关于正倒向随机微分方程和倒向广义自回归条件异方差模型的统计推断
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摘要
2010年3月31日,酝酿四年之久的融资融券交易在深沪交易所首次启动,真正意义上的“买空卖空”机制开始在中国股票市场上演。时隔半月,具有风险对冲和盈利双重功效的股指期货也在中国金融期货交易所正式推出。这一系列金融衍生产品的发放,对于丰富和深化中国资本市场,增强其流动性起到积极的促进作用。而与此相关的学术研究课题也正在受到越来越多的关注。放眼世界,自1973年全球首家期权交易所在美国芝加哥开业,大批的新型金融产品就不断推出以满足衍生品市场的需求。同年,Black和Scholes(1973)([10])提出著名的期权定价公式,Merton(1973)([77])也给出了证券价格的一般均衡模型。从那时起,随机微分方程模型就作为现代金融理论的基础工具,被广泛应用于投资管理,资产定价,风险监控等多个领域。
作为期权定价模型的理想之选,正倒向随机微分方程由Pardoux和Peng(1990)([87])首先提出,其系统理论随后在Ma和Yong(1999)([75])的著作中得到详细阐述。正倒向随机微分方程的一般形式如下本论文的研究对象为一类具有马尔科夫性的正倒向随机微分方程模型,即{Ys}t≤s≤T和{Zs}t≤s≤T是{Xs}t≤s≤T的确定性函数。在期权定价的例子中,证券价格{Xs}t≤s≤T和复制性投资组合的价值{Ys}t≤s≤T都是可观测的,而对冲投资组合价值{Zs}t≤s≤T尽管无法观测,却往往是人们的兴趣所在。其他的研究关注点还包括函数系数b,σ和g.事实上也可以将Z一并视为函数系数。对于一个特定问题,其对应的正倒向随机微分方程模型的具体表达式既不会由金融市场自动给出,也无法由数理金融理论直接提供,因此我们采用模型(1)的非参数形式,在保持灵活性的同时保证了稳健性。
在本论文中,我们考虑对非参数的正倒向随机微分方程模型进行推断。我们利用局部线性平滑方法估计模型中的函数系数,并根据实际情况对结果进行调整。除了对估计的渐近分布做出理论推导,我们还通过数据模拟来考察估计在有限样本中的表现。另外,我们利用两种不同的工具：渐近分布和经验似然,分别为模型的函数系数构造了置信区间。在基于渐近正态性质建立置信区间时,由于渐近方差中包含多个未知的统计量,我们预先给出了渐近方差的相合估计。对于经验似然方法下的置信域构建,我们证明了所定义的对数经验似然比统计量渐近服从χ2分布。
此外,本论文构造了一类新型的时间序列模型,我们称之为倒向广义自回归条件异方差模型。对于所研究的动态机制,这一类模型能够突出强调终端条件对它的影响,而这正是被往常的随机微分方程推断所忽略的。借助于Fan et al(2007)([35])提出的动态加权,我们将新模型下的估计与正倒向随机微分方程模型的估计合并,从而使得最终结果既依赖于终端条件,又具备稳健性,与原来的两种估计相比也更为渐近有效。所以,这不只是对先前结论的简单拓展和改进,更为相关领域的研究带来建设性的建议,并且获得了创新性的进展。
本学位论文共分为五个章节,其主要结论的组织如下：
第一章首先阐述了论文选择模型和推断方法的理论依据,然后介绍了正倒向随机微分方程模型,包括正倒向随机微分方程理论的发展和应用等背景知识,并且通过一个具体的期权定价问题阐释了模型的结构。随后,我们讲述了构建倒向广义自回归条件异方差模型的动机和理论基础,揭示了时间序列模型和随机微分方程模型的内在联系,从而为合并两种模型下的结论做好准备。
第二章主要对正倒向随机微分方程模型的函数系数进行非参数估计。假设{(Ks0+i△,Ys0+i△),i=1,...,n)是初始时刻为s0的过程在等时间间隔上的观测,将其简记为{(Ki△,Yi△),i=1,...,n},并对数据做如下定义
由此,我们给出模型的函数系数在状态点(s,χo)的局部线性估计
