Massimiliano MENZIETTI | Stochastic Finance

cod. 0222400013

STOCHASTIC FINANCE

	0222400013
	DEPARTMENT OF ECONOMICS AND STATISTICS
	EQF7
	STATISTICAL SCIENCES FOR FINANCE
	2024/2025

CURRICULUM BIG DATA ANALYTICS

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2014
	SPRING SEMESTER

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	SECS-S/06	10	60	LESSONS	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS

PROFESSORS

	MARILENA SIBILLO T Curriculum
	MASSIMILIANO MENZIETTI Curriculum

	Objectives
	THE COURSE IN STOCHASTIC FINANCE, IN LINE WITH THE FUNDAMENTAL OBJECTIVES OF THE DEGREE PROGRAM, IS DEVOTED TO THE ANALYSIS AND APPLICATION OF MATHEMATICAL MODELS USED IN FINANCE WHEN THE STOCHASTIC ELEMENT BECOMES PART OF THE QUANTITATIVE DESCRIPTION. IN PARTICULAR, STUDENTS WILL ACQUIRE THE TOOLS OF STOCHASTIC COMPUTATION AND COMPUTER IMPLEMENTATION NECESSARY FOR THE SOLUTION OF COMPLEX PROBLEMS RELATED TO THE FINANCIAL SECTOR. TEACHING FINALIZES FORMAL RIGOR TO AN APPLIED PERSPECTIVE, WITH EMPHASIS ON CRITICAL DISCUSSION OF RESULTS. KNOWLEDGE AND UNDERSTANDING THE TEACHING AIMS TO: - DEEPEN THE KNOWLEDGE OF BASIC FINANCIAL MATHEMATICS ACQUIRED IN THE THREE-YEAR DEGREE COURSES BY EXTENDING IT TO THE STOCHASTIC FIELD; - PRESENT THE MOST RELEVANT STOCHASTIC MODELS IN THE FINANCIAL FIELD; - DEEPEN THE APPLICATION QUALITIES OF THE ANALYSED PROCESSES; - APPLY THE THEORETICAL SKILLS ACQUIRED IN THE FINANCIAL AND INSURANCE FIELDS. ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING THE MATHEMATICAL TOOLS INTRODUCED DURING THE LESSONS WILL ENABLE THE STUDENT TO BE ABLE TO: - UNDERSTAND THE VALUE OF THE STOCHASTIC APPROACH FOR DESCRIBING FINANCIAL PROCESSES; - ACQUIRE A CAPACITY FOR ANALYSIS THAT IS BOTH RIGOROUS AND CRITICAL TO ADEQUATELY DESCRIBE FINANCIAL PROCESSES; - SELECT AND APPLY THROUGH COMPUTER IMPLEMENTATIONS THE MODELS COVERED BY THE COURSE; - DESCRIBE AND COMMENT CONSTRUCTIVELY ON THE RESULTS OBTAINED. TRANSVERSAL SKILLS STUDENTS WILL ACQUIRE ANALYTICAL AND CRITICAL SKILLS THAT WILL ENABLE THEM TO: - DESCRIBE THE BEHAVIOR OF FINANCIAL MAGNITUDES IN HISTORICAL EVOLUTION AND MAKE PREDICTIONS ABOUT THEIR FUTURE EVOLUTION - ABILITY TO SYNTHESIZE THE IMPACT OF UNCERTAINTY WITH QUANTITATIVE INDICATORS - STRUCTURE MATHEMATICAL DESCRIPTIONS FOR INTEREST RATE-DEPENDENT CONTRACTS - CONFIDENTLY USE SPECIALIZED FINANCIAL AND DEMOGRAPHIC SOFTWARE - DEVELOP SKILLS IN DISCLOSURE ACTIVITIES

