Learning objectives
The course aims to provide the basic tools for the evaluation and management of financial instruments. During the course students will be taught the basic notions of probability theory, which are needed to construct and analyze an elementary model of a financial market under uncertainty. The principles of the valuation by arbitrage and the completeness of the market will also be illustrated, contextualized to the market analyzed but keeping in mind that they are still valid and can be extended to more complex models. Finally, students will be shown how to represent preferences for a rational decision maker and optimally select a portfolio of assets, based on the knowledge of their returns and covariances.
At the end of the course the student is expected to be proficient in:
- Knowledge and understanding: the student will be able to know and illustrate the basic notions of probability calculation and the fundamental elements of a mathematical model for financial markets
- Applying knowledge and understanding:: the student will be able to apply the basic notions learned to construct a probabilistic model to describe real world phenomena and to build an elementary market model in conditions of uncertainty
- Making judgments: the student will be able to analyze an elementary market model in conditions of uncertainty and evaluate the best investment strategies
- Communication skills The student will be able to describe the characteristics of a financial market and the financial instruments present in it with appropriate language
- Learning skills: the student will be able to determine the most appropriate investment strategies based on the preferences of a rational investor.
Prerequisites
Basic elements of calculus and financial mathematics
Course unit content
Introduction to probability theory: the various approaches. The axiomatic approach. conditional probability and Bayes'theorem. Random numbers: the discrete case and the continuous case. Random vectors. Basic notions on financial markets. One-period financial market. Fundamental theorems of asset pricing. Pricing of derivatives. Introduction to expected utility theory. Portfolio selection: Markowitz's model.
Full programme
Introduction to probability theory. Classical, empirical and subjective approaches. Axiomatic approach: sample space, sigma-algebra and probability measure. Axioms of probability. conditional probability, Bayes theorem. Random numbers, measurability. Distribution function. Discrete random numbers: probability mass function. Continuous random numbers: density function.
Expectation, variance and standard deviation. Moments of a random number.
Random vectors. Independent random numbers. Covariance and correlation.
Introduction to financial market. A 1-period financial market, with zero e non-zero interest rate.
Law of one price. Arbitrage and completeness. State price densities and risk-neutral probabilities. Fundamental theorems of asset pricing. Derivatives: call and put options. Put-call parity. Forward contracts and forward prices.
Introduction to expected utility theory. Von-Neumann-Morgenstern axioms. Expected Utility theorem. Portfolio selection: Mean-variance principle. Markowitz's model.
Bibliography
E. CASTAGNOLI, Brevissimo Abbecedario di Matematica Finanziaria, downloadable from Elly course page) or available at the “Centro fotocopie” of the Department.
E. CASTAGNOLI, M. CIGOLA, L. PECCATI, Probability. A Brief Introduction, 2° edizione, Egea, 2009
S. PLISKA, Introduction to Mathematical Finance: Discrete Time Models, Black-
well, Malden 1997 (Seconda edizione).
T. BJORK, Arbitrage Theory in Continuous Time, Oxford University Press, Oxford 1999.
Teaching methods
Classes will take place online through the use of the Teams and Elly platforms. They will be both synchronous (via Teams) and asynchronous (uploaded on the Elly page of the course). Asynchronous lessons will be dedicated to a rigourous exposition of the theory, for the acquisition of knowledge.
Synchrounous classes will be mainly dedicated to the presentation of examples and exercises. Students will be encouraged to actively participate to the discussion, to develop the ability to apply knowledge and the acquisition of judgements and learning skills.
For the acquisition of technical language, the meaning of the specific terms used in the course will be illustrated.
At the beginning of the course, one of the reference texts will be uploaded to the Elly platform, as well as exercises and exam topics assigned in previous years.
The teaching materials (slides) used during the lessons will also be uploaded during the course.
To download the material it is necessary to register for the online course.
The links to the videos will be available on Elly until December 31st, 2020.
Assessment methods and criteria
The summative evaluation of the learning will be done through a written test evaluated on a 0-33 scale: the test will be a quiz on Elly, with the use of the Respondus software (alternativley, Teams will be used).
The instructions can be found on the web pages:
http://selma.unipr.it/;
http://selma.unipr.it/wp-content/uploads/Guida-Respondus.pdf;
https://elly2020.sea.unipr.it
The ID will be uploaded on a OneDrive foldere which link will be sent after the deadline for the registration.
The test will consist of 11 multiple choice questions, each of which is worth 3 points. In particular there will be
a problem, structured in 7 questions, aimed at the analysis of an elementary model of
financial market in order to test learning ability, the capacty of applying knowledge to real problems, and the independence of judgment; 2 theoretical questions on the theory of financial markets and 2 theoretical questions on probability theory to ascertain the capacity of
communicate with an appropriate technical language.
In case of return to activity in presence, the summative evaluation of the learning will be done through a written
test evaluated on a 0-33 scale.
During the test, the student is asked to: 1) solve a problem, structured in
4 questions, aimed at the analysis of an elementary model of
financial market (21pt) in order to test learning ability, the capacty of
applying knowledge to real problems, and the independence of
judgment; 2) present the theoretical arguments learned during the
course, by answering two open questions (6pt each) to ascertain the
capacity of
communicate with an appropriate technical language.
Simulation of the test in both forms can be found on Elly.
During the course, an set consisting of 5 exercises on Probability theory will be assigned. These exercises are optional, but their solution will be evaluated on a 0-30 scale.
The students who deliver the solution of the exercises before the first exam are exempt from answering to the probability questions.
A scientific calculator may be used during the test.
The text of the test with its solution will be uploaded to Elly within a week after the test.
The result of the test will be published on Elly within 10 days after the test.
Information about the lenght, the evaluation of the global exam and the awarding of the "lode" can be found in the syllabus of the whole course.
Please note that online registration for the exam is mandatory.
Other information
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