为了描述以上估计的大样本性质,下面的定理给出渐近偏(方)差的具体表达：
定理2.3.1记{Ki△,i=1,...,n}为混合相关的平稳的马尔科夫过程的n个观测值,相关系数满足ρl=|Hl|2,其中Hl是｛Xi△｝的转移概率算子。假设{Xi△,i=1...,n}的概率密度函数p(·)和条件概率密度函数ρl(y|x)在支撑上连续有界。令n→∞,当h→0,△→0时,仍有nh△→∞。则在任意时刻s∈(s0,T),i=1,…,n,有以下结论成立
(a)bh(s,x0)的渐近偏差为进一步假设p’(χ)和(?)3(σ4(s,x))/(?)x3在χ0的邻域内连续,则当nh3→∞,bh(s,x0)的渐近方差为
(b)σh2(s,x0)的渐近偏差为进一步假设p’(χ)和(?)3(σ8(s,x))/(?)x3在χ0的邻域内连续,则当nh3→∞,σh2(s,x0)的渐近方差为
(c) gh(s,x0)的渐近偏差为进一步假设p’(χ)和(?)3(Z4(s,x))/dx3在χ0的邻域内连续,则当nh3→∞,gh(s,x0)的渐近方差为
(d)Zh2(s,x0)的渐近偏差为进一步假设p’(χ)和(?)3(σ8(s,x))/(?)x3在χo的邻域内连续,则当nh3→∞,Z2h(s,x0)的渐近方差为其中μj =∫uj K(u)du, vj =∫ujK2(u)du,λj—∫ujK4(u)du ,核函数K(·)是具有有界支撑的对称的概率密度函数。对任意正整数l,Pl(y*|x)是给定Xi△时'Y*(i+l)△的条件概率密度函数。
通过下一个定理我们进而确定了非参数估计的渐近正态性：
定理2.3.2除了定理2.3.1的条件,进一步假设{Yi△*,i = l,...,n}和{Yi△*,i = l,...,n}为平稳序列,而且存在一列正整数s。满足sn→∞及sn=o{(nh△)1/2}以使得当n→∞时,有(n/h△)1/2|Hsn|2→∞,此时以下渐近正态分布成立
第三章首先解决在估计波动系数的平方时可能产生的负值问题。我们采用重新分配权重的N-W估计来代替原来的局部线性平滑,所得估计结果的表达式为这个估计的实现虽然需要数值计算方法的辅助,但它既可以保证非负结果,又保持了局部线性估计的优点,如适应性和边界自调等。我们在下面的定理中给出了它的渐近分布：
定理3.2.1在定理2.3.1和定理2.3.2的假定及§3.1.3的条件C1-C3下,估计量υ2(χ)渐近服从于正态分布对于υ2(χ)的渐近方差,我们也做出了相合估计：
定理3.2.2除了定理3.2.1的条件,进一步假设对任意δ>0,E[Y8(1+δ)]<∞,则当n→∞时,其中
接下来我们考虑基于渐近正态性质构造置信域。对于模型中倒向方程的函数系数g及Z2,由于估计的渐近方差是未知的统计量,需要先由下面两个定理提供相合估计：
定理3.3.1假设定理2.3.2的条件成立,但混合相关系数需要满足§3.2中C2,并进一步假定E(Y4)<∞,则当n→∞,有其中
定理3.3.2假设定理3.3.1的条件满足,进一步假定E(Z4)<∞,则当n→∞,有其中由此建立起置信水平为1-α的区间估计：其中γ1-α／2是标准正态分布的1-α／2分位数,而Sg(s,x),Sz2(s,x)分别为渐近标准差的估计量：
为了避免类似上述方法的复杂计算,我们转而考虑利用经验似然结合局部线性平滑来构建置信区间,从而获得了函数系数9所对应的对数经验似然比ι(θg),并且有下面的结论：
定理3.4.1假设满足§3.4中的条件C1-C3,并且nh5→0,则ι(θg)渐近收敛于χ12分布。
由此建立9的置信水平为α的区间估计Iα,9(?){θg:l(θg)≤cα)}其中cα为临界值,即P(x12≤cα)=α.
对于Z2也有类似结论：
定理3.4.2假设定理3.41的条件成立,则Z2的对数经验似然比ι(θz2)渐近收敛于χ12分布。
据此建立的置信水平为α的区间估计Iα,X2(?){θZ2:l(θZ2)cα),其中cα满足P(x12≤cα)=α.
