MATHEMATICAL MODELS FOR FINANCIAL MARKETS

1.Knowledge and understanding

 

The purpose of the course is the acquisition of theoretical principles concerning the financial decisions under risk and uncertainty (stochastic dominance, expected utility, Cumulative Prospect Theory, ...), the most important theories of financial portfolios (mean-variance, CHAPM, APT) and the use of some tools for managing financial risk. Beyond the indispensable theoretical knowledge, properly formalized, we also intend to transfer adequate professional skills to deepen the topics covered by an operational point of view.

 

The teaching methodologies are designed to develop students’ professional skills using also multimedia, database access, use of spreadsheets, etc.

The exam is composed of a written test and an oral examination, withe the goal of testing for the student’s knowledge, his understanding of the abstract concepts, and their translation from an operational point of view.

 

During the entire course, knowledge and understanding are tested on a continuous basis, and a fruitful and active participation by students is always stimulated.

 

 

2. Applying knowledge and understanding

Special attention is also paid to operating activities of future graduates, who are facing the problems professionally before mentioned, often under different assumptions or in different contexts, also transversal and interdisciplinary. To this end, teachers use a teaching method with the emphasis to the acquisition operations ("know-how") of the analytical tools and concepts proposed during the teaching of the discipline, aiming to develop critical skills of the student in a continuous process interaction analysis - synthesis, also presenting in the classroom appropriate real cases, guiding the study and analysis with the help of educational tools and technology more appropriate. Teachers care in its review of final learning the actual acquisition of these skills, even proposing and discussing critically and constructively with students drawn from them prepared with these precipue purposes.

 

3. Making judgments

 

The development of a critical ability in the context of the topics covered is a major educational objectives of teaching.

A good acquisition of theoretical knowledge and operational chapabilities in the program of education is not enough for a complete training of the student if such preparation is not accompanied by the acquisition of a thorough, independent, socially and morally responsible for chapacity assessment, setting and resolution of a problem, proposing models that consider more appropriate analysis of financial issues considered. Such awareness serves as a guide to teachers throughout the training of discipline, making them interact with students in a constructive logic, in order to stimulate all phases of teaching, their chapacity for reflection, acquisition and interpretation of the information needed and Data essential, although insufficient or incomplete, for the management of complex issues, the construction and understanding of formal models, both descriptive and prescriptive. The focus is, therefore, training of research of economic and financial information sources, both traditional and modern, more appropriate (consultations of specialized publications, databases, websites, etc.),

 

4. Communication skills

the teaching will put the student in a position to transfer to third parties, even non-specialists, with clarity, precision and language appropriate technical, information, analysis, value judgments, projects and proposals on complex financial issues, that on the job will face.

The student is continually urged to make oral and formally their thoughts in proper arguments and techniques, to draft documents in writing, to prepare presentations

multimedia, individually and in groups, to discuss what has been presented in the classroom, to stimulate a fruitful collaboration on the level of communication. The final exam is an additional chance for reflection and verification of the various communication skills actually achieved by the student.

5. Learning skills

will provide students both an encouragement for a more active participation as possible to the entire educational process and for an improvement in the method of study and the purpose of a more effective learning of the discipline, presenting characteristics precipue in terms of learning by means of an appropriate process inductive - deductive.

Model prices in the financial markets

The cardinal utility theory; axiomatic structure. fundamental theorem.

Intensity of preference. Absolute and relative risk measures.

The certainty equivalent. Applications equivalent course. Risk premium.

Main families of utility functions and their properties. Paradoxes of Allais and Ellsberg. Some experiments by Kahneman and Tversky .

First and second order stochastic dominance.

Portfolio Theory. Mean-variance models. Risk indices.

Only two risky assets. Efficient Frontier: possible cases particularly interesting.

Contraction risk. Portfolio of minimum risk.

Model mean-variance with the most risky assets.

Main theorems on the efficient frontier. Feasible set and shape of the border.

Diversification and risk reduction. Systematic risk.

Diversification and risk reduction. Systematic risk. Minimization of the variance of the portfolio.

Maximizing the risk premium and the consequences on the efficient frontier. The Chapital Market Line. Portfolio Market. Separation theorem.

CAPM assumptions and mathematical derivations of the model. SML. Systematic risk and specific risk. The coefficient '' beta ''. Diversification and risk reduction in the CAPM.

Non-standard versions of the CAPM.

Single Index Model. Multifactorial models.

Options: first definitions and payoffs. Using options for basic risk management.

Hedging strategies by combining options and below.

The spreads. Combinations of options. Put-call parity. Minimum and maximum limits for the value of a put and a call.

Risk-neutral pricing model.

Fixed-Income Market Models and Derivatives.

1. J. Cvitanic, F. Zapatero, “Introduction to the Economics and Mathematics of Financial Markets.’’The MIT Press Cambridge, Massachusetts London, England, 2004.

2. D. Luenberger , “Finanza e Investimenti’’, Apogeo, Milano 2006.

3. J. Berck , P. DeMarzo, “Chapital budgeting”, Addison Wesley (Pearson International Ed.), Milano 2009.

4. J. Berck, P. DeMarzo, “Finanza aziendale 2”, Addison Wesley (Pearson International Ed.), Milano 2008.

5. S. Benninga , “Modelli finanziari”, McGraw-Hill, Milano 2010.

6. E. J. Elton, M. J. Gruber ,“Modern portfolio theory and investment analysis”, Wiley, 2002.

7. J. Hull, Options, Futures, and Other Derivatives, 10th Edition, Pearson, 2018

8. Rosazza Gianin, E., Sgarra, C., Esercizi di finanza matematica, Springer, 2007

9. Castagnoli, E. Matematica dei Mercati Finanziari, Egea, 2017