Generalized Optimal Stopping Problems and Financial Markets

Dennis Wong

• Publisher: Routledge
• Year: 2017
• ISBN:
  9781351445818
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Provides mathematicians and applied researchers with a well-developed framework in which option pricing can be formulated, and a natural transition from the theory of optimal stopping problems to the valuation of different kinds of options. With the introduction of generalized optimal stopping theory, a unifying approach to option pricing is presented.

Chapter 1

Preliminary

In this chapter we present the definitions and concepts needed in later sections. Further details can be found in [13], [17], [22], [27] and [28].

1.1 Probability Space

A probability space is a triple $\left(\text{Ω},\mathcal{F},P\right)$

, where Ω is a set, $\mathcal{F}$

is a σ-field of subsets of Ω and P is a non-negative measure with total mass 1. The system of sets ${\mathcal{F}}^P$

is called the completion of $\mathcal{F}$

if ${\mathcal{F}}^P$

contains all those sets A ⊂ Ω for which there exist A₁, A₂ ∈ $\mathcal{F}$

such that A₁ ⊂ A ⊂ A₂ and P(A₂ – A₁) = 0. The system of sets ${\mathcal{F}}^P$

is a σ-algebra and the probability P can be uniquely extended to sets from ${\mathcal{F}}^P$

. The probability space $\left(\text{Ω},\mathcal{F},P\right)$

is said to be complete if ${\mathcal{F}}^P$

coincides with $\mathcal{F}$

.

An $\mathcal{F}$

-measurable function X : Ω → R^d is called a random variable if d = 1 or a random vector if d ≥ 2. For an arbitrary index set J and arbitrary functions X_α from Ω to R^d or $\overline{R}=R\cup \left\{\pm \infty \right\}$

, the σ-algebra σ {X_α, α ∈ J} is the smallest σ-algebra on Ω such that each X_α is measurable. This is called the σ-algebra generated by the collection {X_α, α ∈ J}. If $\mathcal{G}$

is a sub σ-algebra of $\mathcal{F}$

, the augmentation of $\mathcal{G}$

...

Table of contents

• Cover
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• 1 reliminary
• 2 Optimal Stopping Problems
• 3 Financial Markets
• 4 Options - A General View
• 5 Options - Constrained Exercise Times
• 6 Options - Constrained Portfolios in M