第四章主要讨论倒向广义自回归条件异方差模型。首先,我们通过分解和迭代将原来的广义自回归条件异方差模型转化为依赖于终端条件的新模型,然后参数化模型中的不可观测量,进而利用最小二乘方法对模型进行推断,所得的估计θ具有如下渐近分布：
定理4.3.1在§4.3的条件C1-C4下,有其中∑,Ω和σξ2的具体表达见§4.3中的定义.
最后我们将前面对g和Z2所做的两种估计结果进行下面的合并：其中gs,F和Zs2,F是来自于正倒向随机微分方程模型的非参数估计,gs,G和Zs2,G则为倒向广义自回归条件异方差模型下的估计,动态权重0≤ωs(g),ωs(Z2)≤1满足及以保证合并结果的方差最小化,从而给出比先前的两种估计都更为渐近有效的推断。
第五章对本论文进行总结,并给出了相关研究方向上仍待讨论的问题。
China launched its margin trading and short selling trial program on the Shanghai and Shenzhen stock exchanges on March 31,2010, after four years of preparation. Half a month later, the trading of stock index futures which is both an earning and hedging tool, started at the China Financial Futures Exchange. These innovations are diversifying and deepening the country's capital market while improving liquidity. In the meantime, the related academic research has attracted increasing attention. Ever since 1973 when the world's first options exchange opened in Chicago, a large number of new financial products have been introduced to meet the customers' demands from the derivative markets. In the same year, Black and Scholes (1973) ([10]) provided their celebrated formula for option pricing and Merton (1973) ([77]) proposed a general equilibrium model for security prices. Since then, modern finance has adopted stochastic differential equations as its basic instruments for portfolio management, asset pricing, risk management,etc.
Among these models, the FBSDEs model is an desirable choice for hedg-ing and pricing an option. FBSDEs, short for forward-backward stochastic differential equations, were first introduced by Pardoux and Peng (1990) ([87]) and the related theory was elaborated by Ma and Yong (1999) ([75]). Its general form is as follows The type of FBSDEs model we focus on is the Markovian FBSDEs, namely, both{Ys}t In this dissertation, we consider the inferences for the nonparametric FBSDEs model (1). We employ the local linear smoothing to estimate the functional coefficients and adjust the results according to practical situa-tions. The performance of the estimation is presented both theoretically by the asymptotic distribution and empirically by simulation study. Further, we construct the confidence intervals for the coefficients basing on two different methods:asymptotic distribution and empirical likelihood. For the imple-ment of the former approach, we provide the estimation of limiting variances while for the latter, we establish the asymptotic x2 distributions of the log empirical likelihood ratios.
Besides, we build a new type of time series model called the backward GARCH-M model to emphasize the effect of the terminal condition, which is ignored in the inferences of stochastic differential equation models. We combine the estimators of the new model and that of the FBSDEs model with the dynamic weighting proposed by Fan et al (2007) ([35]). Consequently, the final results dependent on terminal conditions, enjoy the robustness, and asymptotically more efficient compared with previous estimators. This is not only an extension and improvement of earlier conclusions, but also brings the related research field a constructive suggestion and whole new advancement.
This dissertation consists of five chapters. Its main conclusions are pre-sented as follows:
In Chapter 1, After explaining the theory foundation and method se-lection, we first give some introductions to the FBSDEs model, including its background knowledge and applications with illustrative examples.Then we state the motivation to construct the backward GARCH-M model and integrate the estimators of two models above.
Chapter 2 concentrates on the nonparametric estimation of functional coefficients of FBSDEs models. Given the initial calendar time point s0, we rewrite the time series data{(Xs0+i△,Ys0+i△),i=1,…,n)observed at equally spaced time points into the simpler form{(Xi△,Yi△),i=1,…,n), and denote Then the local linear estimators of the functional coefficients at time point s≥so when Xs=x0 are given by
The explicit expressions of the asymptotic biases and variances of the estimators above are provided as follows:
Theorem 2.3.1 Let{Xi△,i=1,…,n}be a sequence of observations on a stationary Markov process withρ-mixing coefficientρl=|Hl|2,where H1 is the transtion probability operator of{Xi△),satisfying|Hl|2→0,as l→∞.Assume that the probability density ,of{xi△,i=1,…,n),denoted by p(·),is bounded and continuous,and that pl(y|x)is bounded by some constant independent of l and continuous in the variables(y,x).Let n→∞,such that h→0,△→0,and nh△→∞,then at any time s∈(s0,T),i=1,…,n.
