Course Details for A.Y. 2016/2017

Course Objectives

Students should acquire a good knowledge of advanced probabilistic tools employed in the modeling of financial markets. In particular they should become able to - learn and understand the first mathematical models involving stochastic calculus techniques; - solve derivatives evaluation problems of medium difficulty; - extend the studied notions to more complex models; - read an advanced text in financial modeling; - implement computations for the studied models.

Course Content

• Complements of stochastic calculus. The martingale representation theorem, Girsanov theorem, existence and uniqueness of the solutions of stochastic differential equations, quadratic variation.
• Discrete market models. First and second fundamental theorem of asset pricing. Evaluation of European and American derivatives.
• Black and Scholes world. European options evaluation, Barrier options, American options, (perpetual puts and critical price). Option evaluation for a general diffusion model: the infinitesimal generator of a diffusion, computation of expectations and partial differential equations The multidimensional model: viability and completeness. Asian options and exchange options.
• Bonds and interst rate models. Zero coupon bonds. Merton model, Vasicek model, Cox Ingersoll Ross model.
• If there is enough time. Weak solutions of stochastic differential equations. weak solutions via Girsanov. Yamada and Watanabe's results. The unidimensional case.
• If there is enough time. Stochastic volatility models. Stein and Stein, Hull and White, Heston models

Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

Prerequisites and Learning Activities

An advanced course in probability and the first part of the integrated course

Assessment Methods and Criteria

oral exam with possible integrations either written or oral

Textbooks

• I. Karatzas, S. Shreve, Brownian motion and stochastic calculus , Springer.    
• A. Pascucci, Calcolo Stocastico per la Finanza , Springer.    
• D. Lamberton, D. Lapeyre, Introduction to stochastic calculus applied to Finance , Chapman and Ha.    
• J. Zhu, Modular pricing of options (Lecture notes in Economics and Mathematical Systems) , Springer. (vol. 493)    
• P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations , Springer.    
• P. Billingsley, Probability and measure , Wiley . 1984.    

Course page updates

• A.Y. 2014/2015
• A.Y. 2015/2016
• A.Y. 2016/2017
• A.Y. 2017/2018
• A.Y. 2018/2019
• A.Y. 2019/2020