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Department of Economics, Social Studies, Applied Mathematics and Statistics

# Quantitative Finance and Insurance

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## PROBABILITY FOR FINANCE

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Course ID
SEM0173
Teachers
Tiziano De Angelis (Lecturer)
Andrea Bovo (Assistant technician)
Alberto Turigliatto (Assistant technician)
Degree course
Finance
Insurance and Statistics
Year
1st year
Teaching period
First semester
Type
Distinctive
Credits/Recognition
6
Course disciplinary sector (SSD)
MAT/06 - probability and statistics
Delivery
Formal authority
Language
English
Attendance
Optional
Type of examination
Written
Prerequisites
A good knowledge of basic calculus (Matematica Generale), of the foundations of probability calculus and statistical infererence (Statistica)
Propedeutic for
Stochastic calculus and mathematical finance
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## Course objectives

This course provides the mathematical foundation of probability theory which is needed to understand and apply modern tools in mathematical finance. We start from basic set theory and build the key tools in measure theory that allow us to make proper sense of the notions of probability, random variables and stochastic processes.

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## Results of learning outcomes

At the end of the course, students are expected to be capable of:

• providing mathematical formulations and solutions to probabilistic problems of applied interest
• understanding the role and implications of the assumptions made about the model
• being able to think about possible and useful generalizations of the  model
• being able to communicate such findings using appropriate and clear mathematical notation and language
• understanding the role of advanced probability theory in the context of mathematical finance

It is also expected that students will have acquired communication skills, through class discussion and developed their learning abilities, through a variety of learning tools (teaching material, class discussion, lab sessions, homeworks and tests)

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## Program

• Review of differential and integral calculus
• Introduction to measure spaces
• Events and random variables
• Independence
• Introduction to Lebesgue integrals
• Expectation and L^p spaces
• Product measure and Fubini’s theorem
• Conditional expectation
• Elements of martingale theory
• Applications: Black-Scholes model in discrete time
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## Course delivery

The course is articulated in 48 hours of formal in-class lecture time and in at least as many hours of at-home work solving practical exercises.

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## Learning assessment methods

The course grade is determined solely on the basis of a written examination. The examination tests the student's ability to do the following:

1. Present briefly the main ideas, concepts and results developed in the course, also explaining intuitively the meaning and scope of the definitions and the arguments behind the validity of the results
2. Use effectively the concepts and the results to answer questions in basic measure theory and stochastic process theory, e.g., computing conditional expectations.

The above is accomplished by asking the student to answer open questions (2-4 questions). Questions can be essay questions or exercises. The minimum exam grade is 18/30, the maximum grade is 30/30 cum laude. More details on the exam can be found on Moodle.

The exam is an open-book exam lasting 2 hours. Use of calculators is permitted.

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## Support activities

Short course on "Essentials of Mathematics" held in September.

Title:
Probability with Martingales
Year of publication:
1991
Publisher:
Cambridge University Press
Author:
David Williams
Required:
No
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Last update: 31/10/2022 14:16
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