THE COURSE IN STOCHASTIC FINANCE, IN LINE WITH THE FUNDAMENTAL OBJECTIVES OF THE DEGREE PROGRAM, IS DEVOTED TO THE ANALYSIS AND APPLICATION OF MATHEMATICAL MODELS USED IN FINANCE WHEN THE STOCHASTIC ELEMENT BECOMES PART OF THE QUANTITATIVE DESCRIPTION. IN PARTICULAR, STUDENTS WILL ACQUIRE THE TOOLS OF STOCHASTIC COMPUTATION AND COMPUTER IMPLEMENTATION NECESSARY FOR THE SOLUTION OF COMPLEX PROBLEMS RELATED TO THE FINANCIAL SECTOR. TEACHING FINALIZES FORMAL RIGOR TO AN APPLIED PERSPECTIVE, WITH EMPHASIS ON CRITICAL DISCUSSION OF RESULTS.
KNOWLEDGE AND UNDERSTANDING
THE TEACHING AIMS TO:
- DEEPEN THE KNOWLEDGE OF BASIC FINANCIAL MATHEMATICS ACQUIRED IN THE THREE-YEAR DEGREE COURSES BY EXTENDING IT TO THE STOCHASTIC FIELD;
- PRESENT THE MOST RELEVANT STOCHASTIC MODELS IN THE FINANCIAL FIELD;
- DEEPEN THE APPLICATION QUALITIES OF THE ANALYSED PROCESSES;
- APPLY THE THEORETICAL SKILLS ACQUIRED IN THE FINANCIAL AND INSURANCE FIELDS.
ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING
THE MATHEMATICAL TOOLS INTRODUCED DURING THE LESSONS WILL ENABLE THE STUDENT TO BE ABLE TO:
- UNDERSTAND THE VALUE OF THE STOCHASTIC APPROACH FOR DESCRIBING FINANCIAL PROCESSES;
- ACQUIRE A CAPACITY FOR ANALYSIS THAT IS BOTH RIGOROUS AND CRITICAL TO ADEQUATELY DESCRIBE FINANCIAL PROCESSES;
- SELECT AND APPLY THROUGH COMPUTER IMPLEMENTATIONS THE MODELS COVERED BY THE COURSE;
- DESCRIBE AND COMMENT CONSTRUCTIVELY ON THE RESULTS OBTAINED.
TRANSVERSAL SKILLS
STUDENTS WILL ACQUIRE ANALYTICAL AND CRITICAL SKILLS THAT WILL ENABLE THEM TO:
- DESCRIBE THE BEHAVIOR OF FINANCIAL MAGNITUDES IN HISTORICAL EVOLUTION AND MAKE PREDICTIONS ABOUT THEIR FUTURE EVOLUTION
- ABILITY TO SYNTHESIZE THE IMPACT OF UNCERTAINTY WITH QUANTITATIVE INDICATORS
- STRUCTURE MATHEMATICAL DESCRIPTIONS FOR INTEREST RATE-DEPENDENT CONTRACTS
- CONFIDENTLY USE SPECIALIZED FINANCIAL AND DEMOGRAPHIC SOFTWARE
- DEVELOP SKILLS IN DISCLOSURE ACTIVITIES

	Prerequisites
	COURSES IN MATHEMATICAL METHODS FOR ECONOMICS, FINANCIAL MATHEMATICS (I MODULE OF THE QUANTITATIVE METHODS COURSE) AND STATISTICS ARE RECOMMENDED

