The aim of the course is to make students familiar with the mathematical and, more in general, with the quantitative methods adopted in describing the dynamics of financial markets and in valuating and hedging financial derivatives. 

This course is connected to the joint-course 095980-MATHEMATICAL FINANCE II (8 cfu). This sheet defines objectives, programs and learning outcomes expected for both courses. 

Both versions of the course have the following common learning goals:

- 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 solve analytically (if possible) 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.).

- Communication skills

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

PART I: The “No Arbitrage Principle” in a continuous time setting. The fundamental Theorems of Asset Pricing. The Geomeric Brownian motion as a limit of a binomial random walk. The Black-Scholes-Merton analysis and the Black-Scholes formula for European Options. The Black-Scholes equation solved via PDE methods and via the Feynman-Kac representation formula. Complete and incomplete market models. Completeness of the Black-Scholes model and its relation with the martingale representation theorem. Hedging in the Black-Scholes setting: the “Greeks”.

PART II(*): Portfolio dynamics and self-financing strategies. The optimal control problems in the Black-Scholes setting: the optimal-consumption-investment problem and the related Hamilton-Jacobi-Bellman equation. The two-funds theorem: the case with no risk-free asset and the case with a risk-free asset.

PART III: Early exercise features and American option pricing. The free boundary problem for the Black-Scholes equation and its formulation as an optimal stopping problem.

PART IV: Exotic and Path-Dependent options: Barrier, Lookback, Asian options valuation for the lognormal model. The multidimensional BS model and the valuation of basket options. Forward and Futures: Black-76 formula for options on futures. Currency derivatives.

PART V: Fixed-Income derivatives. Short rate models and bond valuation: affine models and affine term structures. The Vasicek, Ho-lee, Cox-Ingersoll-Ross, Hull and White Models. Forward rate models: the Heath-Jarrow-Morton approach. The LIBOR market model. Valuation of Caps, Floors and Swaptions in the LIBOR market model.

PART VI: Valuation of forward and futures in stochastic interest rate models.

PART VII(*): Limitations of the Black-Scholes model: empirical evidences on lorgeturns distributions: fat tails, aggregational Gaussianity, volatility clustering and the leverage effect. Qualitative discussion on the volatility smile and the volatility term structure of option prices.

PART VIII(*): Stochastic volatility: the Heston Model; the Jump-Diffusion Merton model and European option pricing.

While the 10 CFU version cover all the above topics,  part II, part VII and part VIII, denoted above with (*), are not considered in the 8 CFU version.

The course will include exercise sessions in which examples and applications of the general theory to specific concrete cases will be provided and discussed.

Basic knowledge of finance (e.g., interest rate, plain vanilla derivatives, difference between assets and bonds), probability and analysis.

The exam will be performed in two steps: a written part, mainly related to exercises and applications, and an oral part mainly related to the general theory. Both written and oral exams are mandatory. The oral exam will be accessible only if the evaluation of the written exam is above or equal to 16/30.  The written exam will be the same for both the 8 cfu and the 10 cfu version. The oral exam will cover all the course topics (therefore for the 8 cfu version, part II, part VII and part VIII, denoted above with (*), are excluded).

The exam has the goal of checking whether the student has acquired the following skills:
- knowledge of concepts of stochastic calculus, which are essential in quantitative finance;
- knowledge of closed-formula to solve a financial problem properly in the classical Black&Scholes framework;
- ability to express mathematical and financial concepts in a clear and rigorous way.