published_or_final_version / Economics and Finance / Doctoral / Doctor of Philosophy

这篇论文研究了当假设的数据分布与实际不符时估计多维乘积误差模型参数的方法，和该模型在预测领域的应用。论文的第一部分讨论了两种在以前的文献中被用来估计该模型的估计方法：最大似然估计法和广义矩估计法。并在对数据做了不同的干扰后比较了这两种方法。比较结果显示这两种方法都易受偏离值的影响。因此论文的第二部分提出了一种新的估计方法：权重经验似然估计法。在模拟实验和使用包含了当前经济危机间断数据的标准普尔指数的实际实验中，对比最大似然估计法和广义矩估计法，权重经验似然函数显示出了对偏离值有更好的抗性。论文的第三部分进一步研究了多维乘积误差模型在预测中的应用。并且这一部分还提出了实波动性的一种新的分解方式。分解得到的两个新的变量可以被多维乘积误差模型所模拟。通过比较标准普尔指数和纳斯达克指数的预测结果，比起以前用来估计实波动性的三种模型，多维乘积向量模型和新的分解方式显示出了更强的预测能力。 / This thesis studies the estimations of vector Multiplicative Error Model (MEM) under different kinds of model mismatches and its application in forecasting. In the first part of the thesis, two estimation methods, Maximum Likelihood (ML) method and Generalized Method of Moments (GMM), which have previously been used on vector MEM, are compared through different situations of data contaminations. From the comparison results it is found that both ML and GMM estimators are suspected to outliers in data. Therefore in the second part of the thesis a novel estimator is proposed: Weighted Empirical Likelihood (WEL) estimator. It is shown to be more robust than ML and GMM estimators in simulations, and also in forecasting realized volatility and bipower volatility of S&P 500 stock index including the current financial crisis period. The forecast ability of vector MEM is further addressed in the third part of the thesis, where an alternative decomposition of realized volatility is proposed, and vector MEM is used to model and forecast the two components of realized volatility. From the realized volatility forecasts of S&P 500, NASDAQ and Dow Jones, this decomposition together with vector MEM are illustrated to have superior performances over three competing models which have been applied on forecasting realized volatility before. / Detailed summary in vernacular field only. / Ding, Hao. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 203-213). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese. / Abstract --- p.i / Acknowledgement --- p.iv / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Outline of the thesis --- p.5 / Chapter 1.2 --- Conclusion --- p.7 / Chapter 2 --- Background study --- p.9 / Chapter 2.1 --- Multiplicative Error Model --- p.9 / Chapter 2.1.1 --- Introduction --- p.9 / Chapter 2.1.2 --- Developments of MEM --- p.12 / Chapter 2.1.3 --- Vector MEM --- p.17 / Chapter 2.2 --- Two functions for multivariate analysis --- p.25 / Chapter 2.2.1 --- Copula function --- p.25 / Chapter 2.2.2 --- Depth function --- p.32 / Chapter 3 --- Two Estimators for Vector MEM --- p.39 / Chapter 3.1 --- Two Stage Maximum Likelihood --- p.40 / Chapter 3.1.1 --- Introduction --- p.41 / Chapter 3.1.2 --- Simulation of two stage ML --- p.44 / Chapter 3.2 --- Maximum Likelihood estimator --- p.48 / Chapter 3.2.1 --- Derivatives of score function --- p.50 / Chapter 3.3 --- GMM estimator --- p.57 / Chapter 3.4 --- Comparing ML and GMM through simulations --- p.60 / Chapter 3.4.1 --- Generation of clean data --- p.61 / Chapter 3.4.2 --- Data contamination --- p.62 / Chapter 3.4.3 --- Optimization --- p.64 / Chapter 3.4.4 --- Resutls on