	Contents
	THE STOCHASTIC FINANCE COURSE CONSISTS OF TWO MODULES: MODULE I - STOCHASTIC MODELS FOR FINANCE - 30 HOURS OF LECTURE WITH SOLUTION OF EXERCISES AND APPLICATIONS FOR EACH TOPIC. CONTENTS: CHARACTERIZATION OF STOCHASTIC PROCESSES IN FINANCE. DISCRETE PROCESSES. CONTINUOUS PROCESSES. BROWNIAN MOTIONS. DIFFERENTIAL APPROACH. ITO'S LEMMA. INTEREST RATE SENSITIVE CONTRACTS: GENERAL EQUATION. VASICEK'S AND COX, INGERSOLL AND ROSS'S MODELS. THE BLACK AND SCHOLES FORMULA. DEFAULT RISK: THE POISSON PROCESS. MODULE II - COMPUTER APPLICATIONS TO STOCHASTIC PROCESSES FOR FINANCE - 30 HOURS OF LECTURE IN COMPUTER LAB. MONTE CARLO SIMULATIONS. GEOMETRIC BROWNIAN MOTION. STOCHASTIC PROCESSES FOR INTEREST RATES EVOLUTION: ESTIMATION AND SIMULATION OF TERM STRUCTURE: VASICEK AND COX INGERSOLL AND ROSS MODELS. INTEREST RATE TERM STRUCTURE ESTIMATION: PRACTICAL CASES AND COMPARATIVE METHODOLOGIES (8 HOURS OF EMPIRICAL APPLICATIONS IN LAB). EUROPEAN AND AMERICAN FINANCIAL OPTIONS: BLACK-SCHOLES-MERTON MODEL AND BINOMIAL MODEL. NUMERICAL APPLICATION WITH OPTIONS: DELTA/GAMMA/VEGA HEDGING; PORTFOLIO INSURANCE; BRENNAN & SCHWARTZ’S EQUITY-LINKED INSURANCE PRICING MODEL (10 HOURS OF EMPIRICAL APPLICATIONS IN LAB). SURVIVAL ANALYSIS AND ACTUARIAL ASSESSMENT OF LIFE INSURANCE CONTRACTS. PACKAGES: DEMOGRAPHY, STMOMO, LIFECONTINGENCIES. PROJECTIONS WITH LEE CARTER MODEL AND EXTENSIONS, CBD MODEL AND EXTENSIONS (12 HOURS OF LESSONS + EMPIRICAL APPLICATIONS IN THE LAB). THE DETAILED COURSE SCHEDULE IS AVAILABLE ON THE TEACHERS’ WEBSITE.

Contents

THE STOCHASTIC FINANCE COURSE CONSISTS OF TWO MODULES:
MODULE I - STOCHASTIC MODELS FOR FINANCE - 30 HOURS OF LECTURE WITH SOLUTION OF EXERCISES AND APPLICATIONS FOR EACH TOPIC.
CONTENTS: CHARACTERIZATION OF STOCHASTIC PROCESSES IN FINANCE. DISCRETE PROCESSES. CONTINUOUS PROCESSES. BROWNIAN MOTIONS. DIFFERENTIAL APPROACH. ITO'S LEMMA. INTEREST RATE SENSITIVE CONTRACTS: GENERAL EQUATION. VASICEK'S AND COX, INGERSOLL AND ROSS'S MODELS. THE BLACK AND SCHOLES FORMULA. DEFAULT RISK: THE POISSON PROCESS.
MODULE II - COMPUTER APPLICATIONS TO STOCHASTIC PROCESSES FOR FINANCE - 30 HOURS OF LECTURE IN COMPUTER LAB.
MONTE CARLO SIMULATIONS. GEOMETRIC BROWNIAN MOTION. STOCHASTIC PROCESSES FOR INTEREST RATES EVOLUTION: ESTIMATION AND SIMULATION OF TERM STRUCTURE: VASICEK AND COX INGERSOLL AND ROSS MODELS. INTEREST RATE TERM STRUCTURE ESTIMATION: PRACTICAL CASES AND COMPARATIVE METHODOLOGIES (8 HOURS OF EMPIRICAL APPLICATIONS IN LAB). EUROPEAN AND AMERICAN FINANCIAL OPTIONS: BLACK-SCHOLES-MERTON MODEL AND BINOMIAL MODEL. NUMERICAL APPLICATION WITH OPTIONS: DELTA/GAMMA/VEGA HEDGING; PORTFOLIO INSURANCE; BRENNAN & SCHWARTZ’S EQUITY-LINKED INSURANCE PRICING MODEL (10 HOURS OF EMPIRICAL APPLICATIONS IN LAB). SURVIVAL ANALYSIS AND ACTUARIAL ASSESSMENT OF LIFE INSURANCE CONTRACTS. PACKAGES: DEMOGRAPHY, STMOMO, LIFECONTINGENCIES. PROJECTIONS WITH LEE CARTER MODEL AND EXTENSIONS, CBD MODEL AND EXTENSIONS (12 HOURS OF LESSONS + EMPIRICAL APPLICATIONS IN THE LAB).
THE DETAILED COURSE SCHEDULE IS AVAILABLE ON THE TEACHERS’ WEBSITE.

