Course 2025-2026 a.y.

• The one-period model of financial markets: basic notation and definitions, law of one price, arbitrage, risk-neutral probabilities.
• The First Fundamental Theorem of Asset Pricing.
• Complete markets and the Second Fundamental Theorem of Asset Pricing.
• Risk-Neutral valuation of derivative securities.
• The multi-period model of financial markets in discrete time. Information structures, stochastic processes, dynamic investment strategies.
• No arbitrage and the martingale property of the discounted gain process. Dynamic completeness. Risk-neutral valuation of derivatives in the multi-period case.
• Continuous-time financial markets: information, continuous-time stochastic processes, price processes and investment strategies. Standard Brownian motions and Stochastic Differential Equations (SDEs).
• The Black-Scholes model: no-arbitrage, completeness and risk-neutral valuation of european-type derivatives, the Black-Scholes formula, the Black-Scholes Partial Differential Equation.
• The Computational Lab, where the above topics will be brought to real-life practice by implementing them by means of Excell spreasheets.

Illustrate and explain:

• The basic notation and definitions of discrete-time models of financial markets: information structures, stochastic processes, prices and dividend processes, dynamic investment strategies, the law of one price, the notions of no arbitrage and of risk-neutral probabilities.
• The First Fundamental Theorem of Asset Pricing.
• The notion of complete markets and the Second Fundamental Theorem of Asset Pricing.
• The connection between No arbitrage and the martingale property of the discounted gain processes.
• The principle of Risk-neutral valuation of derivative securities.
• The modelling of financial markets in continuous-time via Standard Brownian Motions and Stochastic Differential Equations (SDEs).
• The main features of the Black-Scholes model, the Black-Scholes formula and the Black-Scholes Partial Differential Equation.
• The way in which the above theoretical concepts are brought to real-life practice by using Excel Spreadsheets.

Apply the definitions and theoretical results to:

• assess if no arbitrage holds in both discrete and continuos models of financial markets;
• compute risk-neutral probabilities and employ them to evaluate the absence of arbitrage;
• examine the compleness of the market and connect it to no arbitrage;
• compute no arbitrage prices of various examples of derivative securities;
• compute hedging strategies for various examples of derivative securities;
• compute no arbitrage prices and hedging strategies for various derivatives using Excel.

• Lectures
• Practical Exercises

The exercises will be carried out in a series of 4 lectures, the so-called Computational Lab  constituted of numerical examples of applications of the main theoretical topics discussed in the face-to-face lectures. These numerical exercise are particularly useful to allow the students to get working knowledge of the theoretical results discussed in class. 

The final exams consists of both open and closed questions. The closed questions maybe comprehend both multiple choice and multiple answer questions. More specifically, both open and closed questions will test the students' ability to:

• Assess if no arbitrage holds in both discrete and continuos models of financial markets.
• Compute risk-neutral probabilities and employ them to evaluate the absence of arbitrage.
• Examine the compleness of the market and connect it to no arbitrage.
• Compute no arbitrage prices of various examples of derivative securities.
• Compute hedging strategies for various examples of derivative securities.
• Show the understanding of theoretical aspects discussed in the course, such as definitions, statements and proofs.

The question that tests knowledge of the theoretical aspects discussed in the course will be based on a list of topics that students are responsible to know for the exam. The list will be distributed to the students ahead of the exams. The total number of points available in the final exam is 100. These final points are then mapped into the final grade (out of 31) using the curve implemented at Bocconi Graduate School. 

A. BATTAUZ, F. ORTU, F. ROTONDI, Arbitrage Theory in Discrete and Continuous Time, Lecture Notes EGEA, Revised Edition, 2023.