Old Courses

The aim of the course is to provide a deepest introduction to the basic tools and theorems of mathematical calculus. It is taught as a companion courses for second and third year students. Undergraduate courses in the first and second years are oriented to provide a specific set of computational procedures to be applied to the solution of specific problems. The present course might constitute a good complement as it is more concerned with the definition and study of general notions and it is characterized by a theorem-proving approach.

• Teaching load: 40 hours
• Lecturer : BOTTAZZI Giulio
• Teaching Assistant: Pietro Battiston
• Semester: Fall
• Textbook: Notes directly provided by the teacher
• Further references and Optional readings: Principles of Mathematical Analysis, W.A. Rudin, chapters 3, 6 and 9
• Prerequisites: The course does not have any prerequisite. Knowledge of the exponential, logarithm and trigonometric functions and elementary set theory are advisable.

List of topics covered by the lectures (not necessarily in this order)

Prerequisite:

• Sets, equivalence relations and functions
• Order relations, upper and lower bounds, supremum and infimum
• Countable sets
• Real numbers

Introduction to topology, normed and metric spaces:

• Topological spaces
• Compact and connected sets
• Basis of a topology
• Countable spaces
• Continuity
• Notion of distance and induced topology

Sequences in Metric Spaces:

• Cauchy sequences.
• Limit of sequences and sub-sequences.
• Comparison theroem for limit of sequences.
• The number e as a limit of a sequence.

Series:

• Series as limit of partial sums.
• Root, ratio and comparison tests for series.

Functions and Limits on Metric Spaces:

• Notion of limit of function in a metric space.
• Properties of limit (sum, ratio and product of limits)

Limits of Real Functions:

• Some important limits.
• Properties of limits of functions (comparison test)
• The number e as limit of a function.

Continuity of Real Functions:

• Uniform continuity.
• Weiestrass Theorem.

• Teaching load: 63 hours (ECTS credits 9)
• Lecturers: BOTTAZZI Giulio, GIACHINI Daniele
• Teaching Assistant:
• Semester: Fall
• Outline: This course aims to provide a modern theoretical foundation to the most widespread notions and tools in financial economics, including present value theory, real interest rate, arbitrage pricing, portfolio management and mean-variance analysis.
• Objectives: The students should become acquainted with the notion of arbitrage and equilibrium price in different market settings. They should be able to solve the problem of portfolio optimization and mean-variance analysis in rather general terms.
• Description: General equilibrium, no arbitrage in riskless economies, real interest rate and the yield curve. Arbitrage in security markets, state prices, complete and incomplete markets, valuation functional and the fundamental theorem of finance. Choices under uncertainty, expected utility theory and risk aversion. Portfolio choices, optimal portfolio with multiple risky assets. General equilibrium under uncertainty. Pricing kernel and mean-variance analysis. OPTIONAL: behavioral finance, asset prices under ambiguity, evolutionary finance and the market selection hypothesis.
• Textbooks:

  [LRW]

  Principles of Financial Economics, S.F. LeRoy and J. Werner (2nd edition)

  [FBJ]

  Foundations of Financial Markets and Institutions, Fabozzi, Modigliani and Jones

• Further references and Optional readings:

  [MWG]

  Microeconomic Theory, A. Mas-Colell, M. D. Whinston, and J. R. Green

  [CN]

  Quantitative Financial Economics, K. Cuthbertson and D. Nitzsche

  [HR]

  Financial Economics, T. Hens and M. O. Rieger

  [ES]

  Ambiguity and Asset Markets, L. G. Epstein and M. Schneider, NBER paper;

  [Hu]

  Options, Futures and Other Derivatives, J. C. Hull (7th ed.)

  [Ni]

  The Ascent of Money: A Financial History of the World, Niall Ferguson

  [Eco]

  Guide to the Financial Markets, The Economist

• Prerequisites: The course requires a basic knowledge of linear algebra (linear space, linear map, basis, inversion, eigenvectors and eigensystems), calculus (differential analyses of function of many real variables and static optimization) and, to a lesser extent, probability theory. Some basic knowledge of the theory of choice under uncertainty (utility theory) and economic equilibrium are in general given for granted and only cursorily reviewed.
• Keywords: financial economics, general equilibrium, arbitrage, asset pricing, optimization
• Final exam: the get the final grading students are required to pass a final written exam. At least one fake exam will be organized to test your level of preparation. Students get extra points for completing their homework during the course.
• Calendar: check "Financial Economics" entries for the lectures timetable and locations in the calendar.
• Leacures are also available in the Teams platform at the following link Financial Economics

Data for Assignment 4: weekly closing prices of America Corporation (BAC), General Electric Company (GE), Ford Motor Co. (F), Citigroup, Inc. (C), Pfizer Inc. (PFE), Hewlett-Packard Company (HPQ), JPMorgan Chase & Co. (JPM), AT&T, Inc. (T), Wells Fargo & Company (WFC) and J. C. Penney Company, Inc. (JCP)

• Teaching load: 63 hours (ECTS credits 9)
• Lecturers: BOTTAZZI Giulio, GIACHINI Daniele
• Teaching Assistant:
• Semester: Fall
• Outline: This course aims to provide a modern theoretical foundation to the most widespread notions and tools in financial economics, including present value theory, real interest rate, arbitrage pricing, portfolio management and mean-variance analysis.
• Objectives: The students should become acquainted with the notion of arbitrage and equilibrium price in different market settings. They should be able to solve the problem of portfolio optimization and mean-variance analysis in rather general terms.
• Description: General equilibrium, no arbitrage in riskless economies, real interest rate and the yield curve. Arbitrage in security markets, state prices, complete and incomplete markets, valuation functional and the fundamental theorem of finance. Choices under uncertainty, expected utility theory and risk aversion. Portfolio choices, optimal portfolio with multiple risky assets. General equilibrium under uncertainty. Pricing kernel and mean-variance analysis. OPTIONAL: behavioral finance, asset prices under ambiguity, evolutionary finance and the market selection hypothesis.
• Textbooks:

  [LRW]

  Principles of Financial Economics, S.F. LeRoy and J. Werner (2nd edition)

  [FBJ]

  Foundations of Financial Markets and Institutions, Fabozzi, Modigliani and Jones

• Further references and Optional readings:

  [MWG]

  Microeconomic Theory, A. Mas-Colell, M. D. Whinston, and J. R. Green

  [CN]

  Quantitative Financial Economics, K. Cuthbertson and D. Nitzsche

  [HR]

  Financial Economics, T. Hens and M. O. Rieger

  [ES]

  Ambiguity and Asset Markets, L. G. Epstein and M. Schneider, NBER paper;

  [Hu]

  Options, Futures and Other Derivatives, J. C. Hull (7th ed.)

  [Ni]

  The Ascent of Money: A Financial History of the World, Niall Ferguson

  [Eco]

  Guide to the Financial Markets, The Economist

• Prerequisites: The course requires a basic knowledge of linear algebra (linear space, linear map, basis, inversion, eigenvectors and eigensystems), calculus (differential analyses of function of many real variables and static optimization) and, to a lesser extent, probability theory. Some basic knowledge of the theory of choice under uncertainty (utility theory) and economic equilibrium are in general given for granted and only cursorily reviewed.
• Prerequisites suggested reading (relevant chapters in): C. P. Simon, L. E. Blume, Mathematics for Economists, H. R. Varian Microeconomic Analysis.
• Keywords: financial economics, general equilibrium, arbitrage, asset pricing, optimization
• Final exam: the get the final grading students are required to pass a final written exam. At least one fake exam will be organized to test your level of preparation. Students get extra points for completing their homework during the course.
• Calendar: check "Financial Economics" entries for the lectures timetable and locations in the calendar.

List of lectures. To have a guess about future lectures check the previous years but consider that your mileage may vary.

1. (22/09/2020, Bottazzi) Introduction: the structure of the course and its textbooks; the role of financial institutions slides (Textbooks sections: [FBJ] Ch. 1, 2 and 3; [Ni] Ch. 1; [Eco] Ch. 1)
2. (23/09/2020, Bottazzi) Barter and market economies; Two agents, two goods, one date model.
3. (24/09/2020, Bottazzi) Financial equilibrium. Two date model. Spot and forward markets. The role of arbitrage in setting prices.
4. (28/09/2020, Bottazzi) Introduction to present value model. Present value prices and economic equilibrium. Redundant contracts.
5. (29/09/2020, Bottazzi) Real and nominal interest rates; The role of inflation; Pricing of forward contracts. (Textbooks sections: [FBJ] Ch. 4,5,10 and 11 [Hu] Sections from 4.1 to 4.8 all Chapter 5 and section from 6.1 to 6.3 [Ni] Ch. 2)
6. (30/09/2020, Bottazzi) Forward interest rate. Inter-temporal allocation in a one good two date model: the role of preferences and endowments in setting real interest rates.
7. (02/10/2020, Bottazzi) Structure of security markets with uncertainty; complete markets and redundant assets. Left and right matrix inverse. The Law of One Price. (Textbook sections: [LRW] 1.2,1.6)
8. (07/10/2020, Bottazzi) Payoff pricing correspondence; the Law of One Price; state claims and state prices (Textbook sections: [LRW] 2.1, 2.2, 2.3, 2.5)
9. (08/10/2020, Bottazzi) Arbitrage and valuation: strong and weak arbitrage; positive pricing; the payoffs convex cone; Farkas' Lemma; the valuation functional and strong arbitrage; Stiemke's Lemma; the valuation functional and arbitrage; arbitrage and valuation: the fundamental theorem of finance; (Textbook sections: [LRW] 3.2, 3.3, 3.4, 3.5, 4.1, 4.2, 4.3, 5.2,5.3,5.4,5.5) [Assignment 1]
10. (12/10/2020, Bottazzi) Examples of payoff pricing function and no-arbitrage pricing. Discussion of the no-arbitrage condition in a general two date model. Multiplicity of valuation functionals. Riskless return. Risk neutral probabilities; (Textbook sections: [LRW] 4.5, 5.6, 5.7)
11. (13/10/2020, Giachini) Forward and future contracts in a two-date model, definition and pricing. Call and put options in a two-date model, definition and pricing; put-call parity. (see the discussion of forward and future contracts in [Hu] Ch.1-3)
12. (14/10/2020, Giachini) Completing the market with call (or put) options; using option prices to derive the payoff pricing functional. Agent behavior under uncertainty; Expected Utility Theory. Choices under uncertainty: consumption and portfolio choices. (Textbook sections: [MWG] Ch.6 section 6B)
13. (19/10/2020, Giachini) Expected Utility Theory, separable and time invariant utilities; risk aversion; absolute and relative risk aversion coefficients; certainty equivalent; Concave transformation; comparison of utility functions; first and second order stochastic dominance; (Textbook sections: [LRW] 9.1,9.2, 9.3, 9.4, 9.5, 9.6, 9.8 . The proves of the theorems can also be found in [MWG] Ch.6 Sec.B; further examples are discussed in [CN] Ch.1 Sec.3-5)
14. (20/10/2019, Giachini) Exercises and solutions of Assignment 1
15. (21/10/2019, Giachini) Portfolio optimization: first order conditions; arbitrage and optimal portfolio, optimal consumption, prices and returns. (Textbook sections: [LRW] 1.4, 1.5, 2.6, 3.6, 11.1, 11.2; see [CN] Ch.15 for an introductory discussion of the problem)
16. (26/10/2020, Giachini) Portfolio optimization: Example of internal and boundary solutions. Marginal utility and risk-neutral probabilities.
17. (27/10/2020, Giachini) Portfolio optimization with one risky security: impact of wealth on optimal investment; exercises (Textbook sections: [LRW] 12.2).
18. (28/10/2020, Giachini) Exercises and solutions of Assignment 2
19. (2/11/2020, Bottazzi) Inner products, norms, orthogonal vectors, Gram–Schmidt procedure, orthogonal complement, orthogonal projection, Riesz representation theorem. (Textbook sections: [LRW] from 17.2 to 17.8) [Assignment 3]
20. (3/11/2020, Bottazzi) Quadratic problem in a inner product space. Expectation kernel, pricing kernel, mean-variance frontier payoffs, mean variance frontier returns. (Textbook sections: [LRW] 17.9, 17.10, 18.2, 18.3)
21. (9/11/2020, Bottazzi) Minimum variance frontier return; Mean-variance frontier, the generic case. (Textbook sections: [LRW] 18.4)
22. (11/11/2020, Giachini) Exercises and solutions of Assignment 3
23. (9/11/2020, Bottazzi) Mean-variance frontier and orthogonal kernels.
24. (17/11/2020, Bottazzi) Using the mean-variance frontier to solve portfolio optimization problems. Stochastic discount factors; Factor pricing. (Textbook sections: [LRW] 20.2, 20.3) [Assignment 4]
25. (18/11/2020, Giachini) Economic equilibrium; representative agent model; equilibrium sufficient conditions; equilibrium and arbitrage. (Textbook sections: [LRW] 1.7, 1.8, 1.9, 2.4, 3.7)
26. (23/11/2020, Giachini) An example of general equilibrium with heterogeneous agents. The relation between individual wealth and prevailing prices.
27. (24/11/2020, Giachini) Equilibrium and Pareto optimal allocation, first and second welfare theorem. (Textbook sections: [LRW] 15.2, 15.3, 15.4, 15.5 without theorem 15.5.1, 15.6)
28. (25/11/2020, Giachini) Exercises and solutions of Assignment 4 [Assignment 5]
29. (1/12/2019, Bottazzi) CAPM, (Textbook sections: [LRW] from 19.1 to 19.4)
30. (2/12/2019, Giachini) Exercises and solutions of Assignment 5.
31. (4/12/2019, Bottazzi) Mock exam and solutions