	Teaching Methods
	THE COURSE OF STOCHASTIC FINANCE CONSISTS OF 60 HOURS OF DIDACTICS, 30 OF WHICH WILL BE IN A COMPUTER LAB. THE LECTURES WILL BE FRONTAL AND CARRIED OUT WITH CONSTANT ATTENTION TO THE THEORETICAL APPROACH OF THE TOPICS, TO THE APPLICATIONS AND THE INTERPRETATION OF THE RESULTS OBTAINED. COMPATIBLE WITH OTHER TEACHING ACTIVITIES, INTERACTIVE ACTIVITIES WILL BE CONDUCTED ON THE WAY.

	Verification of learning
	THE VERIFICATION OF LEARNING IS ORIENTED TO STIMULATE THE STUDY AND UNDERSTANDING OF THE TOPICS COVERED DURING THE COURSE BY THE STUDENTS, WITH PARTICULAR ATTENTION TO THE ABILITY TO APPLY THE MODELS TO CONCRETE SITUATIONS. MODULE I: THE STUDENT'S FINAL ASSESSMENT WILL CONSIST OF A WRITTEN TEST AND AN ORAL TEST. THE MODULE WILL BE CONSIDERED PASSED IF BOTH THE WRITTEN TEST AND THE ORAL TEST ARE SCORED AT LEAST SUFFICIENTLY (GREATER THAN OR EQUAL TO 18/30). EACH TEST WILL BE GRADED IN 30MI AND THE FINAL GRADE WILL BE THE AVERAGE OF THE GRADES OBTAINED IN THE TWO TESTS. THE MAXIMUM GRADE IS 30/30 CUM LAUDE. COMPATIBLE WITH THE SPECIFICS OF THE COURSE, THERE WILL BE INTERACTIVE ACTIVITIES IN ITINERE. THE WRITTEN TEST WILL HAVE A STRONG APPLICATIVE CHARACTERIZATION OF THE MODELS EXAMINED. INSTEAD, THE ORAL TEST WILL BE THEORETICAL IN NATURE. MODULE II THE FINAL EXAM CONSISTS OF 3 PROJECT WORKS (ONE ON STOCHASTIC MODELS FOR INTEREST RATES, ONE ON OPTIONS VALUATION AND HEDGING STRATEGIES THROUGH OPTIONS, ONE ON MORTALITY PROJECTION MODELS AND LONGEVITY LINKED SECURITIES VALUATION) TO BE PRESENTED IN THE FORM OF A WRITTEN PAPER AND TO BE DISCUSSED ORALLY. THE EVALUATION IS MADE IN THIRTIETHS AND IS SUFFICIENT IF PASSED WITH AT LEAST 18/30. THE MAXIMUM GRADE IS 30/30 WITH HONORS. THESE WORKS WILL BE ORIENTED TO VERIFY THE ABILITY TO IMPLEMENT THE TOOLS ACQUIRED DURING THE MODULE, IN THE CONTEXT OF APPLICATIVE CASES TO BE SOLVED. THE FINAL EVALUATION IN THE WHOLE EXAM WILL BE THE AVERAGE OF THE EVALUATIONS REPORTED IN EACH OF THE TWO MODULES.

