財務數學導論(一)
本課程是由交通大學應用數學系提供。
本課程主要讓學生了解並熟悉研究財務金融方面所需之數學工具。
課程用書：S. E. Shreve: Stochastic Calculus for Finance II: Continuous-Time Models, Springer, 2004.
參考用書：
- T. M. Apostol: Mathematical Analysis, Second Edition
- M. Baxter and A. Rennie: Financial Calculus.
- T. Björk: Arbitrage Theory in Continuous Time.
- K. L. Chung: A Course in Probability Theory, Second Edition.
- F. Delbaen and W. Schachermayer: The Mathematics of Arbitrage.
- J. Elstrodt: Maβ- und Integrationstheorie, Third Edition.
- H. Föllmer and A. Schied: Stochastic Finance. An Introduction in Discrete Time.
- J. Jacod and Ph. Protter: Probability Essentials.
- J. C. Hull: Options, Futures, & Other Derivatives, Sixth Edition.
- I. Karatzas: Lectures on the Mathematics of Finance.
- I. Karatzas and S. E. Shreve: Brownian Motion and Stochastic Calculus, Second Edition.
- I. Karatzas and S. E. Shreve: Method of Mathematical Finance.
- D. Lamberton and B. Lapeyre: Introduction to Stochastic Calculus Applied to Finance.
- B. Øksendal: Stochastic Differential Equations, An Introduction with Applications,Sixth Edition.
- R. T. Rockafellar: Convex Analysis.
- H. L. Royden: Real Analysis, Third Edition.
- A.N. Shiryaev: Probability Theory, Second Edition.
- S. E. Shreve: Stochastic Calculus for Finance I: The Binomial Asset Pricing Model.
- R. L. Wheeden and A. Zygmund: Measure and integral.
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| 授課教師 | 應用數學系 吳慶堂老師 |
|---|---|
| 課程學分 | 3學分 |
| 授課年度 | 99學年度 |
| 授課對象 | 碩士班學生 |
| 預備知識 | 微積分 |
| 課程提供 | 課程影音 課程綱要 課程行事曆 |
| 週次 | 課程內容 | 課程影音 |
|---|---|---|
| Introduction | 線上觀看 | |
| 單元一 Probability Theory 1.1 Probability space | 線上觀看 | |
| 1.2 Random variables | 線上觀看 | |
| 1.3 Expectation | 線上觀看 | |
| 單元二 Discrete-Time Martingales 2.1 Conditional probability and conditional expectation (1/4) | 線上觀看 | |
| 2.1 Conditional probability and conditional expectation (2/4) | 線上觀看 | |
| 2.1 Conditional probability and conditional expectation (3/4) | 線上觀看 | |
| 2.1 Conditional probability and conditional expectation (4/4) | 線上觀看 | |
| 2.2 Discrete time Martingales | 線上觀看 | |
| 2.3 Martingale transform and Doob decomposition | 線上觀看 | |
| 單元三 One-Period Model Introduction | 線上觀看 | |
| 3.1 Portfolios | 線上觀看 | |
| 3.2 Derivative securities | 線上觀看 | |
| 3.3 Absence of arbitrage | 線上觀看 | |
| 3.4 No arbitrage and price system | 線上觀看 | |
| 3.5 Martingale measures (1/2) | 線上觀看 | |
| 3.5 Martingale measures (2/2) | 線上觀看 | |
| 3.6 Pricing | 線上觀看 | |
| 3.7 Complete market model | 線上觀看 | |
| 單元四 Multi-Period Model Introduction | 線上觀看 | |
| 4.1 The market model | 線上觀看 | |
| 4.2 Arbitrage opportunities | 線上觀看 | |
| 4.3 Martingale measures | 線上觀看 | |
| 4.4 Arbitrage-free prices for European contingent claim | 線上觀看 | |
| 單元五 American Contingent Claim 5.1 Stopping time | 線上觀看 | |
| 5.2 American claims | 線上觀看 | |
| 5.3 Arbitrage-free prices | 線上觀看 | |
| 單元六 Measures of Risk Introduction | 線上觀看 | |
| 6.1 Monetary measure of risk | 線上觀看 | |
| 6.2 Coherent and convex risk measures | 線上觀看 | |
| 6.3 Acceptance sets | 線上觀看 | |
| 6.4 Robust representation of coherent risk measure | 線上觀看 | |
| 6.5 Robust representation of convex risk measures | 線上觀看 | |
| Appendix A、Limits of Sequences of Numbers | 線上觀看 | |
| Appendix B、Convergence of Sequences of Functions / Stochastic Processes I | 線上觀看 | |
| Appendix C、Distribution Functions | 線上觀看 | |
| Appendix D、Convergence of Sequences of Functions / Stochastic Processes II | 線上觀看 | |
| Appendix E、Riemann-Stieltjes Integrals | 線上觀看 |
課程目標
本課程主要讓學生了解並熟悉研究財務金融方面所需之數學工具。
課程章節
| 章節 | 主題內容 |
| 課程介紹 | |
| 單元一 Probability Theory | 1.1 Probability space 1.2 Random variables 1.3 Expectation |