• Teaching load: 63 hours (ECTS credits 9)
• Lecturers: BOTTAZZI Giulio, GIACHINI Daniele
• Teaching Assistant:
• Semester: Fall
• Outline: This course aims to provide a modern theoretical foundation to the most widespread notions and tools in financial economics, including present value theory, real interest rate, arbitrage pricing, portfolio management and mean-variance analysis.
• Objectives: The students should become acquainted with the notion of arbitrage and equilibrium price in different market settings. They should be able to solve the problem of portfolio optimization and mean-variance analysis in rather general terms.
• Description: General equilibrium, no arbitrage in riskless economies, real interest rate and the yield curve. Arbitrage in security markets, state prices, complete and incomplete markets, valuation functional and the fundamental theorem of finance. Choices under uncertainty, expected utility theory and risk aversion. Portfolio choices, optimal portfolio with multiple risky assets. General equilibrium under uncertainty. Pricing kernel and mean-variance analysis. OPTIONAL: behavioral finance, asset prices under ambiguity, evolutionary finance and the market selection hypothesis.
• Textbooks:

  [LRW]

  Principles of Financial Economics, S.F. LeRoy and J. Werner (1st edition)

  [FBJ]

  Foundations of Financial Markets and Institutions, Fabozzi, Modigliani and Jones

• Further references and Optional readings:

  [MWG]

  Microeconomic Theory, A. Mas-Colell, M. D. Whinston, and J. R. Green

  [CN]

  Quantitative Financial Economics, K. Cuthbertson and D. Nitzsche

  [HR]

  Financial Economics, T. Hens and M. O. Rieger

  [ES]

  Ambiguity and Asset Markets, L. G. Epstein and M. Schneider, NBER paper;

  [Hu]

  Options, Futures and Other Derivatives, J. C. Hull (7th ed.)

  [Ni]

  The Ascent of Money: A Financial History of the World, Niall Ferguson

  [Eco]

  Guide to the Financial Markets, The Economist

• Prerequisites: The course requires a basic knowledge of linear algebra (linear space, linear map, basis, inversion, eigenvectors and eigensystems), calculus (differential analyses of function of many real variables and static optimization) and, to a lesser extent, probability theory. Some basic knowledge of the theory of choice under uncertainty (utility theory) and economic equilibrium are in general given for granted and only cursorily reviewed.
• Prerequisites suggested reading (relevant chapters in): C. P. Simon, L. E. Blume, Mathematics for Economists, H. R. Varian Microeconomic Analysis.
• Keywords: financial economics, general equilibrium, arbitrage, asset pricing, optimization
• Final exam: the get the final grading students are required to pass a final written exam. At least one fake exam will be organized to test your level of preparation. Students get extra points for completing their homework during the course.
• Calendar: check "Financial Economics" entries for the lectures timetable and locations in the calendar.

List of lectures. To have a guess about future lectures check the previous years but consider that your mileage may vary.

1. (23/09/2019, Bottazzi) Introduction: the structure of the course and its textbooks; the role of financial institutions slides (Textbooks sections: [FBJ] Ch. 1, 2 and 3; [Ni] Ch. 1; [Eco] Ch. 1)
2. (24/09/2019, Bottazzi) Barter and market economies; Two agents, two goods, one date model. Two date model. Spot and forward markets.
3. (25/09/2019, Bottazzi) Financial equilibrium. The role of arbitrage in setting prices. Introduction to present value model.
4. (30/09/2019, Bottazzi) Present value prices and economic equilibrium. Redundant contracts. Real and nominal interest rates; The role of inflation; Pricing of forward contracts. (Textbooks sections: [FBJ] Ch. 4 and 5 [Hu] Sections from 4.1 to 4.8 all Chapter 5 and section from 6.1 to 6.3 [Ni] Ch. 2) [Assignment 1]
5. (01/10/2019, Bottazzi) Forward interest rate. Inter-temporal allocation in a one good two date model: the role of preferences and endowments in setting real interest rates.
6. (02/10/2019, Bottazzi) Structure of security markets with uncertainty; complete markets and redundant assets. Left and right matrix inverse. (Textbook sections: [LRW] 1.6)
7. (07/10/2019, Bottazzi) Payoff pricing correspondence; the Law of One Price; state claims and state prices (Textbook sections: [LRW] 2.1, 2.3, 2.4, 2.5)
8. (08/10/2019, Bottazzi) Arbitrage and valuation: strong and weak arbitrage; positive pricing; the payoffs convex cone; (Textbook sections: [LRW] 3.2, 3.3, 3.4, 3.5, 5.1)
9. (09/10/2019, Bottazzi) Farkas' Lemma; the valuation functional and strong arbitrage; Stiemke's Lemma; the valuation functional and arbitrage; Arbitrage and valuation: the fundamental theorem of finance; multiplicity of valuation functionals; risk neutral probabilities; (Textbook sections: [LRW] 5.2, 5.3, 5.5, 6.3, 6.4, 6.5;6.6, 6.7;) [Assignment 2]
10. (14/10/2019, Giachini) Examples of payoff pricing function and no-arbitrage pricing. Discussion of the no-arbitrage condition in a general two date model.
11. (15/10/2019, Giachini) Solution of Assignment 1.
12. (16/10/2019, Giachini) Valuation of derivatives in a two date model: forward and future (see the discussion of forward and future contracts in [Hu] Ch.1-3)
13. (21/10/2019, Giachini) Valuation of derivatives in a two date model: call and put options. Choices under uncertainty: consumption and portfolio choices, Expected Utility Theory, separable and time invariant utilities; risk aversion; absolute and relative risk aversion coefficients; certainty equivalent; (Textbook sections: [LRW] 9.2, 9.3, 9.4, 9.5, 9.6, 9.8)
14. (23/10/2019, Giachini) Concave transformation; comparison of utility functions; first and second order stochastic dominance; (Textbook sections: the proves of the theorems can also be found in [MWG] Ch.6 Sec.B; further examples are discussed in [CN] Ch.1 Sec.3-5)
15. (28/10/2019, Giachini) Exercises and solutions of Assignment 2
16. (29/10/2019, Giachini) Portfolio optimization: first order conditions; arbitrage and optimal portfolio, optimal consumption, prices and returns. (Textbook sections: [LRW] 1.4, 1.5, 2.6, 3.6, 11.1, 11.2; see [CN] Ch.15 for an introductory discussion of the problem)
17. (30/10/2019, Giachini) Portfolio optimization: Example of internal and boundary solutions. Marginal utility and risk-neutral probabilities. [Assignment 3]
18. (4/11/2019, Giachini) Portfolio optimization with one risky security: impact of wealth on optimal investment; exercises (Textbook sections: [LRW] 12.2).
19. (5/11/2019, Bottazzi) Exercises and solution. Strategies of portfolio optimization.
20. (6/11/2019, Bottazzi) Scalar product, Gram–Schmidt procedure, orthogonal complement and orthogonal projection. Portfolio optimization with several risky securities: from consumption to investment optimization; risk-return trade off; optimal portfolio under fair pricing. (Textbook sections: [LRW] 13.2, 13.3, 13.4)
21. (11/11/2019, Bottazzi) Portfolio optimization with several risky securities: Linear Risk tolerance. (Textbook sections: [LRW] 9.9, 13.6)
22. (12/11/2019, Bottazzi) Expectations kernel and pricing kernel (Textbook sections: [LRW] from 17.1 to 17.10, 18.5)
23. (13/11/2019, Bottazzi) Mean-variance frontier payoffs and mean-variance frontier returns with complete markets. (Textbook sections: 18.1, 18.2 and Th. 18.2.1 with proof 18.4, 18.6; see also the discussion in [CN] Ch.15 Sec.1-2).
24. (19/11/2019, Giachini) Mean-variance frontier payoffs and mean-variance frontier returns with incomplete markets. (Textbook sections: 18.1, 18.2 and Th. 18.2.1 with proof 18.4, 18.6; see also the discussion in [CN] Ch.15 Sec.1-2).
25. (20/11/2019, Giachini) Zero-Covariance Frontier Returns, Sharpe Ratio of mean-variance frontier returns (Textbook sections: 18.4) [Assignment 4, see below for data]
26. (21/11/2019, Giachini) Exercises and solutions of Assignment 3
27. (25/11/2019, Bottazzi) Economic equilibrium; representative agent model; equilibrium sufficient conditions; equilibrium and arbitrage. (Textbook sections: [LRW] 1.7, 1.8, 1.9, 2.4, 3.7) [Assignment 5]
28. (27/11/2019, Bottazzi) An example of general equilibrium with heterogeneous agents. The relation between individual wealth and prevailing prices.
29. (2/12/2019, Bottazzi) Exercises and solutions of Assignment 4.
30. (3/12/2019, Bottazzi) Consumption based security pricing; a first derivation of CAPM;, (Textbook sections: [LRW] 14.1, 14.2, 14.3, 14.5)
31. (4/12/2019, Bottazzi) Equilibrium and Pareto optimal allocation, first and second welfare theorem; (Textbook sections: [LRW] 15.2, 15.3, 15.4, 15.5 without theorem 15.5.1, 15.6 )
32. (10/12/2019, Giachini) Exercises and solutions of Assignment 5.
33. (11/12/2019, Giachini) Mock Exam and solutions.

Data for Assignment 4: weekly closing prices of America Corporation (BAC), General Electric Company (GE), Ford Motor Co. (F), Citigroup, Inc. (C), Pfizer Inc. (PFE), Hewlett-Packard Company (HPQ), JPMorgan Chase & Co. (JPM), AT&T, Inc. (T), Wells Fargo & Company (WFC) and J. C. Penney Company, Inc. (JCP)

The aim of the course is to provide a deepest introduction to the basic tools and theorems of mathematical calculus. Undergraduate courses are oriented to provide a specific set of computational procedures to be applied to the solution of specific problems. The present course might constitute a good complement as it is more concerned with the definition and study of general notions and it is characterized by a theorem-proving approach.

• Teaching load: 40 hours (ECTS credits 4)
• Lecturer : BOTTAZZI Giulio, BATTISTON Pietro
• Semester: Fall
• Textbook: Notes directly provided by the teacher
• Further references and Optional readings: Principles of Mathematical Analysis, W.A. Rudin, chapters 3, 6 and 9
• Prerequisites: The course requires some knowledge of linear algebra (linear space, linear map, basis, determinant, inversion, eigenvectors) and basic calculus.

