Stochastic Calculus

Objectives

The aim of this course is to provide an introduction to the theory of stochastic processes and Itô’s calculus, and to show how these concepts and tools enter into the problems of risk assessment and risk management with which financial practitioners are constantly confronted. The aim is to :
– explain the reasons for using stochastic processes as tools for modeling economic and financial quantities,
– motivate the specific mathematical investment required to handle stochastic processes,
– provide the main mathematical results of the theories discussed, without going into technical details,
– present concrete examples of the use of stochastic processes in risk assessment and management.

Course outline

1. Introduction: Reminder of a few finance concepts (markets, arbitrage, options, etc.).
2. Continuous-time processes and Brownian motion.
3. Stochastic integration and Itô’s formula.
4. Black-Scholes model and option pricing.
5. Stochastic integral with respect to a continuous martingale.
6. Stochastic differential equations. Girsanov’s theorem.
7. Valuation of derivatives.
8. Introduction of jump process models

Prerequisites

A good command of the probabilistic tools studied at Ensai, in particular the 2nd year course on martingales and Lévy processes.