blatt11

Lecture Notes

“FINANCIAL MATHEMATICS I:

Stochastic Calculus, Option Pricing,

Portfolio Optimization”

Holger Kraft

University of Kaiserslautern

Department of Mathematics

Mathematical Finance Group

Summer Term 2005

This Version: August 4, 2005

CONTENTS 1

Contents

1 Introduction 3

1.1 What is an Option? . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Fundamental Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Discrete-time State Pricing with Finite State Space 6

2.1 Single-period Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Description of the Model . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Arrow-Debreu Securities . . . . . . . . . . . . . . . . . . . . . 7

2.1.3 Deflator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.4 Risk-neutral Measure . . . . . . . . . . . . . . . . . . . . . . 10

2.1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Multi-period Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Modeling Information Flow . . . . . . . . . . . . . . . . . . . 14

2.2.2 Description of the Multi-period Model . . . . . . . . . . . . . 16

2.2.3 Some Basic Definitions Revisited . . . . . . . . . . . . . . . . 19

2.2.4 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.5 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Introduction to Stochastic Calculus 23

3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 Why Stochastic Calculus? . . . . . . . . . . . . . . . . . . . . 23

3.1.2 What is a Reasonable Model for Stock Dynamics? . . . . . . 23

3.2 On Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 Cadlag and Caglad . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.2 Modification, Indistinguishability, Measurability . . . . . . . 25

3.2.3 Stopping Times . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Prominent Examples of Stochastic Processes . . . . . . . . . . . . . . 29

3.4 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.5 Completition, Augmentation, Usual Conditions . . . . . . . . . . . . 34

3.6 Ito-Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.7 Ito’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Continuous-time Market Model and Option Pricing 47

CONTENTS 2

4.1 Description of the Model . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Black-Scholes Approach: Pricing by Replication . . . . . . . . . . . . 49

4.3 Pricing with Deflators . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.4 Pricing with Risk-neutral Measures . . . . . . . . . . . . . . . . . . . 57

5 Continuous-time Portfolio Optimization 68

5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2 Martingale Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3 Stochastic Optimal Control Approach . . . . . . . . . . . . . . . . . 73

5.3.1 Heuristic Derivation of the HJB . . . . . . . . . . . . . . . . . 73

5.3.2 Solving the HJB . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.3.3 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.4 Intermediate Consumption . . . . . . . . . . . . . . . . . . . . . . . . 76

5.5 Infinite Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

References 86

1 INTRODUCTION 3

1 Introduction

1.1 What is an Option?

• There exist two main variants: call and put.

• A European call gives the owner the right to buy some asset (e.g. stock, bond,electricity, corn . . . ) at a future date (maturity) for a fixed price (strike price).

• Payoff: maxS(T ) −K; 0, where S(T ) is the asset price at maturity T andK is the strike price.

• An American call gives the owner the right to buy some asset during the wholelife-time of the call.

• Put: replace “buy” with “sell”

• Since the owner has the right but not the obligation, this right has a positivevalue.

1.2 Fundamental Relations

Consider an option written on an underlying which pays no dividends etc. duringthe life-time of the option. In the following we assume that the underlying is a stock.The time-t call and put prices are denoted by Call(t) and Put(t). The time-t priceof a zero-coupon bond with maturity T is denoted by p(t, T ).

Proposition 1.1 (Bounds)

(i) For a European call we get: S(t) ≥ Call(t) ≥ maxS(t)−Kp(t, T ); 0

(ii) For a European put we have: Kp(t, T ) ≥ Put(t) ≥ maxKp(t, T )− S(t); 0

Proof. (i) The relation S(t) ≥ Call(t) is obvious because the stock can be in-terpreted as a call with strike 0. We now assume that Call(t) < maxS(t) −Kp(t, T ); 0. W.l.o.g. S(t)−Kp(t, T ) ≥ 0. Consider the strategy• buy one call,• sell one stock,• buy K zeros with maturity T ,

leading to an initial inflow of S(t)−Call(t)−Kp(t, T ) > 0. At time T , we distinguishtwo scenarios:

1. S(T ) > K: Then the value of the strategy is −S(T ) + S(T )−K +K = 0.

2. S(T ) ≤ K: Then the value of the strategy is −S(T ) + 0 +K ≥ 0.

Hence, the strategy would be an arbitrage opportunity.

(ii) The relation Kp(t, T ) ≥ Put(t) is valid because otherwise the strategy• sell one put,• buy K zeros with maturity T

1 INTRODUCTION 4

would be an arbitrage strategy. The second relation can be proved similarly to therelation for the call in (i). 2

Proposition 1.2 (Put-Call Parity)

The following relation holds: Put(t) = Call(t)− S(t) +Kp(t, T )

Proof. Consider the strategy• buy one put,• sell one call,• buy one stock,• sell K zeros with maturity T .

It is easy to check that the time-T value of this strategy is zero. Since no additionalpayments need to be made, no arbitrage dictates that the time-t value is zero aswell implying the desired result. 2

Remark. This is a very important relationship because it ties together Europeancall and put prices.

Proposition 1.3 (American Call)

Assume that interest rates are positive. Early exercise of an American call is not

optimal, i.e. Callam(t) = Call(t).

Proof. The following relations hold

Callam(t) ≥ Call(t)(∗)≥ maxS(t)−Kp(t, T ); 0 ≥ S(t)−K

where (∗) holds due to Proposition 1.1 (i). Hence, the American call is worth morealive than dead. 2

Remark.

• Unfortunately, this result does not hold for American puts.• If dividends are paid, then the result for the American call breaks down as well.

Proposition 1.4 (American Put)

Assume that interest rates are positive. Then the following relations hold:

(i) Callam(t)− S(t) +Kp(t, T ) ≤ Putam(t) ≤ Callam(t)− S(t) +K

(ii) Put(t) ≤ Putam(t) ≤ Put(t) +K(1− p(t, T )).

Proof. (i) The first inequality follows from the put-call parity and Proposition 1.3.For the second one assume that Putam(t) > Callam(t)−S(t) +K and consider thefollowing strategy:• sell one put,• buy one call,• sell one stock,• invest K Euros in the money market account.1

1The time-t value of the money market account is denoted by M(t). By convention, M(0) = 1.

