# Mathematical finance (2014/2015)

Course code
4S001109
Credits
6
Coordinator
Leonard Peter Bos
MAT/06 - PROBABILITY AND STATISTICS
Language of instruction
English
Teaching is organised as follows:
Activity Credits Period Academic staff Timetable
Teoria 1 1 I sem. Luca Di Persio
Teoria 2 4 I sem. Leonard Peter Bos
Esercitazioni 1 I sem. Luca Di Persio

### Lesson timetable

I sem.
Activity Day Time Type Place Note
Teoria 1 Monday 1:30 PM - 3:30 PM lesson Lecture Hall M
Teoria 1 Thursday 1:30 PM - 3:30 PM lesson Lecture Hall M
Esercitazioni Wednesday 3:30 PM - 5:30 PM lesson Lecture Hall M

### Learning outcomes

The Mathematical Finance course for the internationalized Master's Degree (delivered completely in English) aims to introduce the main concepts of stochastic discrete and continuous time part of the modern theory of 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 and / or interest rates determined by stochastic differential equations driven by Brownian motion. Basic ingredients are the foundation of the theory of continuous-time martingale, Girsanov theorems and the Faynman-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.

### Syllabus

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.

Stochastic Calculus
• Itȏ's integral
• Itȏ-Döblin formula
• Black-Scholes-Merton Equation
• Evolution of Portfolio/Option Values
• Solution to the Black-Scholes-Merton Equation
• Sensitivity analysis

Risk-Neutral Pricing
• 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
• Dividend/continuously-Paying
• Forwards and Futures

Stochastic Differential Equations
• The Markov Property
• Interest Rate Models
• Multidimensional Feynman-Kac Theorems
• Lookback, Asian, American Option

Term structure models
• Affine-Yield Models
• Two-Factor Vasicek Model
• Two-Factor CIR Model
• Heath-Jarrow-Morton (HJM) Model
• HJM Under Risk-Neutral Measure

### Assessment methods and criteria

There will be a written final exam.