2020 Special lectures on current topics in Mathematics W

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Academic unit or major
Graduate major in Mathematics
Shinozaki Yuji 
Course component(s)
Mode of instruction
Day/Period(Room No.)
Intensive (Zoom)  
Course number
Academic year
Offered quarter
Syllabus updated
Lecture notes updated
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Course description and aims

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.

Student learning outcomes

・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

Course taught by instructors with work experience

Applicable How instructors' work experience benefits the course
The lecturer has been working in a financial institute as a quants.


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

Competencies that will be developed

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/Required learning

  Course schedule Required learning
Class 1 Overview of the lecture Details will be provided during each class
Class 2 Quick review of the foundation of Mathematical Finance Details will be provided during each class
Class 3 Construction of the multiple yield curve model under the no arbitrage condition using SPDE Details will be provided during each class
Class 4 Derivation of the generalized XVA formulas using FBSDE 1 Details will be provided during each class
Class 5 Derivation of the generalized XVA formulas using FBSDE 2 Details will be provided during each class



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)

Assessment criteria and methods

Assignments (100%).

Related courses

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

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

None in particular

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