Portfolio Optimization

Portfolio Optimization

Gerhard-Wilhelm Weber1 Erik Kropat2 Zafer-Korcan Görgülü3

1 Institute
of Applied Mathematics
Middle East Technical University
Ankara, Turkey
2 Department of Mathematics

University of Erlangen-Nuremberg
Erlangen, Germany
3 University of the Federal Armed Forces
Munich, Germany

2008

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Outline I

1 The Mean-Variance Approach in a One-Period Model
Introduction

2 The Continuous-Time Market Model
Modeling the Security Prices
Excursion 1: Brownian Motion and Martingales
Continuation: Modeling the Security prices
ˆ
Excursion 2: The Ito Integral
ˆ
Excursion 3: The Ito Formula
Trading Strategy and Wealth Process
Properties of the Continuous-Time Market Model
Excursion 4: The Martingale Representation Theorem

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Outline II

3 Option Pricing
Introduction
Examples
The Replication Principle
Arbitrage Opportunity
Continuation
Partial Differential Approach (PDA)
Arbitrage & Option Pricing

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Outline III

4 Pricing of Exotic Options and Numerical Algorithms
Introduction
Examples
Examples
Equivalent Martingale Measure
Exotic Options with Explicit Pricing Formulae
Weak Convergence of Stochastic Processes
Monte-Carlo Simulation
Approximation via Binomial Trees
The Pathwise Binomial Approach of Rogers and Stapleton

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Outline IV

5 Optimal Portfolios
Introduction and Formulation of the Problem
The martingale method
Optimal Option Portfolios
Excursion 8: Stochastic Control
Maximize expected value in presence of quadratic control costs
Introduction
Portfolio Optimization via Stochastic Control Method

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Outline


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Outline

Introduction

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Introduction

MVA Based on H. M ARKOWITZ
OPM • Decisions on investment strategies only at the beginning of the
period
• Consequences of these decisions will be observed at the end of the
period (−→ no action in between: static model)

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The one-period model

Market with d traded securities
d different securities with positive prices p1 , . . . , pd at time t = 0
Security prices P1 (T ), . . . , Pd (T ) at ﬁnal time t = T not
foreseeable
−→ modeled as non-negative random variables on probability space
(Ω, F , P)

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Securities in a OPM

Returns of Securities
Pi (T )
Ri (T ) := pi (1 ≤ i ≤ d )

Estimated Means, Variances and Covariances
E (Ri (T )) = µi , Cov Ri (T ), Rj (T ) = σij (1 ≤ i ≤ d )

Remark
The matrix
σ := σij i,j∈{1,...,d}

is positive semi-deﬁnite as it is a variance-covariance matrix.

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Securities in a OPM

Each security perfectly divisable
Hold ϕi ∈ R shares of security i (1 ≤ i ≤ d )
Negative position (ϕi < 0 for some i) corresponds to a selling
−→ Not allowed in OPM
−→ No negative positions: pi ≥ 0 (1 ≤ i ≤ d)
−→ No transaction costs

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Budget equation and portfolio return

The Budget Equation
Investor with initial wealth x > 0 holds ϕi ≥ 0 shares of security
i with
ϕi · pi = x
1≤i≤d

The Portfolio Vector π := (π1 , . . . , πd )T
ϕi · pi
πi := (1 ≤ i ≤ d )
x

Portfolio Return
R π := πi · Ri (T ) = π T R
1≤i≤d

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Remark
πi . . . fraction of total wealth invested in security i

ϕi · pi
1≤i≤d x
πi = = =1
x x
1≤i≤d

X π (T ) . . . ﬁnal wealth corresponding to x and π

X π (T ) = ϕi · Pi (T )
1≤i≤d

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Remark (continued)
Portfolio Return
ϕi · pi Pi (T ) X π (T )
Rπ = πi · Ri (T ) = · =
x pi x
1≤i≤d 1≤i≤d

Portfolio Mean and Portfolio Variance

E (R π ) = πi · µi , Var (R π ) = πi · σij · πj
1≤i≤d 1≤i,j≤d

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Selection of a portfolio–criterion

(i) Maximize mean return (choose security of highest mean return)
−→ risky, big ﬂuctuations of return

(ii) Minimize risk of ﬂuction

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Selection of a portfolio–approach by Markowitz (MVA)

Balance Risk (Portfolio Variance) and Return (Portfolio Mean)
(i) Maximize E (R π ) under given upper bound c1 for Var (R π )

 πi ≥ 0 (1 ≤ i ≤ d )



π πi = 1
max E (R ) subject to
π∈Rd  1≤i≤d



Var (R π ) ≤ c1

(ii) Minimize Var (R π ) under given lower bound c2 for E (R π )

 πi ≥ 0

(1 ≤ i ≤ d )

min Var (R π ) subject to πi = 1
π∈Rd 
 1≤i≤d

E (R π ) ≥ c2

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Solution methods

(i) Linear Optimization Problem with quadratic constraint
−→ No standard algorithms, numerical inefficient

(ii) Quadratic Optimization Problem with positive semidefinite
objective matrix σ
−→ efficient algorithms (i.e., G OLDFARB/I DNANI, G ILL/M URRAY)
Feasible region non-empty if c2 ≤ max µi
1≤i≤d
σ positive definite and feasible region non-empty
−→ unique solution (even if one security riskless)

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Relations between the formulations (i) and (ii)

Theorem
Assume:
σ positive deﬁnite
min µi ≤ c2 ≤ max µi c2 ∈ R+
0
1≤i≤d 1≤i≤d

min σ 2 (π) ≤ c1 ≤ max σ 2 (π) c1 ∈ R+
0
πi ≥0, 1≤i≤d πi =1 πi ≥0, 1≤i≤d πi =1

Then
∗
(1) π ∗ solves (i) with Var R π = c1 =⇒ π ∗ solves (ii) with
∗
c2 := E R π
(2) π solves (ii) with E R π = c2 =⇒ π solves (i) with
c1 := Var R π

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The diversiﬁcation effect–example

Holding different Securities reduces Variance
Both security prices ﬂuctuate
randomly
σ11 , σ22 > 0
independent
σ12 = σ21 = 0
0.5
Then for the Portfolio π = we get
0.5
σ11 σ22
Var (R π ) = Var (0.5 · R1 + 0.5 · R2 ) = +
4 4

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The diversiﬁcation effect–example

Holding different Securities reduces Variance
0.5
−→ If σ11 = σ22 then the Variance of Portfolio is half as big
0.5
1 0
as that of or
0 1

−→ Reduction of Variance . . . Diversiﬁcation Effect depends on
number of traded securities

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Example

Mean-Variance Criterion
Investing into seemingly bad security can be optimal. Let be

1 0.1 −0.1
µ= , σ=
0.9 −0.1 0.15

Formulation (ii) becomes (II)

min Var (R π ) = min 2 2
0.1 · π1 + 0.15 · π2 − 0.2 · π1 π2
π π

 π1 , π2 ≥ 0
subject to π1 + π2 = 1

E (R π ) = π1 + 0.9 · π2 ≥ 0.96

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Example

1 0.5
Consider Portfolios and (does not satify
0 0.5
expectation constraint)
T T
Var R (1,0) = 0.1 , E R (1,0) =1
T T
Var R (0.5,0.5) = 0.125 , E R (0.5,0.5) = 0.95

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Example

Ignore expectation constraint and remember
π1 , π2 ≥ 0 π1 + π2 = 1. Hence

min 0.1 · π1 + 0.15 · (1 − π1 )2 − 0.2 · π1 · (1 − π1 )
2
π
2
= min 0.45 · π1 − 0.5 · π1 + 0.15
π

