课程大纲
Objective  
This course is to make students understand and familiar with mathematical methods while studying Finance.
Outline  
Chapter        Overview
        Introduction
1        Probability Theory
       
        •        1.1 Probability space
•        1.2 Random variables
•        1.3 Expectation
2        Discrete-Time Martingales
        •        2.1 Conditional probability and conditional expectation
•        2.2 Discrete time Martingales
•        2.3 Martingale transform and Doob decomposition
3        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
1        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
5        American Contingent Claim
        •        5.1 Stopping time
•        5.2 American claims
•        5.3 Arbitrage-free prices
6        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
       
        •       

Unit        Overview
7        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
8        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
       
9        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
10        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
11        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
12        Hedging
        •        12.1 Hedging strategy for the simple contingent claim
•        12.2 Delta and gamma hedging
•        12.3 Superhedging
•        12.4 Quantile hedging
6        Volatility
        •        13.1 Historical volatility
•        13.2 Implied volatility
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
•        F . Convex Analysis

Textbook  
•        S. E. Shreve: Stochastic Calculus for Finance II: Continuous-Time Models, Springer, 2004.

References
•        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,S ixth 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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