List of topics covered by the lectures.

Functions of several real variables:

• Continuity in Euclidean spaces
• Derivative and differential
• Concavity and convexity
• Local and global maxima and minima
• Implicit function theorem
• Theorems of alternatives
• Constrained maximization
• Khun-Tucker conditions

Riemann and Steltjes integral:

• Finite sum and limits
• Summability conditions
• Properties of the integral

Measure theory:

• Sigma algebra, measurable space, measure
• Measurable function
• Borel sigma algrebra, outer measure, Lebesgue measure
• Lebesgue integral
• Product measure and multiple integral

• Teaching load: 63 hours (ECTS credits 9)
• Lecturers: BOTTAZZI Giulio
• Teaching Assistant: GIACHINI Daniele
• Semester: Fall
• Outline: This course aims to provide a modern theoretical foundation to the most widespread notions and tools in financial economics, including present value theory, real interest rate, arbitrage pricing, portfolio management and mean-variance analysis.
• Objectives: The students should become acquainted with the notion of arbitrage and equilibrium price in different market settings. They should be able to solve the problem of portfolio optimization and mean-variance analysis in rather general terms.
• Description: General equilibrium, no arbitrage in riskless economies, real interest rate and the yield curve. Arbitrage in security markets, state prices, complete and incomplete markets, valuation functional and the fundamental theorem of finance. Choices under uncertainty, expected utility theory and risk aversion. Portfolio choices, optimal portfolio with multiple risky assets. General equilibrium under uncertainty. Pricing kernel and mean-variance analysis. OPTIONAL: behavioral finance, asset prices under ambiguity, evolutionary finance and the market selection hypothesis.
• Textbooks:

  [LRW]

  Principles of Financial Economics, S.F. LeRoy and J. Werner

  [FBJ]

  Foundations of Financial Markets and Institutions, Fabozzi, Modigliani and Jones

• Further references and Optional readings:

  [MWG]

  Microeconomic Theory, A. Mas-Colell, M. D. Whinston, and J. R. Green

  [CN]

  Quantitative Financial Economics, K. Cuthbertson and D. Nitzsche

  [HR]

  Financial Economics, T. Hens and M. O. Rieger

  [ES]

  Ambiguity and Asset Markets, L. G. Epstein and M. Schneider, NBER paper;

  [Hu]

  Options, Futures and Other Derivatives, J. C. Hull (7th ed.)

  [Ni]

  The Ascent of Money: A Financial History of the World, Niall Ferguson

  [Eco]

  Guide to the Financial Markets, The Economist

• Prerequisites: The course requires a basic knowledge of linear algebra (linear space, linear map, basis, inversion, eigenvectors and eigensystems), calculus (differential analyses of function of many real variables and static optimization) and, to a lesser extent, probability theory. Some basic knowledge of the theory of choice under uncertainty (utility theory) and economic equilibrium are in general given for granted and only cursorily reviewed.
• Prerequisites suggested reading (relevant chapters in): C. P. Simon, L. E. Blume, Mathematics for Economists, H. R. Varian Microeconomic Analysis.
• Keywords: financial economics, general equilibrium, arbitrage, asset pricing, optimization
• Final exam: the get the final grading students are required to pass a final written exam. At least one fake exam will be organized to test your level of preparation. Students get extra points for completing their homework during the course.
• Calendar: check "Financial Economics" entries for the lectures timetable and locations in the calendar.

List of lectures. To have a guess about future lectures check the previous years but consider that your mileage may vary.

1. Introduction: the structure of the course and its textbooks; the role of financial institutions slides (Textbooks sections: [FBJ] Ch. 1, 2 and 3; [Ni] Ch. 1; [Eco] Ch. 1)
2. Barter and market economies; Two agents, two goods, one date model.
3. Two date model. Spot and forward markets. Financial equilibrium. The role of arbitrage in setting prices.
4. Present value prices and economic equilibrium. Redundant contracts.
5. Real and nominal interest rates; The role of inflation; Pricing of future contracts. (Textbooks sections: [FBJ] Ch. 4 and 5 [Hu] Sections from 4.1 to 4.8 all Chapter 5 and section from 6.1 to 6.3 [Ni] Ch. 2)
6. Inter-temporal allocation in a one good two date model: the role of preferences and endowments in setting real interest rates. [Assignment 1]
7. Structure of security markets with uncertainty; complete markets and redundant assets (Textbook sections: [LRW] 1.6)
8. Exercises and solutions of Assignment 1
9. Left and right matrix inverse.
10. Payoff pricing correspondence; the Law of One Price; state claims and state prices (Textbook sections: [LRW] 2.1, 2.3, 2.4, 2.5)
11. Arbitrage and valuation: strong and weak arbitrage; positive pricing; the payoffs convex cone; (Textbook sections: [LRW] 3.2, 3.3, 3.4, 3.5, 5.1)
12. Farkas' Lemma; the valuation functional and strong arbitrage; Stiemke's Lemma; the valuation functional and arbitrage; Arbitrage and valuation: the fundamental theorem of finance; multiplicity of valuation functionals; risk neutral probabilities; (Textbook sections: [LRW] 5.2, 5.3, 5.5, 6.3, 6.4, 6.5;) [Assignment 2]
13. Discussion of the no-arbitrage condition in a general two date model. Risk neutral probabilities; (Textbook sections: [LRW] 6.6, 6.7;)
14. Valuation of derivatives in a two date model: forward and future, call and put options (see the discussion of forward and future contracts in [Hu] Ch.1-3)
15. Choices under uncertainty: consumption and portfolio choices, Expected Utility Theory, separable and time invariant utilities; risk aversion; absolute and relative risk aversion coefficients; certainty equivalent; Concave transformation; comparison of utility functions; first and second order stochastic dominance (Textbook sections: [LRW] 9.2, 9.3, 9.4, 9.5, 9.6, 9.8; the proves of the theorems can also be found in [MWG] Ch.6 Sec.B; further examples are discussed in [CN] Ch.1 Sec.3-5)
16. Mean-variance frontier payoffs and mean-variance frontier returns with complete markets. [Assignment 3]
17. Scalar product and orthogonal projection in linear spaces. Expectations kernel and pricing kernel (Textbook sections: [LRW] from 17.1 to 17.10, 18.5)
18. Mean-variance frontier payoffs and mean-variance frontier returns with complete and incomplete markets. (Textbook sections: 18.1, 18.2 and Th. 18.2.1 with proof 18.4, 18.6; see also the discussion in [CN] Ch.15 Sec.1-2).
19. Example on orthogonal projection and mean-variance frontier payoffs.
20. Zero-Covariance Frontier Returns, Sharpe Ratio of mean-variance frontier returns (Textbook sections: 18.4) [Assignment 4, see below for data]
21. Exercises and solutions of Assignment 3
22. Portfolio optimization: first order conditions; arbitrage and optimal portfolio, optimal consumption, prices and returns. (Textbook sections: [LRW] 1.4, 1.5, 2.6, 3.6, 11.1, 11.2; see [CN] Ch.15 for an introductory discussion of the problem)
23. Exercises on optimal portfolios
24. Exercises and solutions of Assignment 4
25. Portfolio optimization with one risky security: impact of wealth on optimal investment; exercises (Textbook sections: [LRW] 12.2).
26. Portfolio optimization with several risky securities: from consumption to investment optimization; risk-return trade off; optimal portfolio under fair pricing. Linear Risk tolerance. (Textbook sections: [LRW] 13.2, 13.3, 13.4, 9.9, 13.6)
27. Economic equilibrium; representative agent model; equilibrium sufficient conditions; equilibrium and arbitrage. (Textbook sections: [LRW] 1.7, 1.8, 1.9, 2.4, 3.7) [Assignment 5]
28. An example of general equilibrium with heterogeneous agents. The relation between individual wealth and prices.
29. Consumption based security pricing; a first derivation of CAPM;, (Textbook sections: [LRW] 14.1, 14.2, 14.3, 14.5)
30. Equilibrium and Pareto optimal allocation, first and second welfare theorem; (Textbook sections: [LRW] 15.2, 15.3, 15.4, 15.5 without theorem 15.5.1, 15.6 )
31. Exercises and solutions of Assignment 5

Data for Assignment 4: weekly closing prices of America Corporation (BAC), General Electric Company (GE), Ford Motor Co. (F), Citigroup, Inc. (C), Pfizer Inc. (PFE), Hewlett-Packard Company (HPQ), JPMorgan Chase & Co. (JPM), AT&T, Inc. (T), Wells Fargo & Company (WFC) and J. C. Penney Company, Inc. (JCP)

The aim of the course is to provide a deepest introduction to the basic tools and theorems of mathematical calculus. It is taught as a companion courses for second and third year students. Undergraduate courses in the first and second years are oriented to provide a specific set of computational procedures to be applied to the solution of specific problems. The present course might constitute a good complement as it is more concerned with the definition and study of general notions and it is characterized by a theorem-proving approach.

• Teaching load: 40 hours
• Lecturer : BOTTAZZI Giulio
• Teaching Assistant: Pietro Battiston
• Semester: Fall
• Textbook: Notes directly provided by the teacher
• Further references and Optional readings: Principles of Mathematical Analysis, W.A. Rudin, chapters 3, 6 and 9
• Prerequisites: The course requires a knowledge of linear algebra (linear space, linear map, basis, determinant, inversion, eigenvectors) and basic calculus.

List of topics covered by the lectures (not necessarily in this order)

Prerequisite:

• Sets, equivalence relations and functions
• Order relations, upper and lower bounds, supremum and infimum
• Countable sets
• Real numbers

Introduction to topology and metric spaces:

• Topological spaces
• Compact and connected sets
• Basis of a topology
• Countable spaces
• Continuity
• Notion of distance and induced topology

Sequences in Metric Spaces:

• Cauchy sequences.
• Limit of sequences and sub-sequences.
• Comparison theroem for limit of sequences.
• The number e as a limit of a sequence.

Series:

• Series as limit of partial sums.
• Root, ratio and comparison tests for series.

Functions and Limits on Metric Spaces:

• Notion of limit of function in a metric space.
• Properties of limit (sum, ratio and product of limits)

Limits of Real Functions:

• Some important limits.
• Properties of limits of functions (comparison test)
• The number e as limit of a function.

Continuity of Real Functions:

• Uniform continuity.
• Weiestrass Theorem.

Differential calculus of real functions:

• Properties of derivatives.
• Continuity and differential.
• Lagrange theorem.
• de L'Hopital rule and application.
• Derivatives of fundamental functions.

• Teaching load: 12 hours
• Lecturer : BOTTAZZI Giulio, GIACHINI Daniele
• Semester: Spring
• Textbook: Notes directly provided by the teachers
• Further references and Optional readings: Principles of Financial Economics, S.F. LeRoy and J. Werner
• Prerequisites: The course requires some knowledge of linear algebra (linear space, linear map, basis, determinant, inversion, eigenvectors), basic calculus, static optimization and expected utility theory.

• Teaching load: 8 hours
• Lecturer : BOTTAZZI Giulio
• Semester: Spring
• Papers:

  [BS2012]

  G. Bottazzi and A. Secchi Productivity, profitability and growth: the empirics of firm dynamics, Structural Change and Economic Dynamics, Volume 23, Issue 4, pp. 325–328, 2012.

  [BPT]

  G. Bottazzi, D. Pirino and F. Tamagni Zipf law and the firm size distribution: a critical discussion of popular estimators, Journal of Evolutionary Economics 25 (3), 585-610, 2015.

  [Hill]

  B. Hill A Simple General Approach to Inference About the Tail of a Distribution. The Ann Stat 3:1163–1174,

  [Simon]

  H.A.Simon On a Class of Skew Distribution Functions, Biometrika, vol. 42, 1955.

  [Champ]

  D.G. Champernowne A Model of Income Distribution, The Economic Journal, vol. 63,1953.

