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.
The course will focus on continuous-time mathematical models for financial markets and it consists of three main parts.
1) Continuous-time modeling The Black and Scholes model and the derivation of the valuation formula, local volatility models and the derivation of the Dupire's formula, stochastic volatility models.
2) Optimization in continuous time models The Merton Problem and some of its variations, utility maximization in complete markets, martingale methods for investment-consumptions problems.
3) Model ambiguity Introduction to the problem of model risk, Skorohod Embedding Problem (SEP) and some of its solutions, introduction to Martingale Optimal Transport in continuous time and connections to SEP, applications to model-independent super-replication and option prices bounds.
Prerequisites for admission
It is highly recommended some knowledge of the foundations of mathematical finance, the theory of probability and stochastic processes.
1. I. Karatzas, S. Shreve: "Methods of Mathematical Finance", Springer. 2. S. Shreve: "Stochastic Calculus for Finance II", Springer. 3. L.C.G. Rogers: "Optimal Investment", Springer. 4. D. Hobson: "The Skorokhod Embedding Problem and Model-Independent Bounds for Option Prices" in Paris-Princeton Lecture Notes. 5. A. Pascucci: "Calcolo stocastico per la finanza" Springer.
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.