# Coursebooks

## Computational finance

#### Lecturer(s) :

Glau Kathrin Beatrice
Pulido Nino Sergio Andres
Statti Francesco

English

#### Summary

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

#### Content

1. Brief introduction to option pricing
Basic stochastic models in finance
Basic tools of stochastic calculus
Monte Carlo simulation based methods

2. Transformation based methods
Affine models
Option pricing via Fourier transforms

3. Density approximation techniques
Polynomial models and calculation of moments
Option pricing via density approximation

4. 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

#### Keywords

financial models, stochastic calculus, option pricing, numerical methods, Matlab, Monte Carlo simulation, PDE, Fourier transform, density approximation techniques, volatility surface

#### Learning Prerequisites

##### Recommended courses

Stochastic processes / stochastic calculus

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.
• Recall the basic concepts behind the theory of option pricing in financial models.
• Choose method for solving a specific pricing problem.

#### 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

60% of the grade is determined by a computer-based final examination. 40% of the grade is determined by take-home exams / graded exercises.

#### Resources

No

##### Bibliography

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

Hilber, Norbert; Reichmann, Oleg; Schwab, Christoph; Winter, Christoph. Computational methods for quantitative finance. Springer, 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.

Glasserman, Paul. Monte Carlo methods in financial engineering. Springer, 2003

Björk, Tomas. Arbitrage theory in continuous time. Third edition, OUP Oxford, 2009.

Shreve, Steven E. Stochastic calculus for finance II: Continuous-Time models, Volume 11. Springer Science & Business Media, 2004.

Lamberton, Damien; Lapeyre, Bernard. Introduction to stochastic calculus applied to finance. Second revised edition. Chapman & Hall/CRC, 2008.

Additional lecture material will be provided by the instructors.

##### Notes/Handbook

• Computational methods in finance / Hirsa
• Computational methods for quantitative finance /  Hilber
• Tools for computational finance / Seydel
• Computational methods for option pricing / Achdou
• Monte Carlo methods in financial engineering / Glasserman
• Arbitrage theory in continuous time /  Björk
• Stochastic calculus for finance II: Continuous-Time models / Shreve
• Introduction to stochastic calculus applied to finance / Lamberton

### Reference week

MoTuWeThFr
8-9
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
Under construction

Lecture
Exercise, TP
Project, other

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• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German