1 INTRODUCTION 5

By assumption, the initial value of this strategy is strictly positive. For t ∈ [0, T ]we distinguish to scenarios:

1. S(t) > K: Then the value of the strategy is

0 + (S(t)−K)− S(t) +KM(t) = K(M(t)− 1) ≥ 0

since interest rates are positive and thus M(t) ≥ 1.

2. S(t) ≤ K: Then the value of the strategy is

−(K − S(t)) + 0− S(t) +KM(t) = K(M(t)− 1) ≥ 0.

Hence, the strategy would be an arbitrage opportunity.

(ii) We have

Put(t) ≤ Putam(t) ≤ Callam(t)− S(t) +K = Call(t)− S(t) +K

= Put(t) + S(t)−Kp(t, T )− S(t) +K = Put(t) +K(1− p(t, T )).

2

Remark. If interest rates are zero, then Put(t) = Putam(t).

2 DISCRETE-TIME STATE PRICING WITH FINITE STATE SPACE 6

2 Discrete-time State Pricing with Finite State

Space

2.1 Single-period Model

2.1.1 Description of the Model

• We consider a single-period model starting today (t = 0) and ending at timet = T . Without loss of generality T = 1.

• Uncertainty at time t = 1 is represented by a finite set Ω = ω1, . . . , ωI ofstates, one of which will be revealed as true. Ω is said to be the state space.

• Each state ωi ∈ Ω can occur with probability P (ωi) > 0. This defines aprobability measure P : Ω→ (0, 1).

• At time t = 0 one can trade in J + 1 assets. The time-0 prices of theseassets are described by the vector S′ = (S0, S1, . . . , SJ). The state-dependentpayoffs of the assets at time t = 1 are described by the payoff matrix

X =

X0(ω1) · · · XJ(ω1)

X0(ω2) · · · XJ(ω2)...

. . ....

X0(ωI) · · · XJ(ωI)

.

• To shorten notations, Xij := Xj(ωi).

• The first asset is assumed to be riskless, i.e. Xi0 = Xk0 = const. for all i,k ∈ 1, . . . , I.

• Therefore, at time t = 0 there exists a riskless interest rate given by

r =X0

S0− 1.

• An investor buys and sells assets at time t = 0. This is modeled via a tradingstrategy ϕ = (ϕ0, . . . , ϕJ)′, where ϕj denotes the number of asset j which theinvestor holds in his portfolio (combination of assets).

Definition 2.1 (Arbitrage)

We distinguish two kinds of arbitrage opportunities:

(i) A free lunch is a trading strategy ϕ with

S′ϕ < 0 and Xϕ ≥ 0.

(ii) A free lottery is a trading strategy ϕ with

S′ϕ = 0 and Xϕ ≥ 0 and Xϕ 6= 0.


Assumptions:

(a) All investors agree upon which states cannot occur at time t = 1.

(b) All investors estimate the same payoff matrix (homogeneous beliefs).

(c) At time t = 0 there is only one market price for each security and each investorcan buy and sell as many securities as desired without changing the marketprice (price taker).

(d) Frictionless markets, i.e.• assets are divisible,• no restrictions on short sales,2

• no transaction costs,• no taxes.

(e) The asset market contains no arbitrage opportunities, i.e. there are neither freelunches nor free lotteries.

2.1.2 Arrow-Debreu Securities

To characterize arbitrage-free markets, the notion of an Arrow-Debreu security iscrucial:

Definition 2.2 (Arrow-Debreu Security)

An asset paying one Euro in exactly one state and zero else is said to be an Arrow-

Debreu security (ADS). Its price is an Arrow-Debreu price (ADP).

Remarks.

• In our model, we have as many ADS as states at time t = 1, i.e. we have I

Arrow-Debreu securities.• The i-th ADS has the payoff ei = (0, . . . , 0, 1, 0, . . . , 0)′, where the i-th entry is

1. The corresponding Arrow-Debreu price is denoted by πi.• Unfortunately, in practice ADSs are not traded.

We can now prove the following theorem characterizing markets which do not con-tain arbitrage opportunities.

Theorem 2.1 (Characterization of No Arbitrage)

Under assumptions (a)-(d) the following statements are equivalent:

• The market does not contain arbitrage opportunities (Assumption (e)).

2A short sale of an assets is equivalent to selling an asset one does not own. An investor can do

this by borrowing the asset from a third party and selling the borrowed security. The net position

is a cash inflow equal to the price of the security and a liability (borrowing) of the security to

the third party. In abstract mathematical terms, a short sale corresponds to buying a negative

amount of a security.


• There exist strictly positive Arrow-Debreu prices, i.e. πi > 0, i = 1, . . . , I, such

that

Sj =I∑

i=1

πi ·Xij .

Proof. Suppose (ii) to hold. We consider a strategy ϕ with Xϕ ≥ 0 and Xϕ 6= 0.Then

S′ϕ(ii)= π′Xϕ

(∗)> 0,

where (∗) holds because π > 0, Xϕ ≥ 0, and Xϕ 6= 0. For this reason, there areno free lunches. The argument to exclude free lotteries is similar. The implication“(i)⇒(ii)” follows from the so-called Farkas-Stiemke lemma (See Mangasarian (1969,p. 32) or Dax (1997)). 2

Remarks.

• The ADPs are the solution of the following system of linear equations:

S = X ′π.

• The hard part of the proof is actually the proof of implication “(i)⇒(ii)”. Forbrevity, this proof is omitted here.

Given our market with prices S and payoff matrix X, it is now of interest how toprice other assets.

Definition 2.3 (Contingent Claim)

In the single-period model, a contingent claim (derivative, option) is some additional

asset with payoff C ∈ IRI .

Definition 2.4 (Replication)

Consider a market with payoff matrix X. A contingent claim with payoff C can be

replicated (hedged) in this market if there exists a trading strategy ϕ such that

C = Xϕ.

Remark. If an asset with payoff C is replicable via a trading strategy ϕ, by theno arbitrage assumption its price SC has to be (“law of one price”)

SC =J∑

j=1

ϕj · Sj .

