几类不确定性期权定价模型及相关问题研究

发布时间：2019-05-18 21:26

【摘要】：本学位论文主要研究期权定价问题。针对Black-Merton-Scholes模型在假设上的不足,我们对其做了多方面的拓展。期权是衍生品的一种,它的一个重要作用就是为投资者的投资组合提供保险功能。近几十年,期权市场发展迅速,其中很主要的一个原因就是我们可以利用模型和方法来对期权的价格和变动趋势做出估量。Black和Scholes在构造期权定价公式的同时,引入了标的资产服从几何布朗运动,常数波动率等几条不符合实际的假设。针对这些假设的不足,本文所做的工作主要集中在两个方面,其中之一就是关于常数波动率假设的放松。一般而言,决定期权价格的因素有以下几点：标的资产的价格、利率、到期时间、敲定价格和波动率。在这里,除波动率以外的其它因素基本上是可以在市场中观察到或者是相对容易估计的。因此关于波动率建模就成了期权定价问题的关键。为了解决这一问题,学者们做了多种尝试,本文所采用的波动率不确定模型就是其中的一种。它最初的想法很简单,如果我们无法描绘出波动率的确切变化,那么,至少我们可以确定它的变化范围,从而给出期权价格的最优区间。这一定价模型的基本思想是风险中性定价,即通过风险中性测度计算期权的价格。但在应用过程中,因为波动率的不确定性,学者们发现他们往往需要面对一族相互奇异的概率测度,这为最终价格的确定带来了困难。且这一困难是在经典的概率框架中很难克服的。故而,在本文中我们采用G-期望框架来研究这一问题。G-期望是一种次线性期望,它是由彭实戈院士在近几年提出的一个理论框架。这一理论框架是对经典概率理论的一个拓展,通过它,我们获得了一个崭新的视角来看待经典概率论。同时这一理论框架是一个植根于不确定性的理论,这一类不确定性源于人们的未知,是一种奈特所说的不确定性。衍生品定价中的波动率不确定模型就可以归为这一类,G-期望为我们研究这一模型提供了一个有力的工具。本文所做的另一方面工作主要是对标的资产价格变化行为模式的放松,也即针对标的资产服从几何布朗运动这一假设。为此我们做了两类拓展,Levy模型和分数布朗运动驱动的模型。Levy模型是一类由Levy过程驱动的市场模型。Levy过程是对布朗运动的一个推广,它是马尔科夫的同时还是半鞅,它的分布可以是连续的亦可以是带跳的,而且某些Levy过程满足“厚尾”性质,这使得Levy模型相对于B-M-S模型在应用上具有很大的灵活性。分数布朗运动在赫斯特指数H≠1/2时不是半鞅,从而使得这类市场模型跳出了半鞅的框架。且当赫斯特指数1H1/2时,资产价格之间的增量正相关,并具有长期记忆性,这就使得分数布朗运动模型能够在一定程度上描述市场的分形结构。不仅如此,对金融变量演进过程的探索强烈依赖于过去的信息。受到Arriojas etal(2007)[2]的启发,我们在如上三类模型中综合考量了时滞效应。本学位论文的主要研究成果集中在以下几方面：其一,我们构造了一类由G-布朗运动B驱动的时滞市场模型,也就是假设股票价格S满足下面的随机时滞微分方程：其中可以被认为是C([-τ,0]；R)值随机过程。{B(t),t≥0}是G-布朗运动{B(t),t≥0}的二次变差过程。这一模型中的波动率在一定范围内变动,故而这是一类波动率不确定模型。受到Arriojas et al(2007)的启发,我们同时也考虑了趋势效应,即过去的股票价格也许会对现在的价格产生影响。在这一部分之中,我们在Peng(2007), Bai-Lin(2010), Ren(2013,2011)的研究基础上,讨论了这一类随机时滞模型的有效性,进而将这一模型应用于期权定价之中。其二,作为与第一部分的比较,我们分别考虑了由Levy过程与分数布朗运动驱动的时滞期权定价模型,在对Levy过程的跳、分数布朗运动的Hurst指数H以及方程系数的一些正则性条件限制下,我们分别找到了等价鞅测度与Follmer-Schweizer的最小测度,从而给出相应的时滞欧式看涨期权的定价公式。其三,作为第一部分与第二部分的例子,我们考虑了由分数布朗运动与Levy过程构成的无时滞的混合市场模型,在Levy过程的跳是幂跳的情况下,我们证明当3/4H1时,该混合市场是完备无套利的并且给出欧式期权的定价公式的显示解。最后,在讨论上面这些市场的期权定价问题时,作为必要的理论基础,我们对G-布朗运动进行了一些理论探索,拓展了Yamada,Yor以及Yan的研究,获得了G-布朗运动的广义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.
【学位授予单位】：东华大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：O211.63;F830

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