Mathematical finance: continuous time

129 Pages·2003·0.551 MB·English

by  Ewald C.-O.

Mathematical Finance
Introduction to continuous time
Financial Market models
Dr. Christian-Oliver Ewald
School of Economics and Finance
University of St.Andrews
Electronic copy of this paper is available at: http://ssrn.com/abstract=976593
Abstract
These are my Lecture Notes for a course in Continuous Time Finance
which I taught in the Summer term 2003 at the University of Kaiser-
slautern. I am aware that the notes are not yet free of error and the
manuscrip needs further improvement. I am happy about any com-
ment on the notes. Please send your comments via e-mail to ce16@st-
andrews.ac.uk.
Electronic copy of this paper is available at: http://ssrn.com/abstract=976593
Working Version
March 27, 2007
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Contents
1 Stochastic Processes in Continuous Time 5
1.1 Filtrations and Stochastic Processes . . . . . . . . . . . . 5
1.2 Special Classes of Stochastic Processes . . . . . . . . . . . 10
1.3 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Black and Scholes’ Financial Market Model . . . . . . . . 17
2 Financial Market Theory 20
2.1 Financial Markets . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Martingale Measures . . . . . . . . . . . . . . . . . . . . . 25
2.4 Options and Contingent Claims . . . . . . . . . . . . . . . 34
2.5 Hedging and Completeness . . . . . . . . . . . . . . . . . 36
2.6 Pricing of Contingent Claims . . . . . . . . . . . . . . . . 38
2.7 The Black-Scholes Formula . . . . . . . . . . . . . . . . . 42
2.8 Why is the Black-Scholes model not good enough ? . . . . 46
3 Stochastic Integration 48
3.1 Semi-martingales . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 The stochastic Integral . . . . . . . . . . . . . . . . . . . . 55
3.3 Quadratic Variation of a Semi-martingale . . . . . . . . . 65
3.4 The Ito Formula . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5 The Girsanov Theorem . . . . . . . . . . . . . . . . . . . . 76
3.6 The Stochastic Integral for predictable Processes . . . . . 81
3.7 The Martingale Representation Theorem . . . . . . . . . 84
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4 Explicit Financial Market Models 85
4.1 The generalized Black Scholes Model . . . . . . . . . . . . 85
4.2 A simple stochastic Volatility Model . . . . . . . . . . . . 93
4.3 Stochastic Volatility Model . . . . . . . . . . . . . . . . . . 95
4.4 The Poisson Market Model . . . . . . . . . . . . . . . . . . 100
5 Portfolio Optimization 105
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2 The Martingale Method . . . . . . . . . . . . . . . . . . . 108
5.3 The stochastic Control Approach . . . . . . . . . . . . . . 119
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Introduction
MathematicalFinanceisthemathematicaltheoryoffinancialmarkets.
It tries to develop theoretical models, that can be used by “practition-
ers” to evaluate certain data from “real” financial markets. A model
cannot be “right” or wrong, it can only be good or bad ( for practical use
). Even “bad” models can be “good” for theoretical insight.
Content of the lecture :
Introduction to continuous time financial market models.
We will give precise mathematical definitions, what we do understand
underafinancialmarket,untilthisletusthinkofafinancialmarketas
some place where people can buy or sell financial derivatives.
During the lecture we will give various examples for financial deriva-
tives. The following definition has been taken from [Hull] :
A financial derivative is a financial contract, whose value at expire is
determined by the prices of the underlying financial assets ( here we
mean Stocks and Bonds ).
We will treat options, futures, forwards, bonds etc. It is not necessary
to have financial background.
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During the course we will work with methods from