0.5
−→ Minimizing Portfolio (No solution of (II) but better than )
0.5

1 5
π= ·
9 4
T
−→ Portfolio Return Variance Var R ( 9 , 9 )
5 4
¯
= 0.001
T
−→ Portfolio Return Mean E R ( 9 , 9 )
5 4
¯
= 0.95

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Example

1.0

π2

0.5

0.4

0.0
0.0 0.5 0.6 1.0
π1

Pairs (π1 , π2 ) satisfying expectation constraint are above the
dotted line
Intersect with line π1 + π2 = 1
−→ Feasible region of MeanVariance Problem (bold line)

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Example

0.15

0.1

Var

0.05

0
0 0.5 0.6 1.0 1.5
π1

Portfolio Return Variance (as function of π1 ) of all pairs satisfying
π1 + π2 = 1
Minimum in feasible region π ∈ [0.6, 1] is attained at π = 0.6
Optimal Portfolio in (II)

→∗ =
− 0.6 ∗ ∗
π with Var R π = 0.012 , E Rπ = 0.96
0.4

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Stock price model

OPM
No assumption on distribution of security returns
Solving MV Problem just needed expectations and covariances

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Stock price model

OPM with just one security (price p1 at time t = 0 )
At time T security may have price d · p1 or u · p1
q: probability of decreasing by factor d
1−q : probability of increasing by factor u (u > d )

Mean and Variance of Return
P1 (T )
E (R1 (T )) =E = q · u + (1 − q) · d
p1
P1 (T )
Var (R1 (T )) = Var = q · u 2 + (1 − q) · d 2
p1
− (q · u + (1 − q) · d )2

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Stock price model

OPM with just one security (price p1 at time t = 0 )
After n periods the security has price

P1 (n · T ) = p1 · u Xn · d n−Xn

with Xn ∼ B(n, p) number of up-movements of price in
n periods

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Comments on MVA

Only trading at initial time t = 0
No reaction to current price changes possible
( −→ static model)
Risk of investment only modeled by variance of return

Need of Continuous-Time Market Models
Discrete-time multi-period models
(many periods −→ no fast algorithms)

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Outline


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Outline

ˆ
ˆ

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Modeling the security prices

Market with d+1 securities
d risky stocks with
prices p1 , p2 , . . . , pd at time t = 0 and
random prices P1 (t), P2 (t), . . . , Pd (t) at times t > 0

1 bond with
price p0 at time t = 0 and
deterministic price P0 (t) at times t > 0.

Assume: Perfectly devisible securities, no transaction costs.

⇒ Modeling of the price development on the time interval [0, T ].

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The bond price

Assume: Continuous compounding of interest at

a constant rate r :
Bond price: P0 (t) = p0 · er ·t for t ∈ [0, T ]

a non-constant, time-dependent and integrable rate r (t):
t
r (s) ds
Bond price: P0 (t) = p0 · e 0 for t ∈ [0, T ]

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The stock price

Stock price = random ﬂuctuations around an intrinsic bond part

2

1.8

1.6

1.4

1.2

1

0.8

0 0.2 0.4 0.6 0.8 1

log-linear model for a stock price
ln(Pi (t)) = ln(pi ) + bi · t + ”randomness”

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The stock price

Randomness is assumed

to have no tendency, i.e., E("randomness") = 0,
to be time-dependent,
to represent the sum of all deviations of
ln(Pi (t)) from ln(pi ) + bi · t on [0, T ],
∼ N (0, σ 2 t) for some σ > 0.

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The stock price

Deviation at time t
Y (t) := ln(Pi (t)) − ln(pi ) − bi · t
with
Y (t) ∼ N (0, σ 2 t)

Properties:
E (Y (t)) = 0,
Y (t) is time-dependent.

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The stock price

Y (t) = Y (δ) + (Y (t) − Y (δ)), δ ∈ (0, t)

Distribution of the increments of the deviation Y (t) − Y (δ)

depends only on the time span t − δ
is independent of Y (s), s ≤ δ

=⇒ Y (t) − Y (δ) ∼ N 0, σ 2 (t − δ)

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The stock price

Existence and properties of the stochastic process

{Y (t)}t∈[0,∞)

will be studied in the excursion on the Brownian motion.

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Outline

ˆ
ˆ

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General assumptions

General assumptions
Let (Ω, F, P) be a complete probability space with sample space Ω,
σ-ﬁeld F and probability measure P.

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Filtration

Definition
Let {Ft }t∈I be a family of sub-σ-fields of F, I be an ordered index set
with Fs ⊂ Ft for s < t, s, t ∈ I. The family {Ft }t∈I is called a filtration.

A filtration describes flow of information over time.
Ft models events observable up to time t.
If a random variable Xt is Ft -measurable, we are able to
determine its value from the information given at time t.

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Stochastic process

Definition
A set {(Xt , Ft )}t∈I consisting of a filtration {Ft }t∈I and a family of
Rn -valued random variables {Xt }t∈I with Xt being Ft -measurable is
called a stochastic process with filtration {Ft }t∈I .

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Remark

Remark
I = [0, ∞) or I = [0, T ].
Canoncial ﬁltration (natural ﬁltration) of {Xt }t∈I :

Ft := FtX := σ{Xs | s ≤ t, s ∈ I}.

Notation: {Xt }t∈I = {X (t)}t∈I = X .

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Sample path

Sample path
For ﬁxed ω ∈ Ω the set

X .(ω) := {Xt (ω)}t∈I = {X (t, ω)}t∈I

is called a sample path or a realization of the stochastic process.

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Identification of stochastic processes

Can two stochastic processes be identified with each other?

Definition
Let {(Xt , Ft )}t∈[0,∞) and {(Yt , Gt )}t∈[0,∞) be two stochastic processes.
Y is a modification of X , if

P{ω | Xt (ω) = Yt (ω)} = 1 for all t ≥ 0.

Definition
Let {(Xt , Ft )}t∈[0,∞) and {(Yt , Gt )}t∈[0,∞) be two stochastic processes.
X and Y are indistinguishable, if

P{ω | Xt (ω) = Yt (ω) for all t ∈ [0, ∞)} = 1.

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Identification of stochastic processes

Remark
X , Y indistinguishable ⇒ Y modification of X .

Theorem
Let the stochastic process Y be a modification of X . If both processes
have continuous sample paths P-almost surely, then X and Y are
indistinguishable.

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Brownian motion

Deﬁnition
The real-valued process {Wt }t≥0 with continuous sample paths and
i) W0 = 0 P-a.s.
ii) Wt − Ws ∼ N (0, t − s) for 0 ≤ s < t
"stationary increments"
iii) Wt − Ws independent of Wu − Wr for 0 ≤ r ≤ u ≤ s < t
"independent increments"
is called a one-dimensional Brownian motion.

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Brownian motion

Remark
By an n-dimensional Brownian motion we mean the Rn -valued process

W (t) = (W1 (t), . . . , Wn (t)),

with components Wi being independent one-dimensional Brownian
motions.

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Brownian motion and filtration

Brownian motion can be associated with

natural filtration

FtW := σ{Ws | 0 ≤ s ≤ t}, t ∈ [0, ∞)

P-augmentation of the natural filtration (Brownian filtration)

Ft := σ{FtW ∪ N | N ∈ F, P(N) = 0}, t ∈ [0, ∞)

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Requirement iii) of a Brownian motion with respect to a filtration
{Ft }t≥0 is often stated as

iii)∗ Wt − Ws independent of Fs , 0 ≤ s < t.

