Digital options, the basic building blocks for valuing complex financial assets, they play an important role in options valuation and hedging. We survey the digital options pricing formula under diffusion processes and jump-diffusion processes. Since the existent first-touch digital options pricing formulas with jump-diffusion processes are all in their Laplace transform of the option value. To inverse the Laplace transforms is critical when doing options valuation. Therefore, we adopt a phase-type jump-diffusion model which is developed by Chen, Lee and Sheu [2007] as our main model, and use FFT inversion to get the first-touch digital option price under (2,2)-factor exponential jump-diffusion processes.

Main objective of this diploma thesis is to point out to the advantages related to the applications of Exotic options and show that we have to be aware of complexities which arise in hedging such products. There exists a quantity of different Exotic options products so the first chapter is dedicated to its basic classification, although not all instruments were included, as some are very specific. According to the application of options, we took out the most used Exotic options. The number one in the Exotic options world, are the Barrier options, followed by Digital options

In mathematical finance, the sensitivities of option prices to various market parameters, also known as the “Greeks”, reflect the exposure to different sources of risk. Computing these is essential to predict the impact of market moves on portfolios and to hedge them adequately. This is commonly done using Monte Carlo simulations. However, obtaining accurate estimates of the Greeks can be computationally costly. Multilevel Monte Carlo offers complexity improvements over standard Monte Carlo techniques. However the idea has never been used for the computation of Greeks. In this work we answer the following questions: can multilevel Monte Carlo be useful in this setting? If so, how can we construct efficient estimators? Finally, what computational savings can we expect from these new estimators? We develop multilevel Monte Carlo estimators for the Greeks of a range of options: European options with Lipschitz payoffs (e.g. call options), European options with discontinuous payoffs (e.g. digital options), Asian options, barrier options and lookback options. Special care is taken to construct efficient estimators for non-smooth and exotic payoffs. We obtain numerical results that demonstrate the computational benefits of our algorithms. We discuss the issues of convergence of pathwise sensitivities estimators. We show rigorously that the differentiation of common discretisation schemes for Ito processes does result in satisfactory estimators of the the exact solutions’ sensitivities. We also prove that pathwise sensitivities estimators can be used under some regularity conditions to compute the Greeks of options whose underlying asset’s price is modelled as an Ito process. We present several important results on the moments of the solutions of stochastic differential equations and their discretisations as well as the principles of the so-called “extreme path analysis”. We use these to develop a rigorous analysis of the complexity of the multilevel Monte Carlo Greeks estimators constructed earlier. The resulting complexity bounds appear to be sharp and prove that our multilevel algorithms are more efficient than those derived from standard Monte Carlo.

本論文推導違約風險下四種新奇選擇權的評價模型及其避險比率，依序為數據選擇權、寬它選擇權、互換選擇權，極值選擇權。並比較無違約風險與違約風險下的評價模型之差異。假若違約風險不存在時，違約風險下各種類型選擇權的評價模型皆會縮減成為無違約風險下所對應的評價模型。避險比率亦為如此。數值範例則印證違約風險下選擇權的價值較無違約風險選擇權的價值低。本論文完成目前尚無任何學術研究於違約風險下四種新奇選擇權的評價模型及避險比率。這是一個重要貢獻。 關鍵詞：違約風險、新奇選擇權、數據選擇權、寬它選擇權、互換選擇權、極值選擇權。 / This paper presents the analytic pricing formula and the hedging ratio of four kinds of exotic options with correlated credit risk. They are Digital options, Quanto Options, Exchange Options and Extreme-value Options, respectively. Furthermore, compare the discrepancy of the models under the condition whether the default risk exists. Finding that if there is no default risk, all models that we derive will reduce to the corresponding models with no default risks, and so do the hedging ratio. Numerical examples certify that the value of the vulnerable options will be lower than that of the ordinary options. All above that finished has not been done by existing researches and it is a chief contribution in this paper. Keywords: Exotic Options, Credit Risk, Digital Options, Quanto Options, Exchange Options, Extreme-value Options, Default Risk.