(a)The asymptotic bias of bh(s,x0)is Assume further that p'(x) and (?)3(σ4(s,x))/(?)x3 are continuous in neigh-borhood of x0 nh3→∞,then the asymptotic variance is
(b)The asymptotic bias ofσh2(s,x0)is given by Assume further that p'(x) and (?)3(σ8(s,x))/(?)x3 are continuous in a neigh-borhood of x0 nh3→∞,then the asymptotic variance is
(c)The asymptotic bias of (s,x0)is Assume further that p'(x) and (?)3(σ4(s,x))/(?)x3 are continuous in neigh-borhood of x0 nh3→∞,then the asymptotic variance is
(d)The asymptotic bias of Z(s,x0)is given by Assume further that p'(x) and (?)3(σ4(s,x))/(?)x3 are continuous in neigh-borhood of x0 nh3→∞,then the asymptotic variance is whereμj=∫ujK(u)du, Vj=∫ujK2(u)du,λj=∫ujK4(u)du, with the kernel function K(-) as a bounded symmetric probability density function with bounded support. For an integer l> 0, Pl(y|x) denotes the conditional probability density of Y*(i+l)△given Xi△.
Furthermore, the asymptotic normality of the nonparametric estimators are also established by
Theorem 2.3.2 Addition to the conditions of Theorem 2.3.1, we as-sume further that the sequence{Yi△*,i= 1,...,n} and{Yi△*,i=1,...,n} are stationary, and there exists a sequence of positive integers sn satisfy-ing sn→∞and sn= o{(nh△)1/2}; such that (n/h△)1/2|HSn|2→∞,as n→∞. Then there is the following asymptotic normality as n n→∞,
In Chapter 3, we first solve the negative-value problem of the squared-volatility estimation by the re-weighted N-W estimator v2(x) in the form of Though numerical skills are required to implement this method, however, it always produces positive results while preserves appealing features as adap-tation and automatically boundary carpentry, etc. And the estimator has the following asymptotic distribution:
Theorem 3.2.1 Under assumptions of Theorem 2.3.1, Theorem 2.3.2 and conditions C1-C3, the asymptotic normality of v2(x) is presented as Further, a consistent estimation for the asymptotic variance of v2(x) is provided by
Theorem 3.2.2 Suppose the conditions of Theorem 3.2.1 hold. Assume further that E[Y8(1+δ)]<∞for someδ>0, then as n→∞, where
Then we consider the confidence intervals based on the asymptotic nor-mality, which is implemented with the support of the following two theorems:
Theorem 3.3.1 The conditions of Theorem 2.3.2 hold except that the mixing coefficient is strengthened as in C2 of§3.2, further assume E(Y4)<∞, then as n→∞, where
Theorem 3.3.2 The conditions of Theorem 3.3.1 hold, and further assume E(Z4)<∞, then as n→∞, where
Therefore, based on the asymptotic normality, the 1-αintervals for the functional coefficients of the FBSDEs model can be constructed respectively by
whereγ1-α/2 is the 1 -α/2-quantile of standard Gaussian distribution, Sg(s,x) and Sz2(s, x) are the estimated asypmtotic standand derivatives for
g(s,x) and Z2(s,x) respectively,
To avoid the computational complexity above, we build the confidence intervals based on empirical likelihood in conjunction with local linear smoothers. For the functional coefficient g, the asymptotic chi-squared distribution of the log empirical likelihood ratio l(θg) can be established by
Theorem 3.4.1 Assume condition C1-C3 and nh5→0 hold, then l(θg) has an asymptotic x12 distribution.
Consequently, the empirical likelihood confidence interval for g with nominal confidence level a is where cαis the critical value, i.e.
Similar conclusion exits for Z2 as follows
Theorem 3.4.2 Assume conditions C1-C3 and nh5→0 hold, then the log empirical likelihood ratio l(θz2) has an asymptotic x12 distribution.
Then the empirical likelihood confidence interval for Z2 at level a is where ca satisfies
In Chapter 4 we first decompose the conventional GARCH-M model and then iterate the decomposition procedure such that the newly proposed model is goal-dependent. Further we apply parameterization technique to unobservable variables in the model and employ least squares method to inferences. The asymptotic property of estimatorθis investigated by the following theorem.