Definition 2.5 (Completeness)

An arbitrage-free market is said to be complete if every contingent claim is replica-

ble.


Theorem 2.2 (Characterization of a Complete Market)

Assume that the financial market contains no arbitrage opportunities. Then the

market is complete if and only if the Arrow-Debreu prices are uniquely determined,

i.e.

Rank(X) = I.

Proof. π is unique.

⇐⇒ X ′π = S has a unique solution.

⇐⇒ Rank (X) = I

⇐⇒ Range (X) = IRI

⇐⇒ Every contingent claim is replicable. 2

Remarks.

• The market is complete if the number of assets with linearly independent payoffsis equal to the number of states.• If the number of assets is smaller than the number of states (J < I), the market

cannot be complete.

2.1.3 Deflator

Main result from the previous subsection: In an arbitrage-free market, thereexist strictly positive ADPs such that

Sj =I∑

i=1

πi ·Xij .

In this subsection: Using the physical probabilities P (ωi), i = 1, . . . , I, arbitrage-free asset prices can be represented as discounted expected values of the correspond-ing payoffs.

Definition 2.6 (Deflator)

In an arbitrage-free market, a deflator H is defined by

H =π

P.

Remarks.

• Since the market is arbitrage-free, there exists at least one vector of ADSs π, butπ and thus the deflator H are only unique if the market is complete.

• The ADS π and the probability measure P can be interpreted as random variableswith realizations πi = π(ωi) and P (ωi).

Proposition 2.3 (Pricing with a Deflator)

In an arbitrage-free market, the asset prices are given by

Sj = EP [H ·Xj ].


Proof.

Sj =I∑

i=1

πi ·Xij =I∑

i=1

P (ωi)π(ωi)P (ωi)

Xj(ωi) = EP

[ πPXj

]= EP [H ·Xj ]

2

Remarks.

• The deflator H is the appropriate discount factor for payoffs at time t = 1.• The deflator H ist a random variable with realizations π(ωi)/P (ωi).• On average, one needs to discount with the riskless rate r because

EP [H] =I∑

i=1

P (ωi) ·π(ωi)P (ωi)

=I∑

i=1

π(ωi) =1

1 + r.

Interpretation of a Deflator:

• In an equilibrium model one can show that the deflator equals the marginalrate of substitution between consumption at time t = 0 and t = 1 of a repre-sentative investor, i.e. (roughly speaking) we have

H =∂u/∂c1∂u/∂c0

,

where u = u(c0, c1) is the utility function of the representative investor.

• Setting u′0 := ∂u/∂c0 and u′1 := ∂u/∂c1 leads to

r =1

EP [H]− 1 =

u′0EP (u′1)

− 1,

which gives the following result:

(i) u′0 > E(u′1) (”‘time preference”’):3 r > 0,

(ii) u′0 < E(u′1): r < 0.

2.1.4 Risk-neutral Measure

Two important observations:

(i) If an investor buys all ADSs, he gets one Euro at time t = 1 for sure, i.e. we get

I∑i=1

πi =1

1 + r⇒

I∑i=1

πi · (1 + r) = 1.

(ii) In an arbitrage-free market, it holds that πi > 0, i = 1, . . . , I.

3Intuitively, time preference describes the preference for present consumption over future

consumption.


Proposition 2.4 (Risk-neutral Measure)

In an arbitrage-free market, there exists a probability measure Q defined via

Q(ωi) = qi := πi · (1 + r)

This measure is said to be the risk-neutral measure or equivalent martingale mea-

sure.

Proof. follows from (i) and (ii). 2

Using the risk-neutral measure, there is a third representation of an asset price:

Proposition 2.5 (Risk-neutral Pricing)

In an arbitrage-free market, asset prices possess the following representation

Sj = EQ

[Xj

R

],

where R := 1 + r.

Proof.

Sj =I∑

i=1

πi ·Xij =I∑

i=1

πi ·R ·Xij

R=

I∑i=1

qi ·Xij

R= EQ

[Xj

R

].

2

Remarks.

• Warning: The risk-neutral measure Q is different from the physical measure P .• Roughly speaking, risk-neutral probabilities are compounded Arrow-Debreu se-

curities.• The notion “risk-neutral measure” comes from the fact that under this measure

prices are discounted expected values where one needs to discount using theriskless interest rate.

To motivate the notion “equivalent martingale measure”, let us recall some basicdefinitions:

Definition 2.7 (Equivalent Probability Measures)

A Probability Measure Q is said to be continuous w.r.t. another probability measure

P if for any event A ⊂ Ω

P (A) = 0 =⇒ Q(A) = 0.

If Q is continuous w.r.t. P and vice versa, then P and Q are said to be equivalent,

i.e. both measures possess the same null sets.

Theorem 2.6 (Radon-Nikodym)

If Q is continuous w.r.t. P , then Q possesses a density w.r.t. Q, i.e. there exists a

positive random variable D = dQdP such that

EQ[Y ] = EP [D · Y ]

for any random variable Y . This density is uniquely determined (P -a.s.).


Definition 2.8 (Stochastic Process in a Single-period Model)

Consider a single-period model with time points t = 0, 1. The finite sequence

Y (t)t=0,1 forms a stochastic process.

Definition 2.9 (Martingale in a Single-period Model)

Consider a single-period model with finite state space. Uncertainty at time t = 1is modeled by some probability measure Q. A real-valued stochastic process is said

to be a martingale w.r.t. Q (short: Q-martingale) if

EQ[Y (1)] = Y (0),

where it is assumed that EQ[|Y (1)|] <∞.

We can now characterize risk-neutral measures as follows:

Proposition 2.7 (Properties of a Risk-neutral Measure)

(i) The discounted price processes Sj , Xj/R are martingales under a risk-neutral

measure Q.

(ii) The physical measure P and a risk-neutral measure Q are equivalent.

Proof. (i) follows from Proposition 2.5.

(ii) Since ADPs are strictly positive, we have Q(ωi) > 0 and P (ωi) > 0 which givesthe desired result. 2

Remarks.