{Ft }t≥0 natural filtration (Brownian filtration)
⇒ iii) and iii)∗ are equivalent.
{Ft }t≥0 arbitrary filtration
⇒ iii) and iii)∗ are usually not equivalent.

Convention
If we consider a Brownian motion {(Wt , Ft )}t≥0 with an arbitrary
filtration we implicitly assume iii)∗ to be fulfilled.

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Existence of the Brownian motion

How can we show the existence of a stochastic process satisfying the
requirements of a Brownian motion?

Construction and existence proofs are long and technical.
Construction based on weak convergence and approximation by
random walks [Billingsley 1968].
Wiener measure, Wiener process.

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Theorem
The Brownian filtration {Ft }t≥0 is right-continuous as well as
left-continuous, i.e., we have

Ft = Ft+ := Ft+ε and Ft = Ft− := σ Fs .
ε>0 s<t

Definition
A filtration {Gt }t≥0 satifies the usual conditions, if it is right-continuous
and G0 contains all P-null sets of F.

General assumption for this section
Let {Ft }t≥0 be a filtration which satisfies the usual conditions.

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Martingales

Deﬁnition
The real-valued process {(Xt , Ft )}t∈I with E |Xt | < ∞ for all t ∈ I
(where I is an ordered index set), is called
a super-martingale, if for all s, t ∈ I with s ≤ t we have

E (Xt |Fs ) ≤ Xs P-a.s. ,

a sub-martingale, if for all s, t ∈ I with s ≤ t we have

E (Xt |Fs ) ≥ Xs P-a.s. ,

a martingale, if for all s, t ∈ I with s ≤ t we have

E (Xt |Fs ) = Xs P-a.s. .

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Interpretation of the martingale concept

Example: Modeling games of chance

Xn : Wealth of a gambler after n-th participation in a fair game

Martingale condition: E (Xn+1 |Fn ) = Xn P-a.s.

⇒ "After the game the player is as rich as he was before"

favorable game = sub-martingale
non-favorable game = super-martingale

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Example: Tossing a fair coin

"Head": Gambler receives one dollar

"Tail": Gambler loses one dollar

⇒ Martingale

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Theorem
A one-dimensional Brownian motion Wt is a martingale.

Remark
Each stochastic process with independent, centered increments is
a martingale with respect to its natural ﬁltration.
The Brownian motion with drift µ and volatility σ

Xt := µt + σWt , µ ∈ R, σ ∈ R

is a martingale if µ = 0, a super-martingale if µ ≤ 0 and a
sub-martingale if µ ≥ 0.

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Theorem
(1) Let {(Xt , Ft )}t∈I be a martingale and ϕ : R → R be a convex
function with E |ϕ(Xt )| < ∞ for all t ∈ I. Then

{(ϕ(Xt ), Ft )}t∈I

is a sub-martingale.
(2) Let {(Xt , Ft )}t∈I be a sub-martingale and ϕ : R → R a convex,
non-decreasing function with E |ϕ(Xt )| < ∞ for all t ∈ I. Then

{(ϕ(Xt ), Ft )}t∈I

is a sub-martingale.

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Remark
(1) Typical applications are given by

ϕ(x) = x 2 , ϕ(x) = |x|.

(2) The theorem is also valid for d -dimensional vectors

X (t) = (X1 (t), . . . , Xd (t))

of martingales and convex functions ϕ : Rd → R.

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Stopping time

Deﬁnition
A stopping time with respect to a ﬁltration {Ft }t∈[0,∞)
(or {Fn }n∈N ) is an F-measurable random variable

τ : Ω → [0, ∞] (or τ : Ω → N ∪ {∞})

with {ω ∈ Ω | τ (ω) ≤ t} ∈ Ft for all t ∈ [0, ∞)
(or {ω ∈ Ω | τ (ω) ≤ n} ∈ Fn for all n ∈ N).

Theorem
If τ1 , τ2 are both stopping times then τ1 ∧ τ2 := min{τ1 , τ2 } is also a
stopping time.

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The stopped process

The stopped process
Let {(Xt , Ft )}t∈I be a stochastic process, let I be either N or [0, ∞),
and τ a stopping time. The stopped process {Xt∧τ }t∈I is deﬁned by

Xt (ω) if t ≤ τ (ω),
Xt∧τ (ω) :=
Xτ (ω) (ω) if t > τ (ω).

Example: Wealth of a gambler who participates in a sequence of
games until he is either bankrupt or has reached a given level of
wealth.

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The stopped filtration

Let τ be a stopping time with respect to a filtration {Ft }t∈[0,∞) .
σ-field of events determined prior to the stopping time τ

Fτ := {A ∈ F | A ∩ {τ ≤ t} ∈ Ft for all t ∈ [0, ∞)}

Stopped filtration
{Fτ ∧t }t∈[0,∞) .

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What will happen if we stop a martingale or a sub-martingale?

Theorem: Optional sampling
Let {(Xt , Ft )}t∈[0,∞) be a right-continuous sub-martingale (or
martingale). Let τ1 , τ2 be stopping times with τ1 ≤ τ2 . Then for all
t ∈ [0, ∞) we have

E (Xt∧τ2 | Ft∧τ1 ) ≥ Xt∧τ1 P-a.s.

(or
E (Xt∧τ2 | Ft∧τ1 ) = Xt∧τ1 P-a.s.).

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Corollary
Let τ be a stopping time and {(Xt , Ft )}t∈[0,∞) a right-continuous
sub-martingale (or martingale). Then the stopped process
{(Xt∧τ , Ft )}t∈[0,∞) is also a sub-martingale (or martingale).

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Theorem
Let {(Xt , Ft )}t∈[0,∞) be a right-continuous process. Then Xt is a
martingale if and only if for all bounded stopping times τ we have

EXτ = EX0 .

→ Characterization of a martingale

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Deﬁnition
Let {(Xt , Ft )}t∈[0,∞) be a stochastic process with X0 = 0. If there is a
non-decreasing sequence {τn }n∈N of stopping times with

P lim τn = ∞ = 1,
n→∞

such that
(n)
Xt := (Xt∧τn , Ft )
t∈[0,∞)

is a martingale for all n ∈ N, then X is a local martingale. The
sequence {τn }n∈N is called a localizing sequence corresponding to X .

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Remark
(1) Each martingale is a local martingale.
(2) A local martingale with continuous paths is called
continuous local martingale.
(3) There exist local martingales which are not martingales.
E (Xt ) need not exist for a local martingale. However, the
expectation has to exist along the localizing sequence t ∧ τn .
The local martingale is a martingale on the random time
intervals [0, τn ].

Theorem
A non-negative local martingale is a super-martingale.

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Theorem: Doob’s inequality
Let {Mt }t≥0 be a martingale with right-continuous paths and
2
E (MT ) < ∞ or all T > 0. Then, we have
2
2
E sup |Mt | ≤ 4 · E (MT ).
0≤t≤T

Theorem
Let {(Xt , Ft )}t∈[0,∞) be a non-negative super-martingale with
right-continuous paths. Then, for λ > 0 we obtain

λ·P ω sup Xs (ω) ≥ λ ≤ E (X0 ).
0≤s≤t

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Outline

ˆ
ˆ

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Continuation: The stock price

log-linear model for a stock price
ln(Pi (t)) = ln(pi ) + bi · t + ”randomness”

Brownian motion {(Wt , Ft )}t≥0 is the appropriate stochastic process to
model the "randomness"

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Market with one stock and one bond (d=1)
ln(P1 (t)) = ln(p1 ) + b1 · t + σ11 Wt
P1 (t) = p1 · exp b1 · t + σ11 Wt

Market with d stocks and one bond (d>1)
m
ln(Pi (t)) = ln(pi ) + bi · t + σij Wj (t), i = 1, . . . , d
j=1
m
Pi (t) = pi · exp bi · t + σij Wj (t) , i = 1, . . . , d
j=1

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Distribution of the logarithm of the stock prices
m
2
ln(Pi (t)) ∼ N ln(pi ) + bi · t, σij · t
j=1

⇒ Pi (t) is log-normally distributed.