Theorem 4.3.1 Under conditions C1-C4, there is
After extending the model transformation method, we combine the es-timators of the backward GARCH-M model and that of the FBSDEs model with the dynamic optimal weights as follows where gs,F and Z2s,F are estimators of the FBSDEs model, gs,G and Z2s,G are estimators of the backward GARCH-M model, and the dynamic weighting schemes 0≤ωs(g),ωs(Z2)≤1 satisfy and which result in more asymptotically efficient estimators than those of previ-ous two methods.
Chapter 5 summarizes the results of the dissertation and discusses some remaining problems.
作为期权定价模型的理想之选,正倒向随机微分方程由Pardoux和Peng(1990)([87])首先提出,其系统理论随后在Ma和Yong(1999)([75])的著作中得到详细阐述。正倒向随机微分方程的一般形式如下本论文的研究对象为一类具有马尔科夫性的正倒向随机微分方程模型,即{Ys}t≤s≤T和{Zs}t≤s≤T是{Xs}t≤s≤T的确定性函数。在期权定价的例子中,证券价格{Xs}t≤s≤T和复制性投资组合的价值{Ys}t≤s≤T都是可观测的,而对冲投资组合价值{Zs}t≤s≤T尽管无法观测,却往往是人们的兴趣所在。其他的研究关注点还包括函数系数b,σ和g.事实上也可以将Z一并视为函数系数。对于一个特定问题,其对应的正倒向随机微分方程模型的具体表达式既不会由金融市场自动给出,也无法由数理金融理论直接提供,因此我们采用模型(1)的非参数形式,在保持灵活性的同时保证了稳健性。
在本论文中,我们考虑对非参数的正倒向随机微分方程模型进行推断。我们利用局部线性平滑方法估计模型中的函数系数,并根据实际情况对结果进行调整。除了对估计的渐近分布做出理论推导,我们还通过数据模拟来考察估计在有限样本中的表现。另外,我们利用两种不同的工具：渐近分布和经验似然,分别为模型的函数系数构造了置信区间。在基于渐近正态性质建立置信区间时,由于渐近方差中包含多个未知的统计量,我们预先给出了渐近方差的相合估计。对于经验似然方法下的置信域构建,我们证明了所定义的对数经验似然比统计量渐近服从χ2分布。
此外,本论文构造了一类新型的时间序列模型,我们称之为倒向广义自回归条件异方差模型。对于所研究的动态机制,这一类模型能够突出强调终端条件对它的影响,而这正是被往常的随机微分方程推断所忽略的。借助于Fan et al(2007)([35])提出的动态加权,我们将新模型下的估计与正倒向随机微分方程模型的估计合并,从而使得最终结果既依赖于终端条件,又具备稳健性,与原来的两种估计相比也更为渐近有效。所以,这不只是对先前结论的简单拓展和改进,更为相关领域的研究带来建设性的建议,并且获得了创新性的进展。
本学位论文共分为五个章节,其主要结论的组织如下：
第一章首先阐述了论文选择模型和推断方法的理论依据,然后介绍了正倒向随机微分方程模型,包括正倒向随机微分方程理论的发展和应用等背景知识,并且通过一个具体的期权定价问题阐释了模型的结构。随后,我们讲述了构建倒向广义自回归条件异方差模型的动机和理论基础,揭示了时间序列模型和随机微分方程模型的内在联系,从而为合并两种模型下的结论做好准备。
第二章主要对正倒向随机微分方程模型的函数系数进行非参数估计。假设{(Ks0+i△,Ys0+i△),i=1,...,n)是初始时刻为s0的过程在等时间间隔上的观测,将其简记为{(Ki△,Yi△),i=1,...,n},并对数据做如下定义