Verification of learning

THE VERIFICATION OF LEARNING IS ORIENTED TO STIMULATE THE STUDY AND UNDERSTANDING OF THE TOPICS COVERED DURING THE COURSE BY THE STUDENTS, WITH PARTICULAR ATTENTION TO THE ABILITY TO APPLY THE MODELS TO CONCRETE SITUATIONS.
MODULE I: THE STUDENT'S FINAL ASSESSMENT WILL CONSIST OF A WRITTEN TEST AND AN ORAL TEST. THE MODULE WILL BE CONSIDERED PASSED IF BOTH THE WRITTEN TEST AND THE ORAL TEST ARE SCORED AT LEAST SUFFICIENTLY (GREATER THAN OR EQUAL TO 18/30). EACH TEST WILL BE GRADED IN 30MI AND THE FINAL GRADE WILL BE THE AVERAGE OF THE GRADES OBTAINED IN THE TWO TESTS. THE MAXIMUM GRADE IS 30/30 CUM LAUDE. COMPATIBLE WITH THE SPECIFICS OF THE COURSE, THERE WILL BE INTERACTIVE ACTIVITIES IN ITINERE.
THE WRITTEN TEST WILL HAVE A STRONG APPLICATIVE CHARACTERIZATION OF THE MODELS EXAMINED. INSTEAD, THE ORAL TEST WILL BE THEORETICAL IN NATURE.
MODULE II THE FINAL EXAM CONSISTS OF 3 PROJECT WORKS (ONE ON STOCHASTIC MODELS FOR INTEREST RATES, ONE ON OPTIONS VALUATION AND HEDGING STRATEGIES THROUGH OPTIONS, ONE ON MORTALITY PROJECTION MODELS AND LONGEVITY LINKED SECURITIES VALUATION) TO BE PRESENTED IN THE FORM OF A WRITTEN PAPER AND TO BE DISCUSSED ORALLY. THE EVALUATION IS MADE IN THIRTIETHS AND IS SUFFICIENT IF PASSED WITH AT LEAST 18/30. THE MAXIMUM GRADE IS 30/30 WITH HONORS. THESE WORKS WILL BE ORIENTED TO VERIFY THE ABILITY TO IMPLEMENT THE TOOLS ACQUIRED DURING THE MODULE, IN THE CONTEXT OF APPLICATIVE CASES TO BE SOLVED.
THE FINAL EVALUATION IN THE WHOLE EXAM WILL BE THE AVERAGE OF THE EVALUATIONS REPORTED IN EACH OF THE TWO MODULES.

	Texts
	MODULE I: S. M. ROSS, 2012, AN ELEMENTARY INTRODUCTION TO MATHEMATICAL FINANCE, CAMBRIDGE UNIVERSITY PRESS P. WILLMOT, 2007, PAUL WILLMOT INTRODUCES QUANTITATIVE FINANCE, THE WILEY FINANCE SERIES MODULO II: J. HULL, 2021, OPTIONS, FUTURES AND OTHER DERIVATIVES, PEARSON E. PITACCO, M. DENUIT, S. HABERMAN, A. OLIVIERI, 2009, MODELLING LONGEVITY DYNAMICS FOR PENSIONS AND ANNUITY BUSINESS. OXFORD UNIVERSITY PRESS. B. REMILLARD, 2013, STATISTICAL METHODS FOR FINANCIAL ENGINEERING, CRC PRESS, TAYLOR FRANCIS GROUP

Texts

MODULE I: S. M. ROSS, 2012, AN ELEMENTARY INTRODUCTION TO MATHEMATICAL FINANCE, CAMBRIDGE UNIVERSITY PRESS
P. WILLMOT, 2007, PAUL WILLMOT INTRODUCES QUANTITATIVE FINANCE, THE WILEY FINANCE SERIES
MODULO II: J. HULL, 2021, OPTIONS, FUTURES AND OTHER DERIVATIVES, PEARSON
E. PITACCO, M. DENUIT, S. HABERMAN, A. OLIVIERI, 2009, MODELLING LONGEVITY DYNAMICS FOR PENSIONS AND ANNUITY BUSINESS. OXFORD UNIVERSITY PRESS.
B. REMILLARD, 2013, STATISTICAL METHODS FOR FINANCIAL ENGINEERING, CRC PRESS, TAYLOR FRANCIS GROUP

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Massimiliano MENZIETTI | Stochastic Finance