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Statistical Inference For FBSDEs And Backward GARCH-M Models

Posted on:2011-03-14Degree:DoctorType:Dissertation
Country:ChinaCandidate:X ChenFull Text:PDF
GTID:1119360302999816Subject:Probability theory and mathematical statistics
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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 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 byTheorem 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 byTheorem 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→∞, whereThen 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→∞, whereTheorem 3.3.2 The conditions of Theorem 3.3.1 hold, and further assume E(Z4)<∞, then as n→∞, whereTherefore, based on the asymptotic normality, the 1-αintervals for the functional coefficients of the FBSDEs model can be constructed respectively bywhereγ1-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 byTheorem 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 followsTheorem 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 satisfiesIn 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.
Keywords/Search Tags:FBSDEs model, Nonparametric estimation, Option pricing, Local linear estimator, Asymptotic distribution, Empirical likelihood, Backward GARCH-M model, Dynamic weighting
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