由此,我们给出模型的函数系数在状态点(s,χo)的局部线性估计
为了描述以上估计的大样本性质,下面的定理给出渐近偏(方)差的具体表达：
定理2.3.1记{Ki△,i=1,...,n}为混合相关的平稳的马尔科夫过程的n个观测值,相关系数满足ρl=|Hl|2,其中Hl是｛Xi△｝的转移概率算子。假设{Xi△,i=1...,n}的概率密度函数p(·)和条件概率密度函数ρl(y|x)在支撑上连续有界。令n→∞,当h→0,△→0时,仍有nh△→∞。则在任意时刻s∈(s0,T),i=1,…,n,有以下结论成立
(a)bh(s,x0)的渐近偏差为进一步假设p’(χ)和(?)3(σ4(s,x))/(?)x3在χ0的邻域内连续,则当nh3→∞,bh(s,x0)的渐近方差为
(b)σh2(s,x0)的渐近偏差为进一步假设p’(χ)和(?)3(σ8(s,x))/(?)x3在χ0的邻域内连续,则当nh3→∞,σh2(s,x0)的渐近方差为
(c) gh(s,x0)的渐近偏差为进一步假设p’(χ)和(?)3(Z4(s,x))/dx3在χ0的邻域内连续,则当nh3→∞,gh(s,x0)的渐近方差为
(d)Zh2(s,x0)的渐近偏差为进一步假设p’(χ)和(?)3(σ8(s,x))/(?)x3在χo的邻域内连续,则当nh3→∞,Z2h(s,x0)的渐近方差为其中μj =∫uj K(u)du, vj =∫ujK2(u)du,λj—∫ujK4(u)du ,核函数K(·)是具有有界支撑的对称的概率密度函数。对任意正整数l,Pl(y*|x)是给定Xi△时'Y*(i+l)△的条件概率密度函数。
通过下一个定理我们进而确定了非参数估计的渐近正态性：
定理2.3.2除了定理2.3.1的条件,进一步假设{Yi△*,i = l,...,n}和{Yi△*,i = l,...,n}为平稳序列,而且存在一列正整数s。满足sn→∞及sn=o{(nh△)1/2}以使得当n→∞时,有(n/h△)1/2|Hsn|2→∞,此时以下渐近正态分布成立
第三章首先解决在估计波动系数的平方时可能产生的负值问题。我们采用重新分配权重的N-W估计来代替原来的局部线性平滑,所得估计结果的表达式为这个估计的实现虽然需要数值计算方法的辅助,但它既可以保证非负结果,又保持了局部线性估计的优点,如适应性和边界自调等。我们在下面的定理中给出了它的渐近分布：
定理3.2.1在定理2.3.1和定理2.3.2的假定及§3.1.3的条件C1-C3下,估计量υ2(χ)渐近服从于正态分布对于υ2(χ)的渐近方差,我们也做出了相合估计：
定理3.2.2除了定理3.2.1的条件,进一步假设对任意δ>0,E[Y8(1+δ)]<∞,则当n→∞时,其中
接下来我们考虑基于渐近正态性质构造置信域。对于模型中倒向方程的函数系数g及Z2,由于估计的渐近方差是未知的统计量,需要先由下面两个定理提供相合估计：
定理3.3.1假设定理2.3.2的条件成立,但混合相关系数需要满足§3.2中C2,并进一步假定E(Y4)<∞,则当n→∞,有其中
定理3.3.2假设定理3.3.1的条件满足,进一步假定E(Z4)<∞,则当n→∞,有其中由此建立起置信水平为1-α的区间估计：其中γ1-α／2是标准正态分布的1-α／2分位数,而Sg(s,x),Sz2(s,x)分别为渐近标准差的估计量：
为了避免类似上述方法的复杂计算,我们转而考虑利用经验似然结合局部线性平滑来构建置信区间,从而获得了函数系数9所对应的对数经验似然比ι(θg),并且有下面的结论：
定理3.4.1假设满足§3.4中的条件C1-C3,并且nh5→0,则ι(θg)渐近收敛于χ12分布。
由此建立9的置信水平为α的区间估计Iα,9(?){θg:l(θg)≤cα)}其中cα为临界值,即P(x12≤cα)=α.
对于Z2也有类似结论：
定理3.4.2假设定理3.41的条件成立,则Z2的对数经验似然比ι(θz2)渐近收敛于χ12分布。
据此建立的置信水平为α的区间估计Iα,X2(?){θZ2:l(θZ2)cα),其中cα满足P(x12≤cα)=α.