clean data --- p.65 / Chapter 3.4.5 --- Results on contaminated data --- p.66 / Chapter 3.5 --- conclusion --- p.69 / Chapter 4 --- Weighted Empirical Likelihood Estimator --- p.77 / Chapter 4.1 --- Introduction --- p.78 / Chapter 4.2 --- Vector multiplicative error model and two estimation methods --- p.83 / Chapter 4.3 --- Weighted Empirical Likelihood --- p.88 / Chapter 4.3.1 --- Inner optimization --- p.93 / Chapter 4.3.2 --- Calculation of weights --- p.97 / Chapter 4.4 --- Simulation study on outliers --- p.101 / Chapter 4.4.1 --- Clean data --- p.103 / Chapter 4.4.2 --- Outliers --- p.105 / Chapter 4.4.3 --- Simulation results --- p.108 / Chapter 4.5 --- Computations of high dimension vector MEM --- p.111 / Chapter 4.5.1 --- The influences of dimension on ML --- p.111 / Chapter 4.5.2 --- The influences of dimension on GMM --- p.113 / Chapter 4.5.3 --- The influences of dimension on WEL --- p.115 / Chapter 4.5.4 --- Simulation --- p.116 / Chapter 4.6 --- Compare weighted empirical likelihood and empirical likelihood --- p.118 / Chapter 4.7 --- Empirical example --- p.121 / Chapter 4.7.1 --- Model --- p.123 / Chapter 4.7.2 --- Forecast comparison criteria --- p.125 / Chapter 4.7.3 --- Results --- p.126 / Chapter 4.8 --- Conclusions --- p.127 / Chapter 5 --- Forecast RV by Vector MEM --- p.142 / Chapter 5.1 --- Introduction --- p.143 / Chapter 5.2 --- Multiplicative jump and vector MEM --- p.148 / Chapter 5.2.1 --- Multiplicative jump --- p.148 / Chapter 5.2.2 --- Vector MEM for jump and continuous components --- p.153 / Chapter 5.3 --- Empirical analysis --- p.156 / Chapter 5.3.1 --- Data summary --- p.157 / Chapter 5.3.2 --- Models --- p.160 / Chapter 5.3.3 --- Forecast comparison criteria --- p.164 / Chapter 5.3.4 --- Before-crisis period --- p.166 / Chapter 5.3.5 --- Crisis period --- p.172 / Chapter 5.3.6 --- Comparing M-jump and log M-jump --- p.176 / Chapter 5.3.7 --- Conclusion on empirical analysis --- p.183 / Chapter 5.4 --- Conclusion --- p.185 / Chapter 6 --- Conclusion and future Work --- p.198 / Bibliography --- p.203

published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

This thesis studies two types of problems in financial derivatives pricing. The first type is the free boundary problem, which can be formulated as a partial differential equation (PDE) subject to a set of free boundary condition. Although the functional form of the free boundary condition is given explicitly, the location of the free boundary is unknown and can only be determined implicitly by imposing continuity conditions on the solution. Two specific problems are studied in details, namely the valuation of fixed-rate mortgages and CEV American options. The second type is the multi-dimensional problem, which involves multiple correlated stochastic variables and their governing PDE. One typical problem we focus on is the valuation of basket-spread options, whose underlying asset prices are driven by correlated geometric Brownian motions (GBMs). Analytic approximate solutions are derived for each of these three problems. / For each of the two free boundary problems, we propose a parametric moving boundary