The aim of the course is to make students familiar with the advanced quantitative methods adopted in describing the dynamics of financial markets, in valuating and hedging financial derivatives, and in optimal investments problems. An implementation of all models and methods in Matlab is an essential part of the program. The goal is to provide students with all the instruments necessary to be able to solve a financial problem, like pricing derivatives, choosing the best model and coding to obtain a numerical solution. A part of the course is also devoted to Fintech applications.

This course is connected to the joint-course 097667 - COMPUTATIONAL FINANCE (8 cfu). This sheet defines objectives, programs and learning outcomes expected for both courses. 

Lectures and coding sessions will allow students to acquire the following competences:

- Knowledge and understanding

know some advanced concept of stochastic calculus, which are essential in quantitative finance;
know how to choose modeling assumptions to solve a financial problem properly.

- Ability in applying knowledge and understanding

know how to write a code/algorithm to solve a financial problem, exploiting the developed theoretical knowledgements.

- Making judgements

find proper modeling assumption to describe a financial asset (like interest rate, volatility, commodities, etc.);
be able to judge the possible financial risks raising from using wrong modeling assumption (like misprice of financial products).

- Communication skills

to be able to express mathematical and financial concepts in a clear and rigorous way.

Advanced Models for Financial Markets and Asset Allocation

Lévy processes: stochastic calculus for jump processes; fondamental properties of Lévy processes and reason why they overcome the Black&Scholes limits; Lévy-Kintchine formula; simulating Lévy processes.

Stochastic Volatility models: Heston, Hull-White, Stein & Stein models. SABR. Stochastic volatility models with jumps.

Asset Allocation: Hamilton-Jacobi-Bellman equation in optimal allocation problems; managing assets' portfolio. [only for the 10 cfu version]

Numerical Methods for Finance

Monte Carlo Simulations: random numbers generator; sampling from uniform and normal distribution; Quasi Monte Carlo methods; simulating continuous processes; simulating jump diffusion processes; variance reduction techniques; pricing European and exotic options; pricing American derivatives: the Longstaff & Schwarz algorithm.

PDE: Finite Differences and Finite Elements discretizations: evaluating barrier and European options via Finite Differences and Finite Elements; pricing American derivatives: the PSOR algorithm. 

FFT: Carr-Madan method for European derivatives. The Convolution method

Bitcoin and Blockchain Technologies  [only for the 10 cfu version]

Students are required to know the following topics.

- stochastic calculus: Wiener process, martingale, filtration.
- finance: option pricing in the Black&Scholes framework.
- coding: Matlab.