# Teaching

## Table of Contents

## Academic Term (Anno Accademico) 2020/2021: financial economics

- 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.

- (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)
- (23/09/2020, Bottazzi) Barter and market economies; Two agents, two goods, one date model.
- (24/09/2020, Bottazzi) Financial equilibrium. Two date model. Spot and forward markets. The role of arbitrage in setting prices.
- (28/09/2020, Bottazzi) Introduction to present value model. Present value prices and economic equilibrium. Redundant contracts.
- (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)
- (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.
- (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)
- (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)
- (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]
- (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)
- (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)
- (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)
- (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)
- (20/10/2019, Giachini) Exercises and solutions of Assignment 1
- (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)
- (26/10/2020, Giachini) Portfolio optimization: Example of internal and boundary solutions. Marginal utility and risk-neutral probabilities.
- (27/10/2020, Giachini) Portfolio optimization with one risky security: impact of wealth on optimal investment; exercises (Textbook sections: [LRW] 12.2).
- (28/10/2020, Giachini) Exercises and solutions of Assignment 2
- (2/11/2020, Bottazzi) Inner products, norms, orthogonal vectors, Gramâ€“Schmidt procedure, orthogonal complement, orthogonal projection, Riesz representation theorem. (Textbook sections: [LRW] from 17.2 to 17.8) [Assignment 3]
- (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)
- (9/11/2020, Bottazzi) Minimum variance frontier return; Mean-variance frontier, the generic case. (Textbook sections: [LRW] 18.4)
- (11/11/2020, Giachini) Exercises and solutions of Assignment 3
- (9/11/2020, Bottazzi) Mean-variance frontier and orthogonal kernels.
- (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]
- (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)
- (23/11/2020, Giachini) An example of general equilibrium with heterogeneous agents. The relation between individual wealth and prevailing prices.
- (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)
- (25/11/2020, Giachini) Exercises and solutions of Assignment 4 [Assignment 5]
- (1/12/2019, Bottazzi) CAPM, (Textbook sections: [LRW] from 19.1 to 19.4)
- (2/12/2019, Giachini) Exercises and solutions of Assignment 5.
- (4/12/2019, Bottazzi) Mock exam and solutions

## Academic Term (Anno Accademico) 2020/2021: advanced calculus

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.

Differential calculus of real functions:

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