# Course Details for A.Y. 2017/2018

#### Name:

**Stochastic models and applications / Stochastic models and applications**

### Basic information

##### Credits:

*:* Master Degree in Mathematics 6 CFU (c)

##### Degree(s):

Master Degree in Mathematics 2^{nd} anno curriculum Generale Elective

##### Language:

English

### Course Objectives

Students should acquire a good knowledge of advanced probabilistic tools employed in the modeling of financial markets.
In particular they should become able to
- learn and understand the first mathematical models involving stochastic calculus techniques;
- solve derivatives evaluation problems of medium difficulty;
- extend the studied notions to more complex models;
- read an advanced text in financial modeling;
- implement computations for the studied models.

### Course Content

- Complements of stochastic calculus. The martingale representation theorem,
Girsanov theorem, existence and uniqueness of the solutions of stochastic differential
equations, quadratic variation.
- Discrete market models. First and second fundamental theorem of asset pricing.
Evaluation of European and American derivatives.
- Black and Scholes world. European options evaluation, Barrier options, American options, (perpetual puts and critical price).
Option evaluation for a general diffusion model: the infinitesimal generator of a diffusion, computation of expectations and partial differential equations
The multidimensional model: viability and completeness.
Asian options and exchange options.
- Bonds and interst rate models. Zero coupon bonds. Merton model, Vasicek model, Cox Ingersoll Ross model.
- If there is enough time. Weak solutions of stochastic differential equations. weak solutions via Girsanov. Yamada and Watanabe's results. The unidimensional case.
- If there is enough time. Stochastic volatility models. Stein and Stein, Hull and White, Heston models

### Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

- Students shuould acquire a good knowledge of advanced probabilistic tools employed in the modeling of
financial markets.
- students should become able to
- learn and understand the first mathematical models involving stochastic calculus techniques;
- solve derivatives evaluation problems of medium difficulty;
- students should become able to
- solve derivatives evaluation problems of medium difficulty;
- extend the studied notions to more complex models;
- Students should become able to expose the main points of financial modeling to an audience of experts and
non experts.
- students should become able to
- read an advanced text in financial modeling;
- implement computations for the studied models.

### Prerequisites and Learning Activities

An advanced course in probability and the first part of the integrated course

### Assessment Methods and Criteria

oral exam with possible integrations either written or oral

### Textbooks

- I. Karatzas, S. Shreve, Brownian motion and stochastic calculus , Springer.
* *
- A. Pascucci, Calcolo Stocastico per la Finanza , Springer.
* *
- D. Lamberton, D. Lapeyre, Introduction to stochastic calculus applied to Finance , Chapman and Ha.
* *
- J. Zhu, Modular pricing of options (Lecture notes in Economics and Mathematical Systems) , Springer. (vol. 493)
* *
- P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations , Springer.
* *
- P. Billingsley, Probability and measure , Wiley . 1984.
* *

### Course page updates

This course page is available (with possible updates) also for the following academic years:

** To read the *** current * information on this course, if it is still available, go to the university course catalogue .
*Course information last updated on: 31 agosto 2016, 14:03*