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Prof. Sergio Bianchi

Contact information:

Term: First Semester

Credits (ECTS): 6

Prerequisites: None

Language of Instruction: English

Class hours:


- Be aware of the main derivative contracts and of their use in financial practice
- Develop an understanding of the main characteristics of stock and derivative markets
- Understand the basics of the mathematical theory used to model the price evolution in financial markets
- Develop an understanding of the basic concepts of the asset pricing theory
- Be able to assess the limits of the standard models



This course aims at giving the students the basic mathematical concepts in derivative asset pricing. It provides a description of the principal assets traded in financial markets and then proceeds with an in-depth discussion of Arbitrage Pricing Theory. The theory is applied to the pricing options and derivative securities in the contexts of binomial probability trees and the Black-Scholes lognormal model. The dynamical hedging of option portfolios and of tailor-made financial derivatives is discussed.



The class will meet for 2 hours (gross of interclass break), twice a week. After an introduction aimed at providing the needed background, students are required to read the materials related to the class and to be prepared prior to coming to class. Classes will consist of a lecture by the instructor, who will address - at the end of each class - all the doubts brought up by the students.



Students are expected to spend at least 2,5 hours of time on academic studies outside of, and in addition to, each hour of class time.

Detailed topics:

Financial derivatives: Forwards, Futures, Options, Swaps, notion of arbitrage.

Probability Essentials: Probability spaces, filtrations as information content, random variables, conditional expectations, Definition and classification of random processes, martingales.

Discrete-time Finance. Pricing by arbitrage, risk-neutral probability measures, valuation of contingent claims, fundamental theorem of asset pricing, Cox-Ross-Rubinstein (CRR) model, pricing and hedging of European and American derivatives in CRR model, general results related to prices of derivatives.

Stochastic Calculus: Brownian motion, martingales, Itô’s formula, Itô integral, risk-neutral measure, SDE; Risk-neutral measure, Girsanov's theorem for change of measure, martingale representation theorems, representation of Brownian martingales.

Continuous-time Finance: Black-Scholes-Merton model of stock prices as geometric Brownian motion, derivation of the Black-Scholes-Merton partial differential equation, the Black-Scholes formula and simple extensions of the model, self-financing strategies and model completeness, risk neutral measures, the fundamental theorems of asset pricing, continuous time optimal stopping and pricing of American options, forwards and futures in Black-Scholes-Merton model.


The instructor will use differentiated forms of assessment to calculate the final grade the students receive for this course.

For the record, these are listed and weighted below. The content, criteria and specific requirements for each assessment category will be explained in greater detail in class.

Class Participation (15%) - Mid-Term Exam (35%) - Final Exam (50%)

Class participation:

This grade will be calculated to reflect students' participation in class discussions, with special emphasis on the capacity to introduce ideas and thoughts dealing with the texts, the ability use language effectively, and to present analysis in intellectual and/or constructive argumentation.

Mid-term Exam:

The midterm is designed to establish and communicate the progress the student is making towards meeting the course learning objectives. The abilities will be tested within two areas of competency: the amount of information the student masters and the accuracy of the information she presents.

Structure: A combination of 15 true/false questions (30%), open question (5%). Prior to the examinations, a comprehensive review will be given during class.

Final Exam:

The final exam is designed to establish the overall progress achieved by the student in meeting the course learning objectives. The abilities will be tested with regards to the amount of information the student masters and to the accuracy of the information she presents.

Structure:  A combination of 15 true/false questions (30%) and two open questions (20%) will be asked. Prior to the examinations, a comprehensive review will be given during class.


Neftci, Salih, 1999, An Introduction to the Mathematics of Financial Derivatives, Academic Press.

Slides distributed during the course.

Recommended readings:

Shreve , Steven E., 2003, Stochastic Calculus for Finance I & II, Springer.

Hull, John C., Options, Futures, and Other Derivatives, 8th edition, Pearson Press.

[Ultima modifica: martedì 12 settembre 2017]