第四章主要讨论倒向广义自回归条件异方差模型。首先,我们通过分解和迭代将原来的广义自回归条件异方差模型转化为依赖于终端条件的新模型,然后参数化模型中的不可观测量,进而利用最小二乘方法对模型进行推断,所得的估计θ具有如下渐近分布：
定理4.3.1在§4.3的条件C1-C4下,有其中∑,Ω和σξ2的具体表达见§4.3中的定义.
最后我们将前面对g和Z2所做的两种估计结果进行下面的合并：其中gs,F和Zs2,F是来自于正倒向随机微分方程模型的非参数估计,gs,G和Zs2,G则为倒向广义自回归条件异方差模型下的估计,动态权重0≤ωs(g),ωs(Z2)≤1满足及以保证合并结果的方差最小化,从而给出比先前的两种估计都更为渐近有效的推断。
第五章对本论文进行总结,并给出了相关研究方向上仍待讨论的问题。
China launched its margin trading and short selling trial program on the Shanghai and Shenzhen stock exchanges on March 31,2010, after four years of preparation. Half a month later, the trading of stock index futures which is both an earning and hedging tool, started at the China Financial Futures Exchange. These innovations are diversifying and deepening the country's capital market while improving liquidity. In the meantime, the related academic research has attracted increasing attention. Ever since 1973 when the world's first options exchange opened in Chicago, a large number of new financial products have been introduced to meet the customers' demands from the derivative markets. In the same year, Black and Scholes (1973) ([10]) provided their celebrated formula for option pricing and Merton (1973) ([77]) proposed a general equilibrium model for security prices. Since then, modern finance has adopted stochastic differential equations as its basic instruments for portfolio management, asset pricing, risk management,etc.
Among these models, the FBSDEs model is an desirable choice for hedg-ing and pricing an option. FBSDEs, short for forward-backward stochastic differential equations, were first introduced by Pardoux and Peng (1990) ([87]) and the related theory was elaborated by Ma and Yong (1999) ([75]). Its general form is as follows The type of FBSDEs model we focus on is the Markovian FBSDEs, namely, both{Ys}t
Besides, we build a new type of time series model called the backward GARCH-M model to emphasize the effect of the terminal condition, which is ignored in the inferences of stochastic differential equation models. We combine the estimators of the new model and that of the FBSDEs model with the dynamic weighting proposed by Fan et al (2007) ([35]). Consequently, the final results dependent on terminal conditions, enjoy the robustness, and asymptotically more efficient compared with previous estimators. This is not only an extension and improvement of earlier conclusions, but also brings the related research field a constructive suggestion and whole new advancement.
This dissertation consists of five chapters. Its main conclusions are pre-sented as follows:
In Chapter 1, After explaining the theory foundation and method se-lection, we first give some introductions to the FBSDEs model, including its background knowledge and applications with illustrative examples.Then we state the motivation to construct the backward GARCH-M model and integrate the estimators of two models above.
Chapter 2 concentrates on the nonparametric estimation of functional coefficients of FBSDEs models. Given the initial calendar time point s0, we rewrite the time series data{(Xs0+i△,Ys0+i△),i=1,…,n)observed at equally spaced time points into the simpler form{(Xi△,Yi△),i=1,…,n), and denote Then the local linear estimators of the functional coefficients at time point s≥so when Xs=x0 are given by
The explicit expressions of the asymptotic biases and variances of the estimators above are provided as follows:
Theorem 2.3.1 Let{Xi△,i=1,…,n}be a sequence of observations on a stationary Markov process withρ-mixing coefficientρl=|Hl|2,where H1 is the transtion probability operator of{Xi△),satisfying|Hl|2→0,as l→∞.Assume that the probability density ,of{xi△,i=1,…,n),denoted by p(·),is bounded and continuous,and that pl(y|x)is bounded by some constant independent of l and continuous in the variables(y,x).Let n→∞,such that h→0,△→0,and nh△→∞,then at any time s∈(s0,T),i=1,…,n.
