In this dissertation, we investigate the spread option pricing problem with stochastic interest rate. First, we will review the basic concept and theories of stochastic calculus, give an introduction of spread options and provide some examples of spread options in different markets. We will also review the market efficiency theory, arbitrage and assumptions that are commonly used in mathematical finance. In Chapter 3, we will review existing spread pricing models and term-structure models such as Vasicek Mode, and the Heath-Jarrow-Morton framework. In Chapter 4, we will use the martingale approach to derive a partial differential equation for the price of the spread option with stochastic interest rate. In Chapter 5, we will study the spread option numerically. We will conclude this dissertation with ideas for future research.

本研究以隨機利率模型應用至淨現值法、邊際年利率法、比較利率法及內部報酬率法等保單成本評價方法中，藉此觀察多期保單年度的價值變化，進而找出保單的報酬率、成本值或指數所呈現的趨勢，並考量在相同的情況下，比較各險種的成本或報酬優劣，最後希望消費者能在合理的隨機利率下更清楚了解保單成本的概念，並基於消費者對於合理的保單成本分析需求，能提供主管機關對於揭露保單成本的規範有更多的參考。 / 本研究發現各種保單在隨機利率變化的情況下，分紅終身壽險於各種成本分析方法中皆有良好的表現，不論是在考量淨成本結構的淨現值法，或是考量儲蓄性質、投資報酬為主的比較利率法以及各年度的邊際報酬利率等方法，整體而言分紅終身壽險對於消費者及保險公司應該是最優質的選擇。 / 再者以內部報酬率法應用隨機利率模型分析年金保險，可得知傳統遞延年金的報酬優於利率變動型年金。另外由於各種成本評價方法所著重的要素不同，想要了解保單完整全面性的評價，透過數個不同性質的保單成本分析方法計算較能呈現客觀且適切評價結果。 / This research is applied in the Stochastic Interest Model to the appraised method of insurance policy cost, such as Net Present Value Method, Marginal Yield Method, Comparative Interest Rate Method and Internal Rate of Return…etc., so as to observe the annual variation of value for different term of insurance policies, and then find out the rate of returns, cost value or trend appeared of index of the insurance policy, and consider it in the same cases to compare the good and bad from the cost or remuneration of every insurance. Hope consumers can finally clearer understand the concept of the insurance policy cost under the rational Stochastic Interest Rate, and on the basis of consumers’ demand for the rational insurance policy cost analysis can offer the competent authority more reference in revealing norms of the insurance policy cost. / In this research discovered that various insurance policies in changing of Stochastic Interest Rate, its Participating Whole-Life Insurance in varied cost analytical methods has good representation, no matter in considering the Net Present Value Method of the net cost structure, or considering Comparative Interest Rate Method of the main nature of deposits or main invest remuneration, and the annual marginal return interest rate…etc., the Participating Whole-Life Insurance should be generally the most high-quality choice to consumer and insurance company. / Moreover, according to the Internal Rate of Return, using the Stochastic Interest Model to analyze the annuity insurance can learn the remuneration of the Traditional Deferred Annuity is superior to the Interest Sensitive Annuity. In addition, as various cost appraised methods focused on different elements, if want to comprehension overall appraisal of insurance policy, then it can represent more objective and appropriate calculation through the analytical method of several different nature insurance policy cost.

In financial markets, spread option is a derivative security with two underlying assets and the payoff of the spread option depends on the difference of these assets. We consider American style spread option which allows the owners to exercise it at any time before the maturity. The complexity of pricing American spread option is that the boundary of the corresponding partial differential equation which determines the option price is unknown and the model for the underlying assets is two-dimensional.In this dissertation, we incorporate the stochasticity to the interest rate and assume that it satisfies the Vasicek model or the CIR model. We derive the partial differential equations with terminal and boundary conditions which determine the American spread option with stochastic interest rate and formulate the associated free boundary problem. We convert the free boundary problem to the linear complimentarity conditions for the American spread option, so that we can go around the free boundary and compute the option price numerically. Alternatively, we approximate the option price using methods based on the Monte Carlo simulation, including the regression-based method, the Lonstaff and Schwartz method and the dual method. We make the comparisons among the option prices derived by the partial differential equation method and Monte Carlo methods to show the accuracy of the result.

The globilisation phenomena is causing an increasing interaction between different markets and sectors. This has led to the evolution of derivative instruments from ”single asset” instruments to complex derivatives that have underlying assets from different markets, sectors and sub-sectors. These are the so-called hybrid products that have multi-assets as underlying instruments. This article focuses on interest rate hybrid products. In this article an analysis of the application of stochastic interest rate models and stochastic volatility models in pricing and hedging interest rate hybrid products will be explored.

