Mathematical Finance 2

A.Y. 2019/2020
Overall hours
Learning objectives
Aim of this course is to cover some of the most important topics of Mathematical Finance in continuous time involving techniques related to Stochastic Calculus and dynamical optimization.
Expected learning outcomes
Pricing and hedging using probabilistic/analytic methods, of financial derivatives in complete/incomplete markets, described by diffusion time-continuous processes.
Resolution of some problems concerning dynamic optimization, using optimal control/stopping methods.
Course syllabus and organization

Single session

Lesson period
Second semester
Course syllabus
The course presents the continuous-time mathematical models that are applied in modern finance and consists of 4 main blocks.

1) Stochastic Processes
On the Brownian Motion (BM) and continuous time martingales. The exponential martingale. The BM paths are not of finite variation. Quadratic variation of the BM. The stochastic integral with respect to the BM and the martingale property. Ito processes and Ito formula. Novikov Lemma and Girsanov Theorem. The predictable representation property. Examples of stochastic differential equations.

2) Continuous time stochastic models and the Black and Scholes model
Description of the BS model and its analysis in the light of the general principles of asset pricing. Verification of the existence of an equivalent martingale measure. The BS formula and its derivation (probabilistic method and analytical method). The BS partial derivatives equation and boundary conditions.
Contingent claims valuation in incomplete markets in continuous time. Evaluation with non-negotiable financial instruments: the partial differential equation and the market price of risk. The problem of calibration: implied volatility and "smile" effect. Limits of the BS model. Outlines of models with local volatility and stochastic volatility.

3) Pricing of exotic and american options
Lookback options and Asiatic options. American Opzions: optimal stopping problem. Pricing of American Perpetual Put Options: probabilistic approach. Pricing of American Put Options and American Call Options with finite expiration.

4) Introduction to stochastic optimization problems.
Examples of deterministic/stochastic optimization problems in Finance. Mathematical formulation of the problem. Admissible controls, the Dynamic Programming Principle, the Hamilton-Jacobi-Bellman equation, the Verification Theorem. Applications to Finance. Merton's problem over a finite horizon, Linear Quadratic Gaussian control. Systemic Risk and optimal control of a financial network finding Nash equilibria.
Prerequisites for admission
It is highly suggested the knowledge of foundations of mathematical finance, the theory of probability and stochastic processes, as presented in courses such as Mathematical Finance 1, Probability and Stochastic Processes.
Teaching methods
Taught lectures
Teaching Resources
Web site:

Reference books
1. T. Bjork: "Arbitrage Theory in Continuous Time", 3rd edition, Oxford University Press, 2009.
2. H. Pham: "Continuous-time Stochastic control and Optimization with Financial Applications", Springer 2009.
3. A. Pascucci: "Calcolo stocastico per la finanza" Springer, 2008.
4. S. Shreve: "Stochastic Calculus for Finance II", Springer, 2004.
Assessment methods and Criteria
The exam consists of an oral discussion in which students will be asked to illustrate some results of the proposed program. Moreover some problems about pricing of financial instruments or dynamic optimization will be proposed, in order to evaluate the knowledge and understanding of the topics covered, as well as the ability to apply them in real finanical models.
The vote ranges out of thirty and will be communicated immediately at the end of the oral test.
Lessons: 42 hours
on appointment
Office 1043, first floor, Math. Dept., Via Saldini 50.
On appointment
Department of Mathematics, office number 1005