2018 Advanced topics in Analysis G

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Academic unit or major
Graduate major in Mathematics
Ninomiya Syoiti 
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Course description and aims

This lecture and its sequel ``Advanced topics in analysis H'' are aimed at those wishing to learn about both the basic mathematical concepts and the overall picture of mathematical finance.

First, we begin from the single and multi-term binomial setting. The following notions, arbitrage pricing, martingale measures, the 1st fundamental theorem, and complete markets are discussed in the setting. Martingales in the discrete time setting play an essential role.

Student learning outcomes

Understanding the important notion and ideas of option pricing theory in discrete time models.
Supplies the basis for the sequel ``Advanced topics in analysis H''.


Mathematical Finance, Martingale(discrete time)

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Blackboard and handouts

Course schedule/Required learning

  Course schedule Required learning
Class 1 Introduction: Pricing and Hedging/binomial models/option prices(1) Details will be provided each class session.
Class 2 Martingale Measures(discrete time)-1: Discrete time market model/Trading strategies/Risk neutral pricing/Black-Scholes formula
Class 3 Martingale Measures(discrete time)-2: Risk neutral pricing/Black-Scholes formula
Class 4 The 1st fundamental theorem(1): The separating hyperplane theorem/construction of martingale measure
Class 5 The 1st fundamental theorem(2): Geometric interpretation of the theoreml/Generalization
Class 6 Complete markets(1): martingale representation/completeness
Class 7 Complete markets(2): Incompleteness
Class 8 American options


None in particular.

Reference books, course materials, etc.

J. Sekine, ``Mathematical Finance'', Baifukan (in Japanese)
D. Williams, ``Probability with Martingales'', Cambridge
R. J. Elliott and P. E. Kopp, ``Mathematics of Financial Markets'', Springer

Assessment criteria and methods

Based on reports. Details will be provided in the class.

Related courses

  • MTH.C361 : Probability Theory
  • MTH.C504 : Advanced topics in Analysis H

Prerequisites (i.e., required knowledge, skills, courses, etc.)

None in particular


None in particular

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