### General Information

Course Code | X_400352 |
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Credits | 6 EC |

Period | P1+2 |

Course Level | 400 |

Language of Tuition | English |

Faculty | Faculty of Science |

Course Coordinator | dr. D. Dobler |

Examiner | dr. D. Dobler |

Teaching Staff |
dr. D. Dobler |

### Practical Information

You need to register for this course yourself

Last-minute registration is available for this course.

Teaching Methods | Lecture |
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Target audiences

This course is also available as:

### Course Objective

After the course, the student will- be aquainted with the notion of pricing derivatives and arbitrage, and

with the basics of discrete stochastic processes, including the concepts

of martingales; be able to price derivatives in the binomial model.

- be familiar with the basics of continuous time stochastic processes,

concepts of measure theory, Brownian motion, martingales; be able to

verify whether the process is a Brownian motion and/or is a martingale.

- be familiar with the notions of stochastic integral Ito and SDE, the

Ito formula; be able to compute stochastic integrals, to use the Ito

formula, and to derive properties of stochastic integrals and the

solutions of SDEs.

- be familiar with the Black-Sholes model; be able to price options by

the no-arbitrage and Girsanov's theorem, to compute the hedging

portfolio and the Greeks, and to make connections to the corresponding

Black-Scholes PDEs.

- be familiar with the extended Black-Sholes models; be able to price

derivatives in those models, to make connection with the corresponding

PDEs, and to apply the Black-Scholes machinary to the interest rates

models.

### Course Content

Financial institutions trade in risk, and it is therefore essential tomeasure and control such risks. Financial instruments such as options,

swaps, forwards, etc. play an important role in risk management, and to

handle them one needs to be able to price them. This course gives an

introduction to the mathematical tools and theory behind risk

management.

A "stochastic process" is a collection of random variables, indexed by a

set T. In financial applications the elements of T model time, and T is

the set of natural numbers (discrete time), or an interval in the

positive real line (continuous time). "Martingales" are processes whose

increments over an interval in the future have zero expectation given

knowledge of the past history of the process. They play an important

role in financial calculus, because the price of an option (on a stock

or an interest rate) can be expressed as an expectation under a

so-called martingale measure. In this course we develop this theory in

discrete and continuous time. Most models for financial processes in

continuous time are based on a special Gaussian process, called Brownian

motion. We discuss some properties of this process and introduce

"stochastic integrals" with Brownian motion as the integrator. Financial

processes can next be modeled as solutions to "stochastic differential

equations". After developing these mathematical tools we turn to finance

by applying the concepts and results to the pricing of derivative

instruments. Foremost, we develop the theory of no-arbitrage pricing of

derivatives, which are basic tools for risk management.

### Teaching Methods

Lectures (13 x 2 hours) and discussion of exercises (13 x 1 hour)### Method of Assessment

Assignments (30%) and written examination/resit examination (70%).There is no resit possibility for the assignments.

### Entry Requirements

Probability (X_400622) and Analysis 1 (X_400005), or their equivalents.### Literature

Lecture notesAdditional literature:

Shreve, "Stochastic Calculus for Finance I: The Binomial Asset Pricing

Model", Springer;

Shreve, "Stochastic Calculus for Finance II: Continuous-time models",

Springer.

### Target Audience

mBA, mBA-D, mMath, mSFM, master Econometrics.### Additional Information

A significant part of the course is used to introduce mathematicalsubjects and techniques like Brownian motion, stochastic integration and

Ito calculus. In view of this, the course is NOT meant for students who

already followed the master course "Stochastic Integration" or

"Stochastic differential equations". On the other hand, after completing

this course, students may be motivated to follow other courses (like the

two mentioned above) where stochastic calculus is treated in a deeper

and more rigorous way.