to approximate the unknown free boundary, so that the original problem transforms into a moving boundary problem which can be solved analytically. The governing parameter of the moving boundary is determined by imposing the first derivative continuity condition on the solution. The analytic form of the solution allows the price and the hedging parameters to be computed very efficiently. When compared against the benchmark finite-difference method, the computational time is significantly reduced without compromising the accuracy. The multi-stage scheme further allows the approximate results to systematically converge to the benchmark results as one recasts the moving boundary into a piecewise smooth continuous function. / For the multi-dimensional problem, we generalize the Kirk (1995) approximate two-asset spread option formula to the case of multi-asset basket-spread option. Since the final formula is in closed form, all the hedging parameters can also be derived in closed form. Numerical examples demonstrate that the pricing and hedging errors are in general less than 1% relative to the benchmark prices obtained by numerical integration or Monte Carlo simulation. By exploiting an explicit relationship between the option price and the underlying probability distribution, we further derive an approximate distribution function for the general basket-spread variable. It can be used to approximate the transition probability distribution of any linear combination of correlated GBMs. Finally, an implicit perturbation is applied to reduce the pricing errors by factors of up to 100. When compared against the existing methods, the basket-spread option formula coupled with the implicit perturbation turns out to be one of the most robust and accurate approximation methods. / 本論文為金融衍生產品定價的兩類問題作出了研究。第一類是自由邊界問題，它可以制定一個受制於自由邊界條件的偏微分方程式(PDE)，雖然當中自由邊界條件的函數形式是已知的，但自由邊界的位置是未知的，只能通過為實際解施加連續性條件作隱式確定。這裡為兩個具體問題進行了研究，分別是固定利率按揭合約(fixed-rate mortgages)定價和方差恆彈性模型的美式期權(CEV American options)定價。第二類是多維問題，它涉及到多個相關隨機變量及他們引申出的多維PDE。這裡為一個典型例子進行了研究，稱為籃子差異期權(basket-spread options)，其基礎資產價格由相關的幾何布朗運動驅動。我們為這三個問題提出了解析近似解。 / 對於上述的自由邊界問題，我們提出了一項參數移動邊界來近似模仿未知的自由邊界，使原來的自由邊界問題轉化為移動邊界問題，從而提出一種解析近似解。控制移動邊界的參數是通過滿足近似解的一階導數連續性條件來定。得到了解析近似解令當中的衍生產品定價和避險參數能有效快速地計算出，相比於有限差分法(finite-difference method)，精度保持了但計算時間顯著降低。再透過應用一個多階段方案，將移動邊界重鑄成一項分段光滑的連續函數，能有系統地將近似解的結果逼近有限差分法的結果。 / 對於上述的多維問題，我們從Kirk(1995)的二維差異期權(spread option)近似解定價公式推廣到多維的籃子差異期權。由於最終的定價公式是封閉形式，所有避險參數也從而得到封閉式近似解。從一些模擬例子顯示出，近似解的定價和避險參數，與通過數值積分法(numerical integration)或蒙地卡羅模擬法(Monte Carlo simulation)獲得的基準值比較，只有小於百分之一的誤差。此外，透過利用一種期權價格和相關基礎變量的概率分佈關係，我們進一步推論出一項籃子差異變量的近似解分佈函數，這可應用到任何多維幾何布朗運動的線性組合變量分佈。最後，我們提出一種隱式攝動方法，把定價誤差減少高達一百倍，跟現有的近似解定價方法相比，這是其中一種最健全和準確的籃子差異期權定價方法。 / Lau, Chun Sing = 自由邊界和多維的金融衍生產品定價問題 : 解析近似解 / 劉振聲. / Thesis Ph.D. Chinese University of Hong Kong 2014. / Includes bibliographical references (leaves 174-186). / Abstracts also in Chinese. / Title from PDF title page (viewed on 12, September, 2016). / Lau, Chun Sing = Zi you bian jie he duo wei de jin rong yan sheng chan pin ding jia wen ti : jie xi jin si jie / Liu Zhensheng. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only.

