2012-08-012024-05-17https://scholars.lib.ntu.edu.tw/handle/123456789/695873摘要：衍生性金融商品之價值決定於標的物如股票，而標的物價格則為一連續時間形隨機程序。衍生性金融商品之定價問題等同於在適當機率測度空間(稱為風險中立機率測度空間)下，計算其收益的數學期望值。 亞式選擇權為受到歡迎的路徑相關型衍生性金融商品，不但在資本市場上十分受到歡迎，而且在學術界亦受到廣泛地研究。雖然已有不少逼近演算法存在，但它們執行時間通常非最佳。雖然連續觀察型亞式選擇權較離散觀察型亞式選擇權的研究多，但離散觀察型亞式選擇權較常見(也可能是唯一的)。之前我們已求得連續觀察型亞式選擇權之最佳演算法，本計畫將求得離散觀察型亞式選擇權之最佳演算法，演算法會建立在Lagrange乘數上以求得每個格子點的狀況數。 障礙型選擇權在資本市場上十分受到歡迎，障礙可為連續觀察型、或離散觀察型兩種。本計畫將針對一大類的混合連續觀察及離散觀察型之障礙型選擇權，得到公式解。此類的障礙型選擇權甚至可選擇多種障礙結構與還錢方式(rebate)，且公式解將考慮多種還錢方式。此結果使過去文獻上的多個結果得到統一解，也可以激勵新形式的衍生性金融商品之設計。 含多個標的物之複雜性衍生性金融商品，通常沒有公式解，因此其價格必須用例如格子或蒙地卡羅模擬等數字方法求得。若要計價可轉債或其他利率相關衍生性金融商品，研究者多用含利率及股價的二元模型所產生的二元格子，不幸地，當利率之值沒有事先給定上限時，文獻上已知的二元格子會有不合法的機率。因為絕大多數之隨機利率模型皆無此上限值，解決不合法機率的問題變得非常重要。本計畫將求得文獻上第一個保證合法機率的二元格子，且如果利率模型的利率值成長率為多項式時，則二元格子大小的成長率亦為多項式，而如果利率模型的利率值成長率超過多項式時，則二元格子大小的成長率亦超過多項式。此結果一項重要的推論為，如果利率模型為受歡迎的對數常態模型時，任何二元格子大小的成長率必超過多項式，因此我們的二元格子為最佳解。 本計畫所有演算法都將寫成程式，並與文獻上其他演算法在效率、精確度、及收歛性率做徹底的比較。<br> Abstract: Derivatives are financial instruments whose payoff depends on some underlying asset such as stock. The underlying asset is described by a continuous-time stochastic process. It is known that derivative pricing is equivalent to calculating the expected value of its payoff function under a suitable probability measure called the risk-neutral probability measure. Asian options are popular path-dependent financial derivatives in the capital markets and intensively studied in the literature. Although many approximation pricing algorithms have been proposed, they usually have suboptimal running time. Among Asian options, the continuously monitored type is more intensively studied than the discretely monitored type. But the discretely monitored Asian option is more popular (if not the only type) in practice. Previously, we had obtained optimal algorithms for the continuously monitored Asian option. This proposal intends to obtain the most efficient tree algorithms for the discrete Asian options. The algorithm relies on the Lagrange multipliers to choose the number of states for each node of the tree. Barrier-type options are very popular in the market. Barrier can be discretely monitored or discretely monitored. This proposal plans to present a closed-form solution to a broad class of options with a serial mixture of discrete and continuous barriers. These options can choose from a variety of barrier structures and rebate schemes. A few commonly used rebate schemes are also considered. The analytical result unifies many previous results and may inspire the design of new exotic options. Complex financial instruments with multiple state variables often have no analytical formulas and therefore must be priced by numerical methods, like trees. For pricing convertible bonds and many other interest rate-sensitive products, research has focused on bivariate trees for models with two state variables: stock price and interest rate. Unfortunately, when the interest rate component allows rates to grow without bounds in magnitude, those trees may generate invalid transition probabilities. As the overwhelming majority of stochastic interest rate models share this property, a solution to the problem becomes important. This proposal plans to present the first bivariate tree that guarantees valid probabilities. The proposed bivariate tree grows (super)polynomially in size if the interest rate model allows rates to grow (super)polynomially. Furthermore, we plan to show that any valid constant-degree bivariate tree must grow superpolynomially in size with lognormal interest rate models, which form a very popular class of interest rate models. Therefore, our bivariate tree can be said to be optimal. In all cases, extensive and exhaustive numerical evaluations will be performed to compare our algorithms to those in the literature in terms of speed, accuracy, and convergence rate.衍生性金融商品演算法蒙地卡羅模擬亞式選擇權障礙選擇權格子二元模型複雜度derivativealgorithmMonte Carlo simulationAsian optionbarrier optiontreebivariate modelcomplexity