Scientific Disciplinary Sector (SSD)
SECS-S/06 - MATHEMATICAL METHODS OF ECONOMICS, FINANCE AND ACTUARIAL SCIENCES
Primo semestre dal Oct 3, 2022 al Jan 27, 2023.
The aim of the first part of the course is to present the tools and topics of classical financial mathematics (compounding regimes, mortgages, bonds, immunization). The second part of the lecture provides an in-depth introduction to modern financial mathematics and stochastic methods in discrete time (stochastic processes and martingales in discrete time) that are useful in view of more advanced lectures on the topic. Students will have the opportunity to learn the terminology and the concepts that are useful for the understanding and use the techniques of classical and modern mathematical finance. For some topics, software examples using the Java programming language will be provided (Finmath library). The lecture provides important examples of applications of concepts from the lectures on probability.
Prerequisites and basic notions
Calculus, Linear Algebra, Probability. Extra notions on probability will be provided.
Part 1: classical financial mathematics - Main Reference: Scandolo
1) Basic financial operations, simple interest, interest in advance, compounding of interest, exponential regime.
2) Annuities and amortization: non-elementary investment and financing, annuities with constant rates, annuities with installments following a geometric progression, amortization, common amortization clauses, amortization with viariable interest rate.
3) Choice without uncertainty: return for elementary and generic investment, choice criteria for investment and financing operations.
4) Bonds: classification, zero coupon bonds, fixed coupon bonds. Term structure: yield curve, complete and incomplete markets.
5) Immunization: Maculay’s duration and convexity, immunized portfolios.
Part 2: mathematical finance in the presence of uncertainty - Main references: Föllmer Schied and Pascucci Runggaldier.
6) Probability theory refresher: probability spaces, independence, Radon-Nikodym theorem, expectation, conditional expectation, martingales, convergence of random variables.
7) Arbitrage theory in one period: foundations and fundamental theorem of asset pricing, contingnt claimds, market completeness.
8) Arbitrage theory in multiperiod models: fundamental on multiperiod models, absence of arbitrage, European contingent claims, binomial model (Cox-Ross Rubinstein).
9) American contingent claims: foundataions, valuation and hedging, arbitrage free prices and replicability in general markets.
Time permitting: Preferences and risk aversion: expected utility criterion (St. Petersburgh paradox), von Neumann Morgenstern axioms, stochastic dominance, mean variance criterion and static portfolio optimization, CAPM.
Visualizza la bibliografia con Leganto, strumento che il Sistema Bibliotecario mette a disposizione per recuperare i testi in programma d'esame in modo semplice e innovativo.
Learning assessment procedures
Intermediate test + 90 minute final exam.
Alternatively, a 2 Hour written exam for those who are not giving the intermediate test.
The tests will contain both exercises and theoretical questions (statements to be proved)
- Knowing and understanding the fundamental concepts of basic financial mathematics in a deterministic setting
- Knowing and understanding the fundamental concepts of modern financial mathematics in a stochastic setting
- Obtaining adequate analytical and abstraction skills.
- Knowing how to apply the above knowledge to solve problems and exercise, demonstrating a good level of mathematical rigour.
Mathematical rigour both in the proofs and in the exercises. Correctedness of the calculations.
Criteria for the composition of the final grade
For students taking the intermediate test
25% intermediate exam 75% final exam (9 ECTS case)
100% final exam otherwise.