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Lemma
m
1 2
Let bi := bi + 2 σij for i = 1, . . . , d .
j=1

(1) E (Pi (t)) = pi · ebi t .
m
(2) Var (Pi (t)) = pi2 · exp(2bi t) · exp 2
σij t −1 .
j=1
m
1
(3) Xt := a · exp cj Wj (t) − cj2 t with a, cj ∈ R, j = 1, . . . , m
2
j=1
is a martingale.

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Interpretation of the stock price model

The stock price model

m
1 2
Pi (t) = pi · exp(bi t) · exp σij Wj (t) − σij t ,
2
j=1

Pi (0) = pi , i = 1, . . . , d .

The stock price is the product of
the mean stock price pi · exp(bi t) and
a martingale with expectation 1, namely
m
1 2
exp σij Wj (t) − σij t
2
j=1

which models the stock price around its mean value.
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Interpretation of the stock price model

Vector of mean rates of stock returns

b = (b1 , . . . , bd )T

Volatility matrix  
σ11 . . . σ1m
 . . 
σ= .
.
..
. . 
.
σd1 . . . σdm
Pi (t) is a geometric Brownian motion with drift bi and volatility
σi. = (σi1 , . . . , σim )T .

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Summary: Stock prices

Bond price and stock prices
P0 (t) = p0 · ert
Bond price
P0 (0)= p0
m
1 2
Pi (t) = pi · exp(bi t) · exp σij Wj (t) − σ t
2 ij Stock prices
j=1
Pi (0) = pi , i = 1, . . . , d .

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Extension

Extension: Model with non-constant, time-dependent, and integrable
rates of return bi (t) and volatilities σ(t).

Stock prices:
t m
1 2
Pi (t) = pi · exp bi (s) − σij (s) ds
2
0 j=1

m t

· exp σij (s) dWj (s)
j=1 0

t
Problem: σij (s) dWj (s)
0
ˆ
⇒ Stochastic integral (Ito integral)

76 / 477

Outline

ˆ
ˆ

77 / 477

ˆ
The Ito integral

Is it possible to deﬁne the stochastic integral
t

Xs (ω) dWs (ω)
0

ω-wise in a reasonable way?

78 / 477

ˆ
The Ito integral

Theorem
P-almost all paths of the Brownian motion {Wt }t∈[0,∞) are nowhere
differentiable.

⇒ A deﬁnition of the form
t t
dWs (ω)
Xs (ω) dWs (ω) = Xs (ω) ds
ds
0 0

is impossible.

79 / 477

ˆ
The Ito integral

Theorem
With the definition
2n
Zn (ω) := W i (ω) − W i−1 (ω) , n ∈ N, ω ∈ Ω
2n 2n
i=1

we have
n→∞
Zn (ω) − − ∞
−→ P-a.s. ,
i.e., the paths Wt (ω) of the Brownian motion admit infinite variation on
the interval [0, 1] P-almost surely.
The paths Wt (ω) of the Brownian motion have infinite variation on each
non-empty finite interval [s1 , s2 ] ⊂ [0, ∞) P-almost surely.

80 / 477

General assumptions

General assumptions for this section
Let (Ω, F, P) be a complete probability space equipped with a filtration
{Ft }t satisfying the usual conditions. Further assume that on this
space a Brownian motion {(Wt , Ft )}t∈[0,∞) with respect to this filtration
is defined.

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Simple process

Deﬁnition
A stochastic process {Xt }t∈[0,T ] is called a simple process if there exist
real numbers 0 = t0 < t1 < . . . < tp = T , p ∈ N, and bounded random
variables Φi : Ω → R, i = 0, 1, . . . , p, with

Φ0 F0 -measurable, Φi Fti−1 -measurable, i = 1, . . . , p

such that Xt (ω) has the representation
p
Xt (ω) = X (t, ω) = Φ0 (ω) · 1{0} (t) + Φi (ω) · 1(ti−1 ,ti ] (t)
i=1

for each ω ∈ Ω.

82 / 477

Simple process

Remark
Xt is Fti−1 -measurable for all t ∈ (ti−1 , ti ].
The paths X (., ω) of the simple process Xt are left-continuous step
functions with height Φi (ω) · 1(ti−1 ,ti ] (t).

1

0.9

0.8

0.7

0.6

X(.,ω) 0.5

0.4

0.3

0.2

0.1

0
0 t t2 t3 T
1
t

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Stochastic integral

Deﬁnition
For a simple process {Xt }t∈[0,T ] the stochastic integral I.(X ) for
t ∈ (tk , tk +1 ] is deﬁned according to
t

It (X ) := Xs dWs := Φi (Wti − Wti−1 ) + Φk +1 (Wt − Wtk ),
0 1≤i≤k

or more generally for t ∈ [0, T ]:
t

It (X ) := Xs dWs := Φi (Wti ∧t − Wti−1 ∧t ).
0 1≤i≤p

84 / 477

Stochastic integral

Theorem: Elementary properties of the stochastic integral
Let X := {Xt }t∈[0,T ] be a simple process. Then we have

(1) {It (X )}t∈[0,T ] is a continuous martingale with respect to {Ft }t∈[0,T ] .
In particular, we have E (It (X )) = 0 for all t ∈ [0, T ].
t 2 t
(2) E Xs dWs =E 2
Xs ds for t ∈ [0, T ].
0 0

t 2 T
(3) E sup Xs dWs ≤4·E 2
Xs ds .
0≤t≤T 0 0

85 / 477

Stochastic integral

Remark
(1) By (2) the stochastic integral is a square-integrable stochastic
process.
(2) For the simple process X ≡ 1 we obtain
t

1 dWs = Wt
0

and
t t
2
E dWs = E (Wt2 ) =t= ds.
0 0

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Stochastic integral

Remark
(1) Integrals with general boundaries:
T T t

Xs dWs := Xs dWs − Xs dWs for t ≤ T .
t 0 0

For t ≤ T , A ∈ Ft we have
T T

1A (ω) · Xs (ω) · 1[t,T ] (s) dWs = 1A (ω) · Xs (ω) dWs .
0 t

(2) Let X , Y be simple processes, a, b ∈ R. Then we have

It (aX + bY ) = a · It (X ) + b · It (Y ) (linearity)

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Measurability

Definition
A stochastic process {(Xt , Gt )}t∈[0,∞) is called measurable if the
mapping

[0, ∞) × Ω → Rn
(s, ω) → Xs (ω)

is B([0, ∞)) ⊗ F-B(Rn )-measurable.

Remark
Measurability of the process X implies that X (., ω) is
B([0, ∞))-B(Rn )-measurable for a fixed ω ∈ Ω. Thus, for all t ∈ [0, ∞),
t
i = 1, . . . , n, the integral Xi2 (s) ds is defined.
0

88 / 477

Measurability

Deﬁnition
A stochastic process {(Xt , Gt )}t∈[0,∞) is called progressively
measurable if for all t ≥ 0 the mapping

[0, t] × Ω → Rn
(s, ω) → Xs (ω)

is B([0, t]) ⊗ Gt -B(Rn )-measurable.

