Financial mathematics (2019/2020)

Course code
Name of lecturer
Alessandro Gnoatto
Alessandro Gnoatto
Number of ECTS credits allocated
Academic sector
Language of instruction
I semestre dal Oct 1, 2019 al Jan 31, 2020.

Lesson timetable

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Learning outcomes

This course presents the basic models for the analysis and evaluation of financial operations, both under conditions of certainty and randomness. The main goal of the course is to equip the student with the ability to model and solve some basic mathematical problems, commonly encountered in the financial practice.


Part 1: classical financial mathematics - Main Reference: Scandolo

1) Basic financial operations, simple interest, interest in advance, compounding of interest, exponential regime.

2) Market rates. Some sketch of the classical theory with some warnings regarding the multiple curve phenomenon.

3) 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.

4) Choice without uncertainty: return for elementary and generic investment, choice criteria for investment and financing operations.

5) Bonds: classification, zero coupon bonds, fixed coupon bonds.

6) Term structure: yield curve, complete and incomplete markets.

7) Immunization: Maculay’s duration and convexity, immunized portfolios.

Part 2: mathematical finance in the presence of uncertainty - Main references: Föllmer Schied and Pascucci Runggaldier.

8) Probability theory refresher: probability spaces, independence, Radon-Nikodym theorem, expectation, conditional expectation, martingales, convergence of random variables.

9) Preferences and risk aversion: expected utility criterion (St. Petersburgh paradox), von Neumann Morgenstern axioms, stochastic dominance, mean variance criterion and static portfolio optimization, CAPM.

10) Arbitrage theory in one period: foundations and fundamental theorem of asset pricing, contingnt claimds, market completeness.

11) Arbitrage theory in multiperiod models: fundamental on multiperiod models, absence of arbitrage, European contingent claims, binomial model (Cox-Ross Rubinstein).

12) American contingent claims: foundataions, valuation and hedging, arbitrage free prices and replicability in general markets.

Reference books
Author Title Publisher Year ISBN Note
Pascucci, A. Runggaldier, W. J. Finanza matematica. Teoria e problemi per modelli multiperiodali (Edizione 1) Springer 2009 978-8-847-01441-1
Scandolo Giacomo Matematica Finanziaria Amon 2013
Scandolo Giacomo Matematica finanziaria - Esercizi Amon 2013
Föllmer, H. Schied, A. Stochastic Finance: An Introduction in Discrete Time (Edizione 4) De Gruyter 2016 978-3-110-46344-6

Assessment methods and criteria

Two-hour written exam. The exam consists of practical and theoretical exercises, including the proof of certain claims. The exam aims to verify the student's ability to identify the correct resolution, knowledge of basic financial laws and sophisticated assessment models, and the ability to apply acquired knowledge to concrete cases in new and variable contexts. The exam aims also to assess the level of understanding of the theoretical aspects of the lecture.