財務數學導論（二）

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

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 Ito 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. 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