  [BS2006a]

  G.Bottazzi and A.Secchi Explaining the Distribution of Firms Growth Rates Rand Journal of Economics, 37, pp. 234-263, 2006

  [BS2006b]

  G.Bottazzi and A.Secchi Gibrat's Law and Diversification Industrial and Corporate Change, 15, pp. 847-875, 2006

  [BS2003]

  G.Bottazzi, A.Secchi Why are distributions of firm growth rates tent-shaped ? Economic Letters vol. 80 pp.415-420, 2003

List of lectures:

1. The object of study. Pareto distribution, scale invariance and fat tails. Order statistics, Ranyi transfromation and Hill estimator ([BS2012],[BPT],[Hill]).
2. Gibrat process, strong and weak form. Sub-diffusive model. Characteristic function, cumulant generating function and central limit theorem.
3. Simon model of firm growth. Champernow model and the generating function method ([Simon,[Champ]]).
4. Birth and death models and the diversification structure of firms, Distribution of growth rates, generalization of the central limit theorem and Laplace distribution ([BS2006b],[BS2003],[BS2006a]). handout

• Teaching load: 63 hours (ECTS credits 9)
• Lecturers: BOTTAZZI Giulio
• Teaching Assistant: GIACHINI Daniele
• Semester: Fall
• Outline: This course aims to provide a modern theoretical foundation to the most widespread notions and tools in financial economics, including present value theory, real interest rate, arbitrage pricing, portfolio management and mean-variance analysis.
• Objectives: The students should become acquainted with the notion of arbitrage and equilibrium price in different market settings. They should be able to solve the problem of portfolio optimization and mean-variance analysis in rather general terms.
• Description: General equilibrium, no arbitrage in riskless economies, real interest rate and the yield curve. Arbitrage in security markets, state prices, complete and incomplete markets, valuation functional and the fundamental theorem of finance. Choices under uncertainty, expected utility theory and risk aversion. Portfolio choices, optimal portfolio with multiple risky assets. General equilibrium under uncertainty. Pricing kernel and mean-variance analysis. OPTIONAL: behavioral finance, asset prices under ambiguity, evolutionary finance and the market selection hypothesis.
• Textbooks:

  [LRW]

  Principles of Financial Economics, S.F. LeRoy and J. Werner

  [FBJ]

  Foundations of Financial Markets and Institutions, Fabozzi, Modigliani and Jones

• Further references and Optional readings:

  [MWG]

  Microeconomic Theory, A. Mas-Colell, M. D. Whinston, and J. R. Green

  [CN]

  Quantitative Financial Economics, K. Cuthbertson and D. Nitzsche

  [HR]

  Financial Economics, T. Hens and M. O. Rieger

  [ES]

  Ambiguity and Asset Markets, L. G. Epstein and M. Schneider, NBER paper;

  [Hu]

  Options, Futures and Other Derivatives, J. C. Hull (7th ed.)

  [Ni]

  The Ascent of Money: A Financial History of the World, Niall Ferguson

  [Eco]

  Guide to the Financial Markets, The Economist

  see the syllabus for references to specific parts of these books.

• Prerequisites: The course requires a basic knowledge of linear algebra (linear space, linear map, basis, inversion, eigenvectors and eigensystems), calculus (differential analyses of function of many real variables and static optimization) and, to a lesser extent, probability theory. Some basic knowledge of the theory of choice under uncertainty (utility theory) and economic equilibrium are in general given for granted and only cursorily reviewed.
• Prerequisites suggested reading (relevant chapters in): C. P. Simon, L. E. Blume, Mathematics for Economists, H. R. Varian Microeconomic Analysis.
• Keywords: financial economics, general equilibrium, arbitrage, asset pricing, optimization
• Final exam: the get the final grading students are required to pass a final written exam. At least one fake exam will be organized to test your level of preparation. Students get extra points for completing their homework during the course.
• Calendar: check "Financial Economics" entries for the lectures timetable and locations

List of lectures. To have a guess about future lectures check the previous years but consider that your mileage may vary.

1. Introduction: the structure of the course and its textbooks; the role of financial institutions slides (Textbooks sections: [FBJ] Ch. 1, 2 and 3; [Ni] Ch. 1; [Eco] Ch. 1)
2. The notion of economic equilibrium; Two agents, two goods, one date model.
3. Two date model. Spot and forward markets. Financial equilibrium. The role of arbitrage in setting prices.
4. Present value prices and economic equilibrium. Redundant contracts.
5. Real and nominal interest rates; The role of inflation; Pricing of future contracts. (Textbooks sections: FBJ] Ch. 4 and 5 [Hu] Sections from 4.1 to 4.8 all Chapter 5 and section from 6.1 to 6.3 [Ni] Ch. 2) [Assignment 1]
6. Inter-temporal allocation in a one good two date model: the role of preferences and endowments in setting real interest rates. Structure of security markets with uncertainty; complete markets and redundant assets (Textbook sections: [LRW] 1.6)
7. Exercises and solutions of Assignment 1
8. Left and right matrix inverse. Payoff pricing correspondence; the Law of One Price; state claims and state prices (Textbook sections: [LRW] 2.1, 2.3, 2.4, 2.5)
9. Arbitrage and valuation: strong and weak arbitrage; positive pricing; the payoffs convex cone; (Textbook sections: [LRW] 3.2, 3.3, 3.4, 3.5, 5.1)
10. Farkas' Lemma; the valuation functional and strong arbitrage; Stiemke's Lemma; the valuation functional and arbitrage; Arbitrage and valuation: the fundamental theorem of finance; multiplicity of valuation functionals; risk neutral probabilities; (Textbook sections: [LRW] 5.2, 5.3, 5.5, 6.3, 6.4, 6.5, 6.6, 6.7;) [Assignment 2]
11. Choices under uncertainty: consumption and portfolio choices, Expected Utility Theory, separable and time invariant utilities; risk aversion; absolute and relative risk aversion coefficients; certainty equivalent; Concave transformation; comparison of utility functions; first and second order stochastic dominance (Textbook sections: [LRW] 9.2, 9.3, 9.4, 9.5, 9.6, 9.8; the proves of the theorems can also be found in [MWG] Ch.6 Sec.B; further examples are discussed in [CN] Ch.1 Sec.3-5)
12. Exercises and solutions of Assignment 2
13. Valuation of derivatives in a two date model: forward and future (see the discussion of forward and future contracts in [Hu] Ch.1-3) [Assignment 3]
14. Portfolio optimization: first order conditions; arbitrage and optimal portfolio, optimal consumption, prices and returns. (Textbook sections: [LRW] 1.4, 1.5, 2.6, 3.6, 11.1, 11.2; see [CN] Ch.15 for an introductory discussion of the problem)
15. Portfolio optimization with one risky security: impact of wealth on optimal investment; exercises (Textbook sections: [LRW] 12.2).
16. Exercises and solutions of Assignment 3.
17. Put and call options. The role of options in completing the market. Exercise: an useful picture and the Octave/Matlab file; (Textbook sections: [LRW] 15.4).
18. Portfolio optimization with several risky securities: from consumption to investment optimization; risk-return trade off; optimal portfolio under fair pricing. (Textbook sections: [LRW] 13.2, 13.3, 13.4, 9.9, 13.6)
19. Portfolio optimization with several risky securities: Linear Risk tolerance, recap on Euclidean spaces, Orthogonal projection and scalar product. [Assignment 4]
20. Expectations kernel and pricing kernel (Textbook sections: [LRW] from 17.1 to 17.10, 18.5)
21. Exercises and solutions of Assignment 4.
22. Again on expectations kernel and pricing kernel (Textbook sections: [LRW] from 17.1 to 17.10, 18.5)
23. Mean-variance frontier payoffs and mean-variance frontier returns with complete markets.
24. Mean-variance frontier payoffs and mean-variance frontier returns with complete and incomplete markets. Zero-Covariance Frontier Returns, Sharpe Ratio of mean-variance frontier returns (Textbook sections: 18.1, 18.2 and Th. 18.2.1 with proof 18.4, 18.6; see also the discussion in [CN] Ch.15 Sec.1-2). [Assignment 5 see below for data]
25. Economic equilibrium; representative agent model; equilibrium sufficient conditions; equilibrium and arbitrage. (Textbook sections: [LRW] 1.7, 1.8, 1.9, 2.4, 3.7)
26. Consumption based security pricing; a first derivation of CAPM;, the social planner problem; equilibrium and Pareto optimality, first and second welfare theorems; (Textbook sections: [LRW] 14.2, 14.3, 14.5, 15.2, 15.3, 15.4, 15.5, without theorem 15.5.1)
27. Exercises and solutions of Assignment 4.
28. Simulated exam
29. Correction and discussion of the exercises of the simulated exam

Data for Assignment 5: weekly closing prices of America Corporation (BAC), General Electric Company (GE), Ford Motor Co. (F), Citigroup, Inc. (C), Pfizer Inc. (PFE), Hewlett-Packard Company (HPQ), JPMorgan Chase & Co. (JPM), AT&T, Inc. (T), Wells Fargo & Company (WFC) and J. C. Penney Company, Inc. (JCP)

The aim of the course is to provide a deepest introduction to the basic tools and theorems of mathematical calculus. Undergraduate courses in the first and second year are oriented to provide a specific set of computational procedures to be applied to the solution of specific problems. The present course might constitute a good complement as it is more concerned with the definition and study of general notions and it is characterized by a theorem-proving approach.

• Teaching load: 40 hours (ECTS credits 3)
• Lecturer : BOTTAZZI Giulio, BATTISTON Pietro
• Semester: Fall
• Textbook: Notes directly provided by the teacher
• Further references and Optional readings: Principles of Mathematical Analysis, W.A. Rudin, chapters 3, 6 and 9
• Prerequisites: The course requires some knowledge of linear algebra (linear space, linear map, basis, determinant, inversion, eigenvectors) and basic calculus.

List of topics covered by the lectures.

Functions of several real variables:

• Continuity in Euclidean spaces
• Derivative and differential
• Concavity and convexity
• Local and global maxima and minima
• Implicit function theorem
• Constrained maximization
• Khun-Tucker conditions

Riemann integral:

• Finite sum and limits
• Summability conditions
• Properties of the integral

Measure theory:

• Sigma algebra, measurable space, measure
• Measurable function
• Borel sigma algrebra, outer measure, Lebesgue measure
• Lebesgue integral
• Multiple integral

• Teaching load: 10 hours
• Lecturers: BOTTAZZI Giulio
• Outline: The course introduces some theoretical aspects of the study of the growth dynamics of business firms. Basic stochastic processes and the classical models are reviewed.

• References and Optional readings:

  [Kal]

  On the Gibrat Distribution, M. Kalecki, Econometrica, 1945.

  [Cham]

  A model of Income Distribution, D.G. Champernowne, The Economic Journal, 1953.

  [Simon]

  On a class od skew distribution functions, H.A. Simon, Biometrika, 1955.

  [Mitz]

  A Brief History of Generative Models for Power Law and Lognormal Distributions, Michael Mitzenmacher, Internet Math, 2003.

  [New]

  Power laws, Pareto distributions and Zipf's law, M. E. J. Newman, Contemporary Physics, 2004.

  [Bot1]

  Explaining the distribution of firms gwoth rates, G. Bottazzi and A. Secchi, RAND journal of economics, 2006.

  [Bot2]

  Gibrat's Law and Diversification, G. Bottazzi and A. Secchi, Industrial and Corporate Change, 2006.

  Note that these references do not contain a complete coverage of all the topics reviewed in the course.

List of lectures.