隨著金融市場的開放，越來越多的金融商品被開發出來以迎合市場參予者的需求，利率衍生性金融商品是一種以利率為標的的一種新金融商品，而這種新金融商品的交易量也是相當的可觀。我們在設計金融商品的第一步就是要去定價，在現實社會中利率是隨機波動的而不是像在B-S的選擇權公式中是固定的。隨機利率模型的用途就是在描述利率隨機波動的行為，進而對利率衍生性金融商品定價。本文嘗試以隨機利率模型估計台灣公債市場的殖利率曲線，而殖利率曲線的建立對於固定收益證券及其衍生性金融商品的定價是很重要的。在台灣大部分的利率模型的研究都是利用模擬的方式做比較，這也許是因為資料取得上的問題，本文利用CKLS(1992)所提出的方式以GMM(Generalized Method of Moment)的估計方法，利用隨機利率模型估計出台灣公債市場的殖利率曲線。本文中將三種隨機利率模型做比較他們分別為: Vasicek model (Vasicek 1977),、隨機均數的Vasicek 模型 (BDFS 1998) ，以及隨機均數與隨機波動度的Vasicek 模型 (Chen,Lin 1996). 後面兩個模型是首次出現在台灣的研究文獻中。在本文的附錄中將提出如何利用偏微分方程式(PDE)的方法求解出這三個模型的零息債券價格的封閉解(Closed-Form Solution)。文中利用台灣商業本票的價格當作零息債券價格的近似值，再以RMSE (Root mean squared Price Prediction Error)作為利率模型配適公債市場價格能力的指標。本文的主要貢獻在於嘗試以隨機利率模型估計出台灣公債市場的殖利率曲線，以及介紹了兩種首次在台灣研究文獻出現的利率模型，並且詳細推導其債券價格的封閉解，這對於想要建構一個新的隨機利率模型的研究人員而言，這是一個相當好的一個練習。 / With the growth in the area of financial engineering, more and more financial products are designed to meet demands of the market participants. Interest rate derivatives are those instruments whose values depend on interest rate changes. These derivatives form a huge market worth several trillions of dollars. The first step to design or develop a new financial product is pricing. In the real world interest rate is not a constant as in the B-S option instead it changes over time. Stochastic interest rate models are used for capturing the volatile behavior of interest rate and valuing interest rate derivatives. Appropriate models are necessary to value these instruments. Here we want to use stochastic interest rate models to construct the yield curve of Taiwan Government Bond (TGB) market. It is important to construct yield curve for pricing some financial instruments such as interest rate derivatives and fixed income securities. 　In Taiwan Although most of the research surrounding interest rate models is intended towards studying their usefulness in valuing and hedging complex interest rate derivatives by simulation. But just a few papers focus on empirical study. Maybe this is due to the problems for data collection. In this paper we want to use stochastic interest models to construct the yield curve of Taiwan’s Government Bond market. The estimation method that we use in this paper is GMM (Generalized Method of Moment) followed CKLS (1992). I introduce three different interest rate model, Vasicek model (Vasicek 1977), Vasicek with stochastic mean model (BDFS 1998) and Vasicek with stochastic mean and stochastic volatility model (Chen,Lin 1996). The last two models first appear in Taiwan’s research. In the Chapter 3, I will introduce these models in detail and in the appendix of my thesis I will show how to use PDE approach to derive each model’s zero coupon bond price close-form solution. In this paper we regard Taiwan CP (cmmercial Paper) rates as a proxy of short rate to estimate the parameters of each model. Finally we use these models to construct the yield curve of Taiwan Government Bonds market and to tell which model has the best fitting bond prices performance. Our metric of performance for these models is RMSE (Root mean squared Price Prediction Error). The main contribution of this study is to construct the yield curve of TGB market and it is useful to price derivatives and fixed income securities and I introduce two stochastic interest rates models, which first appear in Taiwan’s research. I also show how to solve the PDE for a bond price and it is a useful practice for someone who wants to construct his/her own model.

In this thesis, we present two different approaches for the stochastic volatility and stochastic interest rate model with jump and analyze the performance of four alternative models. In the first approach, suggested by Scott, the closed form solution for prices on European call stock options are developed by deriving characteristic functions with the help of martingale methods. Here, we study the asset price process and give in detail the derivation of the European call option price process. The second approach, suggested by Bashki-Cao-Chen, describes the closed form solution of European call option by deriving the partial integro-differential equation. In this one we g ive the derivations of both asset price dynamics and the European call option price process. Finally, in the application part of the thesis, we examine the performance of four alternative models using General Electric Stock Option Data. These models are constructed by using the theoretical results of the second approach.

