Fiche de cours

Computational finance

MATH-472

Fiche de cours

Enseignant(s) :

Kressner Daniel
Nobile Fabio
Pulido Nino Sergio Andres

Langue:

English

Summary

Participants of this course will be exposed to computational techniques frequently used in mathematical finance applications. Emphasis will be put on the implementation and practical aspects.

Content

1. Transformation based methods
Derivatives pricing via Fourier transforms and the saddlepoint method.
2. Option pricing via PDE models
Finite difference approximation of Black-Scholes PDE.
American options and free boundary problems.
Jump-diffusion processes and integro-differential equations.
3. Numerical optimization
Model calibration in financial applications, portfolio optimization.
Linear optimization techniques.
Gradient descent techniques and constrained optimization.

Important: This course is concerned with computational tools used in mathematical finance. It is not to be understood as an introduction into mathematical finance.

Keywords

derivatives pricing, numerical methods, Matlab, optimization, PDE, Fourier transform, saddle point approximation, calibration, volatility surface

Learning Prerequisites

Required courses

Stochastic processes / stochastic calculus

Recommended courses

Numerical Analysis
Introduction to Finite Elements
Derivatives

Important concepts to start the course

Basic background in numerical analysis, linear algebra, and differential equations.
Command of Matlab.

Learning Outcomes

By the end of the course, the student must be able to:

• Choose method for solving a specific pricing or calibration problem.
• Implement numerical algorithms.
• Interpret the results of a computation.
• Recall the advantages and limitations of different methods.
• Assess / Evaluate the performance of several financial models.
• Compare the results from different pricing algorithms.

Transversal skills

• Use a work methodology appropriate to the task.

Teaching methods

Ex cathedra lecture, exercises in the classroom and with computer.

Expected student activities

Attendance of lectures.
Completing exercises.
Solving problems on the computer.

Assessment methods

Computer-based final examination. 20% of the grade are determined by take-home exams / graded exercises.

Resources

Bibliography

Hirsa, Ali. Computational methods in finance. Chapman & Hall/CRC Financial Mathematics Series. CRC Press, Boca Raton, FL, 2013.

Seydel, Rüdiger U. Tools for computational finance. Fourth edition. Universitext. Springer-Verlag, Berlin, 2009.

Achdou, Yves; Pironneau, Olivier Computational methods for option pricing. Frontiers in Applied
Mathematics, 30. SIAM, Philadelphia, PA, 2005.

Additional lecture material will be provided by the instructors.

Notes/Handbook

• Tools for computational finance / Seydel
• Computational methods for option pricing / Achdou
• Computational methods in finance / Hirsa