1. Uncertain in the definition of measured quantities. The Pareto distribution and its properties: scale-free and fat tails. The central limit theorem for the sum of independent random variables. Characteristic function and cumulant generating functions. ([New] and [Mitz]).
2. The asymptotic of Gibrat process: the case of auto-regressive growth rates and the dependence on size ([Kal]).
3. From the Simon model to the Yule distribution. The Champernowne model. ([Simon] and [Cham]).
4. Laplace distribution and firm growth rates. ([Bot1])
5. Poisson process, Yule process and firm diversification. ([Bot2])

• Teaching load: 63 hours (ECTS credits 9)
• Lecturers: BOTTAZZI Giulio and GIACHINI Daniele
• Teaching Assistant: SANTI Caterina
• Semester: Fall
• Outline: This course aim to provide a modern theoretical foundation to the most widespread notions and tools in modern finance, including present value theory, real interest rate, arbitrage pricing, portfolio allocation and mean-variance analysis.
• Objectives: The students should become acquainted with the notion of arbitrage and equilibrium price in different market settings. They should be able to solve the problem of portfolio optimization and mean-variance analysis in rather general terms.
• Description: General equilibrium, no arbitrage in riskless economies, real interest rate and the yield curve. Arbitrage in security markets, state prices, complete and incomplete markets, valuation functional and the fundamental theorem of finance. Choices under uncertainty, expected utility theory and risk aversion. Portfolio choices, optimal portfolio with multiple risky assets. General equilibrium under uncertainty. Pricing kernel and mean-variance analysis. OPTIONAL: behavioral finance, asset prices under ambiguity, evolutionary finance and the market selection hypothesis.
• Textbooks:

  [LRW]

  Principles of Financial Economics, S.F. LeRoy and J. Werner

  [FBJ]

  Foundations of Financial Markets and Institutions, Fabozzi, Modigliani and Jones

• Further references and Optional readings:

  [MWG]

  Microeconomic Theory, A. Mas-Colell, M. D. Whinston, and J. R. Green

  [CN]

  Quantitative Financial Economics, K. Cuthbertson and D. Nitzsche

  [HR]

  Financial Economics, T. Hens and M. O. Rieger

  [ES]

  Ambiguity and Asset Markets, L. G. Epstein and M. Schneider, NBER paper;

  [Hu]

  Options, Futures and Other Derivatives, J. C. Hull (7th ed.)

  [Ni]

  The Ascent of Money: A Financial History of the World, Niall Ferguson

  [Eco]

  Guide to the Financial Markets, The Economist

  see the syllabus for references to specific parts of these books.

• Prerequisites: The course requires a knowledge of linear algebra (linear space, linear map, basis, inversion, eigenvectors and eigensystems), calculus (differential analyses of function of many real variables and static optimization) and, to a lesser extent, probability theory. Some basic knowledge of the theory of choice under uncertainty (utility theory) and economic equilibrium are in general given for granted and only cursorily reviewed.
• Prerequisites suggested reading (relevant chapters): C. P. Simon, L. E. Blume, Mathematics for Economists, H. R. Varian Microeconomic Analysis.
• Keywords: financial economics, general equilibrium, arbitrage, asset pricing, optimization
• Final exam: the get the final grading students are required to pass a final written exam. At least one trial exam will be organized to test your level of preparation. Students get extra points for completing their homework during the course.
• Calendar: check "Financial Economics" entries for the lectures timetable and locations

List of lectures. To have a guess about actual lectures check the previous years but consider that your mileage may vary.

1. Introduction: the structure of the course and its textbooks; the role of financial institutions slides (Textbooks sections: [FBJ] Ch. 1, 2 and 3; [Ni] Ch. 1; [Eco] Ch. 1)
2. The notion of economic equilibrium; Two agents, two goods, one date model. The role of prices.
3. Two date model. Spot and forward markets. Financial equilibrium.
4. Present value prices and economic equilibrium. Redundant contracts; the role of arbitrage in setting prices.
5. Real and nominal interest rate; The role of inflation; Pricing of future contracts. (Textbooks sections: FBJ] Ch. 4 and 5 [Hu] Sections from 4.1 to 4.8 all Chapter 5 and section from 6.1 to 6.3 [Ni] Ch. 2) [Assignment 1]
6. Inter-temporal allocation in a one good two date model: the role of preferences and endowments.
7. Structure of security markets with uncertainty; complete market and redundant assets; Left and right matrix inverse; (Textbook sections: [LRW] 1.6)
8. Exercises and solutions of Assignment 1.
9. Payoff pricing correspondence; the Law of One Price; state claims and state prices (Textbook sections: [LRW] 2.1, 2.3, 2.4, 2.5)
10. Arbitrage and valuation: strong and weak arbitrage; positive pricing; the payoffs convex cone; Farkas' Lemma; the valuation functional and strong arbitrage; (Textbook sections: [LRW] 3.2, 3.3, 3.4, 3.5, 5.1)
11. Stiemke's Lemma; the valuation functional and arbitrage; Arbitrage and valuation: the fundamental theorem of finance; multiplicity of valuation functionals; risk neutral probabilities; (Textbook sections: [LRW] 5.2, 5.3, 5.5, 6.3, 6.4, 6.5, 6.6, 6.7;) [Assignment 2]
12. Valuation of derivatives in a two date model: forward and future
13. Put and call options. the role of options in completing the market (see the discussion of forward and future contracts in [Hu] Ch.1-3)
14. Exercises and solutions of Assignment 2
15. Choices under uncertainty: consumption and portfolio choices, Expected Utility Theory, separable and time invariant utilities; risk aversion; absolute and relative risk aversion coefficients; certainty equivalent; (Textbook sections: [LRW] 9.2, 9.3, 9.4, 9.5, 9.6, 9.8)
16. Concave transformation; comparison of utility functions; first and second order stochastic dominance; (Textbook sections: the proves of the theorems can also be found in [MWG] Ch.6 Sec.B; further examples are discussed in [CN] Ch.1 Sec.3-5)
17. Portfolio optimization: first order conditions; arbitrage and optimal portfolio, optimal consumption, prices and returns. (Textbook sections: [LRW] 1.4, 1.5, 2.6, 3.6, 11.1, 11.2; see [CN] Ch.15 for an introductory discussion of the problem)
18. Portfolio optimization with one risky security: impact of wealth on optimal investment; exercises (Textbook sections: [LRW ]12.2)
19. Portfolio optimization with several risky securities: from consumption to investment optimization; risk-return trade off; optimal portfolio under fair pricing; (Textbook sections: [LRW] 13.2, 13.3, 13.4, 9.9, 13.6)
20. Portfolio optimization with several risky securities: Linear Risk tolerance [Assignment 4]
21. Constrained utility maximization. [Assignment 3]
22. Orthogonal projection and scalar product, expectations kernel, pricing kernel (Textbook sections: from 17.1 to 17.10, 18.5)
23. Mean-variance frontier payoffs and mean-variance frontier returns with complete and incomplete markets.
24. Zero-Covariance Frontier Returns, Sharpe Ratio of mean-variance frontier returns (Textbook sections: 18.1, 18.2 and Th. 18.2.1 with proof 18.4, 18.6; see also the discussion in [CN] Ch.15 Sec.1-2)
25. Exercises and solutions of Assignment 4
26. Economic equilibrium; Example with heterogeneous agents, log-utilities and Arrow securities.
27. Economic equilibrium; representative agent model; equilibrium sufficient conditions; equilibrium and arbitrage. (Textbook sections: [LRW] 1.7, 1.8, 1.9, 2.4, 3.7)
28. Consumption based security pricing; a first derivation of CAPM;, the social planner problem; equilibrium and Pareto optimality, first and second welfare theorems; (Textbook sections: [LRW] 14.2, 14.3, 14.5, 15.2, 15.3, 15.4, 15.5, without theorem 15.5.1)
29. Introduction to otimal betting and Kelly rule
30. Exercises and solutions of Assignment 5
31. Simulated exam
32. Correction and discussion of the exercises of the simulated exam

Data for Assignment 5: weekly closing prices of America Corporation (BAC), General Electric Company (GE), Ford Motor Co. (F), Citigroup, Inc. (C), Pfizer Inc. (PFE), Hewlett-Packard Company (HPQ), JPMorgan Chase & Co. (JPM), AT&T, Inc. (T), Wells Fargo & Company (WFC) and J. C. Penney Company, Inc. (JCP)

The aim of the course is to provide a deepest introduction to the basic tools and theorems of mathematical calculus. It is taught as a companion courses for second and third year students. Undergraduate courses in the first and second years are oriented to provide a specific set of computational procedures to be applied to the solution of specific problems. The present course might constitute a good complement as it is more concerned with the definition and study of general notions and it is characterized by a theorem-proving approach.

• Teaching load: 40 hours (ECTS credits 3)
• Lecturer : BOTTAZZI Giulio
• Semester: Spring
• Textbook: Notes directly provided by the teacher
• Further references and Optional readings: Principles of Mathematical Analysis, W.A. Rudin, chapters 3, 6 and 9
• Prerequisites: The course requires a knowledge of linear algebra (linear space, linear map, basis, determinant, inversion, eigenvectors) and basic calculus.

List of topics covered by the lectures (not necessarily in this order)

Prerequisite:

• Sets, equivalence relations and functions
• Order relations, upper and lower bounds, supremum and infimum
• Countable sets
• Real numbers

Introduction to topology and metric spaces:

• Topological spaces
• Compact and connected sets
• Basis of a topology
• Countable spaces
• Continuity
• Notion of distance and induced topology

Sequences in Metric Spaces:

• Cauchy sequences.
• Limit of sequences and sub-sequences.
• Comparison theroem for limit of sequences.
• The number e as a limit of a sequence.

Series:

• Series as limit of partial sums.
• Root, ratio and comparison tests for series.

Functions and Limits on Metric Spaces:

• Notion of limit of function in a metric space.
• Properties of limit (sum, ratio and product of limits)

Limits of Real Functions:

• Some important limits.
• Properties of limits of functions (comparison test)
• The number e as limit of a function.

Continuity of Real Functions:

• Uniform continuity.
• Weiestrass Theorem.

Real functions:

• Properties of derivatives.
• Continuity and differential.
• Lagrange theorem.
• de L'Hopital rule and application.
• Derivatives of fundamental functions.

Functions of several real variables:

• Continuity in Euclidean spaces
• Derivative and differential
• Concavity and convexity
• Local and global maxima and minima

• Teaching load: 24 hours (ECTS credits 9)
• Lecturer: BOTTAZZI Giulio
• Semester: Spring
• Prerequisites: Basic mathematical skills. A previous basic knowledge of economic notions is helpful.
• References: S.O. Hansson, Decision Theory: a brief introduction, M.D. Resnik, Choices, P.L. Berstain, Against the Gods: The Remarkable Story of Risk,

This is the first part of a two part course taught in the MISS master programme in Political Science.

The course is aimed at familiarizing students with the concepts of risk, uncertainty and security in micro and macro economics and their present and historical role in economic reasoning. After the introduction of the basic mathematical tools, the course cover the basics of decision making under uncertainty, its normative and positive aspects and its role in modern economic institutions. Through discussion of notable examples the students will see the application of these concepts to practical issues

By the end of this part of the course students should:

• Understand the problem of decision theory under uncertainty,
• Be able to apply the notion of expected payoff and expected utility to practical example,
• Be able to distinguish between the normative content of decision theory and the positive findings about human decision process in uncertain environment.

List of lectures:

1. Short history of risk (1/2)
2. Short history of risk (2/2)
3. Expected payoff in games of chance (1/2)
4. Expected payoff in games of chance (2/2)
5. Utility theory: non satiating hypothesis, decreasing marginal utility
6. Expected utility theory, certainty equivalent
7. The role of wealth and preferences in decision making
8. Paradoxes and behavioral approach (1/2)
9. Paradoxes and behavioral approach (2/2)
10. Individual probability and Bayesian updating (1/2)
11. Individual probability and Bayesian updating (2/2)
12. Trial exam (with solutions)

• Teaching load: 63 hours (ECTS credits 9)
• Lecturers: BOTTAZZI Giulio and GIACHINI Daniele
• Semester: Spring
• Outline: This course aim to provide a modern theoretical foundations to the most widespread notions and tools in modern finance, including present value theory, real interest rate, arbitrage pricing, portfolio allocation and mean-variance analysis.
• Objectives: The students should become acquainted with the notion of arbitrage and equilibrium prices in different market settings. They should be able to solve the problem of portfolio optimization and mean-variance analysis in rather general terms.
• Description: General equilibrium, no arbitrage in riskless economies, real interest rate and the interest rate curve. Arbitrage in security markets, state prices, complete and incomplete markets, valuation functional and the fundamental theorem of finance. Choices under uncertainty, expected utility theory and risk aversion. Portfolio choices, optimal portfolio with multiple risky assets. General equilibrium under uncertainty. Pricing kernel and mean-variance analysis. OPTIONAL: behavioral finance, asset prices under ambiguity, evolutionary finance and the market selection hypothesis.
• Textbooks:

  [LRW]

  Principles of Financial Economics, S.F. LeRoy and J. Werner

  [FBJ]

  Foundations of Financial Markets and Institutions, Fabozzi, Modigliani and Jones

• Further references and Optional readings:

  [MWG]

  Microeconomic Theory, A. Mas-Colell, M. D. Whinston, and J. R. Green

  [CN]

  Quantitative Financial Economics, K. Cuthbertson and D. Nitzsche

  [HR]

  Financial Economics, T. Hens and M. O. Rieger

  [ES]

  Ambiguity and Asset Markets, L. G. Epstein and M. Schneider, NBER paper;

  [Hu]

  Options, Futures and Other Derivatives, J. C. Hull (7th ed.)

