The first part of the course aims to provide a good knowledge of stochastic calculus and no arbitrage principles that constitute the foundations in the pricing of financial derivatives. We first discuss the Wiener process, then we move to the construction of stochastic integrals. We also introduce the concept of a martingale measure and its connection with the Fundamental Theorem of Asset Pricing. The second part of the course aims to introduce students to the main numerical methods for the estimation of stochastic processes and to the numerical evaluation of contingent claims. The main topics presented are: Monte Carlo simulation, parameter estimation of stochastic processes, model selection and calibration.
Expected learning outcomes
At the end of the course, students should have acquired the fundamentals of stochastic calculus and the main numerical methods for the evaluation of contingent claims. Students should be able to produce scripts in the R programming language for the estimation of a stochastic process that describes the asset price dynamics and evaluate numerically contingent claims based on no arbitrage principles.
Binomial tree model for option pricing Introduction to continuous time stochastic processes Wiener process Stochastic differential equations Ito stochastic integral and Ito formula Black and Scholes Model Simulation of stochastic differential equations Monte Carlo approach to option pricing Parameter estimation from discretely observed stochastic differential equations Historical and implied volatility Calibration Methods.
Prerequisites for admission
The students must have some preliminary knowledge of calculus, standard financial mathematics, linear algebra and optimization. Elementary Probability and Integration
classroom and laboratories
Bjork, T. (2009) Arbitrage Theory in Continuous Time. Oxford University press, 2009 Iacus, S.M. (2011) Option Pricing and Estimation of Financial Models with R, John Wiley & Sons, Ltd., Chichester, 472 page, ISBN: 978-0-470-74584-7 Iacus, S.M. (2008) Simulation and Inference for Stochastic Differential Equations: with R examples, Springer Series in Statistics, Springer NY, 300 pages, ISBN: 978-0-387-75838-1
Assessment methods and Criteria
I module: Written exam composed of practical exercises and theoretical questions + assignment