89 / 477

Measurability

Remark
(1) If the real-valued process {(Xt , Gt )}t∈[0,∞) is progressively
measurable and bounded, then for all t ∈ [0, ∞) the integral
t
Xs ds is Gt -measurable.
0

(2) Every progressively measurable process is measurable.
(3) Each measurable process possesses a progressively measurable
modiﬁcation.

90 / 477

Measurability

Theorem
If all paths of the stochastic process {(Xt , Gt )}t∈[0,∞) are
right-continuous (or left-continuous), then the process is progressively
measurable.

Theorem
Let τ be a stopping time with respect to the ﬁltration {Gt }t∈[0,∞) . If the
stochastic process {(Xt , Gt )}t∈[0,∞) is progressively measurable, then
so is the stopped process {(Xt∧τ , Gt )}t∈[0,∞) . In particular, Xt∧τ is Gt
and Gt∧τ -measurable.

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Extension of the stochastic integral to
L2 [0, T ]-processes

Deﬁnition

L2 [0, T ] := L2 [0, T ], Ω, F, {Ft }t∈[0,T ] , P

:= {(Xt , Ft )}t∈[0,T ] real-valued stochastic process
T

{Xt }t progressively measurable, E Xt2 dt < ∞
0

T
Norm on L2 [0, T ]: X 2
T := E Xt2 dt .
0

92 / 477


· 2 L2 -norm on the probability space
T
[0, T ] × Ω, B([0, T ]) ⊗ F, λ ⊗ P .

· 2 semi-norm ( X −Y 2 = 0 ⇒ X = Y ).
T T

X equivalent to Y :⇔ X = Y a.s. λ ⊗ P.

93 / 477


ˆ
Ito isometry
Let X be a simple process. The mapping X → I.(X ) induces by
T T
2
2 2 2
I.(X ) LT := E Xs dWs =E Xs ds = X T
0 0

a norm on the space of stochastic integrals.

⇒ I.(X ) linear, norm-preserving (= isometry)
ˆ
⇒ I.(X ) Ito isometry

94 / 477


Use processes X ∈ L2 [0, T ] approximated by a sequence X (n) of
simple processes.
I.(X (n) ) is a Cauchy-sequence with respect to · LT .

To show: I.(X (n) ) is convergent, limit independent of X (n) .
Denote limit by
I(X ) = Xs dWs .

95 / 477


J(.)
C
X ∈ L2 [0, T ] _ _ _ _ _ _ _/ J(X ) ∈ M2
O O
· T · LT

X (n) / I(X (n) )
I(.)

simple process stochastic integral
for simple processes

96 / 477


Theorem
An arbitrary X ∈ L2 [0, T ] can be approximated by a sequence of
simple processes X (n) .
More precisely: There exists a sequence X (n) of simple processes with
T
(n) 2
lim E Xs − Xs ds = 0.
n→∞
0

97 / 477


Lemma
Let {(Xt , Gt )}t∈[0,∞) be a martingale where the filtration {Gt }t∈[0,∞)
satisfies the usual conditions. Then the process Xt possesses a
right-continuous modification {(Yt , Gt )}t∈[0,∞) such that
{(Yt , Gt )}t∈[0,∞) is a martingale.

98 / 477


ˆ
Construction of the Ito integral for processes in L2 [0, T ]
There exists a unique linear mapping J from L2 [0, T ] into the space of
continuous martingales on [0, T ] with respect to {Ft }t∈[0,T ] satisfying
(1) X = {Xt }t∈[0,T ] simple process
⇒ P Jt (X ) = It (X ) for all t ∈ [0, T ] = 1
t
(2) E Jt (X )2 = E 2 ˆ
Xs ds Ito isometry
0

Uniqueness: If J, J ′ satisfy (1) and (2), then for all X ∈ L2 [0, T ] the
processes J ′ (X ) and J(X ) are indistinguishable.

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Deﬁnition
For X ∈ L2 [0, T ] and J as before we deﬁne by
t

Xs dWs := Jt (X )
0

ˆ
the stochastic integral (or Ito integral) of X with respect to W .

100 / 477


Theorem: Special case of Doob’s inequality
Let X ∈ L2 [0, T ]. Then we have
t T
2
2
E sup Xs dWs ≤4·E Xs ds .
0≤t≤T
0 0

101 / 477

Multi-dimensional generalization of the stochastic integral
{(W (t), Ft )}t : m-dimensional Brownian motion
with W (t) = (W1 (t), . . . , Wm (t))
{(X (t), Ft )}t : Rn,m -valued progressively measurable process with
Xij ∈ L2 [0, T ].
ˆ
Ito integral of X with respect to W :
 t 
m
 X1j (s) dWj (s)
 
t  j=1 
 0 
 .
. 
X (s) dW (s) :=  . , t ∈ [0, T ]
 
0  m t 
 
 Xnj (s) dWj (s)
j=1 0

102 / 477

Further extension of the stochastic integral

Deﬁnition

H 2 [0, T ] := H 2 [0, T ], Ω, F, {Ft }t∈[0,T ] , P

:= {(Xt , Ft )}t∈[0,T ] real-valued stochastic process

{Xt }t progressively measurable,
T

Xt2 dt < ∞ P-a.s.
0

103 / 477

Processes X ∈ H 2 [0, T ]
do not necessarily have a ﬁnite T -norm
→ no approximation by simple processes as for
processes in L2 [0, T ]
can be localized with suitable sequences of stopping times
Stopping times (with respect to {Ft }t ):
t
2
τn (ω) := T ∧ inf 0 ≤ t ≤ T Xs (ω) ds ≥ n , n ∈ N
0

Sequence of stopped processes:
(n)
Xt (ω) := Xt (ω) · 1{τn (ω)≥t}

⇒ X (n) ∈ L2 [0, T ] ⇒ Stochastic integral already deﬁned.
104 / 477


Stochastic integral:

It (X ) := It (X (n) ) for 0 ≤ t ≤ τn

Consistence property:

It (X ) = It (X (m) ) for 0 ≤ t ≤ τn (≤ τm ), m ≥ n

⇒ It (X ) well-deﬁned for X ∈ H 2 [0, T ]

105 / 477


Stopping times:
n→∞
τn − − +∞ P-a.s.
−→

⇒ It (X ) local martingale with localizing sequence τn .

⇒ Stochastic integral is linear and possesses continuous paths.

106 / 477

Outline

ˆ
ˆ

107 / 477

ˆ
The Ito formula

Let (Ω, F, P) be a complete probability space equipped with a filtration
{Ft }t satisfying the usual conditions. Further, assume that on this
space a Brownian motion {(Wt , Ft )}t∈[0,∞) with respect to this filtration
is defined.

108 / 477

ˆ
The Ito formula

Deﬁnition
Let {(Wt , Ft )}t∈[0,∞) be an m-dimensional Brownian motion.
ˆ
(1) {(X (t), Ft )}t∈[0,∞) is a real-valued Ito process if for all t ≥ 0 it
admits the representation
t t

X (t) = X (0) + K (s) ds + H(s) dW (s)
0 0
t m t

= X (0) + K (s) ds + Hj (s) dWj (s) P-a.s.
0 j=1 0

109 / 477

ˆ
The Ito formula

X (0) F0 -measurable,
{K (t)}t∈[0,∞) , {H(t)}t∈[0,∞) progressively measurable with

t t

|K (s)| ds < ∞, Hi2 (s) ds < ∞ P-a.s.
0 0

for all t ≥ 0, i = 1, . . . , m.

