Mathematical Finance

Core Theory, Problems and Statistical Algorithms

Nikolai Dokuchaev

• Publisher: Routledge
• Year: 2007
• Print ISBN: 9780367475017
• eBook ISBN: 9781134121977
• Topic: Business
• Subtopic: Business General
• Index: Business

Written in a rigorous yet logical and easy to use style, spanning a range of disciplines, including business, mathematics, finance and economics, this comprehensive textbook offers a systematic, self-sufficient yet concise presentation of the main topics and related parts of stochastic analysis and statistical finance that are covered in the majori

1
Review of probability theory

In probability theory based on Kolmogorov’s probability axioms, the model of randomness is the following. It is assumed that there exists a set Ω, and it is assumed that subsets A ⊆ Ω are random events. Some value P(A) ∈ [0, 1] is attached to any event as the probability of an event, and P(Ω)=1. To make this model valid, some axioms about possible classes of events are accepted such that the expectation can be interpreted as an integral.

1.1 Measure space and probability space

σ-algebra of events

Let Ω be a non-empty set. We denote by 2^Ω the set of all subsets of Ω.

Example 1.1 Let Ω={a, b}, then 2^Ω={∅, {a}, {b}, Ω}.

Definition 1.2 A system of subsets Ƒ ᑕ 2^Ω is called an algebra of subsets of Ω if

(i)Ω ∈ Ƒ ;

(ii) If A ∈ Ω then Ω\A ∈ Ƒ :

(iii) If A_1, A₂,..., A_n ∈ Ƒ, then

∈ Ƒ .

Note that (i) and (ii) imply that the empty set ∅ always belongs to an algebra.

Definition 1.3 A system of subsets Ƒ ᑕ 2^Ω is called a σ-algebra of subsets of Ω if

(i) It is an algebra of subsets;

(ii) If A_1, A₂,... ∈ Ƒ (i.e

ᑕ Ƒ), then

A_i ∈ Ƒ .

Definition 1.4 Let Ω be a set, let Ƒ be a σ-algebra of subsets, and let

: Ƒ → [0, + ∞ ]be a mapping.

(i) We said that µ is a σ-additive measure if µ

=

for any A_1, A₂,... ∈ Ƒ such that A_i ∩ A_j = Ø if i ≠ j . In that case, the triplet ( Ω, Ƒ,

) is said to be a measure space.

(ii) If µ(Ω)<+∞, then the measure µ is said to be finite.

(iii) If µ(Ω)=1, then the measure µ is said to be a probability measure.

To make notations more visible, we shall use the symbol P for the probability measures.

Definition 1.5 Consider a measure space (Ω, Ƒ, µ) . Assume that some property holds for all (ω ∈ Ω₁, where Ω₁ ∈ Ƒ is such that μ(Ω\Ω₁)=0. We say that this property holds a.e. (almost everywhere). In the case of a probability measure, we say that this property holds with probability 1, or a.s. (almost surely).

In probability theory based on Kolmogorov’s probability axioms, the following definition is accepted.

Definition 1.6 A measure space (Ω, Ƒ, P) is said to be a probability space if P is a probability measure, i.e., P(Ω)=1. Elements ω ∈ Ωare said to be elementary events, and sets A ∈ Ƒ are said to be events (or random events). Correspondingly, Ƒ is the σ - algebra of events.

Under these axioms, A∩B means the event ‘A and B’ (or A·B), and A∪B means the event ‘A or B’ (or A+B), where A and B are events.

A random event A={ω} is a set of elementary events.

Example 1.7 For Ω=[0, 1], a probability measure may be defined such that P((a, b])=P([a, b))=P((a, b))=P([a,b])=b−a
for all intervals, where 0≤ a<b ≤1. Clearly, the set of all intervals does not form an algebra and, therefore, it does not form a σ-algebra. The question arises for which σ-algebra this measure can be defined. A natural solution is to take the minimal σ-algebra that contains all open intervals. It is the so-called Borel σ-...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Preface
• Acknowledgments
• 1: Review of probability theory
• 2: Basics of stochastic processes
• 3: Discrete time market models
• 4: Basics of Ito calculus and stochastic analysis
• 5: Continuous time market models
• 6: American options and binomial trees
• 7: Implied and historical volatility
• 8: Review of statistical estimation
• 9: Estimation of models for stock prices
• Legend of notations and abbreviations
• Selected answers and key figures
• Bibliography