Learning objectives

The course deals with the problem of how to evaluate financial derivative securities, showing how stochastic modeling of the dynamics of the underlying securities is operationally useful.
The basic principles of the risk neutrality approach are illustrated and practical results particularly used in quantitative finance are recalled.
It is then shown how the same concepts have been extended and analogous results have been obtained for more realistic models.

Prerequisites and basic notions

Important knowledge for a profitable learning: linear systems, real functions of one or more real variables (in particular: continuous functions, composition of functions, derivatives, plots), basic concepts of financial mathematics (interest rate, return on an investment, difference between bonds and stocks, discount factor), fundamental concepts of probability theory (sigma algebra, random variables, probability and expected values, independence, conditional expected values, equivalent probability measures, probability density, Gaussian law)

Preparatory courses: Mathematics, Financial Mathematics, Statistics, Probability
Skills necessary for profitable learning: willingness and ability to conduct logical reasoning rigorously, to always justify passages and conclusions

Program

1. Preliminaries
Bonds, stocks
The problem of pricing derivatives
2. Equity prices
Pricing under uncertainty: stochastic models
Risk neutral pricing, risk aversion
3. Pricing derivatives
Risk hedging
Options prices and forward contracts
On the basis of the audience the following models will be treated: discrete single/multi-period models or models in continuous time and with continuous trajectories with constant or stochastic volatility or models in continuous time and with jumps.
4. The problem of selecting/estimating a model.

Bibliography

When and where

reading course

Learning assessment procedures

The exam consists of a series of presentations of assigned material to be studied and understood, and a written report containing the commented proof of a theoretical result.
For the exam to be passed it is necessary that the grade obtained is at least D.

Assessment

The student is required to demonstrate a critical and in-depth knowledge of the topics covered in the course. The concepts must not be exposed in a mechanical but reasoned way, connections may be required.
The synthetic but exhaustive, rigorous and direct exposition immediately at the heart of the matter will be particularly appreciated. Vague, imprecise, poorly detailed or incorrect statements will be penalized

Criteria for the composition of the final grade

50% for the presentations, 50% for the written proof