
Course unit
MATHEMATICS FOR FINANCIAL RISK AND DERIVATIVES
EPP6077357, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2019/20
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Core courses 
SECSS/06 
Mathematics for Economics, Actuarial Studies and Finance 
9.0 
Course unit organization
Period 
First semester 
Year 
1st Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Lecture 
9.0 
63 
162.0 
No turn 
Prerequisites:

A solid knowledge of basic mathematics, probability and statistics (in particular: linear algebra, differential calculus, discrete and continuous random variables, mean, variance, correlation, normal distribution) 
Target skills and knowledge:

This course provides a rigorous introduction to asset pricing and quantitative finance. The first part of the course covers fundamental topics such as the time value of money and classical portfolio theory. The second part of the course consists of an introduction to financial derivatives, hedging, pricing, and the modeling of asset prices, first in a discretetime setting and then in a continuoustime setting. The relevant tools of stochastic calculus will be presented together with their financial applications. 
Examination methods:

Written exam 
Assessment criteria:

100% final written exam 
Course unit contents:

A Brief Introduction to Financial Markets
– Exchangetraded markets and OTC markets
– Hedgers, speculators and arbitrageurs
The Monetary Value of Time
– Types of interest rates
– Bond pricing, yields and duration
– Net Present Value (NPV) and Internal Rate of Return (IRR)
Financial Derivatives
– Forwards and futures
– Options: European and American, vanilla and exotic
– The noarbitrage principle and the law of one price
– Static strategies with options
– PutCall parity
– General noarbitrage bounds
The Binomial Model
– Absence of arbitrage
– Dynamic trading and hedging
– Pricing by hedging
– Riskneutral probability measures
– Pricing by riskneutral valuation
– American options: optimal exercise, hedging and pricing
– Law of large numbers and limit behavior
Brownian Motion and Stochastic Calculus
– An introduction to stochastic processes
– Definition and basic properties of the Brownian motion
– Martingales and Markov processes
– The stochastic integral
– Itô's formula
– Stochastic differential equations
The BlackScholesMerton (BSM) Model
– Geometric Brownian motion
– Dynamic trading and hedging in continuous time
– Absence of arbitrage
– Riskneutral probability measures
– Changes of measure (the Girsanov theorem)
– The FeynmanKač formula
– The BlackScholes pricing formula
– PDE approach to the BSM model
– The Greeks
– Implied volatility 
Planned learning activities and teaching methods:

Lectures. 
Additional notes about suggested reading:

Slides of the lectures and reference textbooks. 
Textbooks (and optional supplementary readings) 

John C. Hull, Options, Futures, and Other Derivatives (10th Edition). : Pearson, 2017.

Tomas Björk, Arbitrage Theory in Continuous Time 3°ed. Oxford: Oxford University Press, 2009.

Barucci Emilio, Fontana Claudio, Financial markets theoryequilibrium, efficiency and information 2°ed. London: Springer, 2017.

Pascucci, Andrea, Calcolo stocastico per la finanza. Milano: Springer, 2009.

Innovative teaching methods: Software or applications used
 Moodle (files, quizzes, workshops, ...)