當保險契約被發行時，保險公司必須被要求盡可能的具備承擔未來不可知的風險的能力。本文將死亡風險、利率風險、脫退風險以及流動性風險引入，並針對生死合險契約進行盈餘分析。在此以 Vasicek (1977) 所提出之隨機利率模型、根據被保險人理性行為作為基礎之脫退模型以及引入簡化後的 Longstaff、Mithal與Nies (2005)流動性風險債券價格來描述各種風險。根據上述模型假設下計算保費及準備金，遂以蒙地卡羅模擬法量化源於各種風險之盈餘。最後，本文計算保險公司之盈餘對各風險參數之敏感度分析，並計算各期破產與發生流動性問題之可能性。 / Once insurance contracts are issued, the insurers should be capable to deal with the unknown conditions in the future as possible. In this paper, we analyze the impact of mortality, interest rate, surrender and liquidity risks on the surplus of endowment contract. We model the interest rate risk by Vasicek model, the surrender rate based on the rational behavior of policyholders and introduce the discounted price of zero coupon bonds as the liquidity risk. Under such assumptions, we compute the premium and reserve, demonstrate the simulated insurance surplus, and finally exhibit the statistics of the surplus from different sources. The simulated results show the sensitivity of the surplus to the parameters of the risks. At the same time, we also show the probabilities of insolvency and illiquidity of the insurer before the maturity date of the contract due to the fluctuating surrender rate and liquidity risk resulting from the stochastic interest rate.

One of the major issue for life insurance companies is proper and consistent valuation of liabilities. This thesis introduces the standard estimation methods used in practice and discussed the alternative methods, which might help to speed up these calculations. It studies two possible methods of acceleration of calcu- lations in life insurance: analytic function and cluster analysis. The outcome of these work is comparison of discussed methods applied on generated life insur- ance portfolio. All methods were applied on two possible insurance products. Comparison of the results is based on the calculation precision and time needed to process the liabilities of the insurance company's portfolio. 1

Finding the optimal capital structure has been a relevant subject for many decades. Therehas for a long time been a discrepancy between observed leverage ratio and those proposedby theory, with many different theories suggested and developed throughout time. One ofthose theories is the Contingent Claims Analysis (CCA). Based initially on Black & Scholes’option-pricing theory and formulas, and pioneered by Merton, the CCA-methodology hasthroughout the years been developed further and moved from pricing liabilities todetermining capital structures. The research and development on CCA-models have for thepast years mostly been on a theoretical level and less about its practical applicability. Thosefew applications that have been made were based on the U.S. market and companies.Ju and Ou-Yang developed one of the most recent CCA-methodologies in 2006,abbreviated as the JOY-model in this study. What distinguishes this model is its ability toshow the non-monotone relation of debt maturity and debt face amount through the morecomplex tradeoffs between tax benefits, bankruptcy costs and transactions cost. With a fewchanges made to it, and with almost all data from the Swedish market and companies, theJOY-model yields higher leverage ratios than what the 5 analyzed companies have today.The optimal leverage ratio, defined as debt value/firm value ranges from 10 – 40% and theoptimal debt maturity period is at 4 – 6 years. Out of all the model parameters, the long-runmean of the stochastic risk-free interest rate has the biggest impact on the final results. TheJOY-model and CCA in general are complex and resource intense models that need certainimprovements. Nonetheless, its overall potential is still promising.

This study applies the Vasicek stochastic interest rate model in order to determine optimal capital structures for listed firms. A Swedish interest rate data set is used to estimate Vasicek model parameter that are reliable and independent of initial start values. These interest rate parameters are then used in a capital structure model which is evaluated through a sensitivity analysis and a firm-specific analysis which is applied to listed Swedish firms. The tax benefits of debt must be balanced against transaction costs and bankruptcy costs when determining optimal leverage ratio and optimal debt maturity. The results imply that firms should primarily focus on the long-term mean parameter of the interest rate process, the volatility of its firm value, the transaction cost of issuing debt and the effective corporate tax rate when choosing a capital structure. The capital structure model is well-founded in previous research and yields results which align with empirical data quite well. The conclusions of this study has implications for corporate finance, as the Vasicek model provides a better understanding of the stochastic nature of interest rates and its influence in determining optimal capital structures. / Denna studie tillämpar Vasiceks stokastiska räntemodell för att bestämma optimala kapitalstrukturer för noterade företag. Vasicekmodellen anpassas till ett Svensk dataset över räntor för att estimera parametrar. Dessa parametrar används sedan i en kapitalstruktursmodell för att analysera modellens känslighet för variationer i parametrar samt för att härleda optimala kapitalstrukturer för en rad Svenska noterade bolag. Skattefördelarna av skuld måste balanseras mot transaktionskostnader av skuldemissioner och konkurskostnader vid bestämning av optimal skuldsättningsgrad och optimal skuldlöptid. Resultaten antyder att företag bör primärt fokusera på det långsiktiga medelvärdet av den stokastiska ränteprocessen, volatiliteten av bolagsvärdet, transaktionskostnaden av skuldemissioner och effektiva bolagsskatten när de väljer en kapitalstruktur. Denna studie har implikationer för finansieringsteori då Vasiceks modell närmare modellerar räntors stokastiska dynamiker och dess påverkan på bestämning av bolags kapitalstrukturer.