本论文主要研究两类风险估计问题：一类是美式期权价格关于模型参数的敏感性估计， 另一类是投资组合的风险估计。针对这两类问题，我们相应地提出了高效的蒙特卡洛模拟方法。这构成了本文的两个主要部分。 / 第二章是本文的第一部分。在这章中，我们将美式期权的敏感性估计问题提成了更具一般性的估计问题：如果一个随机最优化问题依赖于某些模型参数， 我们该如何估计其最优目标函数关于参数的敏感性。在该问题中， 由于最优决策关于模型参数可能不连续，传统的无穷小扰动分析方法不能直接应用。针对这个困难，我们提出了一种广义的无穷小扰动分析方法，得到敏感性的无偏估计。 我们的方法显示, 在估计敏感性时， 其实并不需要样本路径关于参数的可微性。这是我们在理论上的新发现。另一方面， 该方法可以非常容易的应用于美式期权的敏感性估计。在实际应用中敏感性的无偏估计可以直接嵌入流行的美式期权定价算法，从而同时得到期权价格和价格关于模型参数的敏感性。包括高维问题和多种不同的随机过程模型在内的数值实验， 均显示该估计在计算上具有显著的优越性。最后，我们还从理论上刻画了美式期权的近似最优执行策略对敏感性估计的影响，给出了误差上界。 / 第三章是本文的第二部分。在本章中，我们研究投资组合的风险估计问题。该问题也可被推广成一个一般性的估计问题：如何估计条件期望在作用上一个非线性泛函之后的期望。针对该类估计问题，我们提出了一种多层模拟方法。我们的估计量实际上是一些简单嵌套估计量的线性组合。我们的方法非常容易实现，并且可以被广泛应用于不同的问题结构。理论分析表明我们的方法适用于不同维度的问题并且算法复杂性低于文献中现有的方法。包括低维和高维的数值实验验证了我们的理论分析。 / This dissertation mainly consists of two parts: a generalized infinitesimal perturbation analysis (IPA) approach for American option sensitivities estimation and a multilevel Monte Carlo simulation approach for portfolio risk estimation. / In the first part, we develop efficient Monte Carlo methods for estimating American option sensitivities. The problem can be re-formulated as how to perform sensitivity analysis for a stochastic optimization problem when it has model uncertainty. We introduce a generalized IPA approach to resolve the difficulty caused by discontinuity of the optimal decision with respect to the underlying parameter. The unbiased price-sensitivity estimators yielded from this approach demonstrate significant advantages numerically in both high dimensional environments and various process settings. We can easily embed them into many of the most popular pricing algorithms without extra simulation effort to obtain sensitivities as a by-product of the option price. This generalized approach also casts new insights on how to perform sensitivity analysis using IPA: we do not need pathwise differentiability to apply it. Another contribution of this chapter is to investigate how the estimation quality of sensitivities will be affected by the quality of approximated exercise times. / In the second part, we propose a multilevel nested simulation approach to estimate the expectation of a nonlinear function of a conditional expectation, which has a direct application in portfolio risk estimation problems under various risk measures. Our estimator consists of a linear combination of several standard nested estimators. It is very simple to implement and universally applicable across various problem settings. The results of theoretical analysis show that the algorithmic complexities of our estimators are independent of the problem dimensionality and are better than other alternatives in the literature. Numerical experiments, in both low and high dimensional settings, verify our theoretical analysis. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Liu, Yanchu. / "December 2012." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 89-96). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Abstract --- p.i / Abstract in Chinese --- p.iii / Acknowledgements --- p.v / Contents --- p.vii / List of Tables --- p.ix / List of Figures --- p.xii / Chapter 1. --- Overview --- p.1 / Chapter 2. --- American Option Sensitivities Estimation via a Generalized IPA Approach --- p.4 / Chapter 2.1. --- Introduction --- p.4 / Chapter 2.2. --- Formulation of the American Option Pricing Problem --- p.10 / Chapter 2.3. --- Main Results --- p.14 / Chapter 2.3.1. --- A Generalized IPA Approach in the Presence of a Decision Variable --- p.16 / Chapter 2.3.2. --- Unbiased First-Order Sensitivity Estimators --- p.21 / Chapter 2.4. --- Implementation Issues and Error Analysis --- p.23 / Chapter 2.5. --- Numerical Results --- p.26 / Chapter 2.5.1. --- Effects of Dimensionality --- p.27 / Chapter 2.5.2. --- Performance under Various Underlying Processes --- p.29 / Chapter 2.5.3. --- Effects of Exercising Policies --- p.31 / Chapter 2.6. --- Conclusion Remarks and Future Work --- p.33 / Chapter 2.7. --- Appendix --- p.35 / Chapter 2.7.1. --- Proofs of the Main Results --- p.35 / Chapter 2.7.2. --- Likelihood Ratio Estimators --- p.43 / Chapter 2.7.3. --- Derivation of Example 2.3 --- p.49 / Chapter 3. --- Multilevel Monte Carlo Nested Simulation for Risk Estimation --- p.52 / Chapter 3.1. --- Introduction --- p.52 / Chapter 3.1.1. --- Examples --- p.53 / Risk Measurement of Financial Portfolios --- p.53 / Derivatives Pricing --- p.55 / Partial Expected Value of Perfect Information --- p.56 / Chapter 3.1.2. --- A Standard Nested Estimator --- p.57 / Chapter 3.1.3. --- Literature Review --- p.59 / Chapter 3.1.4. --- Summary of Our Contributions --- p.61 / Chapter 3.2. --- The Multilevel Approach --- p.63 / Chapter 3.2.1. --- Motivation --- p.63 / Chapter 3.2.2. --- Multilevel Construction --- p.65 / Chapter 3.2.3. --- Theoretical Analysis --- p.67 / Chapter 3.2.4. --- Further Improvement by Extrapolation --- p.69 / Chapter 3.3. --- Numerical Experiments --- p.72 / Chapter 3.3.1. --- Single Asset Setting --- p.73 / Chapter 3.3.2. --- Multiple Asset Setting --- p.74 / Chapter 3.4. --- Concluding Remarks --- p.77 / Chapter 3.5. --- Appendix: Technical Assumptions and Proofs of the Main Results --- p.79 / Bibliography --- p.89
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_328174 |
| Date | January 2013 |
| Contributors | Liu, Yanchu., Chinese University of Hong Kong Graduate School. Division of Systems Engineering and Engineering Management. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography |
| Format | electronic resource, electronic resource, remote, 1 online resource (xiii, 96 leaves) : ill. |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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