Introduction to Financial Mathematics II
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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| Instructor(s) | Department of Applied Mathematics Prof. Ching-Tang Wu |
|---|---|
| Course Credits | 3 Credits |
| Academic Year | 99 Academic Year |
| Level | Graduate Students |
| Prior Knowledge | Calculus |
| Related Resources | Course Video Course Syllabus Course Calendar |
| Week | Course Content | Course Video |
|---|---|---|
| 單元七 Coutinuous-time Martingales 7.1 Stochastic process (1/2) | Watch Online | |
| 7.1 Stochastic process (2/2) | Watch Online | |
| 7.2 Uniform integrability | Watch Online | |
| 7.3 Martingale theory in continuous-time | Watch Online | |
| 7.4 Local martingales | Watch Online | |
| 7.5 Doob-Meyer decomposition | Watch Online | |
| 7.6 Semimartingales | Watch Online | |
| 單元八 Brownian Motions 8.1 Scaled random walk | Watch Online | |
| 8.2 Brownian motions | Watch Online | |
| 8.3 The Brownian sample paths | Watch Online | |
| 8.4 Exponential martingales | Watch Online | |
| 8.5 d-dimensional Brownian motions | Watch Online | |
| 單元九 Stochastic Integrals 9.1 Construction of stochastic integrals with respect to martingales (1/3) | Watch Online | |
| 9.1 Construction of stochastic integrals with respect to martingales (2/3) | Watch Online | |
| 9.1 Construction of stochastic integrals with respect to martingales (3/3) | Watch Online | |
| 9.2 Stochastic integrals with respect to semimartingales | Watch Online | |
| 9.3 Stochastic integrals with respect to local martingales | Watch Online | |
| 9.4 Itô formula (1/2) | Watch Online | |
| 9.4 Itô formula (2/2) | Watch Online | |
| 9.5 Integration by parts | Watch Online | |
| 9.6 Martingale representation theorem | Watch Online | |
| 9.7 Change of Measures | Watch Online | |
| 9.8 Girsanov theorem | Watch Online | |
| 9.9 Local times | Watch Online | |
| 單元十 Stochastic Differential Equations 10.1 Examples and some solution methods (1/2) | Watch Online | |
| 10.1 Examples and some solution methods (2/2) | Watch Online | |
| 10.2 An existence and uniqueness result | Watch Online | |
| 10.3 Weak and strong solutions | Watch Online | |
| 10.4 Feynman-Kac theorem | Watch Online | |
| 單元十一 Continuous-Time Models 11.1 Market portfolios and arbitrage | Watch Online | |
| 11.2 Equivalent local martingale measures | Watch Online | |
| 11.3 Completeness | Watch Online | |
| 11.4 Pricing for attainable contingent claim | Watch Online | |
| 11.5 Black-Scholes-Merton formula | Watch Online | |
| 11.6 The Greeks | Watch Online | |
| 11.7 Parity rrelations | Watch Online | |
| 單元十二 Hedging 12.1 Hedging strategy for the simple contingent claim 12.2 Delta and gamma hedging | Watch Online | |
| Appendix F、Characteristic Functions | Watch Online | |
| Appendix G、Differntial Equations | Watch Online | |
| Appendix H、Convex Analysis | Watch Online |
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 |
| 單元七 Continuous-Time Martingales | 7.1 Stochastic processes 7.2 Uniform integrability 7.3 Martingale theory in continuous-time 7.4 Local martingales 7.5 Doob-Meyer decomposition 7.6 Semimartingales |
| 單元八 Brownian Motions | 8.1 Scaled random walk 8.2 Brownian motions 8.3 The Brownian sample paths 8.4 Exponential martingales 8.5 d-dimensional Brownian motions |
| 單元九 Stochastic Integrals | 9.1 Construction of stochastic integrals with respect to martingales 9.2 Stochastic integrals with respect to semimartingales 9.3 Itô formula 9.4 Integration by parts 9.5 Martingale representation theorem 9.6 Girsanov theorem 9.7 Local times |
| 單元十 Stochastic Differential Equations | 10.1 Examples and some solution methods 10.2 An existence and uniqueness result 10.3 Weak and strong solutions 10.4 Feynman-Kac theorem |
| 單元十一 Continuous-Time Models | 11.1 Market portfolios and arbitrage 11.2 Equivalent local martingale measures 11.3 Completeness 11.4 Pricing for attainable contingent claim 11.5 Black-Scholes-Merton formula 11.6 Parity relations 11.7 The greeks |
| 單元十二 Hedging | 12.1 Hedging strategy for the simple contingent claim 12.2 Delta and gamma hedging 12.3 Superhedging 12.4 Quantile hedging |
| 單元六 Volatility | 13.1 Historical volatility 13.2 Implied volatility |
| Appendix | F . Convex Analysis |
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 |
| 單元七 Continuous-Time Martingales | 7.1 Stochastic processes 7.2 Uniform integrability 7.3 Martingale theory in continuous-time 7.4 Local martingales 7.5 Doob-Meyer decomposition 7.6 Semimartingales |
| 單元八 Brownian Motions | 8.1 Scaled random walk 8.2 Brownian motions 8.3 The Brownian sample paths 8.4 Exponential martingales 8.5 d-dimensional Brownian motions |
| 單元九 Stochastic Integrals | 9.1 Construction of stochastic integrals with respect to martingales 9.2 Stochastic integrals with respect to semimartingales 9.3 Itô formula 9.4 Integration by parts 9.5 Martingale representation theorem 9.6 Girsanov theorem 9.7 Local times |
| 單元十 Stochastic Differential Equations | 10.1 Examples and some solution methods 10.2 An existence and uniqueness result 10.3 Weak and strong solutions 10.4 Feynman-Kac theorem |
| 單元十一 Continuous-Time Models | 11.1 Market portfolios and arbitrage 11.2 Equivalent local martingale measures 11.3 Completeness 11.4 Pricing for attainable contingent claim 11.5 Black-Scholes-Merton formula 11.6 Parity relations 11.7 The greeks |