  [Ni]

  The Ascent of Money: A Financial History of the World, Niall Ferguson

  [Eco]

  Guide to the Financial Markets, The Economist

  see the syllabus for references to specific parts of these books.

• Prerequisites: The course requires a knowledge of linear algebra (linear space, linear map, basis, inversion, eigenvectors and eigensystems), calculus (differential analyses of function of many real variables and static optimization) and, to a lesser extent, probability theory. Some basic knowledge of the theory of choice under uncertainty (utility theory) and economic equilibrium are in general given for granted and only cursorily reviewed.
• Prerequisites suggested reading: C. P. Simon, L. E. Blume, Mathematics for Economists, H. R. Varian Microeconomic Analysis.
• Keywords: financial economics, general equilibrium, arbitrage, asset pricing, optimization
• Final exam: the get the final grading students are required to pass a final written exam. At least one trial exam will be organized to test your level of preparation. Students get extra points for completing their homework during the course.

List of lectures:

1. Lecture 1: Introduction: the structure of the course and its textbooks; the role of financial institutions slides (Textbooks sections: [FBJ] Ch. 1, 2 and 3; [Ni] Ch. 1; [Eco] Ch. 1)
2. Lecture 2: The notion of economic equilibrium; Two agents, two goods, one date model. The role of prices.
3. Lecture 3: Two date model. Spot and forward markets. Financial equilibrium.
4. Lecture 4: Present value prices and economic equilibrium. Redundant contracts; the role of arbitrage in setting princes.
5. Exercises and assignment solutions
6. Lecture 5: Real and nominal interest rate; The role of inflation; (Textbooks sections: [FBJ] Ch. 4 and 5)
7. Lecture 6: The role of preferences and endowments in the definition of interest rates. Pricing of future contracts (Textbooks sections: [Hu] Sections from 4.1 to 4.8 all Chapter 5 and section from 6.1 to 6.3 [Ni] Ch. 2)
8. Lecture 7: A second look at intertemporal allocation in a one good two date model: the role of impatience and optimism in the equity market. The idea of the "Wealth of nations" and the first theorem of welfare.
9. Lecture 8: Seminar by Dr. Giachini on "market sentiment". Structure of security markets with uncertainty; complete markets and redundant assets; (Textbook sections: [LRW] 1.1, 1.2)
10. Exercises and assignment solutions
11. Lecture 9: Structure of security markets: Left and right matrix inverse; payoff pricing correspondence; the Law of One Price; state claims and state prices (Textbook sections: [LRW] 1.6, 2.1, 2.3, 2.4, 2.5)
12. Lecture 10: Arbitrage and valuation: strong and weak arbitrage; positive pricing; the payoffs convex cone; Farkas' Lemma; the valuation functional and strong arbitrage; (Textbook sections: [LRW] 3.2, 3.3, 3.4, 3.5, 5.1)
13. Exercises and assignment solutions
14. Lecture 11: Stiemke's Lemma; the valuation functional and arbitrage; Arbitrage and valuation: the fundamental theorem of finance; multiplicity of valuation functional; (Textbook sections: [LRW] 5.2, 5.3, 5.5, 6.3, 6.4, 6.5, 6.6, 6.7;)
15. Lecture 12: values bound; risk neutral probabilities; Valuation of derivatives in a two date model: forward and future; put and call options. (see the discussion of forward and future contracts in [Hu] Ch.1-3)
16. Lecture 13: example of market securities; the role of options in completing the market
17. Lecture 14: Choices under uncertainty: consumption and portfolio choices, Expected Utility Theory, separable and time invariant utilities; risk aversion; absolute and relative risk aversion coefficients; certainty equivalent; concave transformation; comparison of utility functions (Textbook sections: [LRW] 9.2, 9.3, 9.4, 9.5, 9.6, 9.8; the proves of the theorems can also be found in [MWG] Ch.6 Sec.B; further examples are discussed in [CN] Ch.1 Sec.3-5)
18. Exercises and assignment solutions
19. Lecture 15: Portfolio optimization: first order conditions; arbitrage and optimal portfolio, optimal consumption, prices and returns. (Textbook sections: [LRW] 1.4, 1.5, 2.6, 3.6, 11.1, 11.2; see [CN] Ch.15 for an introductory discussion of the problem)
20. Lecture 16: Portfolio optimization with one risky security: impact of wealth on optimal investment; exercises (Textbook sections: [LRW ]12.2)
21. Lecture 17: Portfolio optimization with several risky securities: from consumption to investment optimization; risk-return trade off; optimal portfolio under fair pricing; (Textbook sections: [LRW] 13.2, 13.3, 13.4, 9.9, 13.6)
22. Lecture 18: refresh of main results about inner product, orthogonal complement, Gram-Schmidt orthogonalization, direct sum of orthogonal spaces, orthogonal projection, scalar product representation of functional. Expectations inner product (Textbook sections: from 17.1 to 17.8)
23. Lecture 19: expectations kernel, pricing kernel, fair pricing, beta pricing mean-variance frontier payoffs and mean-variance frontier returns when market is complete (Textbook sections: 17.9, 17.10, 18.5)
24. Lecture 20: mean-variance frontier payoffs and mean-variance frontier returns in the general case; Zero-Covariance Frontier Returns, Sharpe Ratio of mean-variance frontier returns (Textbook sections: 18.1, 18.2 and Th. 18.2.1 with proof 18.4, 18.6; see also the discussion in [CN] Ch.15 Sec.1-2)
25. Lecture 21: Economic equilibrium; representative agent model; equilibrium sufficient conditions; equilibrium and arbitrage (Textbook sections: [LRW] 1.7, 1.8, 1.9, 2.4, 3.7)
26. Lecture 22: Economic equilibrium: example with heterogeneous agents, log-utilities and Arrow securities. Exercises.
27. Lecture 23: Consumption based security pricing; a first derivation of CAPM;, the social planner problem; equilibrium and Pareto optimality, first and second welfare theorems; (Textbook sections: [LRW] 14.2, 14.3, 14.5, 15.2, 15.3, 15.4, 15.5, without theorem 15.5.1)
28. Simulated exam
29. Correction and discussion of the exercises of the simulated exam
30. Further exercises for the final exam

Data for Assignment 5: weekly closing prices of America Corporation (BAC), General Electric Company (GE), Ford Motor Co. (F), Citigroup, Inc. (C), Pfizer Inc. (PFE), Hewlett-Packard Company (HPQ), JPMorgan Chase & Co. (JPM), AT&T, Inc. (T), Wells Fargo & Company (WFC) and J. C. Penney Company, Inc. (JCP)

The course presents an introduction to measure theory and Lebesgue integration. The relationship between measure and probability theory will serve as a background example for the first part of the course. Differences between Riemann and Lebesgue integration will be illustrated. The final part of the course is devoted to multiple integrals.

• Teaching load: 20 hours (ECTS credits 2)
• Lecturer : BOTTAZZI Giulio
• Semester: Fall
• Textbooks:

  [RB]

  "Real Analysis for Graduate Students" by Richard F. Bass

• Prerequisites: The course requires a basic knowledge of calculus. Previous knowledge of topology and Riemann integration is useful but not necessary.

Syllabus:

• Measure theory
  • Measurable spaces and their properties
  • Measurable spaces and probabilities
  • Measure and topology. Borel sigma algrebra
  • Product measure
  • Measurable functions
• Lebesgue integration
  • From external measure to Lebesgue measure
  • Lebesgue and Riemann integrals
  • Multiple integrals

Lecture notes will be made available to all students. The textbook can be used as a further reference.

• Teaching load: 63 hours (ECTS credits 9)
• Lecturers: BOTTAZZI Giulio and GIACHINI Daniele
• Semester: Spring
• Outline: This course aim to provide a modern theoretical foundations to the most widespread notions and tools in modern finance, including present value theory, real interest rate, arbitrage pricing, portfolio allocation and mean-variance analysis.
• Objectives: The students should become acquainted with the notion of arbitrage and equilibrium prices in different market settings. They should be able to solve the problem of portfolio optimization and mean-variance analysis in rather general terms.
• Description: General equilibrium, no arbitrage in riskless economies, real interest rate and the interest rate curve. Arbitrage in security markets, state prices, complete and incomplete markets, valuation functional and the fundamental theorem of finance. Choices under uncertainty, expected utility theory and risk aversion. Portfolio choices, optimal portfolio with multiple risky assets. General equilibrium under uncertainty. Pricing kernel and mean-variance analysis. OPTIONAL: behavioral finance, asset prices under ambiguity, evolutionary finance and the market selection hypothesis.
• Textbooks:

  [LRW]

  Principles of Financial Economics, S.F. LeRoy and J. Werner

  [FBJ]

  Foundations of Financial Markets and Institutions, Fabozzi, Modigliani and Jones

• Further references and Optional readings:

  [MWG]

  Microeconomic Theory, A. Mas-Colell, M. D. Whinston, and J. R. Green

  [CN]

  Quantitative Financial Economics, K. Cuthbertson and D. Nitzsche

  [HR]

  Financial Economics, T. Hens and M. O. Rieger

  [ES]

  Ambiguity and Asset Markets, L. G. Epstein and M. Schneider, NBER paper;

  [Hu]

  Options, Futures and Other Derivatives, J. C. Hull (7th ed.)

  [Ni]

  The Ascent of Money: A Financial History of the World, Niall Ferguson

  references are provided below to specific parts of these books.