An important problem in mathematical finance is to develop option pricing models that are able to capture implied volatility “smile” or “skew” commonly observed in financial markets. Many existing models are based on time-homogeneous Markov processes and they often have difficulty in calibrating implied volatilities across both strikes and maturities. In this dissertation, we develop two parsimonious and analytically tractable option pricing models to evaluate VIX options and crack spread options, respectively. Our modeling approach is based on additive subordination, which is a natural generalization of classical Bochner’s subordination. Probabilistically, additive subordination corresponds to a stochastic time change with respect to an independent additive subordinator. To model the VIX dynamics, we timechange a non-affine mean-reverting 3/2 diffusion with an independent additive subordinator to capture its empirical features, such as mean reversion and jumps, as well as upward-sloping implied volatility skew in VIX options. Moreover, we develop a parsimonious and analytically tractable two-factor model for crude oil and its refined product to evaluate crack spread option, where each factor is an additive subordinate Cox-Ingersoll-Ross process. This model captures key empirical features of individual commodities, such as mean-reversion and jumps, as well as of their spread. Analytical formulas for related options prices under each model are derived via an eigenfunction expansion approach. Empirical results show that our models have great flexibility in calibrating implied volatilities across strikes and maturities of each underlying with excellent performance. Our results suggest that additive subordination is a useful technique that allows one to construct a large family of jump-diffusions and/or pure jump processes with rich time- and state-dependent local characteristics, which are suited for parsimoniously reproducing empirical features with analytical tractability. / 金融數學中的一個重要問題是建立能夠捕獲金融市場普遍觀察到的隱含波動率微笑現象的期權定價模型。許多現存的模型基於時間齊次的馬爾可夫過程且這些模型一般難以同時校準具有各種執行價格和到期時間的隱含波動率。在此博士論文中，我們建立了兩個簡潔且易於分析的期權定價模型，分別用於定價VIX期權和裂變價差期權。我們的建模方法基於additivesubordination，該方法是經典的Bochner的Subordination方法的自然延伸。從概率論上講，additive subordination定義了一個關於additive subordinator的隨機時間變換。為了對VIX的動態變化建模，我們對一個具有非仿射均值回复的3/2擴散過程進行時間變換來捕獲VIX的相關性質，如均值回复和跳躍，以及VIX期權中的向上偏的隱含波動率曲線。進一步，我們對原油和其成品油創建了一個簡潔的且易於分析的俩因子模型來定價裂變價差期權，其中每一個因子都是一個additive subordinate Cox-Ingersoll-Ross過程。這個模型可以捕獲每個油品價格的關鍵屬性，如均值回复和跳躍，以及其他之間的價差。每個模型下的相應期權價格的解析公式通過特偵函數展開的方式求解得到。實證研究表明我們的模型具有較好的靈活性，在校準每個期權品種的隱含波動率曲面方面都具有非常好的表現。我們的研究結果也表明additive subordination是一個非常有用的方法。它可以用於創建一大類具有時間和狀態相依特偵的跳躍擴散或純跳過程，這些過程可用於簡便的建模一些實證特徵且便於分析。 / Li, Jing. / Thesis Ph.D. Chinese University of Hong Kong 2015. / Includes bibliographical references (leaves 129-142). / Abstracts also in Chinese. / Title from PDF title page (viewed on 13, September, 2016). / Detailed summary in vernacular field only.

published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

In the first essay, this thesis provides a new methodology for pricing the fixed income derivatives using the arbitrage-free Heath-Jarrow-Morton model (hereafter HJM model). While, most previous empirical implementations of HJM model like that by Amin and Morton (1994) are focused on one-factor model only, the essay attempts to extend the test to a two-factor model that could further capture the subtleties of the forward rate process. The two-factor Poisson-Gaussian version of HJM model derived by Das (1999) that incorporates a jump component as the second state variables is used to value the actively traded Eurodollar futures call option under the jump diffusion lattice. The one-factor and two-factor models are compared with five volatility functions to evaluate the degree of pricing improvement by the inclusion of one more state variable. / The essay also addresses the critical issues on the volatility structure of forward rates that affect the pricing performance of option under the HJM framework. Three new volatility specifications are constructed to estimate the traded options. The first volatility function is the humped & curvature adjusted model that allows for humped shape in volatility structure and better adjustment to the curvature of the term structure. The second is the humped & proportional model that exhibits humped volatility feature and is proportional to the forward rate. The third function is the linear exponential model that is extended from Vasicek's exponential model. They are compared with two other volatility structures developed by previous researchers on their pricing performances. The alternative models are examined from the perspectives of in-sample fit, out-of-sample pricing and hedging. / The second essay develops an approach for estimating the Value-at-Risk (hereafter VaR) with jumps using the Monte Carlo simulation method. It is by far the first paper to estimate VaR using the HJM model. The paper takes the framework of the Poisson Gaussian version of HJM model (hereafter, HJM jump-diffusion model) from Das (1999). The model is incorporated with a jump component to capture the kurtosis effect in the daily price changes. As a result, the HJM jump-diffusion model allows for the fat tailed and skewed distribution of return in most financial markets. The simulation process is expedited by using variance reduction method. The model is used for calculating the VaR of a portfolio consisting of the fixed income derivatives. The accuracy of the VaR estimates is examined statistically at the VaR at confidence level of both 95 and 99 percent. / This thesis is a collection of two essays that explore issues related to the pricing and the risk management of fixed income securities and derivatives in US markets. In the context of the pricing of derivatives, the arbitrage-free pricing approach is adopted. For the issue of risk management, the estimation of Value-at-Risk is presented. / by Ze-To Yau Man. / Source: Dissertation Abstracts International, Volume: 62-09, Section: A, page: 3138. / Supervisors: Jia He; Ying-foon Chow. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (p. 145-151). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest dissertations and theses, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Chan, Tsz Him. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (p. 94-96). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Monte Carlo method in Finance --- p.6 / Chapter 2.1 --- Principle of MC and pricing theory --- p.6 / Chapter 2.2 --- An illustrative example --- p.9 / Chapter 3 --- Discretization method --- p.15 / Chapter 3.1 --- The Euler scheme and Milstein scheme --- p.16 / Chapter 3.2 --- Convergence of Mean Square Error --- p.19 / Chapter 4 --- Quasi Monte Carlo method --- p.22 / Chapter 4.1 --- Basic idea of QMC --- p.23 / Chapter 4.2 --- Application of QMC in Finance --- p.29 / Chapter 4.3 --- Another illustrative example --- p.34 / Chapter 5 --- Our Methodology --- p.42 / Chapter 5.1 --- Measure decomposition --- p.43 / Chapter 5.2 --- QMC in SDE simulation --- p.51 / Chapter 5.3 --- Towards a workable algorithm --- p.58 / Chapter 6 --- Numerical Result --- p.69 / Chapter 6.1 --- Case I Generalized Wiener Process --- p.69 / Chapter 6.2 --- Case II Geometric Brownian Motion --- p.76 / Chapter 6.3 --- Case III Ornstein-Uhlenbeck Process --- p.83 / Chapter 7 --- Conclusion --- p.91 / Bibliography --- p.96

Cheng, Ping. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (p. 91-96). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background --- p.1 / Chapter 1.1.1 --- Accumulator in a Nutshell --- p.1 / Chapter 1.1.2 --- Criticism over Accumulators --- p.3 / Chapter 1.1.3 --- Significance of Research over Accumulators --- p.4 / Chapter 1.1.4 --- Contribution of this Research --- p.5 / Chapter 1.2 --- Literature Review --- p.6 / Chapter 1.2.1 --- Literature on Option Pricing Theory --- p.6 / Chapter 1.2.2 --- Literature on Occupation Time Derivatives and Accumulators --- p.9 / Chapter 1.2.3 --- Accumulators as Occupation Time Deriva- tives --- p.14 / Chapter 1.3 --- Structure of this Thesis --- p.16 / Chapter 2 --- Theoretical Foundation --- p.17 / Chapter 2.1 --- Black Scholes Framework --- p.18 / Chapter 2.1.1 --- The Model --- p.18 / Chapter 2.1.2 --- Girsanov's