針對金融時序資料變異數不齊一的性質，隨機波動模型除了提供於ARCH族外的另一選擇；且由於其設定隱含波動本身亦為一個隨機波動函數，藉由設定隨時間改變且自我相關的條件變異數，使得隨機波動模型較ARCH族來得有彈性且符合實際。傳統上處理隨機波動模型的參數估計往往需要面對到複雜的多維積分，此問題可藉由貝氏分析裡的馬可夫鏈蒙地卡羅法解決。本文主要的探討標的，即在於利用馬可夫鏈蒙地卡羅法估計美元/新台幣匯率隨機波動模型參數。除原始模型之外，模型的擴充分為三部分：其一為隱含波動的二階自我回歸模型；其二則為藉由基本模型的修改，檢測匯率市場上的槓桿效果；最後，我們嘗試藉由加入scale mixture的方式以驗證金融時序資料中常見的厚尾分配。

本篇論文將厚尾分配（Heavy-Tailed Distribution）應用在財務及保險精算上。本研究主要有三個部分：第一部份是用厚尾分配來重新建構Lee-Carter模型（1992），發現改良後的Lee-Carter模型其配適與預測效果都較準確。第二部分是將厚尾分配建構於具有世代因子（Cohort Factor）的Renshaw and Haberman模型（2006）中，其配適及預測效果皆有顯著改善，此外，針對英格蘭及威爾斯（England and Wales）訂價長壽交換（Longevity Swaps），結果顯示此模型可以支付較少的長壽交換之保費以及避免低估損失準備金。第三部分是財務上的應用，利用Schmidt等人（2006）提出的多元仿射廣義雙曲線分配（Multivariate Affine Generalized Hyperbolic Distributions; MAGH）於Boyle等人（2003）提出的低偏差網狀法（Low Discrepancy Mesh; LDM）來定價多維度的百慕達選擇權。理論上，LDM法的數值會高於Longstaff and Schwartz（2001）提出的最小平方法（Least Square Method; LSM）的數值，而數值分析結果皆一致顯示此性質，藉由此特性，我們可知道多維度之百慕達選擇權的真值落於此範圍之間。 / The thesis focus on the application of heavy-tailed distributions in finance and actuarial science. We provide three applications in this thesis. The first application is that we refine the Lee-Carter model (1992) with heavy-tailed distributions. The results show that the Lee-Carter model with heavy-tailed distributions provide better fitting and prediction. The second application is that we also model the error term of Renshaw and Haberman model (2006) using heavy-tailed distributions and provide an iterative fitting algorithm to generate maximum likelihood estimates under the Cox regression model. Using the RH model with non-Gaussian innovations can pay lower premiums of longevity swaps and avoid the underestimation of loss reserves for England and Wales. The third application is that we use multivariate affine generalized hyperbolic (MAGH) distributions introduced by Schmidt et al. (2006) and low discrepancy mesh (LDM) method introduced by Boyle et al. (2003), to show how to price multidimensional Bermudan derivatives. In addition, the LDM estimates are higher than the corresponding estimates from the Least Square Method (LSM) of Longstaff and Schwartz (2001). This is consistent with the property that the LDM estimate is high bias while the LSM estimate is low bias. This property also ensures that the true option value will lie between these two bounds.