2018 Advanced topics in Analysis H

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

In this lecture, the topics discussed in ``Advanced topics in analysis G'' are developed in continuous time models.

The following notions such as various European type options, American options, bonds and their term structures are discussed. As Ito calculus and stochastic differential equations are essential to the continuous time theory of mathematical finance, we prepare these mathematical notions first.

Student learning outcomes

Understanding the following notions:
Ito calculus and basic knowledge of stochastic differential equations, european and american option pricing, Ito formula for the most general form, basics of interest rate term structure theory.


Ito calculus, stochastic differential equation, option pricing, interest rate term structure

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 Continuous time stochastic processes: Martingale Details will be provided each class session.
Class 2 Stochastic Integral: Stochastic Integral/Ito formula
Class 3 Stochastic Differential Equations
Class 4 Option pricing(1): Cameron-Martin Maruyama Girsanov theorem/Martingale representation
Class 5 Option Pricing(2): Equivalent martingale measure/Black-Scholes formula
Class 6 American Options
Class 7 Interest rate: Interest rate market/Term structure
Class 8 Some advanced topics


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
T. Bjork, ``Arbitrage Theory in Continuous Time'', Oxford

Assessment criteria and methods

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

Related courses

  • MTH.C361 : Probability Theory
  • MTH.C503 : Advanced topics in Analysis G

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

None in particular


None in particular

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