measure and the Follower-Schweizer, so that the corresponding time-delay European model is given. As an example of the first and second parts, we consider the mixed market model with time-delay composed of the fractional Brownian motion and the Levy process, and in the case of the jump in the Levy process is a power-jump, We prove that when 3/ 4H1, the hybrid market is complete and free of arbitrage and gives a European Finally, in the discussion of the option pricing problem of these markets, we have made some theoretical exploration on the G-Brownian motion, and expanded the research of Yamada, Yor and Yan, and obtained the generalized Ito of the G-Brownian motion.
【學(xué)位授予單位】：東華大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2014
【分類號】：O211.63;F830

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