STOCHASTIC MODELS FOR DETIVATES
The course aims at providing general and issue-specific knowledge of pricing models for European and American derivatives. The emphasis is on finite time models; firstly, necessary mathematical tools are introduced, then, the general methodology is proposed. Finally, particular and well known models are given, such as binomial and trinomial models.
Expected learning outcomes
Knowledge and understanding: the student should be able to understand the themes and problems related to the theory of derivative pricing in financial markets; he should also know the main tools from the theory of stochastic processes that are used in the pricing theory.
Applying knowledge and understanding: the student should be able to apply the acquired knowledge to concrete problems in specific models. To this purpose, the teacher will illustrate some different examples and specific cases of financial markets and derivatives during the lessons; for non-attending students, assistance time will be provided.
Making judgements: the student should be able to use the acquired knowledge also in an autonomous way, by also applying them to specific issues and problems that are more general or different with respect to those illustrated by the teacher.
Communication: the student should be able to answer in a clear and detailed way to the questions of the written examination and to those of the oral examination.
Lifelong learning skills: the student should be able to show a good learning ability, by widening, for example, his/her knowledge with use of relevant bibliographic references, The student should also be able to tackle the pricing problems independently from the specific model considered.
Some basic knowledge of mathematics and statistics, previously acquired by the students in basic courses of their undergraduate programs. For students coming from different first-level degree programs, an integration including a relevant bibliographic reference will be provided.
I module: Stochastic Calculus
The basic tools from the theory of stochastic processes are introduced. The notion of sub/super martingale are presented. The Doob decomposition theorem is finally analysed.
II module:Pricing models for european derivatives
The discrete market models are introduced and the first and the second fundamental theorem of asset pricing are presented. The pricing problem for European derivatives is studied by computing equivalent martingale measures and replicating portfolios. The binomial and the trinomial models are then introduced. Some notion of portfolio optimization are also given in this framework. Finally some preliminary results on continuous models and on the Black and Scholes formula are given. (24 hours)
III module: American derivatives
In the third part of the course, American derivatives are proposed. The Snell envelope and the price process are constructed and optimal stopping times are studied. (8 hours)
During the lessons the issues mentioned in the study program will be discussed and presented together with applications and examples such as specific cases of derivative contracts. Additional teaching and support material is made available through the e-learning online platform Moodle, where notes of the course as well as additional exercises solved and explained in detail.
- Pascucci e Runggaldier (2009) Finanza Matematica, Springer.
- Roman (2012), Introduction to the Mathematics of Finance. Springer.
- Notes by Giuseppe De Marco.
The assessment is based on a written and an oral examination. The written examination consists in the resolution of 3 problems/exercises in 1 hour and 30 minutes. The questions are composed in order to evaluate the actual achievement of the objectives on part the students, but, at the same time, the reasoning ability and the capability to apply the theoretical lessons received. The oral examination has the purpose to evaluate the depth in understanding general theoretical knowledge. In their answers the students should be able to clearly show and illustrate the fundamental concepts acquired during their studies