Computational Finance and its Object Oriented Implementation (with Application to Interest-Rates and Hybrid Models)

Content

This lecture discusses the interplay of Theory, Modelling, Numerical Methods and Implementation in Mathematical Finance.

All aspects learn from each other: one needs to understand the theory to build models and good implementations. Studying numerical experiments gives deeper theoretical insight. Using the computer to understand math can be fun!

We discuss how to build an industry-grade implementation of our models and allow future extensions while being efficient. We discuss practical applications in the financial industry.

The lecture tries to be as self-contained as possible, but we will use some numerical methods developed in a previous course. We will start with a short recapitulation of the numerical methods needed for those who did not follow the last lecture. It is possible to consider most of these parts as “given” (“black box”). Don’t panic: We will assist you.

The lecture will discuss the theory and application of some prominent methods and models from mathematical finance. We focus on interest rate and hybrid models with high relevance for the financial industry. In an excursion, we consider a climate model (DICE), extend it and combine it with our interest rate models.We will then use our implication to gain a deeper understanding of the theoretical properties of the model.

If time permits, we conclude the lecture by discussing running our models in a cloud.

Tentative Agenda

• Recapitulation (Monte-Carlo Method, SDEs, etc.)
• Interest Rates, Linear Interest Rates Products
• Multiple Interest Rate Curve Modelling and its Implementation
• Interest Rate Options, Convexity Adjustment
• Discrete Term-Structure Models (formerly known as LIBOR Market Models) and their Implementation
  • Model Calibration
  • Valuation of Complex Derivatives
  • Object Oriented Implementation, Definition of Model Interfaces
• Short Rate Models and their Implementation
• Climate Models
• Numerical Methods in the Context of Mathematical Finance
  • Model Calibration: Optimization and Root-Finding
  • Correlation Modelling: Prinzipal Component Analysis and Factor Reduction, Regression Methods
• IT Implementation of Models, e.g. in the Cloud

The lecture covers the object oriented implementation of the algorithms and using modern software development tools. As part of the implementation of the models and the valuation algorithms, the lecture will discuss some of the latest standards in software development.

• revision control systems (git)
• unit-testing (junit)
• build management (maven, gradle)
• continuous integration (TravisCI, Jenkins)

Target Participants: Master students in Mathematics or Financial and Insurance Mathematics.

Pre-requisites: The lecture requires some basic knowledge on stochastic processes. The knowledge of an object oriented programming language is advantageous. Although the lecture tries to be ”self-contained” whenever feasible, the knowledge of the previous courses (”Numerical Methods in Mathematical Finance” or ”Introduction to Interest Rates and the LIBOR Market Model” and our ”Introduction to Java”) will be useful.

Applicable credits: Students may apply the credits from this course to:

• WP38 or WP43 for the Master Finanz- und Versicherungsmathematik PO 2011
• WP15 or WP23 for the Master Finanz- und Versicherungsmathematik PO 2019
• WP14 or WP22 for the Master Finanz- und Versicherungsmathematik PO 2021
• WP31 or WP33 for the Master Mathematik
• Diplomhauptprüfung Mathematik (AM), Diplomhauptprüfung Wirtschaftsmathematik (Kernfach C).

[1] Fries, Christian P.: MathematicalFinance: Theory, Modeling, Implementation.Wiley, 2007. ISBN 0-470-04722-4.
[2] Brigo, Damiano; Mercurio, Fabio: Interest Rate Models - Theoryand Practice. Springer-Verlag, Berlin, 2001. ISBN 3-540-41772-9.
[3] Baxter, Martin W.; Rennie, Andrew J.O.: Financial Calculus: An introductionto derivative pricing. Cambridge University Press, Cambridge, 2001. ISBN 0-521-55289-3.
[4] Eckel, Bruce: Thinking in Java. Prentice Hall, 2003. ISBN 0-130-27363-5.
[5] Hunt, P.J.; Kennedy, J.E.: Financial Derivatives in Theory and Practice. John Wiley&Sons, 2000. ISBN 0-471-96717-3.
[5] Oksendal, Bernt K.: Stochastic differential equations: an introduction with applications. Springer-Verlag, 2000. ISBN 3-540-64720-6.

Active participation in the exercise courses, thinking through the problems and correcting your solutions is the best preparation for the exam. Exercise sheets will be uploaded during the course. The written solutions to theory-related exercises need not be submitted, but if you wish them to be corrected, please submit your exercise solutions.

During these sessions, the students may work on the weekly exercise problems or the midterm project and ask help from the instructor. Students who are participating in the lecture are highly advised to attend these supplementary exercises also because some topics for the lectures and exercises may be reviewed when further clarification is needed.

Support will be provided on the IT tools, for e.g.:
• Eclipse IDE
• Git repositories
• The finmath Java library
• Working with Obba and spreadsheets

Details about the exam will be announced soon.