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 abilities:
Lay the cornerstone of probability theory: Deeply understand the core concepts such as random events, chances and commonly used chances distribution, and establish a stable mathematical foundation for understanding the randomness of the financial market.
Master random process analysis: Be familiar with the definitions, properties and their applications in describing the dynamic changes of financial asset prices.
Proficient in random micro-scoring tools: learn and master the basic calculation of random micro-scoring (such as Ito Lemma), and be able to use it to promote pricing model and risk management analysis of financial derivative commodities.
Strengthen theories and practice links: Cultivate the ability to apply abstract financial mathematical theories to solve real-world financial problems, and improve the comprehensive skills of problem analysis and model application through case studies.
2. Course Content
This course will systematically introduce the core theories and application tools of financial math, mainly covering the following parts:
Foundations of Probability Theory:
Random events, sample space and chances.
Conditional probability, Beth's theorem and the independence of events.
Random variations, expected values, variations, and important probability distributions (such as normal distributions, relatives and normal distributions) and their meaning in finance.
Random Processes:
The definition, classification and basic nature of random processes.
The mathematical structure of Brown’s movement (Vina process) and its core role in simulating asset price paths.
Introduction to concepts such as the Random Walk Model, the Markov process and the Martingale process and their financial implications.
The application of random processes in financial time sequence modeling such as stock prices and interest rates.
Random Micro Score (Stochastic Calculus):
Ito Integral concept and calculation.
Itô's Lemma and its key role in solving random differential equations.
The establishment and application of random differential equations (SDEs) in financial models (such as describing asset price evolution).
Applications in Derivative Valuation:
Use random micro points to promote the Black-Scholes selection of the right price model.
The principle of risk-neutral pricing.
Price concepts for various strange choice rights or other structural derivatives.
Avoidance strategies and risk management technologies such as Delta Hedging.
Case Studies and Practical Problem Solving:
Analyze real market data or simulated situations, and apply the theoretical analysis to conduct derivatives evaluation and risk analysis.
Through specific problem-solving processes, deepen understanding of theoretical arguments and improve practical operation 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)
| 評分項目 Grading Method | 配分比例 Grading percentage | 說明 Description |
|---|---|---|
| 第一次小考第一次小考 First small exam |
50 | 可以用AI |
| 第二次小考第二次小考 The second small exam |
30 | 只能帶書 |
| 期末考期末考 Final exam |
20 | 純考試無法帶任何東西。 |