Web 2.0是一種架構在知識環境上的網路，使用者可以協同參與網站內容、工具和應用軟體的使用、發展與管理。為了創造更多使用者價值，各式各樣的Web 2.0新服務因應而生，例如：Wikipedia和Facebook，就是從這種服務導向的技術架構中衍生出來，進而提供有效的知識創造與社群互動。同時，由於Web 2.0平台具有協同合作與互動等特性，使用者的知識分享意願亦會受到平台中的知識與流程(即Web 2.0選擇權)所影響，因此，使用者參與和分享是十分重要的議題。 為了有效管理這類社群網路平台的知識創造與持續創新相關活動，本研究進行了兩階段的調查。在第一階段中，主要探索Web 2.0對於合作學習的知識創造服務。本研究分析並分類1,042個 Web 2.0 網站，進而定義出兩種知識創造服務平台(經驗交流型與智慧累進型) ，以及五種Web 2.0服務模式，包含交換模式、整合模式、組織模式、協同合作模式與開放模式，由於這些服務模式具體描繪出現有Web 2.0的不同應用，因此能清楚地闡釋Web 2.0科技的運用模式與價值建立。 第二階段本研究根據認知理論，調查知識貢獻者的分享動機，包括認知有用性、社群認同感、利他主義傾向、認知娛樂性和Web 2.0自我效能，以及Web 2.0選擇權對於分享意圖的調節效果。本研究對象來自於全球性樣本，即第一階段中兩大類型Web 2.0平台的貢獻者，經驗交流型平台蒐集了568位有效樣本；智慧累進型平台則有694位有效樣本。結果發現，Web 2.0選擇權會顯著地影響使用者的知識分享動機。對於經驗交流型平台而言，高度的知識與流程選擇權會顯著地調節知識貢獻者的社群認同感、利他主義傾向以及Web 2.0自我效能對於知識分享的意圖；對於智慧累進型平台而言，低度的知識與流程選擇權會顯著地調節知識貢獻者的社群認同感與認知有用性對於知識分享的意圖，而高度的流程選擇權則會調節貢獻者的認知娛樂性對於分享的意願。因此，組織在提供Web 2.0服務時，無論是為了人際交流目的或是知識累積目的，皆需要妥善地利用Web 2.0選擇權，才能有效建立良善的知識分享循環之平台。 / Web 2.0 is a network on which individuals contribute to the development and distribution of content, tools, and software applications over the Internet. To provide different values to the users, various innovative Web 2.0 services, such as Wikipedia and Facebook, have emerged from this omnibus technological concept, and the Web 2.0 platform has been praised for providing an effective platform for knowledge creation and group innovation. For a cyclical flow of knowledge generation and reinvention, user participation and contribution become the most critical part of the platform. Meanwhile, due to the collaborative and interactive features of the platform, this user intention can be highly affected by the Web 2.0 options—i.e., the reach and richness of knowledge and processes enabled on the Web 2.0 platforms. In order to effectively leverage this kind of social network platform for knowledge creation and innovation sustainment, this study has two stages. In Stage one, it explores Web 2.0 knowledge-creating services for collaborative learning. More than 1,000 Web 2.0 application sites were analyzed and classified, and two types of knowledge-creating services with five service models were identified. The two services are called experience-socialization and intelligence-proliferation, and the service models are coined with the names of Exchanger, Aggregator, Organizer, Collaborator, and Liberator. These service models represent the diversity of existing Web 2.0 applications and provide a base for better communications of the operating patterns and value propositions of the Web 2.0 technology. In Stage two, based on cognitive theories, it examines the determinants of knowledge sharing and the moderating effects of Web 2.0 options on the sharing intention. Motivators such as perceived usefulness, community identification, altruism tendency, perceived enjoyment, and Web 2.0 self-efficacy were verified by frequent contributors of all types. A global sample from contributors of two major types of Web 2.0 platforms, experience-socialization platforms (N = 568) and intelligence-proliferation platforms (N = 694), were collected. The results confirm that user motivations to contribute are moderated by Web 2.0 options. For contributors to experience-socialization platforms high knowledge and process options can affect their sense of community identity, altruism tendency, and Web 2.0 self-efficacy towards sharing intention. For contributors of intelligence-proliferation platform, low knowledge and process options affect the user’s sense of community identity and perceived usefulness that shapes their intention to share. By the same token, high process options affect contributors’ feeling of joyfulness towards contributing on the platform. Organizations providing Web 2.0 services for either socialization purposes or knowledge retention purposes need to leverage Web 2.0 options properly in order to build effective platforms and nurture a strong, growing stream of knowledge contribution.