(2) n-dimensional Ito process X = (X (1) , . . . , X (n) )
ˆ
ˆ
= vector with components being real-valued Ito processes.

110 / 477

ˆ
The Ito formula

Remark
Hj ∈ H 2 [0, T ] for all T > 0.
ˆ
The representation of an Ito process is unique up to
indistinguishability of the representing integrands Kt , Ht .
Symbolic differential notation:

dXt = Kt dt + Ht dWt

111 / 477

ˆ
The Ito formula

Deﬁnition
ˆ
Let X and Y be two real-valued Ito processes with
t t

X (t) = X (0) + K (s) ds + H(s) dW (s),
0 0
t t

Y (t) = Y (0) + L(s) ds + M(s) dW (s).
0 0

Quadratic covariation of X and Y :

m t

X,Y t := Hi (s) · Mi (s) ds.
i=1 0

112 / 477

ˆ
The Ito formula

Deﬁnition
Quadratic variation of X

X t := X , X t .

Notation
ˆ
Let X be a real-valued Ito process, and Y a real-valued, progressively
measurable process. We set
t t t

Y (s) dX (s) := Y (s) · K (s) ds + Y (s) · H(s) dW (s)
0 0 0

if all integrals on the right-hand side are deﬁned.

113 / 477

ˆ
The Ito formula
ˆ
Theorem: One-dimensional Ito formula
ˆ
Let Wt be a one-dimensional Brownian motion, and Xt a real-valued Ito
process with
t t

Xt = X0 + Ks ds + Hs dWs .
0 0

Let f ∈ C 2 (R). Then, for all t ≥ 0 we have
t t
′ 1
f (Xt ) = f (X0 ) + f (Xs ) dXs + f ′′ (Xs ) d X s
2
0 0
t t
1
= f (X0 ) + f (Xs ) · Ks + · f ′′ (Xs ) · Hs ds +
′ 2
f ′ (Xs )Hs dWs
2
0 0

114 / 477

ˆ
The Ito formula

Remark
ˆ
The Ito formula differs from the fundamental theorem of calculus
by the additional term
t
1
f ′′ (Xs ) d X s .
2
0

The quadratic variation X t ˆ
is an Ito process.
Differential notation:
1 ′′
df (Xt ) = f ′ (Xt ) dXt + · f (Xt ) d X t .
2

115 / 477

ˆ
The Ito formula

Lemma
Let X be a martingale with |Xs | ≤ C for all s ∈ [0, t] P-a.s.
Let π = {t0 , t1 , . . . , tm }, t0 = 0, tm = t, be a partition of [0, t] with

π := max |tk − tk −1 |.
1≤k ≤m

Then we have
m 2
2
(1) E Xtk − Xtk −1 ≤ 48 · C 4
k =1
m
4 π →0
(2) X continuous ⇒ E Xtk − Xtk −1 − − → 0.
−−
k =1

116 / 477

′
ˆ
Some applications of Ito s formula

′
ˆ
Some applications of Ito s formula I
(1) Xt = t :
Representation:
t t

Xt = 0 + 1 ds + 0 dWs .
0 0

For f ∈ C 2 (R) we have
t

f (t) = f (0) + f ′ (s) ds.
0

⇒ Fundamental theorem of calculus is a special case of
Ito′ s formula.
ˆ

117 / 477

′
ˆ
Some applications of Ito s formula II

ˆ′
(2) Xt = h(t) :
For h ∈ C 1 (R) Ito′ s formula implies the chain rule
ˆ

t t
′
Xt = h(0) + h (s) ds + 0 dWs
0 0

t

⇒ (f ◦ h)(t) = (f ◦ h)(0) + f ′ (h(s)) · h′ (s) ds.
0

118 / 477

′
ˆ
Some applications of Ito s formula III

′
ˆ
(3) Xt = Wt , f (x) = x 2 :
Due to
t t

Wt = 0 + 0 ds + 1 dWs
0 0

we obtain
t t t
1
Wt2 = 2 · Ws dWs + · 2 ds = 2 · Ws dWs + t
2
0 0 0

⇒ Additional term "t"
(→ nonvanishing quadratic variation of Wt ).

119 / 477

ˆ
The Ito formula
ˆ
Theorem: Multi-dimensional Ito formula
ˆ
X (t) = X1 (t), . . . , Xn (t) n-dimensional Ito process with

t m t

Xi (t) = Xi (0) + Ki (s) ds + Hij (s) dWj (s), i = 1, . . . , n
0 j=1 0

where W (t) = W1 (t), . . . , Wm (t) is an m-dimensional Brownian motion.
Let f : [0, ∞) × Rn → R be a C 1,2 -function. Then, we have

f (t, X1 (t), . . . , Xn (t)) = f (0, X1 (0), . . . , Xn (0))
t n t

+ ft (s, X1 (s), . . . , Xn (s)) ds + fxi (s, X1 (s), . . . , Xn (s)) dXi (s)
0 i=1 0

n t
1
+ · fxi xj (s, X1 (s), . . . , Xn (s)) d Xi , Xj s .
2
i,j=1 0

120 / 477

Product rule or partial integration
Corollary: Product rule or partial integration
ˆ
Let Xt , Yt be one-dimensional Ito processes with
t t

Xt = X0 + Ks ds + Hs dWs ,
0 0
t t

Yt = Y0 + µs ds + σs dWs .
0 0

Then we have t t t

Xt · Yt = X0 · Y0 + Xs dYs + Ys dXs + d X,Y s
0 0 0
t t

= X0 · Y0 + Xs µs + Ys Ks + Hs σs ds + Xs σs + Ys Hs dWs .
0 0

121 / 477

The stock price equation
Simple continuous-time market model (1 bond, one stock).
Stock price inﬂuenced by a one-dimensional Brownian motion
Price of the stock at time t:

P(t) = p · exp b − 1 σ 2 t + σWt
2

Choose
t t
1 2
Xt = 0 + b− 2σ ds + σ dWs , f (x) = p · ex
0 0

ˆ
Ito formula implies
t t
1 2 1 2
f (Xt ) = p + f (Xs )(b − 2 σ ) + 2 f (Xs ) ·σ ds + f (Xs ) · σ dWs
0 0

122 / 477


t t

P(t) = p + P(s) · b ds + P(s) · σ dWs
0 0

Remark
The stock price equation is valid for time-dependent b and σ, if
t t
1 2
Xt = b(s) − 2 σ (s) ds + σ(s) dWs .
0 0

123 / 477


The stock price equation in differential form

dP(t) = P(t) b dt + σ dWt

P(0) = p

124 / 477


Theorem: Variation of constants
Let {(W (t), Ft )}t∈[0,∞) be an m-dimensional Brownian motion.
Let x ∈ R and A, a, Sj , σj be progressively measurable, real-valued
processes with
t

|A(s)| + |a(s)| ds < ∞ for all t ≥ 0 P-a.s.
0
t

Sj2 (s) + σj2 (s) ds < ∞ for all t ≥ 0 P-a.s. .
0

125 / 477


Then the stochastic differential equation
m
dX (t) = A(t) · X (t) + a(t) dt + Sj (t)X (t) + σj (t) dWj (t)
j=1
X (0) = x

possesses a unique solution with respect to λ ⊗ P :
t m
1
X (t) = Z (t) · x + a(u) − Sj (u)σj (u) du
Z (u)
0 j=1

m t
σj (u)
+ dWj (u)
Z (u)
j=1 0

126 / 477


Hereby is
t t
1 2
Z (t) = exp A(u) − 2 · S(u) du + S(u) dW (u)
0 0

the unique solution of the homogeneous equation

dZ (t) = Z (t) A(t) dt + S(t)T dW (t)
Z (0) = 1.

