The main aim of this thesis is in analyzing and empirically testing the various valuation models and hedging schemes of foreign exchange barrier options and their robustness with respect to changing of market conditions. The purpose of the main empirical section is to get a detailed understanding of the static and dynamic performance of the analyzed models for the barrier options payoff mainly in the extreme market conditions, where we performed a benchmarking of the various hedging schemes. As a by-product, we analyzed the accomplishment of some of the model assumptions in real world setting, and the model dependency of the barrier options.

Diplomová práce primárně řeší zda jsou exotické opce vhodné pro zajištění kurzových rizik a přináší návrh vhodné aplikace exotických opcí. Práce je zaměřena na úzkou skupinu exotických opcí, tzv. Path-Dependent opce. Tři často používané typy těchto opcí jsou analyzovány a testovány jak mezi sebou tak pro lepší porovnání i s klasickou vanilla opcí. Hlavním výstupem diplomové práce je návrh vhodného využití testovaných exotických opcí.

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.

本論文一共分為兩大主題，分別為匯率連結商品之評價與分析，及違約事件相關係數之估計。在結構型金融商品於市場上熱賣之後，金融業者紛紛投入財務工程領域，競相推出類似的產品。然而，自1971年世界各國開始實行浮動匯率制度之後，匯率風險較以往提高不少，因此各種不同設計的外匯衍生性商品開始不斷地問世。有鑑於此，本文希望藉由分析市場上的匯率商品：「新加坡華僑銀行一年期匯率連結結構型存款」，讓發行商和投資人了解結構型商品設計的要點與風險所在。在此商品中，本文利用多變數蒙地卡羅模擬法求出商品的近似價格，除了看發行商是否有利可尋之外，也提供發行商可行之避險策略。同時，本文也透過商品條款分析與情境分析，讓投資人了解其獲利所在與將面臨的風險。 / 由於近年來信用事件層出不窮，顯示出信用風險控管的重要性，信用衍生性商品也因而開始蓬勃發展。目前信用衍生性商品以信用違約交換為最大宗，擔保債權憑證(Collateralized Debt Obligations, CDO)為其次。由於一籃子信用衍生性商品和擔保債權憑證涉及多檔標的資產，在評價時，公司之間的違約相關係數是個重要因子，因此本文在另一個主題上，透過Robert Jarrow與Donald van Deventer(2005)提出的違約相關係數之估計方法，以簡約模型估計違約相關係數，利用台灣公司資料做實證分析，期許對連結多項標的資產之信用衍生性商品之評價有所幫助。