Special lectures on current topics in Mathematics W

Academic unit or major

Graduate major in Mathematics

Instructor(s)

Yuji Shinozaki

Class Format

Lecture (Zoom)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

Intensive (Zoom)

Class

-

Course Code

MTH.E654

Number of credits

200

Course offered

2020

Offered quarter

4Q

Syllabus updated

Jul 10, 2025

Language

Japanese

Syllabus

Course overview and goals

In this course, some of the mathematical problems which have recently become important for the financial institutions will be described in a general and mathematically rigorous manner. The main aims of this course are to introduce some practical aspects of the mathematical finance and to present the mathematical formulations of practically important financial problems.
✔ Course description and aims
3. Are the course description and aims given using separate paragraphs? 4. Are they written from an instructor’s perspective?
In this lecture, following a brief introduction of some practical backgrounds two important topics after the financial crisis will be formulated: the multiple yield curve modeling and the XVA (short for ``X-Value Adjustment’’). As for the multiple yield curve modeling we will first overview the practical constraints due to the changes in the market environments and the derivative transaction terms, and then introduce the OIS discounting and the initial yield curve construction. After that, the generalized HJM model, which is capable of capturing the dynamics of various market bases under the arbitrage free condition, will be formulated by using the framework of the stochastic partial differential equations. Furthermore, we derive some popular interest rate models from the generalized HJM model and review their features. Regarding the XVA pricing, after a short discussion on the concept and the practical importance of XVA we will formulate XVAs based on the replication theory using the forward backward stochastic differential equations/non-linear partial differential equations. Finally, we will derive some practical formulas of XVAs from a general equation. As long as time allows we will review the CCP basis modeling and some numerical techniques of the forward backward stochastic differential equations.

Course description and aims

・Understand how the probability theory and the mathematical finance are used in the financial institution
・Understand recent developments of mathematical finance in response to practical requirements such as financial crisis
・Be able to survey the recent hot topics of mathematical finance
・Get conscious about linkages of pure mathematics to the real world

Student learning outcomes

実務経験と講義内容との関連 (又は実践的教育内容)

The lecturer has been working in a financial institute as a quants.

Keywords

Arbitrage free pricing theory, Replication theory, HJM model, OIS discount, Multiple yield curve, Stochastic partial differential equation, XVA, Forward-backward stochastic differential equation

Competencies

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

Class flow

This is a standard lecture course with the presentation slides and black boards. There will be some assignments.

Course schedule/Objectives

Study advice (preparation and review)

Textbook(s)

none

Reference books, course materials, etc.

・A general HJM framework for multiple yield curve modelling (Christa Cuchiero, Claudio Fontana, Alessandro Gnoatto, 2016)
・Arbitrage-free XVA (Maxim Bichuch, Agostino Capponi, Stephan Sturm, 2016)

Evaluation methods and criteria

Assignments (100%).

Related courses

• MTH.C361 ： Probability Theory
• MTH.C507 ： Advanced topics in Analysis G1
• MTH.C508 ： Advanced topics in Analysis H1

Prerequisites

None in particular