127 / 477


Remark
The process {(X (t), Ft )}t∈[0,∞) solves the stochastic differential
equation in the sense that X (t) satisﬁes
t

X (t) = x + A(s) · X (s) + a(s) ds
0
m t

+ Sj (s) X (s) + σj (s) dWj (s)
j=1 0

for all t ≥ 0 P-almost surely.

128 / 477

Outline

ˆ
ˆ

129 / 477

General assumptions

(Ω, F, P) be a complete probability space,
{(W (t), Ft )}t∈[0,∞) m-dimensional Brownian motion.
Dynamics of bond and stock prices:
t

P0 (t) = p0 · exp r (s) ds bond
0
t m
1 2
Pi (t) = pi · exp bi (s) − σij (s) ds
2
0 j=1

m t

+ σij (s) dWj (s) stock
j=1 0
for t ∈ [0, T ], T > 0, i = 1, . . . , d .

130 / 477

General assumptions (continued)

General assumptions for this section (continued)
r (t), b(t) = (b1 (t), . . . , bd (t))T , σ(t) = (σij (t))ij
progressively measurable processes with respect to {Ft }t ,
component-wise uniformly bounded in (t, ω).

σ(t)σ(t)T uniformly positive deﬁnite,
i.e., it exists K > 0 with

x T σ(t)σ(t)T x ≥ Kx T x

for all x ∈ Rd and all t ∈ [0, T ] P-a.s.

Deterministic rate of return r (t) is not required
r (t) can be a stochastic process
⇒ bond is no longer riskless.

131 / 477

Bond and stock prices

Bond and stock prices are unique solutions of the stochastic
differential equations

dP0 (t) = P0 (t) · r (t) dt bond
P0 (t) = p0

m
dPi (t) = Pi (t) bi (t) dt + σij (t) dWj (t) , i = 1, . . . , d
j=1

Pi (0) = pi stock

ˆ
⇒ Representations of prices as Ito processes

132 / 477

Possible actions of investors

(1) Investor can rebalance his holdings
→ sell some securities
→ invest in securities
⇒ Portfolio process / trading strategy.

(2) Investor is allowed to consume parts of his wealth
⇒ Consumption process.

133 / 477

Requirements on a market model

(1) Investor should not be able to foresee events
→ no knowledge of future prices.
(2) Actions of a single investor have no impact on the stock prices
(small investor hypothesis).
(3) Each investor has a ﬁxed initial capital at time t = 0.
(4) Money which is not invested into stocks has to be invested in
bonds.
(5) Investors act in a self-ﬁnancing way
(no secret source or sink for money).

134 / 477

Requirements on a market model

(6) Securities are perfectly divisible.
(7) Negative positions in securities are possible
bond → credit
stock → we sold some stock short.
(8) No transaction costs.

135 / 477

Negative bond positions and credit interest rates

Negative bond positions and credit interest rates

Assume: Interest rate r (t) is constant
Negative bond position = it is possible to borrow money for the
same rate as we would get for investing in bonds.
Interest depends on the market situation ((t, ω) ∈ [0, T ] × Ω), but
not on positive or negative bond position.

136 / 477

Mathematical realizations of some requirements

Market with 1 bond and d stocks

Time t = 0: – Initial capital of investor: x > 0
– Buy a selection of securities
T
ϕ(0) = ϕ0 (0), ϕ1 (0), . . . , ϕd (0)

Time t > 0: – Trading strategy: ϕ(t)

(1) ⇒ trading strategy is progressively measurable
with respect to {Ft }t
Decisions on buying and selling are made on basis of information
available at time t (→ modelled by {Ft }t )
(5) ⇒ only self-ﬁnancing trading strategies should be used.

137 / 477

Discrete-time example: self-ﬁnancing strategy

Market with 1 riskless bond and 1 stock

Two-period model for time points t = 0, 1, 2.
Number of shares of bond and stock at time t:

(ϕ0 (t), ϕ1 (t))T ∈ R2

Consumption of investor at time t: C(t)
Wealth at time t: X (t)
Bond/stock prices at time t: P0 (t), P1 (t)
Initial conditions: C(0) = 0, X (0) = x

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t =0
Investor uses initial capital to buy shares of bond and stock

X (0) = x = ϕ0 (0) · P0 (0) + ϕ1 (0) · P1 (0).

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t =1
Security prices have changed, investor consumes parts of his wealth

Current wealth:

X (1) = ϕ0 (0) · P0 (1) + ϕ1 (0) · P1 (1) − C(1).

Total:

X (1) = x + ϕ0 (0) · P0 (1) − P0 (0) + ϕ1 (0) · P1 (1) − P1 (0) − C(1)

Wealth = initial wealth + gains/losses - consumption

Invest remaining capital at the market:

X (1) = ϕ0 (1) · P0 (1) + ϕ1 (1) · P1 (1).

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t =2
Invest remaining capital at the market

Wealth:
X (2) = ϕ0 (2) · P0 (2) + ϕ1 (2) · P1 (2).

Wealth = total wealth of shares held

Total:
2
X (2) = x + ϕ0 (i − 1) · (P0 (i) − P0 (i − 1))
i=1
+ϕ1 (i − 1) · (P1 (i) − P1 (i − 1))
2
− C(i).
i=1

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Self-ﬁnancing trading strategy:
wealth before rebalancing - consumption = wealth after rebalancing

Condition:

ϕ0 (i) · P0 (i) + ϕ1 (i) · P1 (i)
= ϕ0 (i − 1) · P0 (i) + ϕ1 (i − 1) · P1 (i) − C(i)

⇒ Useless in continuous-time setting
(securities can be traded at each time instant /
"before" and "after" cannot be distinguished)

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Continuous-time setting

Wealth process corresponding to strategy ϕ(t):
t t t

X (t) = x + ϕ0 (s) dP0 (s) + ϕ1 (s) dP1 (s) − c(s) ds
0 0 0

ˆ
⇒ Price processes are Ito processes.

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Trading strategy and wealth processes

Deﬁnition
(1) A trading strategy ϕ with
T
ϕ(t) := ϕ0 (t), ϕ1 (t), . . . , ϕd (t)

is an Rd+1 -valued progressively measurable process with respect
to {Ft }t∈[0,T ] satisfying

T

|ϕ0 (t)| dt < ∞ P-a.s.
0

d T
2
ϕi (t) · Pi (t) dt < ∞ P-a.s. for i = 1, . . . , d .
j=1 0

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Deﬁnition
The value
d
x := ϕi (0) · pi
i=0

is called initial value of ϕ.

(2) Let ϕ be a trading strategy with initial value x > 0.
The process
d
X (t) := ϕi (t)Pi (t)
i=0

is called wealth process corresponding to ϕ with
initial wealth x.

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Deﬁnition
(3) A non-negative progressively measurable process c(t) with
respect to {Ft }t∈[0,T ] with

T

c(t) dt < ∞ P-a.s.
0

is called consumption (rate) process.