• Proposition 2.7 is the reason why Q is said to be an equivalent martingale

measure.• From Propositions 2.3 and 2.5, we get

EQ

[Xj

R

]= Sj = EP [H ·Xj ].

Therefore, we obtainEQ[Xj ] = EP [ H ·R︸︷︷︸

=dQ/dP

·Xj ],

i.e. H = (dQ/dP )/R. Analogously, one can show dP/dQ = 1/(H ·R).

2.1.5 Summary

Theorem 2.8 (No Arbitrage)

Consider a financial market with price vector S and payoff matrix X. If the as-

sumptions (a)-(d) are satisfied, then the following statements are equivalent:

(i) The financial market contains no arbitrage opportunities (assumption (e)).

(ii) There exist strictly positive Arrow-Debreu prices πi, i = 1, . . . , I, such that

Sj =I∑

i=1

πi ·Xij .


(iii) There exists a state-dependent discount factor H, the so-called deflator, such

that for all j = 0, . . . , JSj = EP [H ·Xj ].

(iv) There exists an equivalent martingale measure Q such that the discounted

price processes Sj , Xj/R are martingales under Q, i.e. for all j = 0, . . . , J

Sj = EQ

[Xj

R

]

Theorem 2.9 (Completeness)

Consider a financial market with price vector S and payoff matrix X. If the as-

sumptions (a)-(e) are satisfied, then the following statements are equivalent:

(i) The financial market is complete, i.e. each additional asset can be replicated.

(ii) The Arrow-Debreu prices πi, i = 1, . . . , I, are uniquely determined.

(iii) There exists only one deflator H.

(iv) There exists only one equivalent martingale Q.

Remarks.

• In a complete market, the price of any additional asset is given by the price ofits replicating portfolio.• The value of this portfolio can be calculated applying the following theorem.

Theorem 2.10 (Pricing of Contingent Claims)

Consider an arbitrage-free and complete market with price vector S and payoff

matrix X. Let C ∈ IRI be the payoff of a contingent claim. Its today’s price SC

can be calculated by using one of the following formulas:

(i) SC = C ′π =I∑

i=1

πi · C(ωi) (Pricing with ADSs).

(ii) SC = EP [H · C] (Pricing with the deflator).

(iii) SC = EQ

[CR

](Pricing with the equivalent martingale measure).

Proof. In an arbitrage-free and complete market, there exist ADPs which areuniquely determined. The no-arbitrage requirement together with the definition ofan ADP gives formula (i). Formulas (ii) and (iii) follow from (i) using H = π/P

and Q(ωi) = πi ·R. 2


2.2 Multi-period Model

In the previous section we have modeled only two points in time:

t = 0: The investor chooses the trading strategy.

t = T : The investor liquidates his portfolio and gets the payoffs.

This has the following consequences:• We have disregarded the information flow between 0 and T .• Every trading strategy was a static strategy (”‘buy and hold”’), i.e. intermediate

portfolio adjustments was not allowed.

In this section

• we will model the information flow between 0 and T .→ Filtration• and the investor can adjust his portfolio according to the arrival of new informa-

tion about prices.→ dynamic (self-financing) trading strategy.

2.2.1 Modeling Information Flow

Motivation

Consider the dynamics of a stock in a two-period binomial model:

121

110

100

90

81

99

1

PPPPPPPq

1

PPPPPPPq

1

PPPPPPPq

t = 0 t = 0.5 t = 1

• Due to the additional trading date at time t = 0.5, the investor gets more infor-mation about the stock price at time t = 1.• At t = 0 it is possible that the stock price at time t = 1 equals 81, 99 or 121.• If, however, the investor already knows that at time t = 0.5 the stock price equals

110 (90), then at time t = 1 the stock price can only take the values 121 or 99(99 or 81).• It is one of the goals of this section to model this additional piece of information!

Reference Example

To demonstrate how one can model the information flow, we consider the followingexample:


ω1

ω1, ω2

ω1, ω2, ω3, ω4

ω3, ω4

ω4

ω2, ω3

1

PPPPPPPq

1

PPPPPPPq

1

PPPPPPPq

t = 0 t = 1 t = 2

The state space thus equals: Ω = ω1, ω2, ω3, ω4.

Example for the interpretation of the states.

ω1 : DAX stands at 5000 at t = 1 and at 6000 at time t = 2.ω2 : DAX stands at 5000 at t = 1 and at 4000 at time t = 2.ω3 : DAX stands at 3000 at t = 1 and at 4000 at time t = 2.ω4 : DAX stands at 3000 at t = 1 and at 2000 at time t = 2.

First Approach to Model the Flow of Information

Definition 2.10 (Partition)

A partition of the state space is a set A1, . . . , AK of subsets Ak ⊂ Ω of a state

space Ω such that

• the subsets Ak are disjoint, i.e. Ak ∩Al = ∅,• Ω = A1 ∪ . . . ∪AK .

Example. ω1, ω2, ω3, ω4.

Definition 2.11 (Information Structure)

An information structure is a sequence Ftt of partitions F0, . . ., FT such that

• F0 = ω1, . . . , ωI (”‘ At the beginning each state can occur.”’),

• FT = ω1, . . . , ωI (”‘ At the end, one knows for sure which

state has occurred. ”’),

• the partitions become finer over time.

Example.

F0 = ω1, ω2, ω3, ω4.

F1 = ω1, ω2, ω3, ω4.

F2 = ω1, ω2, ω3, ω4.

Remarks.

• Problem: This intuitive definition cannot be generalized to the continuous-timecase.• However, it provides a good intuition of what information flow actually means.


Second Approach to Model the Flow of Information: Filtration

Definition 2.12 (σ-Field)

A system F of subsets of the state space Ω is said to be a σ-field if

(i) Ω ∈ F ,

(ii) A ∈ F ⇒ AC ∈ F ,

(iii) for every sequence Ann∈IN with An ∈ F we have∞⋃

n=1An ∈ F .

Remark. For a finite σ-field, it is sufficient if finite (instead of infinite) unions ofevents belong to the σ-field (see requirement (iii) of Definition 2.12).

