一、課程目標 (Course Objectives)
本課程旨在使學生達成以下核心能力：

奠定機率理論基石： 深刻理解隨機事件、機率及常用機率分佈等核心概念，為理解金融市場的隨機性建立穩固的數學基礎。
掌握隨機過程分析： 熟悉如布朗運動、隨機漫步等重要隨機過程的定義、性質及其在描述金融資產價格動態變化中的應用。
精通隨機微積分工具： 學習並掌握隨機微積分（如伊藤引理）的基本運算，並能運用其進行金融衍生性商品的定價模型推導與風險管理分析。
強化理論與實務連結： 培養將抽象的財務數學理論應用於解決實際金融問題的能力，並透過案例研究提升問題解析與模型應用的綜合技能。
二、課程內涵 (Course Content)
本課程將系統性地介紹財務數學的核心理論與應用工具，主要涵蓋以下幾個部分：

機率論基礎 (Foundations of Probability Theory):

隨機事件、樣本空間與機率公設。
條件機率、貝氏定理與事件的獨立性。
隨機變數、期望值、變異數以及重要的機率分佈（如常態分佈、對數常態分佈）及其在金融中的意義。
隨機過程 (Stochastic Processes):

隨機過程的定義、分類與基本性質。
布朗運動（維納過程）的數學構造及其在模擬資產價格路徑中的核心角色。
隨機漫步模型、馬可夫過程及鞅過程等概念簡介及其金融意涵。
隨機過程在股價、利率等金融時間序列建模中的應用。
隨機微積分 (Stochastic Calculus):

伊藤積分 (Itô Integral) 的概念與計算。
伊藤引理 (Itô's Lemma) 及其在隨機微分方程求解中的關鍵作用。
隨機微分方程 (SDEs) 在金融模型中的建立與應用（如描述資產價格演變）。
衍生性商品評價應用 (Applications in Derivative Valuation):

運用隨機微積分推導Black-Scholes選擇權定價模型。
風險中性定價原理。
各種奇異選擇權或其他結構性衍生品的定價概念。
避險策略與Delta Hedging等風險管理技術。
案例研究與實務問題研討 (Case Studies and Practical Problem Solving):

分析真實市場數據或模擬情境，應用所學理論進行衍生品評價與風險分析。
透過具體問題的解決過程，加深對理論的理解並提升實務操作技能。

1. Course Objectives
This course aims to enable students to achieve the following core competencies:

Lay the cornerstone of probability theory: Deeply understand core concepts such as random events, probability and commonly used probability distributions, and establish a solid mathematical foundation for understanding the randomness of financial markets.
Master stochastic process analysis: Be familiar with the definition and properties of important stochastic processes such as Brownian motion and random walk and their application in describing dynamic changes in financial asset prices.
Proficient in stochastic calculus tools: Learn and master the basic operations of stochastic calculus (such as Ito's Lemma), and be able to use it to derive pricing models and risk management analysis for financial derivatives.
Strengthen the connection between theory and practice: Cultivate the ability to apply abstract financial mathematical theories to solve practical financial problems, and improve comprehensive skills in problem analysis and model application through case studies.
2. Course Content
This course will systematically introduce the core theories and application tools of financial mathematics, mainly covering the following parts:

Foundations of Probability Theory:

Random events, sample spaces and postulates of probability.
Conditional probability, Bayesian theorem and independence of events.
Random variables, expected values, mutations, and important probability distributions (such as normal distribution, lognormal distribution) and their significance in finance.
Stochastic Processes:

The definition, classification and basic properties of stochastic processes.
The mathematical construction of Brownian motion (Wiener process) and its central role in simulating asset price paths.
An introduction to concepts such as random walk model, Markov process and martingale process and their financial implications.
The application of stochastic processes in financial time series modeling such as stock prices and interest rates.
Stochastic Calculus:

The concept and calculation of Itô Integral.
Itô's Lemma and its key role in the solution of stochastic differential equations.
The construction and application of stochastic differential equations (SDEs) in financial models (e.g. describing asset price evolution).
Applications in Derivative Valuation:

Use stochastic calculus to derive the Black-Scholes option pricing model.
Risk-neutral pricing principle.
Pricing concepts for various exotic options or other structured derivatives.
Hedging strategies and risk management techniques such as Delta Hedging.
Case Studies and Practical Problem Solving:

Analyze real market data or simulated scenarios, and apply the theories learned to conduct derivatives evaluation and risk analysis.
Through the process of solving specific problems, you can deepen your understanding of theory and improve your practical skills.

1. Shreve, S.E., Stochastic Calculus for Finance II Continuous-Time Models, Springer-Verlag, NY, 2004. (Textbook)
2. Musiela, M. and Rutkowski, M., Martingale Methods in Financial Modelling, Springer-Verlag, NY, 1997. (Reference)

1. Shreve, S.E., Stochastic Calculus for Finance II Continuous-Time Models, Springer-Verlag, NY, 2004. (Textbook)
2. Musiela, M. and Rutkowski, M., Martingale Methods in Financial Modelling, Springer-Verlag, NY, 1997. (Reference)