Theorem --- p.20 / Chapter 2.1.3 --- Simulation --- p.21 / Chapter 2.2 --- Heston Framework --- p.22 / Chapter 2.2.1 --- Motivation to Extend to Heston Model --- p.22 / Chapter 2.2.2 --- The Model --- p.23 / Chapter 2.2.3 --- The Monte Carlo Method --- p.25 / Chapter 3 --- Pricing under Black-Scholes Framework --- p.30 / Chapter 3.1 --- Structure One (Suspension Feature) --- p.30 / Chapter 3.1.1 --- Introduction --- p.30 / Chapter 3.1.2 --- Model --- p.32 / Chapter 3.1.3 --- Sensitivity Tests --- p.35 / Chapter 3.1.4 --- Simulation Results --- p.38 / Chapter 3.2 --- Structure Two (Knock-out Feature) --- p.40 / Chapter 3.2.1 --- Introduction --- p.40 / Chapter 3.2.2 --- Model --- p.41 / Chapter 3.2.3 --- Sensitivity Tests --- p.45 / Chapter 3.2.4 --- Simulation Results --- p.48 / Chapter 3.3 --- Structure Three (Knock-out & Double Commit- ment Feature) --- p.50 / Chapter 3.3.1 --- Introduction --- p.50 / Chapter 3.3.2 --- Model --- p.51 / Chapter 3.3.3 --- Sensitivity Tests --- p.56 / Chapter 3.3.4 --- Simulation Results --- p.58 / Chapter 4 --- Extension: Pricing under Heston Framework --- p.59 / Chapter 4.1 --- Structure One --- p.59 / Chapter 4.1.1 --- Pricing of the Contract --- p.59 / Chapter 4.2 --- Structure Two and Three --- p.61 / Chapter 4.2.1 --- Simulation Results --- p.62 / Chapter 4.3 --- Heston Parameters Estimates --- p.63 / Chapter 5 --- Discussion --- p.66 / Chapter 5.1 --- Volatility of Accumulators --- p.66 / Chapter 5.2 --- Instability in the Model Parameters --- p.69 / Chapter 5.3 --- Premium over Accumulators --- p.71 / Chapter 5.4 --- Return of the Accumulator Products --- p.72 / Chapter 6 --- Future Work & Conclusion --- p.75 / Chapter 6.1 --- Future Work --- p.75 / Chapter 6.2 --- Conclusion --- p.76 / Chapter A --- Other Parameters Estimation --- p.77 / Chapter B --- Sample Contracts --- p.80 / Chapter B.1 --- Equity Accumulator --- p.80 / Chapter B.2 --- Commodity Accumulator --- p.80 / Chapter B.3 --- FX-Linked Accumulation --- p.80 / Bibliography --- p.91

This thesis comprises three essays that explore several theoretical and empirical features of affine term structure models. In the first essay, we focus on the ability of continuous-time affine term structure models to capture time variability in the second conditional moment. Using data on US Treasury yields, we conclude that affine term structure models are much better at extracting time-series volatility from the cross-section of yields than argued in the literature. These models have nonetheless difficulty capturing volatility dynamics at the short end of the maturity spectrum, perhaps indicating some form of segmentation between long-maturity and short-maturity bonds. These results are robust to the choice of sample period, interpolation method and estimation method. In the second essay, we propose the use of the unscented Kalman filter technique for the estimation of affine term structure models using non-linear instruments. We focus on swap rates and show that the unscented Kalman filter leads to important reductions in bias and gains in precision. The use of the unscented Kalman filter results in substantial improvements in out-of-sample forecasts. Our findings suggest that the unscented Kalman filter may prove to be a good approach for a number of problems in fixed income pricing in which the relationship between the state vector and the observations is nonlinear, such as the estimation of term structure models using interest rate derivatives or coupon bonds, and the estimation of quadratic term structure models. The third essay provides a tractable framework for pricing defaultable securities with recovery risk. Pricing solutions are explored for a large family of discrete-time affine processes and a five-factor Gaussian model is estimated on BBB and B Standard and Poor's yield indices. This rich econometric setup allows the model to simultaneously capture two important stylized facts of defaultable securities: The positive correlation between the loss given default and the intensity of default, and the negative correlation between the intensity of default and the risk-free interest rate.