23695 - Advanced Mathematics for Financial Markets

Course Unit Page


This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

Quality education

Academic Year 2019/2020

Learning outcomes

At the end of the course the student masters the main concepts of financial mathematics. The course will cover the stochastic dynamics of asset prices assumed under the efficient market theory, the concept of arbitrage-free pricing and replicating strategies, leading to the PDE approach to pricing.

Course contents

  1. Stochastic calculus principles: stochastic process, discrete and continuous martingale, diffusion and Ito's process, Markov 's process, exponential martingale and probability changing, Girsanov's theorem, stochastic integration and Ito's lemma, SDE and PDE, Kolmogorov's PDE, Feynman-Kac's theorem;
  2. Plain vanilla contingent claims's pricing and hedging: forward and future, european and american options, pricing and hedging by arbitrage, self-financing portfolio, CRR's model, BS's model, volatility analysis and smile effect, arbitrage model for Ito's market, market premium and market numeraire, BS formula for exchange options, complete and incomplete markets;
  3. Domestic-Foreign arbitrage and exotic options: Black's model, quantos and compos, digital options, regular and reverse barrier options, loockback options and options on running minimum (maximum) of underlying asset.


  • Romagnoli S., Mathematical Finance-Theory, 2019, Esculapio.
  • Romagnoli S., Mathematical Finance-Practice, 2019, Esculapio.
  • Financial calculus-An introduction to derivative pricing, Baxter-Rennie, Cambridge university press, 1997.
  • Elementary stochastic calculus with finance in view, Mikosch, World scientific, Singapore 1999.
  • Introduction to stochastic calculus applied to finance, Lamberton-Lapeyre, Chapman and Hall, London 1996.
  • Teaching methods

    Theoretical lessons will be support by applied examples of discussed models to incite students to find themselves the explicit solutions of the theoretical problems applying the correct mathematical instruments.

    Assessment methods

    The learning test consists in a written exam to solve in 2 hours. This exam is composed by 3 execises which are structured into 2 questions. During the exam it is permitted to use the calculator but it is not allowed to consult books or notes. It is attributed on average 10 points to each exercise. The students pass the exam with a score not lower than 18 points.

    The students can ask also for an oral exam about all the programme of the course. The final grade will be the average of the oral and the witten exam's grade.

    Teaching tools

    Teaching tools will be blackboard and slides.

    Office hours

    See the website of Silvia Romagnoli