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
The educational activities will be conducted through oral lessons, accompanied by exercise sessions.
During the lessons the theoretical contents will be exposed in a rigorous manner In the exercise sessions there will be a wide discussion of examples and exercises, with particular attention to those of a more financial nature. The participation of the students will be requested in the resolution of these exercises.
On the Elly platform one of the reference texts will be uploaded at the beginning of the course, as well as exercises and tests assigned in previous years. To download the material, registration to the online course is required.
Assessment methods and criteria
The summative evaluation of the learning will be done through a written test evaluated on a 0-32 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 (20pt) 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.
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 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 appeal is mandatory.
Other information
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