Probability theory, Stochastic Analysis and Partial Differ-
ential Equations.
TheStochasticAnalysisandPartialDifferentialEquationsmethods
arepartofthecourse,theProbabilityTheorymethodsshouldbeknown
from courses like Probability Theory and Prama Stochastik.
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Chapter 1
Stochastic Processes in
Continuous Time
Given the present, the price S of a certain stock at some future time t
t
is not known. We cannot look into the future. Hence we consider this
price as a random variable. In fact we have a whole family of random
variables S , for every future time t. Let’s assume, that the random
t
variables S are defined on a complete probability space (Ω, ,P), now
t
F
it is time 0 , 0 t < and the σ-algebra contains all possible infor-
≤ ∞ F
mation. Choosingsubσ-algebras containingalltheinformation
t
F ⊂ F
up to time t, it is natural to assume that S is measurable, that is
t t
F
the stock price S at time t only depends on the past, not on the future.
t
We say that (S ) is adapted and that S is a stochastic pro-
t t [0, ) t t
∈ ∞ F
cess. Throughout this chapter we assume that (Ω, ,P) is a complete
F
probability space. If X is a topological space, then we think of X as a
measurable space with its associated Borel σ-algebra which we denote
as (X).
B
1.1 Filtrations and Stochastic Processes
Let us denote with I any subset of R.
Definition 1.1.1. Afamily( ) ofsubσ-algebrasof suchthat
t t I s
F ∈ F F ⊂
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whenever s < t is called a filtration of .
t
F F
Definition 1.1.2. A family (X , ) consisting of -measurable Rn-
t t t I t
F ∈ F
valuedrandomvariablesX on(Ω, ,P)andaFiltration( ) iscalled
t t t I
F F ∈
an n-dimensional stochastic process.
ThecasewhereI = Ncorrespondstostochasticprocessesindiscrete
time(seeProbabilityTheory,chapter19). Sincethissectionisdevoted
to stochastic processes in continuous time, from now on we think of I
as a connected subinterval of R .
0
≥
Often we just speak of the stochastic process X , if the reference to the
t
filtration ( ) is clear. Also we say that (X ) is ( ) adapted. If
t t I t t I t t I
F ∈ ∈ F ∈
no filtration is given, we mean the stochastic process (X , ) where
t t t I
F ∈
= X = σ(X s I,0 s t)
Ft Ft s| ∈ ≤ ≤
is the σ-algebra generated by the random variables X up to time t.
s
Given a stochastic process (X , ) , we can consider it as function
t t t I
F ∈
of two variables
X : Ω I Rn , (ω,t) X (ω).
t
× → 7→
On Ω I we have the product σ-algebra (I) and for
× F ⊗B
I := s I s t (1.1)
t
{ ∈ | ≤ }
we have the product σ-algebras (I ).
t t
F ⊗B
Definition 1.1.3. The stochastic process (X , ) is called measur-
t t t I
F ∈
ableiftheassociatedmapX : Ω I Rn from(1.1)is( (I))/ (Rn)
× → F⊗B B
measurable. It is called progressively measurable, if for all t I the
∈
restriction of X to Ω I is ( (I )/ (Rn) measurable.
t t t
× F ⊗B B
In this course we will only consider measurable processes. So from
now on, if we speak of a stochastic process, we mean a measurable
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stochastic process.
Working with stochastic processes the following space is of fundamen-
tal importance :
(Rn)I := Map(I,Rn) = ω : I Rn (1.2)
{ → }
i.e. the maps from I to Rn. For any t I we have the so called evalua-
∈
tion map
ev : (Rn)I Rn
t
→
ω ω(t)
7→
Definition 1.1.4. The σ-algebra on (Rn)I
σ := σ(ev s I)
cyl s
| ∈
generated by the evaluation maps is called the σ-algebra of Borel
cylinder sets.
:= σ := σ(ev s I,s t)
t cyl,t s
F | ∈ ≤
defines a filtration of σ . Whenever we consider (Rn)I as a measur-
cyl
able space, we consider it together with this σ-algebra and this filtra-
tion.
The space (Rn)I has some important subspaces :
C(I,Rn) := ω : I Rn ω is continuous (1.3)
{ → | }
C (I,Rn) := ω : I Rn ω is right-continuous (1.4)
+
{ → | }
C (I,Rn) := ω : I Rn ω is left-continuous (1.5)
− { → | }
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