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Definition
A pair (ϕ, c) consisting of a trading strategy ϕ and a consumption rate
process c is called self-financing if the corresponding wealth process
X (t) satisfies

d t t

X (t) = x + ϕi (s) dPi (s) − c(s) ds P-a.s.
i=0 0 0

current wealth = initial wealth + gains/losses - consumption

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Remark
We have
t t

ϕ0 (s) dP0 (s) = ϕ0 (s) P0 (s) r (s) ds
0 0
t t

ϕi (s) dPi (s) = ϕi (s) Pi (s) bi (s) ds
0 0
m t

+ ϕi (s) Pi (s) σij (s) dWj (s), i = 1, . . . , d .
j=1 0

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Self-financing portfolio process

Definition
Let (ϕ, c) be a self-financing pair consisting of a trading strategy and a
consumption process with corresponding wealth process X (t) > 0
P-a.s. for all t ∈ [0, T ]. Then the Rd -valued process

T ϕi (t) · Pi (t)
π(t) = π1 (t), . . . , πd (t) with πi (t) =
X (t)

is called a self-financing portfolio process corresponding to the
pair (ϕ, c).

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Portfolio processes

Remark
(1) The portfolio process denotes the fractions of total wealth invested
in the different stocks.
(2) The fraction of wealth invested in the bond is given by

ϕ0 (t) · P0 (t)
1 − π(t)T 1 = , where 1 := (1, . . . , 1)T ∈ Rd .
X (t)

(3) Given knowledge of wealth X (t) and prices Pi (t), it is possible for
an investor to describe his activities via a self-ﬁnancing pair (π, c).
→ Portfolio process and trading strategy are
equivalent descriptions of the same action.

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The wealth equation

The wealth equation

dX (t) = [r (t) X (t) − c(t)] dt

+ X (t) π(t)T (b(t) − r (t) 1) dt + σ(t) dW (t)

X (0) = x

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Alternative definition of a portfolio process

Definition
The progressively measurable Rd -valued process π(t) is called a
self-financing portfolio process corresponding to the consumption
process c(t) if the corresponding wealth equation possesses a unique
solution X (t) = X π,c (t) with
T
2
X (t) · πi (t) dt < ∞ P-a.s. for i = 1, . . . , d .
0

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Admissibility

Definition
A self-financing pair (ϕ, c) or (π, c) consisting of a trading strategy ϕ or
a portfolio process π and a consumption process c will be called
admissible for the initial wealth x > 0, if the corresponding wealth
process satisfies

X (t) ≥ 0 P-a.s. for all t ∈ [0, T ].

The set of admissible pairs will be denoted by A(x).

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An example

Portfolio process:
π(t) ≡ π ∈ Rd constant
Consumption rate:
c(t) = γ · X (t), γ > 0
Wealth process corresponding to (π, c) :

X (t)

Investor rebalances his holdings in such a way that the fractions of
wealth invested in the different stocks and in the bond remain
constant over time.
Consumption rate is proportional to the current wealth of the
investor.

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An example

Wealth equation:

dX (t) = [r (t) − γ] X (t) dt
+ X (t)π T (b(t) − r (t) 1) dt + σ(t) dW (t)
X (0) = 0

Wealth process:
t
1 T
X (t) = x · exp r (s) − γ + π T b(s) − r (s) · 1 − π σ(s) 2
ds
2
0
t

+ π T σ(s) dW (s)
0

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Outline

ˆ
ˆ

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Properties of the continuous-time market model

Assumptions:
Dimension of the underlying Brownian motion
= number of stocks
Past and present prices are the only sources of information for the
investors
⇒ Choose Brownian ﬁltration {Ft }t∈[0,T ]

Aim: Final wealths X (T ) when starting with initial capital of x.

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General assumption / notation

General assumption for this section
d =m

Notation
t

γ(t) := exp − r (s) ds
0
θ(t) := σ −1 (t) b(t) − r (t) 1
t t
T 1 2
Z (t) := exp − θ(s) dW (s) − θ(s) ds
2
0 0
H(t) := γ(t) · Z (t)

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Properties of the continuous-time market model

b, r uniformly bounded
σσ T uniformly positive deﬁnite
⇒ θ(t) 2 uniformly bounded
Interpretation of θ(t): Relative risk premium for stock investment.
Process H(t) is important for option pricing.
H(t) is positive, continuous, and progressively measurable with
respect to {Ft }t∈[0,T ] .
H(t) is the unique solution of the SDE

dH(t) = −H(t) r (t) dt + θ(t)T dW (t)
H(0) = 1.

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Completeness of the market

Theorem: Completeness of the market
(1) Let the self-ﬁnancing pair (π, c) consisting of a portfolio process π
and a consumption process c be admissible for an initial wealth of
x ≥ 0, i.e.,
(π, c) ∈ A(x).
Then the corresponding wealth process X (t) satisﬁes
t

E H(t) X (t) + H(s)c(s) ds ≤ x for all t ∈ [0, T ].
0

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Theorem: Completeness of the market
(2) Let B ≥ 0 be an FT -measurable random variable and c(t) a
consumption process satisfying
T

x := E H(T ) B + H(s)c(s) ds < ∞.
0

Then there exists a portfolio process π(t) with (π, c) ∈ A(x) and
the corresponding wealth process X (t) satisﬁes

X (T ) = B P-a.s.

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H(t) can be regarded as the appropriate discounting process that
determines the initial wealth at time t = 0
T

E H(s) · c(s) ds + E (H(T ) · B)
0

which is necessary to attain future aims.
(1) puts bounds on the desires of an investor given his initial
capital x ≥ 0.
(2) proves that future aims which are feasible in the sense of part
(1) can be realized.
(2) says that each desired final wealth in t = T can be attained
exactly via trading according to an appropriate self-financing pair
(π, c) if one possesses sufficient initial capital
(completeness/complete model).
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Remark
1/H(t) is the wealth process corresponding to the pair

π(t), c(t) = σ −1 (t)T θ(t), 0

with initial wealth x := 1/H(0) = 1
and ﬁnal wealth B:= 1/H(T ).

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Outline

ˆ
ˆ

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Excursion 4: The martingale representation theorem

General assumptions
(Ω, F, P) complete probability space.
{(Wt , Ft )}t∈[0,∞) m-dimensional Brownian motion.
{Ft }t Brownian filtration.

Definition
A real-valued martingale {(Mt , Ft )}t∈[0,T ] with respect to the Brownian
filtration {Ft }t is called a Brownian martingale.

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The martingale representation theorem

Martingale representation theorem
Let {(Mt , Ft )}t∈[0,T ] be a square-integrable Brownian martingale, i.e.,

EMt2 < ∞ for all t ∈ [0, T ].

Then there exists a progressively measurable Rm -valued process Ψ(t)
with
T
2
E Ψ(t) dt <∞
0

and
t

Mt = M0 + Ψ(s)T dW (s) P-a.s. .
0

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Corollary
Let {(Mt , Ft )}t∈[0,T ] be a local martingale with respect to the Brownian
ﬁltration {Ft }t . Then there exists a progressively measurable
Rm -valued process Ψ(t) with
T
2
Ψ(t) dt < ∞
0

and
t

Mt = M0 + Ψ(s)T dW (s) P-a.s.
0

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Remark
Each local martingale with respect to the Brownian ﬁltration can
ˆ
be represented as an Ito process.
ˆ
Each Brownian martingale can be represented as an Ito process.

⇒ Quadratic variation and quadratic covariation are deﬁned.

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Outline

3 Option Pricing

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Outline

3 Option Pricing
Introduction
Examples
The Replication Principle
Arbitrage Opportunity
Continuation
Partial Differential Approach (PDA)
Arbitrage & Option Pricing

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Portfolio Optimization

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