Example. ∅, ω1, ω2, ω3, ω4,Ω.

Definition 2.13 (Filtration)

Consider a state space F . A family Ftt of sub-σ-fields Ft, t ∈ I, is said to be a

filtration if Fs ⊂ Ft, s ≤ t and s, t ∈ I. Here I denotes some ordered index set.

Remark. In our discrete time model, we can always use filtrations with the fol-lowing properties (P(Ω) denotes the power-set of Ω):• F0 = ∅,Ω (”‘At the beginning, one has no information.”’),• FT = P(Ω) (”‘At the end one is fully informed.”’),• F0 ⊂ F1 ⊂ . . . ⊂ FT (”‘The state of information increases over time.”’).

Example. F0 = ∅,Ω, F1 = ∅, ω1, ω2, ω3, ω4,Ω, F2 = P(Ω).

Some facts about filtrations:

• Filtrations are used to model the flow of information over time.• At time t, one can decide if the event A ∈ Ft has occurred or not.• Taking conditional expectation E[Y |Ft] of a random variable Y means that one

takes expectation on the basis of all information available at time t.

2.2.2 Description of the Multi-period Model

Properties of the Model:

• Trading takes place at T + 1 dates, t = 0, . . . , T .• Investors can trade in J + 1 assets with price processes (S0(t), . . . , SJ(t)), t =

0, . . . , T .• Sj(t) denotes the price of the j-th asset at time t.• The range of Sj(t) is finite.• Trading of an investor is modeled via his trading strategy ϕ(t) = (ϕ0(t), . . . , ϕJ(t))′,

where ϕj(t) denotes the number of asset j at time t in the investor’s portfolio.• A trading strategy needs to be predictable, i.e. on the basis of all information

about prices S(t−1) at time t−1 the investor selects his portfolio which he holdsuntil time t.


• The value of a trading strategy at time t is given by:

Vϕ(t) =

S(t)′ϕ(t) =

∑Jj=0 Sj(t)ϕj(t) fur t = 1, . . . , T

S(0)′ϕ(1) =∑J

j=0 Sj(0)ϕj(1) fur t = 0

Definition 2.14 (Contingent Claim)

A contingent claim leads to a state-dependent payoff C(t) at time t, which is mea-

surable w.r.t. Ft.

Remarks.

• A typical contingent claim is a European call with payoff C(T ) = maxS(T ) −K; 0, where S(T ) denotes the value of the underlying at time T and K the strikeprice.• American options are no contingent claims in the sense of Definition 2.14, but

one can generalize this definition accordingly.• Recall the following paradigm: Arbitrage-free pricing of a contingent claim

means finding a hedging (replication) strategy of the corresponding payoff.• In the single-period model: The price of the replication strategy is then the price

of the contingent claim.• In the multi-period model we need an additional requirement: The replication

strategy should not require intermediate inflows or outflows of cash.• Only in this case, the initial costs are equal to the fair price of the contingent

claim.• This leads to the notion of a self-financing trading strategy.

Definition 2.15 (Self-financing Trading Strategy)

A trading strategy ϕ(t) = (ϕ1(t), . . . , ϕJ(t)) is said to be self-financing if for t =1, . . . , T − 1

S(t)′ϕ(t) = S(t)′ϕ(t+ 1) ⇐⇒J∑

j=0

Sj(t) · ϕj(t) =J∑

j=0

Sj(t) · ϕj(t+ 1).

Remarks.

• To buy assets, the investor has to sell other ones.• For this reason, the portfolio value changes only if prices change and not if the

numbers of assets in the portfolio change.

Notation:

• Change of the value of strategy ϕ at time t:

∆Vϕ(t) := Vϕ(t)− Vϕ(t− 1)

• Price changes at time t

∆S(t) = (∆S0(t), . . . ,∆SJ(t)) := (S0(t)− S0(t− 1), . . . , SJ(t)− SJ(t− 1))


• Gains process of the trading strategy ϕ at time t:

ϕ(t)′∆S(t) =J∑

j=0

ϕj(t)∆Sj(t)

• Cumulative gains process of the trading strategy ϕ at time t:

Gϕ(t) :=t∑

τ=1

ϕ(τ)′∆S(τ) =t∑

τ=1

J∑j=0

ϕj(τ)∆Sj(τ)

Proposition 2.11 (Characterization of a Self-financing Trading Strategy)

A self-financing trading strategy ϕ(t) has the following properties:

(i) The value of ϕ changes only due to price changes, i.e.

∆Vϕ(t) = ϕ(t)′∆S(t).

(ii) The value of ϕ is equal to its initial value plus the cumulative gains, i.e.

Vϕ(t) = Vϕ(0) +Gϕ(t) = Vϕ(0) +t∑

τ=1

ϕ(τ)′∆S(τ).

Proof. (i) For a self-financing strategy ϕ, we obtain

∆Vϕ(t) = Vϕ(t)− Vϕ(t− 1)

=J∑

j=0

Sj(t)ϕj(t)−J∑

j=0

Sj(t− 1)ϕj(t− 1)

(∗)=

J∑j=0

Sj(t)ϕj(t)−J∑

j=0

Sj(t− 1)ϕj(t)

=J∑

j=0

ϕj(t)(Sj(t)− Sj(t− 1)

)

=J∑

j=0

ϕj(t)∆Sj(t)

= ϕ(t)′∆S(t).

The equality (∗) is valid because ϕ is self-financing.

(ii) Any (not necessary self-financing) trading strategy satisfies

Vϕ(t) = Vϕ(0) +t∑

τ=1

∆Vϕ(τ).

Since ϕ is assumed to be self-financing, by (i), we obtain

Vϕ(t) = Vϕ(0) +t∑

τ=1

∆Vϕ(τ).

(i)= Vϕ(0) +

t∑τ=1

ϕ(τ)′∆S(τ)

= Vϕ(0) +Gϕ(t).


2

Remarks.

• In continuous-time models, the relation (i) is still valid if ∆ is replace by d.• Unfortunately, the intuitive relation

S(t)′ϕ(t) = S(t)′ϕ(t+ 1)

has no continuous-time equivalent.

