Scientific Disciplinary Sector (SSD)
MAT/06 - PROBABILITY AND STATISTICS
I sem. dal Oct 3, 2016 al Jan 31, 2017.
Academic Year 2016/2017
The Mathematical Finance course for the internationalized Master's Degree ( completely taught in English) aims to introduce the main concepts of discrete as well as continuous time, stochastic approach to the theory of modern financial markets.
In particular, the fundamental purpose of the course is to provide the mathematical tools characterizing the setting of Itȏ stochastic calculus for the determination, the study and the analysis of models for options, interest rates models, financial derivatives, etc., determined by stochastic differential equations driven by Brownian motion and/or impulsive random noises.
Basic ingredients are the foundation of the theory of continuous-time martingale, Girsanov theorems and the Feynman–Kac theorem and their applications to the theory of option pricing with specific examples in equities, also considering path-dependent options, and within the framework of interest rates models.
Great attention will be also given to the practical study and realisation of concrete models characterising the modern approach to both the risk managment and option pricing frameworks, also by mean of numerical computations and computer oriented lessons.
It is important to emphasize how the Stochastic Systems course is organized in such a way that students can concretely complete and further develop their own:
°ability to establish profound connections with non-mathematical disciplines, both in terms of motivation of mathematical research and of the application of the results of such surveys;
° capacity of analysis, synthesis and abstraction;
° specific computational and computer skills;
° ability to understand texts, even advanced, of Mathematics in general and Applied Mathematics in particular;
• ability to develop mathematical models for physical and natural sciences, while being able to analyze its limits and actual applicability, even from a computational point of view;
° skills concerning how to develop mathematical and statistical models for the economy and financial markets;
° capacity to extract qualitative information from quantitative data;
° knowledge of programming languages or specific software.
The MathFin course will be enriched by the contributions of Michele Bonollo e Luca Spadafora, for the details of their respective parts, please see below.
[ Luca Di Persio ]
Discrete time models
• Contingent claims, value process, hedging strategies, completeness, arbitrage
• Fundamental theorems of Asset Pricing (in discrete time)
The Binomial model for Assset Pricing
• One period / multiperiod Binomial model
• A Random Walk (RW) interlude (scaled RW, symmetric RW, martingale property and quadratic variation of the symmetric RW, limiting distribution)
• Derivation of the Black-Scholes formula (continuous-time limit)
Brownian Motion (BM)
• review of the main properties of the BM: filtration generated by BM, martingale property, quadratic variation, volatility, reflection properties, etc.
• Itȏ integral
• Itȏ-Döblin formula
• Black-Scholes-Merton Equation
• Evolution of Portfolio/Option Values
• Solution to the Black-Scholes-Merton Equation
• Sensitivity analysis
• Risk-Neutral Measure and Girsanov's Theorem
• Pricing under the Risk-Neutral Measure
• Fundamental Theorems of Asset Pricing
• Existence/uniqueness of the Risk-Neutral Measure
• Forwards and Futures
[ Luca Spadfora ]
*Theory Review: distributions, the moments of a distribution, statistical estimators, Central Limit Theorem (CLT), mean, variance and empirical distributions.
*Elements of Extreme Value Theory: what is the distribution of the maximum?
Numerical studies: statistical error of the sample mean, CLT at work, distributions of extreme values.
*How can we measure risk? Main risk measures: VaR and Expected Shorfall
*How to model risk: historical, parametric and Montecarlo methods
*We have a risk model: does it works? The backtesting methodology
*Empirical studies a) empirical behavior and stylized facts of historical series
*Empirical studies b) Implementation of risk models
*Empirical studies c) Implementation of risk models backtesting
[ Miche Bonollo ]
*** Tools for derivatives pricing
* Functionals of brownian motions: fist hitting time, occupation time, local time, min-MAX distribution review
* Application 1: range accrual payoff
* Application 2: worst of and Rainbow payoff
*** Credit portfolio models
* The general framework. The credit portfolio data
* Gaussian Creidit Metrics - Merton model
* The quantile estimation problem with Montecarl approach. L-Estimators, Harrel-Davis
A. F. McNeil, R. Frey, P. Embrechts, Quantitative Risk Management:Concepts, Techniques and Tools, Princeton University Press, 2015.
J. -P. Bouchaud, M. Potter, Theory of Financial Risk - From Statistical Physics to Risk Management, University Press, Cambridge, 2000.
R. Cont, P. Tankov, Financial Modelling With Jump Processes, Chapman and Hall, CRC Press, 2003.
E. J. Gumbel, Statistics of Extremes, Dover Publications, Mineola (NY), 2004.
M.Yor et al, "Exponential Functionals of Brownian Motion and related Processes", Springer.
Shreve, Steven , Stochastic Calculus for Finance II: Continuous-Time Models
Shreve, Steven , Stochastic Calculus for Finance I: The Binomial Asset Pricing Model
|M.Yor et al
||Exponential Functionals of Brownian Motion and related Processes
|R. Cont, P. Tankov
||Financial Modelling With Jump Processes
||Chapman and Hall, CRC Press
|A. F. McNeil, R. Frey, P. Embrechts
||Quantitative Risk Management:Concepts, Techniques and Tools
||Princeton University Press
|E. J. Gumbel
||Statistics of Extremes
|| Dover Publications, Mineola (NY)
|S. E. Shreve
||Stochastic Calculus for Finance II: Continuous-Time Models
||Springer, New York
|S. E. Shreve
||Stochastic Calculus for Finance I: The Binomial Asset Pricing Model
||Springer, New York
|J. -P. Bouchaud, M. Potter
||Theory of Financial Risk - From Statistical Physics to Risk Management
||University Press, Cambridge
Academic Year 2016/2017
Final Exam : the exam will consists in an oral session, to be given with prof. L. Di Persio, which will be targeted on the theory behind all the arguments treated in the whole course, hence including the parts developed by M. Bonollo and L. Spadafora.
Moreover each student will be called to develop a case study within a list of projects proposed by both M. Bonollo and L. Spadafora, according with the notions that will have been addressed during their respective parts [ see the Course Program section ].
The final vote is expressed out of 30: in particular:
° The doctors Bonollo and Spadafora will communicate to prof. Of Persio a report on the goodness of the project presented by the single student;
° professor. Di Persio will use the previous report, along with the outcome of the oral examination he conducted, to decide on a final grade expressed out of 30.
It is important to emphasize how the skills acquired by students at the end of the course will enable them to:
- carry out high-profile technical and / or professional tasks, both mathematically oriented and of
computational type, both in laboratories and / or research organizations, in the fields of finance, insurance, services, and public administration, both individually and in groups;
° read and understand advanced texts of math and applied sciences, even at the level of advanced research;
• to use high-tech computing and computing tools with the utmost ease of implementation algorithms and models illustrated in the course, as well as to acquire further information;
- to know in depth the demonstration techniques used during the course in order to be able to exploit them to solve problems in different mathematical fields, also by taking the necessary tools and methods, from seemingly distant contexts, thus mathematically formalizing problems expressed in languages of other scientific disciplines as well as economical ones, using, adapting and developing advanced models.