• Prerequisites: The course requires a knowledge of linear algebra (linear space, linear map, basis, inversion, eigenvectors and eigensystems), calculus (differential analyses of function of many real variables and static optimization) and, to a lesser extent, probability theory. Some basic knowledge of the theory of choice under uncertainty (utility theory) and economic equilibrium are in general given for granted and only cursorily reviewed.
• Prerequisites suggested reading: C. P. Simon, L. E. Blume, Mathematics for Economists, H. R. Varian Microeconomic Analysis.
• Keywords: financial economics, general equilibrium, arbitrage, asset pricing, optimization

List of lectures so far (to have a guess about future lectures check the list of previous years but consider that your mileage may vary):

• Lecture 1: Introduction: the structure of the course and its textbooks; the role of financial institutions slides (Textbooks sections: [FBJ] Ch. 1, 2 and 3; [Ni] Ch. 1)
• Lecture 2: The notion of economic equilibrium; Two agents, two goods, one date model. The role of prices.
• Lecture 3: Two date model. Spot and forward market. Financial equilibrium. Present value prices and economic equilibrium. Redundant contracts; the role of arbitrage in setting princes;
• Lecture 4: Real and nominal interest rate; The role of inflation; (Textbooks sections: [FBJ] Ch. 4 and 5)
• Lecture 5: A second look at intertemporal allocation in a one good two date model: the role of preferences and endowments. Pricing of future contracts (Textbooks sections: [Hu] Sections from 4.1 to 4.8 and from 6.1 to 6.3 [Ni] Ch. 2)
• Lecture 6: Structure of security markets with uncertainty; complete markets and redundant assets; (Textbook sections: [LRW] 1.1, 1.2)
• Lecture 7: Structure of security markets: Left and right matrix inverse; payoff pricing correspondence; the Law of One Price; state claims and state prices (Textbook sections: [LRW] 1.6, 2.1, 2.3, 2.4, 2.5)
• Lecture 8: Exercises on the payoff pricing functional. The case of complete markets and the case of no redundant assets.
• Lecture 9-10: Arbitrage and valuation: strong and weak arbitrage; positive pricing; the payoffs convex cone; Farkas' Lemma; the valuation functional and strong arbitrage; Stiemke's Lemma; the valuation functional and arbitrage; (Textbook sections: [LRW] 3.2, 3.3, 3.4, 3.5, 5.1)
• Lecture 11: Arbitrage and valuation: the fundamental theorem of finance; multiplicity of valuation functional; values bound; risk neutral probabilities; (Textbook sections: [LRW] 5.2, 5.3, 5.5, 6.3, 6.4, 6.5, 6.6, 6.7;)
• Lecture 12: Valuation of derivatives in a two date model: forward and future; put and call options. Practical example: from option price to risk-neutral measure, see the picture and the Octave/Matlab file; (see the discussion of forward and future contracts in [Hu] Ch.1-3)
• Lecture 13-14: Choices under uncertainty: consumption and portfolio choices, Expected Utility Theory, separable and time invariant utilities; risk aversion; absolute and relative risk aversion coefficients; certainty equivalent; concave transformation; comparison of utility functions (Textbook sections: [LRW] 9.2, 9.3, 9.4, 9.5, 9.6, 9.8; the proves of the theorems can also be found in [MWG] Ch.6 Sec.B; further examples are discussed in [CN] Ch.1 Sec.3-5)
• Lecture 15: Portfolio optimization: first order conditions; arbitrage and optimal portfolio, optimal consumption, prices and returns. (Textbook sections: [LRW] 1.4, 1.5, 2.6, 3.6, 11.1, 11.2; see [CN] Ch.15 for an introductory discussion of the problem)
• Lecture 16: Portfolio optimization with one risky security: impact of wealth on optimal investment; exercises (Textbook sections: [LRW ]12.2)
• Lecture 17: Portfolio optimization with several risky securities: from consumption to investment optimization; risk-return trade off; optimal portfolio under fair pricing; (Textbook sections: [LRW] 13.2, 13.3, 13.4, 9.9, 13.6)
• Lecture 18: Economic equilibrium; representative agent model; equilibrium sufficient conditions; equilibrium and arbitrage (Textbook sections: [LRW] 1.7, 1.8, 1.9, 2.4, 3.7)
• Lecture 19: Economic equilibrium: example with heterogeneous agents, log-utilities and Arrow securities. Exercises.
• Lecture 20: Consumption based security pricing; the social planner problem; equilibrium and Pareto optimality, first and second welfare theorems; (Textbook sections: [LRW] 14.2, 14.3, 14.5, 15.2, 15.3, 15.4, 15.5, without theorem 15.5.1)

The material is available upon request.

The course presents an introduction to differential and difference equations. A particular emphasis is put on the analysis of linear systems, whose study is then exploited to obtain results about high order equations and for the stability analysis of non-linear systems.

• Teaching load: 30 hours (ECTS credits 3)
• Lecturer : BOTTAZZI Giulio
• Semester: Fall
• Textbooks:

  [HHW]

  Discrete and continuous dynamical systems" by Heijnen, Hommes and Wagener

  [MB]

  Istituzioni di matematica by Bertsch (in Italian)

• Further references and Optional readings:

  [HS]

  Differential Equations, Dynamical Systems and Linear Algebra by Hirsh and Smale;

  [GB]

  Advanced Calculus by Bottazzi.

  [RD]

  A first course in Chaotic Dynamical Systems, by Devaney

• Prerequisites: The course requires a knowledge of linear algebra (linear space, linear map, basis, determinant, inversion, eigenvectors), basic calculus and the algebra of complex numbers.

Syllabus and textbook references:

• Introduction to Differential Equations
  • Complex numbers: algebra, Euler representation and the exponential function. ([MB] Ch.12)
  • The simplest differential problem and the Torricelli-Barrow theorem
  • Cauchy problems and Cauchy-Lipshitz sufficient conditions; counter examples with no solution or multiple solutions
  • Differential equations in one dimension: separation of variables. ([MB] Ch.13 Sec. 1)
  • Linear equations: the method of the variation of constant; general and particular solutions. ([MB] Ch.13 Sec. 2)
  • The Malthus growth model, technological adoption and qualitative analysis. ([MB] Ch.13 Sec. 3-5)
  • Higher order equations and systems of equations
• Linear system with constant coefficients
  • Evolution in time; linear systems ([HHW] Ch. 1,7)
  • Asymptotic behavior; the single eigenvalue case; power and exponential of a matrix
  • The diagonalisable case: real eigenvalues ([HHW] Ch. 2,8)
  • The diagonalisable case: complex eigenvalues ([HHW] Ch. 3,9).
  • The Jordan case: general solution ([HHW] Ch. 4,10).
  • Generalized eigenvalues and direct sum decomposition of a linear space
  • Nonhomogeneous equation: variation of constants.
• Non-linear system.
  • Null-clines and phase portrait ([HHW] Ch. 5,11).
  • Linearization around equilibrium point and stability analysis ([HHW] Sec 5.3,5.4 and Ch. 14)
• Examples
  • The Goodwin model
  • Curnot duopoly ([HHW] Sec. 14.3)
  • Bounded rationality and price fluctuation
• Numerical exercises with Octave
  • Matrix manipulation download
  • 1 and 2 dimensional systems download
  • chaotic dynamics, bifurcation plot and Lyapunov exponents download
  • exercises of [RD], Ch. 3 download

The aim of the course is to provide a deepest introduction to the basic tools and theorems of mathematical calculus applied to economic and finance. It is taught as a companion courses for second and third year students. Undergraduate courses in the first and second years are oriented to provide a specific set of computational procedures to be applied to the solution of specific problems. The present course consitutes a good complement. It is more concerned with the definition and study of general notions and it is characterized by a theorem-proving approach.

• Teaching load: 40 hours (ECTS credits 3)
• Lecturer : BOTTAZZI Giulio
• Semester: Fall
• Textbook: Notes directly provided by the teacher
• Further references and Optional readings: Principles of Mathematical Analysis, W.A. Rudin, chapters 3, 6 and 9
• Prerequisites: The course requires a knowledge of linear algebra (linear space, linear map, basis, determinant, inversion, eigenvectors) and basic calculus.
• Topics: see the course outline.
• Exams: midterm exam final exam

This course is a companion course for first year students. It is intended as a deeper introduction to linear algebra. Almost all results are explicitly derived.

• Teaching load: 30 hours (ECTS credits 3)
• Lecturer : SETTEPANELLA Simona
• Semester: Spring
• Prerequisites: Basic calculus.
• Topics: see the course outline.

• Teaching load: 42 hours (ECTS credits 6)
• Lecturers: BOTTAZZI Giulio
• Semester: Spring
• Outline: This course aim to provide a modern theoretical foundations to the most widespread notions and tools in modern finance, including present value theory, real interest rate, arbitrage pricing, portfolio allocation and mean-variance analysis.
• Objectives: The students should become acquainted with the notion of arbitrage and equilibrium prices in different market settings. They should be able to solve the problem of portfolio optimization and mean-variance analysis in rather general terms.
• Description: General equilibrium, no arbitrage in riskless economies, real interest rate and the interest rate curve. Arbitrage in security markets, state prices, complete and incomplete markets, valuation functional and the fundamental theorem of finance. Choices under uncertainty, expected utility theory and risk aversion. Portfolio choices, optimal portfolio with multiple risky assets. General equilibrium under uncertainty. Pricing kernel and mean-variance analysis. OPTIONAL: behavioral finance, asset prices under ambiguity, evolutionary finance and the market selection hypothesis.
• Textbooks:

  [LRW]

  Principles of Financial Economics, S.F. LeRoy and J. Werner

  [FBJ]

  Foundations of Financial Markets and Institutions, Fabozzi, Modigliani and Jones

• Further references and Optional readings:

  [MWG]

  Microeconomic Theory, A. Mas-Colell, M. D. Whinston, and J. R. Green

  [CN]

  Quantitative Financial Economics, K. Cuthbertson and D. Nitzsche

  [HR]

  Financial Economics, T. Hens and M. O. Rieger

  [ES]

  Ambiguity and Asset Markets, L. G. Epstein and M. Schneider, NBER paper;

  [Hu]

  Options, Futures and Other Derivatives, J. C. Hull (7th ed.)

  [Ni]

  The Ascent of Money: A Financial History of the World, Niall Ferguson

  references are provided below to specific parts of these books.

• Prerequisites: The course requires a knowledge of linear algebra (linear space, linear map, basis, inversion, eigenvectors and eigensystems), calculus (differential analyses of function of many real variables and static optimization) and, to a lesser extent, probability theory. Some basic knowledge of the theory of choice under uncertainty (utility theory) and economic equilibrium are in general given for granted and cursorily overwieved.
• Prerequisites suggested reading: C. P. Simon, L. E. Blume, Mathematics for Economists, H. R. Varian Microeconomic Analysis.

• Keywords: financial economics, general equilibrium, arbitrage, asset pricing, optimization

  List of lectures so far (to have a guess about future lectures check the list of previous years but consider that your mileage may vary):

• Lecture 1: Introduction: the structure of the course and its textbooks; the role of financial institutions slides (Textbooks sections: [FBJ] Ch. 2 and 3; [Ni] Ch. 1)
• Lecture 2: The notion of economic equilibrium; Review of statitic optimization; Two agents, two goods, one date model;
• Lecture 3: Two agents, two goods and two date model: redundant contracts; the role of arbitrage in setting princes; The real interest rate;
• Lecture 4: Real and nominal interest rate; The role of inflation; (Textbooks sections: [FBJ] Ch. 4 and 5)
• Lecture 5: A second look at intertemporal allocation in a one good two date model: the role of preferences and endowments.
• Lecture 6: The yield curve for pricing future cash flows: bond, anuities, perpetuities and mortgages (Textbooks section: [FBJ] Ch. 10 and 11 [Hu] Sections from 4.1 to 4.8 and from 6.1 to 6.3 [Ni] Ch. 2)
• Lecture 7: Structure of security markets with uncertainty; complete markets and redundant assets; (Textbook sections: [LRW] 1.1, 1.2)
• Lecture 8: Structure of security markets: Left and right matrix inverse; payoff pricing correspondence; the Law of One Price; state claims and state prices (Textbook sections: [LRW] 1.6, 2.1, 2.3, 2.4, 2.5)
• Lecture 9: Arbitrage and valuation: strong and weak arbitrage; positive pricing; the payoffs convex cone; Farkas' Lemma; the valuation functional and strong arbitrage; Stiemke's Lemma; the valuation functional and arbitrage; (Textbook sections: [LRW] 3.2, 3.3, 3.4, 3.5, 5.1)
• Lecture 10: Arbitrage and valuation: the fundamental theorem of finance; multiplicity of valuation functional; values bound; risk neutral probabilities (Textbook sections: [LRW] 5.2, 5.3, 5.5, 6.3, 6.4, 6.5, 6.6, 6.7; see the discussion of forward and future contracts in [Hu] Ch.1-3)
• Lecture 11: Choices under uncertainty: consumption and portfolio choices, Expected Utility Theory, separable and time invariant utilities; risk aversion; absolute and relative risk aversion coefficients; certainty equivalent; concave transformation; comparison of utility functions (Textbook sections: [LRW] 9.2, 9.3, 9.4, 9.5, 9.6, 9.8; the proves of the theorems can also be found in [MWG] Ch.6 Sec.B; further examples are discussed in [CN] Ch.1 Sec.3-5)
• Lecture 12: Portfolio optimization: first order conditions; arbitrage and optimal portfolio, optimal consumption, prices and returns. (Textbook sections: [LRW] 1.4, 1.5, 2.6, 3.6, 11.1, 11.2; see [CN] Ch.15 for an introductory discussion of the problem)
• Lecture 13: Portfolio optimization with one risky security: impact of wealth on optimal investment; exercises (Textbook sections: [LRW ]12.2)
• Lecture 14: Portfolio optimization with several risky securities: from consumption to investment optimization; risk-return trade off; optimal portfolio under fair pricing; linear risk tolerance (LRT); optimal portfolio with LRT utility function (Textbook sections: [LRW] 13.2, 13.3, 13.4, 9.9, 13.6)
• Lecture 15: Economic equilibrium; representative agent model; equilibrium sufficient conditions; equilibrium and arbitrage (Textbook sections: [LRW] 1.7, 1.8, 1.9, 2.4, 3.7)
• Lecture 16: Economic equilibrium: example with heterogeneous agents, log-utilities and Arrow securities. Exercises.
• Lecture 17: Consumption based security pricing; a first derivation of CAPM; the social planner problem; equilibrium and Pareto optimality, first and second welfare theorems; (Textbook sections: [LRW] 14.2, 14.3, 14.5, 15.2, 15.3, 15.4, 15.5, without theorem 15.5.1)
• Lecture 18: Practical example: from option price to invariant measure, see the picture here; mean-variance efficient frontier from empirical data, download open and close weekly price data on NYSE for Bank of America Corporation (BAC), General Electric Company (GE), Ford Motor Co. (F), Citigroup, Inc. (C), Pfizer Inc. (PFE), Hewlett-Packard Company (HPQ), JPMorgan Chase & Co. (JPM), AT&T, Inc. (T), Wells Fargo & Company (WFC) and J. C. Penney Company, Inc. (JCP); (The software used is octave. Data courtesy of yahoo!finance)
• Lecture 19: CAPM (Textbook sections: chapter 19 all, theorems without proof; ; see also the discussion in [CN] Ch.15 Sec. 3-5)
• Lecture 20: Exercises
• Lecture 21: Trial exam