2.2.3 Some Basic Definitions Revisited

The notion of a self-financing trading strategy is also important to generalize thefollowing definitions to the multi-period setting:

Definition 2.16 (Arbitrage)

We distinguish two kinds of arbitrage opportunities:

(i) A free lunch is a self-financing trading strategy ϕ with

Vϕ(0) < 0 und Vϕ(T ) ≥ 0.

(ii) A free lottery is a self-financing trading strategy ϕ with

Vϕ(0) = 0 und Vϕ(T ) ≥ 0 und Vϕ(T ) 6= 0.

Definition 2.17 (Replication of a Contingent Claim)

A contingent claim with payoff C(T ) is said to be replicable if there exists a self-

financing trading strategy ϕ such that

C(T ) = Vϕ(T ).

The trading strategy ϕ is said to be a replication (hedging) strategy for the claim

C(T ).

Remarks.

• By definition, a replication strategy of a claim leads to the same payoff as theclaim.• In general, replication strategies are dynamic strategies, i.e. at every trading

date the hedge portfolio needs to be adjusted according to the arrival of newinformation about prices.

The definition of a complete market remains the same:

Definition 2.18 (Completeness)

An arbitrage-free market is said to be complete if each contingent claim can be

replicated.


2.2.4 Assumptions

(a) All investors agree upon which states cannot occur at times t > 0.

(b) All investors agree upon the range of the price processes S(t).

(c) At time t = 0 there is only one market price for each security and each investorcan buy and sell as many securities as desired without changing the marketprice (price taker).

(d) Frictionless markets, i.e.• assets are divisible,• no restrictions on short sales,4

• no transaction costs,• no taxes.

(e) The asset market contains no arbitrage opportunities, i.e. there are neither freelunches nor free lotteries.

2.2.5 Main Results

Conventions:

• The 0-th asset is a money market account, i.e. it is locally risk-free with

S0(0) = 1 und S0(t) =t∏

u=1

(1 + ru), t = 1, . . . , T,

where ru denotes the interest rate for the period [τ − 1, τ ].• The discounted (normalized) price processes are defined by

Zj(t) :=Sj(t)S0(t)

.

• The money market account is thus used as a numeraire.

Definition 2.19 (Martingal)

A real-valued adapted process Y (t)t, t = 0, . . . , T , with E[|Y (t)|] < ∞ is a mar-

tingale w.r.t. a probability measure Q if for s ≤ t

EQ[Y (t)|Fs] = Y (s).

Remark. A martingale is thus a process being on average stationary.

Important Facts:4A short sale of an assets is equivalent to selling an asset one does not own. An investor can do

this by borrowing the asset from a third party and selling the borrowed security. The net position

is a cash inflow equal to the price of the security and a liability (borrowing) of the security to

the third party. In abstract mathematical terms, a short sale corresponds to buying a negative

amount of a security.


• A multi-period model consists of a sequence of single-period models.• Therefore, a multi-period model is arbitrage-free if all single-period models are

arbitrage-free (See, e.g., Bingham/Kiesel (2004), Chapter 4).

For these reasons, the following three theorems hold:

Theorem 2.12 (Characterization of No Arbitrage)

If assumptions (a)-(d) are satisfied, the following statements are equivalent:

(i) The financial market contains no arbitrage opportunities (assumption (e)).

(ii) There exist a state-dependent discount factor H, the so-called deflator, such

that for all time points s, t ∈ 0, . . . , T with s ≤ t and all assets j = 0, . . . , Jwe have

Sj(s) =EP [H(t)Sj(t)|Fs]

H(s),

i.e. all prices can be represented as discounted expectations under the physical

measure.

(iii) There exists a risk-neutral probability measure Q such that all discounted

price processes Zj(t) are martingales under Q, i.e. for all time points s, t ∈0, . . . , T with s ≤ t and all assets j = 1, . . . , J we have

Zj(s) = EQ[Zj(t)|Fs] ⇐⇒ Sj(s)S0(s)

= EQ

[Sj(t)S0(t)

∣∣∣Fs

]Remarks.

• The equivalence “No Arbitrage ⇔ Existence of Q” is known as Fundamental

Theorem of Asset Pricing.• Unfortunately, in continuous-time models this equivalence breaks down.• However, the implication “Existence of Q ⇒ No Arbitrage”’ is still valid,

Theorem 2.13 (Pricing of Contingent Claims)

Consider a financial market satisfying assumptions (a)-(e) and a contingent claim

with payoff C(T ). Then we obtain for s, t ∈ 0, . . . , T with s ≤ t:

(i) The replication strategy ϕ for the claim is unique, i.e. the arbitrage-free price

of the claim is uniquely determined by

C(s) = Vϕ(s).

(ii) Using the deflator, we can calculate the claim’s time-s price

C(s) =EP [H(t)C(t)|Fs]

H(s).

(iii) The following risk-neutral pricing formula holds:

C(s) = S0(s) · EQ

[ C(t)S0(t)

∣∣∣Fs

].


Remarks.

• Result (i) says that the today’s price of a claim is uniquely determined by itsreplication costs, i.e. C(0) = Vϕ(0).• From (iii) we get

C(s)S0(s)

= EQ

[ C(t)S0(t)

∣∣∣Fs

],

i.e. the discounted price processes of contingent claims are Q-martingales as well.Note that one needs to discount with the money market account.• If interest rates are constant, then from (iii)

C(0) = EQ

[ C(T )(1 + r)T

]=

EQ[C(T )](1 + r)T

.

• Pricing of contingent claims boils down to computing expectations under therisk-neutral measure.

Theorem 2.14 (Completeness)

Consider a financial market satisfying assumptions (a)-(e). Then the following

statements are equivalent

(i) The market is complete.

(ii) There exists a unique deflator H.

(iii) There exists a unique martingale measure Q.

2.2.6 Summary

The following results are important:• It is always valid that a financial market contains no arbitrage opportunities if

an equivalent martingale measure exists.• In our discrete model, the converse is also valid: If the market contains no arbi-

trage opportunities, there exists a martingale measure.• If a martingale measureQ exists, the discounted price processes areQ-martingales.• If a unique martingale measure exists, the market contains is arbitrage-free and

complete.• If a unique martingale measure exists, the today’s price of a contingent claim is

given by the following Q-expectation

C(0) = EQ

[ C(T )S0(T )

].

