The Brownian Motion: an Introduction

The Brownian motion is one of the most famous and important stochastic process.

Let’s start with a bit of history.

Robert Brown discovered the Brownian motion unintentionally while observing movements of grains of pollen through a microscope in 1827. He noticed that pollen seeds suspended in water moved in an irregular zigzag and random manner.

Louis Bachelier was the first person to model it. He used it to model the dynamic of stock prices and value stock options in his thesis in 1900. Five years later, Albert Einstein modeled the trajectory of atoms subject to shocks with random motion and obtained a Gaussian density.

A stochastic process W(t) is a Brownian motion, also called Wiener process if:

1) W(0) is equal to zero

2) It has a continuous path

3) It has independent and stationary increments

4) the increments follow a Gaussian distribution. W(t+u) – W(t) follows a gaussian distribution with zero mean and variance u.

The concept of martingale is essential in financial mathematics for the determination of the prices of financial products. A stochastic process is a martingale if the conditional expectation of S(t+s) given all the information known at t is equal to S(t). Below are several examples of martingale involving Wiener processes:

– W(t) is a martingale

– W(t)^2 – t is also a martingale

– exp(lambda x W(t) – 1/2 x lambda^2 x t) is a martingale

An important result is that the quadratic variation of a Brownian motion is equal to t with a probability 1. It is the cornerstone of Ito and stochastic calculus. We will not go through the demonstration here, we illustrate it with simulations.

Presentation

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List of Quant Next Courses

We summarize below quantitative finance training courses proposed by Quant Next. Courses are 100% digital, they are composed of many videos, quizzes, applications and tutorials in Python.

Complete training program:

Options, Pricing, and Risk Management Part I: introduction to derivatives, arbitrage free pricing, Black-Scholes model, option Greeks and risk management.

Options, Pricing, and Risk Management Part II: numerical methods for option pricing (Monte Carlo simulations, finite difference methods), replication and risk management of exotic options.

Options, Pricing, and Risk Management Part III:  modelling of the volatility surface,  parametric models with a focus on the SVI model, and stochastic volatility models with a focus on the Heston and the SABR models.

A la carte:

Monte Carlo Simulations for Option Pricing: introduction to Monte Carlo simulations, applications to price options, methods to accelerate computation speed (quasi-Monte Carlo, variance reduction, code optimisation).

Finite Difference Methods for Option Pricing: numerical solving of the Black-Scholes equation, focus on the three main methods: explicit, implicit and Crank-Nicolson.

Replication and Risk Management of Exotic Options: dynamic and static replication methods of exotic options with several concrete examples.

Volatility Surface Parameterization: the SVI Model: introduction on the modelling of the volatility surface implied by option prices, focus on the parametric methods, and particularly on the Stochastic Volatility Inspired (SVI) model and some of its extensions.

The SABR Model: deep dive on on the SABR (Stochastic Alpha Beta Rho) model, one popular stochastic volatility model developed to model the dynamic of the forward price and to price options.

The Heston Model for Option Pricing: deep dive on the Heston model, one of the most popular stochastic volatility model for the pricing of options.