Course Details for A.Y. 2016/2017
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 2nd 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