• Teaching load: 42 hours (ECTS 6)
• Lecturers: BOTTAZZI Giulio
• Semester: Spring
• Outline: Choices under uncertainty, expected utility theory and risk aversion, equilibrium and arbitrage, state prices, complete and incomplete markets, arbitrage and portfolio choices, optimal portfolio with multiple risky assets, equilibrium prices, mean-variance analysis. OPTIONAL: behavioral finance, asset prices under ambiguity, evolutionary finance and the market selection hypothesis.
• Description: The aim of the course is to provide an intermediate treatment of the theory of speculative markets. After a review of decision theory under uncertainty, the notion of arbitrage and equilibrium price are introduced and developed for different market settings. The problem of portfolio optimization and mean-variance analysis is discussed in a rather general framework. Depending on the remaining time, the course might include a short introduction to behavioral and evolutionary finance.
• Textbook: Principles of Financial Economics, S.F. LeRoy and J. Werner
• Further references and Optional readings:

  [MWG]

  Microeconomic Theory, A. Mas-Colell, M. D. Whinston, and J. R. Green

  [CN]

  Quantitative Financial Economics, K. Cuthbertson and D. Nitzsche

  [HR]

  Financial Economics, T. Hens and M. O. Rieger

  [ES]

  Ambiguity and Asset Markets, L. G. Epstein and M. Schneider, NBER paper;

  [Hu]

  Options, Futures and Other Derivatives, J. C. Hull

  references are provided below to specific parts of these books.

• Prerequisites: The course requires a knowledge of linear algebra (linear space, linear map, basis, inversion, eigenvectors and eigensystems), calculus (differential analyses of function of many real variables and static optimization) and, to a lesser extent, probability theory. Previous knowledge of consumer theory and economic equilibrium can be useful.
  • Prerequisites suggested reading: C. P. Simon, L. E. Blume, Mathematics for Economists, H. R. Varian Microeconomic Analysis.
• Keywords: financial economics, arbitrage, asset pricing, portfolio optimization

• Final valuation: Written exam

  Tentative list of lectures (this is just a guess based on previous year's experience, your mileage may vary)

• Lecture 1: Structure of security markets: complete markets and redundant assets; (Textbook sections: 1.1, 1.2)
• Lecture 2: Structure of security markets: Left and right matrix inverse; payoff pricing correspondence; the Law of One Price; state claims and state prices (Textbook sections: 1.6, 2.1, 2.3, 2.4, 2.5)
• Lecture 3: Arbitrage and valuation: strong and weak arbitrage; positive pricing; the payoffs convex cone; Farkas' Lemma; the valuation functional and strong arbitrage; Stiemke's Lemma; the valuation functional and arbitrage; (Textbook sections: 3.2, 3.3, 3.4, 3.5, 5.1)
• Lecture 4: Arbitrage and valuation: the fundamental theorem of finance; multiplicity of valuation functional; values bound; forward and future contracts; put and call options; risk neutral probabilities (Textbook sections: 5.2, 5.3, 5.5, 6.3, 6.4, 6.5, 6.6, 6.7; see the discussion of forward and future contracts in [Hu] Ch.1-3)
• Lecture 5: Choices under uncertainty: consumption and portfolio choices, Expected Utility Theory, separable and time invariant utilities; risk aversion; absolute and relative risk aversion coefficients; certainty equivalent; concave transformation; comparison of utility functions (Textbook sections: 9.2, 9.3, 9.4, 9.5, 9.6, 9.8; the proves of the theorems can also be found in [MWG] Ch.6 Sec.B; further examples are discussed in [CN] Ch.1 Sec.3-5)
• Lecture 6: Choices under uncertainty: comparing lotteries; state-wise dominance; first order stochastic dominance and strictly increasing utility functions; second order stochastic dominance and risk-averse utility functions (Textbook sections: 10.6, 10.7; the proves of the theorems can also be found in [MWG] Ch.6 Sec.D)
• Lecture 7: refresh of main results about inner product, orthogonal complement, Gram-Schmidt orthogonalization, direct sum of orthogonal spaces, orthogonal projection, scalar product representation of functional. Expectations inner product (Textbook sections: from 17.1 to 17.8)
• Lectur 8: expectations kernel, pricing kernel, fair pricing, beta pricing (Textbook sections: 17.9, 17.10, 18.5)
• Lecture 9: mean-variance frontier payoffs and mean-variance frontier returns when market is complete
• Lecture 10: mean-variance frontier payoffs and mean-variance frontier returns in the general case; Zero-Covariance Frontier Returns, Sharpe Ratio of mean-variance frontier returns (Textbook sections: 18.1, 18.2 and Th. 18.2.1 with proof 18.4, 18.6; see also the discussion in [CN] Ch.15 Sec.1-2)
• Lecture 11: exercises on the first part; exam simulation
• Lecture 12: Portfolio optimization: first order conditions; arbitrage and optimal portfolio, optimal consumption, prices and returns. Portfolio optimization with one risky security: first order conditions; approximate quadratic solution (Textbook sections: 1.4, 1.5, 2.6, 3.6, 11.1, 11.2, 11.3, 11.4, 11.5; see [CN] Ch.15 for an introductory discussion of the problem)
• Lecture 13: Portfolio optimization with one risky security: impact of wealth on optimal investment; exercises (Textbook sections: 12.2)
• Lecture 14: Portfolio optimization with several risky securities: from consumption to investment optimization; risk-return trade off; optimal portfolio under fair pricing; linear risk tolerance (LRT); optimal portfolio with LRT utility function (Textbook sections: 13.2, 13.3, 13.4, 9.9, 13.6)
• Lecture 15: Economic equilibrium; representative agent model; equilibrium sufficient conditions; equilibrium and arbitrage (Textbook sections: 1.7, 1.8, 1.9, 2.4, 3.7)
• Lecture 16: Economic equilibrium: example with heterogeneous agents, log-utilities and Arrow securities. Exercises.
• Lecture 17: Consumption based security pricing; a first derivation of CAPM; the social planner problem; equilibrium and Pareto optimality, first and second welfare theorems; (Textbook sections: 14.2, 14.3, 14.5, 15.2, 15.3, 15.4, 15.5, without theorem 15.5.1)
• Lecture 18: Practical example: from option price to invariant measure, see the picture here; mean-variance efficient frontier from empirical data, download open and close weekly price data on NYSE for Bank of America Corporation (BAC), General Electric Company (GE), Ford Motor Co. (F), Citigroup, Inc. (C), Pfizer Inc. (PFE), Hewlett-Packard Company (HPQ), JPMorgan Chase & Co. (JPM), AT&T, Inc. (T), Wells Fargo & Company (WFC) and J. C. Penney Company, Inc. (JCP); (The software used is octave. Data courtesy of yahoo!finance)
• Lecture 19: CAPM (Textbook sections: chapter 19 all, theorems without proof; ; see also the discussion in [CN] Ch.15 Sec. 3-5)
• Lecture 20: Exercises
• Lecture 21: Trial exam

• Teaching load: 30 hours
• Lecturer: Prof. Giulio Bottazzi, Dr. Davide Pirino
• Semester: Spring
• References:

  [CM]

  The Theory of Stochastic Processes, Cox and Miller, Taylor & Francis, 1977

  [Fe]

  An introduction to Probability Theory and its Application, Vol. 1, Feller, Wiley and Sons, 1968

List of lectures (for some topic I provide a reference where further material is discussed):

• Lecture 1: Bernoulli trials and combinatorics; Symmetric random walk, r.w. as a sum of Bernulli variables and transition proabilities ([Fe] Ch.3 Sec.1); The Kolmogorov theorem on stochastic processes, stationary processes, Markov processes.
• Lecture 2: The reflection principle. Probability of return to the origin and of first return. Stirling approximation ([Fe] Ch.3 Secs. 2-3).
• Lecture 3: LABORATORY: Normal distribtuion and error function. The normal approximation to the random walk and the arcsine law ([Fe] Ch.3 Sec. 4). Random number generators and the computation of probabilities via Monte Carlo experiments.

See course outline

• Teaching load: 20 hours
• Lecturer: Dr. Mauro Sylos Labini
• Semester: Fall

• References: Economia del lavoro, Borjas G., Brioschi editore 2010.

  See course outline

The course of mathematical methods is part of the Sant'Anna undergraduate and post-graduate teaching offer. Is is structured in different but complementary modules. Different lecturers are assigned to different modules. Undergraduate students are allowed to chose what modules to follow. It is assumed that students already understand the basic notion of calculus, as they are typically taught in high schools or fist year courses at the Faculty of Economics (see Prerequisite below). During the course, students will be assigned home works in the form of exercises or elaborations on specific topics. In order to follow the development of the lectures, it is in general important that these assignments are duly performed. Please report any problem to the teachers. If deemed necessary, extra teaching hours, albeit in a limited amount, can be organized. One or more copies of the reference textbooks cited below will be available at the Sant'Anna Library and students are not required to buy a copy. In any case the essential material for the different modules will be provided directly by the teachers.

You can check your knowledge of the prerequisite with the training exercices.

This module is part of the Advanced Microeconomic course in the first year program of the Laurea Magistralis (Master degree) in Economics of the University of Pisa, organized in partnership with Scuola Sant'Anna.

• Teaching load: 10 hours
• Lecturers: Prof. Giulio Bottazzi
• courses start: 07/12/2010
• courses end: 16/03/2011
• final exams: 07/06/2011,…

Syllabus:

• von Neumann and Morgenstern expected utility theory (EUT): space of lotteries; completeness, transitivity and continuity axioms; linearity of EU
• EUT with state dependent actions
• introduction to random variables and Stieltjes integral
• the EUT for continuous monetary outcomes
• certainty equivalence and risk premium
• definition of risk aversion; convexity and risk aversion; Arrow-Pratt characterization of utility functions: absolute and relative risk aversion coefficients
• 2 stage optimal portfolio management

Exercises:

These exercises supplement the material already distributed during the lectures. They are intended as examples in preparation of the final written exam. Here are the solutions.

Find below a list of final exam with solutions

07/06/2011 solution

27/06/2011 solution

14/07/2011

12/09/2011