3 INTRODUCTION TO STOCHASTIC CALCULUS 23

3 Introduction to Stochastic Calculus

3.1 Motivation

3.1.1 Why Stochastic Calculus?

• In the previous section, the investor was able to trade at finitely many datest = 0, . . . , T only.

• In a continuous-time model, trading takes place continuously, i.e. at any datein the time period [0, T ].

• Consequently, price dynamics are modeled continuously.

• In continuous-time models asset dynamics are modeled via stochastic differ-ential equations.

Example “Black-Scholes Model”. Two investment opportunities:• money market account: riskless asset with constant interest rate r modeled via

an ordinary differential equation (ODE):

dM(t) = M(t)rdt, M(0) = 1.

• stock: risky asset modeled via a stochastic differential equation (SDE)

dS(t) = S(t)[µdt+ σdW (t)

], S(0) = s0,

where W is a Brownian motion.

In this section we answer the following questions:

Q1 What is a Brownian motion?

Q2 What is the meaning of dW (t)? → Stochastic Integral

Q3 What are stochastic differential equations and do explicit solutions exist? →Variation of Constant

Q4 How does the dynamics of contingent claims look like? → Ito Formula

3.1.2 What is a Reasonable Model for Stock Dynamics?

Model for the Money Market Account

The solution of the ODE B′(t) = r ·B(t) with B(0) = 1 is

B(t) = b0 exp(rt)


and thusln(B(t)) = ln(b0) + r · t.

Model for the Stock

The result for the money market account motivates the following model for thestock dynamics:

ln(S(t)) = ln(s0) + α · t+ “randomness”

Notation: Y (t) = “randomness”, i.e. Y (t) = ln(S(t))− ln(s0)− α · t

Reasonable properties of Y :

1) no tendency, i.e. E[Y (t)] = 0,

2) increasing with time t, i.e. Var(Y (t)) increases with t,

3) no memory, i.e. for Y (t) = Y (s)+ [Y (t)−Y (s)] the difference [Y (t)−Y (s)]• depends only on t− s and• is independent of Y (s),

Properties which would make life easier:

4) normally distributed, i.e. Y (t) ∼ N (0, σ2t), σ > 0,

5) continuous.

3.2 On Stochastic Processes

Definition 3.1 (Stochastic Process)

A stochastic processX defined on (Ω,F) is a collection of random variables Xt)t∈I ,i.e. σXt ⊂ F .

Remark. As otherwise stated, our stochastic processes take values in the mea-surable space (IRn,B(IRn)), i.e. the n-dimensional Euclidian space equipped withσ-field of Borel sets. Therefore, every Xt is (F ,B(IRn))-measurable.

3.2.1 Cadlag and Caglad

Definition 3.2 (Cadlag and Caglad Functions)

Let f : [0, T ] → IRn be a function with existing left and right limits, i.e. for each

t ∈ [0, T ] the limits

f(t−) = limst

f(s), f(t+) = limst

f(s)

exist. The function f is cadlag if f(t) = f(t+) and caglad if f(t) = f(t−).

A cadlag function cannot jump around too widly as the next proposition shows.


Proposition 3.1

(i) A cadlag function f can have at most a countable number of discontinuities, i.e.

t ∈ [0, T ] : f(t) 6= f(t−) is countable.

(ii) For any ε > 0, the number of discontinuities on [0, T ] larger than ε is finite.

Proof. See Fristedt/Gray (1997).

3.2.2 Modification, Indistinguishability, Measurability

Recall the following definition:

Definition 3.3 (Filtration)

Consider a σ-field F . A family Ft of sub-σ-fields Ft, t ∈ I, is said to be a

filtration if Fs ⊂ Ft, s ≤ t and s, t ∈ I. Here I denotes some ordered index set.

Definition 3.4 (Adapted Process)

A process X is said to be adapted to the filtration Ftt∈I if every Xt is measurable

w.r.t. Ft, i.e. σXt ⊂ Ft.

Remarks.

• The notation Xt,Ftt∈I indicates that the process X is adapted to Ftt∈I .• For fixed ω ∈ Ω, the realization Xt(ω)t is said to be a path ofX. For this reason,

a stochastic process can be interpreted as function-valued random variable.

Definition 3.5 (Modification, Indistinguishability)

Let X and Y be stochastic processes.

(i) Y is a modification of X if

P (ω : Xt(ω) = Yt(ω)) = 1 for all t ≥ 0.

(ii) X and Y are indistinguishable if

P (ω : Xt(ω) = Yt(ω) for all t ≥ 0) = 1,

i.e. X and Y have the same paths P -a.s.

Example. Consider a positive random variable τ with a continuous distribution.Set Xt ≡ 0 and Yt = 1t=τ. Then Y is a modification of X, but P (Xt =Yt for all t ≥ 0) = 0.

Proposition 3.2 (Modification, Indistinguishability)

Let X and Y be stochastic processes.

(i) If X and Y are indistinguishable, then Y is a modification of X and vice versa.

(ii) If X and Y have right-continuous paths and Y is a modification of X, then X

and Y are indistinguishable.

(iii) Let X be adapted to Ft. If Y is a modification of X and F0 contains all

P -null sets of F , then Y is adapted to Ftt.


Proof. (i) is obvious.

(ii) Since Y is a modification of X, we have

P (ω|Xt(ω) = Yt(ω) for all t ≥ [0,∞) ∩Q ) = 1.

By the right-continuity assumption, the claim follows.

(iii) Exercise. 2

If X is an (adapted) stochastic process, then every Xt is (F ,B(IRn))-measurable((Ft,B(IRn))-measurable). However, X is actually a function-valued random vari-able, and so, for technical reasons, it is often convenient to have some joint measur-ability properties.