(a)The asymptotic bias of bh(s,x0)is Assume further that p'(x) and (?)3(σ4(s,x))/(?)x3 are continuous in neigh-borhood of x0 nh3→∞,then the asymptotic variance is
(b)The asymptotic bias ofσh2(s,x0)is given by Assume further that p'(x) and (?)3(σ8(s,x))/(?)x3 are continuous in a neigh-borhood of x0 nh3→∞,then the asymptotic variance is
(c)The asymptotic bias of (s,x0)is Assume further that p'(x) and (?)3(σ4(s,x))/(?)x3 are continuous in neigh-borhood of x0 nh3→∞,then the asymptotic variance is
(d)The asymptotic bias of Z(s,x0)is given by Assume further that p'(x) and (?)3(σ4(s,x))/(?)x3 are continuous in neigh-borhood of x0 nh3→∞,then the asymptotic variance is whereμj=∫ujK(u)du, Vj=∫ujK2(u)du,λj=∫ujK4(u)du, with the kernel function K(-) as a bounded symmetric probability density function with bounded support. For an integer l> 0, Pl(y|x) denotes the conditional probability density of Y*(i+l)△given Xi△.
Furthermore, the asymptotic normality of the nonparametric estimators are also established by
Theorem 2.3.2 Addition to the conditions of Theorem 2.3.1, we as-sume further that the sequence{Yi△*,i= 1,...,n} and{Yi△*,i=1,...,n} are stationary, and there exists a sequence of positive integers sn satisfy-ing sn→∞and sn= o{(nh△)1/2}; such that (n/h△)1/2|HSn|2→∞,as n→∞. Then there is the following asymptotic normality as n n→∞,
In Chapter 3, we first solve the negative-value problem of the squared-volatility estimation by the re-weighted N-W estimator v2(x) in the form of Though numerical skills are required to implement this method, however, it always produces positive results while preserves appealing features as adap-tation and automatically boundary carpentry, etc. And the estimator has the following asymptotic distribution:
Theorem 3.2.1 Under assumptions of Theorem 2.3.1, Theorem 2.3.2 and conditions C1-C3, the asymptotic normality of v2(x) is presented as Further, a consistent estimation for the asymptotic variance of v2(x) is provided by
Theorem 3.2.2 Suppose the conditions of Theorem 3.2.1 hold. Assume further that E[Y8(1+δ)]<∞for someδ>0, then as n→∞, where
Then we consider the confidence intervals based on the asymptotic nor-mality, which is implemented with the support of the following two theorems:
Theorem 3.3.1 The conditions of Theorem 2.3.2 hold except that the mixing coefficient is strengthened as in C2 of§3.2, further assume E(Y4)<∞, then as n→∞, where
Theorem 3.3.2 The conditions of Theorem 3.3.1 hold, and further assume E(Z4)<∞, then as n→∞, where
Therefore, based on the asymptotic normality, the 1-αintervals for the functional coefficients of the FBSDEs model can be constructed respectively by
whereγ1-α/2 is the 1 -α/2-quantile of standard Gaussian distribution, Sg(s,x) and Sz2(s, x) are the estimated asypmtotic standand derivatives for
g(s,x) and Z2(s,x) respectively,
To avoid the computational complexity above, we build the confidence intervals based on empirical likelihood in conjunction with local linear smoothers. For the functional coefficient g, the asymptotic chi-squared distribution of the log empirical likelihood ratio l(θg) can be established by
Theorem 3.4.1 Assume condition C1-C3 and nh5→0 hold, then l(θg) has an asymptotic x12 distribution.
Consequently, the empirical likelihood confidence interval for g with nominal confidence level a is where cαis the critical value, i.e.
Similar conclusion exits for Z2 as follows
Theorem 3.4.2 Assume conditions C1-C3 and nh5→0 hold, then the log empirical likelihood ratio l(θz2) has an asymptotic x12 distribution.
Then the empirical likelihood confidence interval for Z2 at level a is where ca satisfies
In Chapter 4 we first decompose the conventional GARCH-M model and then iterate the decomposition procedure such that the newly proposed model is goal-dependent. Further we apply parameterization technique to unobservable variables in the model and employ least squares method to inferences. The asymptotic property of estimatorθis investigated by the following theorem.
Theorem 4.3.1 Under conditions C1-C4, there is
After extending the model transformation method, we combine the es-timators of the backward GARCH-M model and that of the FBSDEs model with the dynamic optimal weights as follows where gs,F and Z2s,F are estimators of the FBSDEs model, gs,G and Z2s,G are estimators of the backward GARCH-M model, and the dynamic weighting schemes 0≤ωs(g),ωs(Z2)≤1 satisfy and which result in more asymptotically efficient estimators than those of previ-ous two methods.
Chapter 5 summarizes the results of the dissertation and discusses some remaining problems.
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