財務數學導論（一）

Abstract

課程首頁

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。

課程概述 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

單元

課程內容

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. Bjork: 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. Follmer 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. Oksendal: 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%

授課對象：碩士班學生

預備知識：微積分