Mathematical and Quantitative Finance Chapter 1

Mathematical and Quantitative Finance: Chapter 1 Theory

• Chapter 1: Covers Wiener Process, Itô's Lemma, Stochastic Differential Equations, Geometric Brownian Motion, a-stable Lévy Motion, Poisson Process, and computer simulations.

Contents

• Chapter 1: Wiener Process and Itô's Lemma: Defines the Wiener Process, its behavior for various drift and diffusion values, Itô Process, Itô's Lemma, Stochastic Differential Equations, Geometric Brownian Motion, a-stable Lévy Motion, Poisson Process, and visualization methods.
• References: Chapter 14 of Options, Futures, and Other Derivatives, 9th edition by J. Hull.

Objectives

• Define Wiener Process: Students should be able to define Wiener process and explain the influence of drift and diffusion coefficients on the behavior of its trajectories.
• Manipulate stochastic integrals: Students should be proficient in manipulating stochastic integrals.
• Apply Itô's Lemma: Ability to apply Itô's Lemma.
• Stochastic differential equations: Knowledge of stochastic differential equations for Geometric Brownian Motion.
• Simulations: Proficiency in simulating solutions of stochastic differential equations using Wiener process/Standard Brownian Motion, a-stable Lévy Motion, and Compensated Poisson Process.

Continuous-time stochastic process

• Stochastic Process in Continuous Time: A collection of random variables indexed by time t.

Wiener Process/Standard Brownian Motion

• Definition: A stochastic process that satisfies specific conditions, including change in the Wiener process over an interval following normal distributions with mean 0 and standard deviation √(t-s).
• Independent Increments: Sequential increments are independent random variables.
• Continuous Function: The process {Z} is generally a continuous function of time.
• Zero Initial Value: Z0 = 0 with probability 1.

Wiener Process/Standard Brownian Motion (continued)

• Change in Z over an Interval: The change in Z between time 0 and T (ZT - Zo) has a specific mean and variance properties.
• Approximation as Sum of Small Changes: ZT - Zo can be represented as a sum of small changes in {Z} across many small intervals within the interval [0, T].
• Short-hand Notation: A short-hand notation for the change of Zt – Zs is used.

Generalized Wiener Process

• Stochastic Differential Equation (SDE): The process satisfies a stochastic differential equation, where a (and b) are constants and {Z} is the Wiener process.
• Term dZt and dXt: dZ_t represents a change in the Wiener process, and dX_t represents a change in the generalized Wiener process over a time dt interval.
• Distribution of dXt: dX_t is normally distributed in small interval with mean a dt and variance b² dt.
• Integral representation of the SDE: Showing how the process can be described using integration.

Generalized Wiener Process (continued)

• Distribution of XT-Xo: The change in the generalized Wiener process over the interval [0, T] (XT - Xo) has specific statistical properties, including mean and standard deviation.
• Constants: 'a' is the expected drift rate, 'b' is the diffusion rate or variance rate.
• Short-hand Notation: A brief, symbolic representation for Xt - Xs is used to convey the behavior of the process.

ITô Process

• Stochastic Differential Equation: Describes a stochastic process that satisfies a particular SDE.
• Functions of Two Variables: The terms a(,) and b(,) are functions that depend on both current state and time.
• Integral Representation of the Ito Process: Showing how the process is expressed via integration.

Ito's Lemma

• Process behavior for G(Xt,t) and its SDE: Describing the SDE for G_t = G(X_t ,t), a function of Xt and t.
• SDE for Gt and its related constants: The SDE for G_t and its associated expected drift rate and variance rate associated with the stochastic process described by G_t.

Ito's Lemma (continued)

• Multiplication rules for dt and dz: Definition of terms in Ito's lemma relating to the multiplication among time-differential and stochastic-differential (dt, dz)

Generalized Wiener Process (continued)

• Numerical Simulation pseudo-code: A conceptual step-by-step procedure outlining how to numerically simulate the generalized Wiener Process.

a-Stable Lévy Motion

• Self-similar Property: The process exhibits self-similarity with a specific parameter (H).
• Increments: The increments (La,ß;t – La,ß;s) follows a stable distribution.

Geometric Brownian Motion

• Stock price modeling: A popular model for stock price behavior, defined by its stochastic differential equation.
• Expected drift (and variance) rate: Describes the average changes and variability of the stock price.

Compensated Poisson Process

• General definition: A type of process without a drift term, frequently used in stochastic modeling (in particular, Poisson processes).

R Code for Simulations

• R packages and functions: Identifies specific R packages and functions useful for mathematical computations and visualizations.