| 單元二 Discrete-Time Martingales | 2.1 Conditional probability and conditional expectation 2.2 Discrete time Martingales 2.3 Martingale transform and Doob decomposition |
| 單元三 One-Period Model | Introduction 3.1 Portfolios 3.2 Derivative securities 3.3 Absence of arbitrage 3.4 No arbitrage and price system 3.5 Martingale measures 3.6 Pricing 3.7 Complete market model |
| 單元四 Multi-Period Model | Introduction 4.1 The market model 4.2 Arbitrage opportunities 4.3 Martingale measures 4.4 Arbitrage-free prices for European contingent claim |
| 單元五 American Contingent Claim | 5.1 Stopping time 5.2 American claims 5.3 Arbitrage-free prices |
| 單元六 Measures of Risk | Introduction 6.1 Monetary measure of risk 6.2 Coherent and convex risk measures 6.3 Acceptance sets 6.4 Robust representation of coherent risk measure 6.5 Robust representation of convex risk measures |
| Appendix | A. Limits of Sequences of Numbers B. Convergence of Sequences of Functions and Stochastic Processes I C. Distribution Functions D. Convergence of Sequences of Functions and Stochastic Processes II E. Riemann-Stieltjes Integrals |
課程書目
S. E. Shreve: Stochastic Calculus for Finance II: Continuous-Time Models, Springer, 2004.
參考書目
T. M. Apostol: Mathematical Analysis, Second Edition
M. Baxter and A. Rennie: Financial Calculus.
T. Björk: Arbitrage Theory in Continuous Time.
K. L. Chung: A Course in Probability Theory, Second Edition.
F. Delbaen and W. Schachermayer: The Mathematics of Arbitrage.
J. Elstrodt: Maβ- und Integrationstheorie, Third Edition.
H. Föllmer and A. Schied: Stochastic Finance. An Introduction in Discrete Time.
J
. Jacod and Ph. Protter: Probability Essentials.
J. C. Hull: Options, Futures, & Other Derivatives, Sixth Edition.
I. Karatzas: Lectures on the Mathematics of Finance.
I. Karatzas and S. E. Shreve: Brownian Motion and Stochastic Calculus, Second Edition.
I. Karatzas and S. E. Shreve: Method of Mathematical Finance.
D. Lamberton and B. Lapeyre: Introduction to Stochastic Calculus Applied to Finance.
B. Øksendal: Stochastic Differential Equations, An Introduction with Applications,Sixth Edition.
R. T. Rockafellar: Convex Analysis.
H. L. Royden: Real Analysis, Third Edition.
A.N. Shiryaev: Probability Theory, Second Edition.
S. E. Shreve: Stochastic Calculus for Finance I: The Binomial Asset Pricing Model.
R. L. Wheeden and A. Zygmund: Measure and integral.
評分標準
| 項目 | 百分比 |
| 平時成績(作業) | 40% |
| 期中考 | 30% |
| 期末考 | 30% |
本課程行事曆提供課程進度與考試資訊參考。
| 章節 | 主題內容 |
| 課程介紹 | |
| 單元一 Probability Theory | 1.1 Probability space 1.2 Random variables 1.3 Expectation |
| 單元二 Discrete-Time Martingales | 2.1 Conditional probability and conditional expectation 2.2 Discrete time Martingales 2.3 Martingale transform and Doob decomposition |
| 單元三 One-Period Model | Introduction 3.1 Portfolios 3.2 Derivative securities 3.3 Absence of arbitrage 3.4 No arbitrage and price system 3.5 Martingale measures 3.6 Pricing 3.7 Complete market model |
| 單元四 Multi-Period Model | Introduction 4.1 The market model 4.2 Arbitrage opportunities 4.3 Martingale measures 4.4 Arbitrage-free prices for European contingent claim |
| 單元五 American Contingent Claim | 5.1 Stopping time 5.2 American claims 5.3 Arbitrage-free prices |
| 單元六 Measures of Risk | Introduction 6.1 Monetary measure of risk 6.2 Coherent and convex risk measures 6.3 Acceptance sets 6.4 Robust representation of coherent risk measure 6.5 Robust representation of convex risk measures |
| Appendix | A. Limits of Sequences of Numbers B. Convergence of Sequences of Functions and Stochastic Processes I C. Distribution Functions D. Convergence of Sequences of Functions and Stochastic Processes II E. Riemann-Stieltjes Integrals |