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
Part I: Discrete time models (40 hours) a) Stochastic Calculus: Finite probability spaces, random variables, filtrations and finite stochastic processes. (super/sub-)martingales and Doob’s decomposition theorem. b) Market models in finite time, equivalent martingales measures, European derivatives and self-financing replicating portfolios. The two foundamentals theorems of asset pricing. Pricing of European derivatives via martingale measures and self-financing replicating portfolios. The binomial and the trinomial models. c) American derivatives: The Snell Envelope, the price process, optimal stopping times.
Part II: Continuous time models (32 hours) a) Infinite probability spaces, sigma-algebras and probability measures, random variables, stochastic processes, filtrations, adapted and martingale processes. Brownian motion as limit of binomial model and properties. Geometric Brownian motion. b) Ito Integral and its properties. Ito formula. Stochastic differential equations. Martingale processes and their properties. Markets and no arbitrage principle. Black and Scholes equation. Solution of parabolic PDE. Feymann-Kac representation theorem and applications to the Black and Scholes equation. Equivalent martingale measure and derivative pricing. Martingale representation theorems. Radon-Nikodyn derivatives. Girsanov theorems and applications to derivative pricing
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) Mathematical Finance, Springer.
- Roman (2012), Introduction to the Mathematics of Finance. Springer.
- Notes by Giuseppe De Marco.
The assessment is based on two different parts. The first one consists in the resolution of 2/3 problems/exercises in 1 hour and 15 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 second part 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