Stochastic calculus

Objectives and Learning Outcomes

The aim of this course is the provide solid working knowledge of stochastic differential equations and stochastic calculus since stochastic differential equations are used to model the behavior of financial assets and stochastic calculus is the fundamental tool for understanding and manipulating these models. At the end of the course, participants should be able to solve different types of linear stochastic differential equations, understand and apply Ito’s Lemma for scalar and vector stochastic processes, understand the connection between stochastic differential equations and partial differential equation of the heat equation type and perform many of needed calculations using both pen and paper as well as using symbolic and numerical computations.

Lectures

Introduction to Brownian motion. Continuous martingales and their properties. Levy construction of the Brownian process. The reflection principle and scalling. Distribution of first hitting times, maximum and minimum for a Brownian motion. Ito’s calculus. Non-differentiability of stock price processes. Quadratic variation. Stochastic Ito integrals. Ito’s formula with scalar and vector processes. Integration by parts and the stochastic Fubini theorem. Defining general Ito processes. Change of measure and Girsanov’s theorem. Exponential martingales. Martingale representation theorem. Stochastic differential equations: weak and strong solutions, existence and uniqueness of solutions. The Feyman – Kac representation. The Black-Scholes model.

Exercises

Performing pen-and-paper based calculations to solidify the notions from lectures. Solving various linear stochastic differential equations in closed form (with additive and multiplicative noise). Determining whether a particular process is a martingale with respect to a particular filtration. Performing changes of measures and applying Girsanov theorem for various continuous time stochastic processes. Applying Feyman-Kac representation to derivate partial differential equations for pricing various types of derivative securities. Showing equivalence between the Black-Scholes partial differential equations and the canonical heat equation. Computer-based exercises implemented in Python: Comparison of Riemann, Riemann-Stiltjes and Ito Integrals on various examples. Using SymPy to perform Ito’s Lemma and conditional expectation calculations. Simulation of general Ito processes.