Definition 3.6 (Measurable Stochastic Process)

A stochastic process is said to measurable, if for each A ∈ B(IRn) the set (t, ω) :Xt(ω) ∈ A belongs to the product σ-field B(IRn)⊗F , i.e. if the mapping

(t, ω) 7→ Xt(ω) : ([0,∞)× Ω,B([0,∞))⊗F)→ (IRn,B(IRn))

is measurable.

Definition 3.7 (Progressively Measurable Stochastic Process)

A stochastic process is said to progressively measurable w.r.t. the filtration Ft, if

for each t ≥ 0 and A ∈ B(IRn) the set (s, ω) ∈ [0, t] × Ω : Xs(ω) ∈ A belongs to

the product σ-field B([0, t])⊗Ft, i.e. if the mapping

(s, ω) 7→ Xs(ω) : ([0, t]× Ω,B([0, t])⊗Ft)→ (IRn,B(IRn))

is measurable for each t ≥ 0.

Remark. Any progressively measurable process is measurable and adapted (Exer-cise). The following proposition provides the extent to which the converse is true.

Proposition 3.3

If a stochastic process is measurable and adapted to the filtration Ft, then it has

a modification which is progressively measurable w.r.t. Ft.

Proof. See Karatzas/Shreve (1991, p. 5).

Fortunately, if a stochastic process is left- or right-continuous, a stronger result canbe proved:

Proposition 3.4

If the stochastic process X is adapted to the filtration Ft and right- or left-

continuous, then X is also progressively measurable w.r.t. Ft.

Proof. See Karatzas/Shreve (1991, p. 5).

Remark. If X is right- or left-continuous, but not necessarily adapted to Ft,then X is at least measurable.


3.2.3 Stopping Times

Consider a measurable space (Ω,F). An F- measurable random variable τ : Ω →I ∪ ∞ where I = [0,∞) or I = IN is said to be a random time. Two specialclasses of random times are of particular importance.

Definition 3.8 (Stopping Time, Optional Time)

(i) A random time is a stopping time w.r.t. the filtration Ftt∈I if the event

τ ≤ t ∈ Ft for every t ∈ I.

(ii) A random time is an optional time w.r.t. the filtration Ftt∈I if the event

τ < t ∈ Ft for every t ∈ I.

Propositions 3.5 and 3.6 show how both concepts are connected.

Proposition 3.5

If τ is optional and c > 0 is a positive constant, then τ + c is a stopping time.

Proof. Exercise.

Definition 3.9 (Continuity of Filtrations)

(i) A filtration Gt is said to be right-continuous if

Gt = Gt+ :=⋂ε>0

Gt+ε

(ii) A filtration Gt is said to be left-continuous if

Gt = Gt− := σ

⋃s<t

Gs

Remark. Let X be stochastic process.• Loosely speaking, left-continuity of FX

t means that Xt can be guessed by ob-serving Xs, 0 ≤ s < t.• Right-continuity means intuitively that if Xs has been observed for 0 ≤ s ≤ t,

then one cannot learn more by peeking infinitesimally far into the future.

Proposition 3.6

Every stopping time is optional, and both concepts coincide if the filtration is right-

continuous.

Proof. Consider a stopping time τ . Then τ < t =⋃

t>ε>0T ≤ t − ε andT ≤ t−ε ∈ Ft−ε ⊂ Ft, i.e. τ is optional. On the other hand, consider an optionaltime τ w.r.t. a right-continuous filtration Ft. Since τ ≤ t =

⋂t+ε>u>tT < u,

we have τ ≤ t ∈ Ft+ε for every ε > 0 implying τ ≤ t =⋂

ε>0 Ft+ε = Ft+ = Ft.2

Remark. In our applications, we will always work with right-continuous filtrations.5

5See Definition 3.18.


Proposition 3.7

(i) Let τ1 and τ2 be stopping times. Then τ1∧τ2 = minτ1, τ2, τ1∨τ2 = maxτ1, τ2,τ1 + τ2, and ατ1 with α ≥ 1 are stopping times.

(ii) Let τnn∈IN be a sequence of optional times. Then the random times

supn≥1

τn, infn≥1

τn, limn→∞τn, limn→∞τn

are all optional. Furthermore, if the τn’s are stopping times, then so is supn≥1 τn.

Proof. Exercise.

Proposition and Definition 3.8 (Stopping Time σ-Field)

Let τ be a stopping time of the filtration Ft. Then the set

A ∈ F : A ∩ τ ≤ t ∈ Ft for all t ≥ 0

is a σ-field, the so-called stopping time σ-field Fτ .

Proof. Exercise.

Proposition 3.9 (Properties of a Stopped σ-Field)

Let τ1 and τ2 be stopping times.

(i) If τ1 = t ∈ [0,∞), then Fτ1 = Ft.

(ii) If τ1 ≤ τ2, then Fτ1 ⊂ Fτ2 .

(iii) Fτ1∧τ2 = Fτ1 ∩ Fτ2 and the events

τ1 < τ2, τ2 < τ1, τ1 ≤ τ2, τ2 ≤ τ1, τ1 = τ2

belong to Fτ1 ∩ Fτ2 .

Proof. Exercise.

Definition 3.10 (Stopped Process)

(i) If X is a stochastic process and τ is random time, we define the function Xτ on

the event τ <∞ by

Xτ (ω) := Xτ(ω)(ω)

If X∞(ω) is defined for all ω ∈ Ω, then Xτ can also be defined on Ω, by setting

Xτ (ω) := X∞(ω) on τ =∞.

(ii) The process stopped at time τ is the process Xt∧τt∈[0,∞).

Proposition 3.10 (Measurability of a Stopped Process)

(i) If the process X is measurable and the random time τ is finite, then the function

Xτ is a random variable.

(ii) If the processX is progressively measurable w.r.t. Ftt∈[0,∞) and τ is a stopping

time of this filtration, then the random variable Xτ , defined on the set τ <∞ ∈Fτ , is Fτ -measurable, and the stopped process Xτ∧t,Ftt∈[0,∞) is progressively

measurable.


Proof. Karatzas/Shreve (1991, p. 5 and p. 9).

Remark. The following result is obvious: If X is adapted and cadlag and τ is astopping time, then

Xt∧τ = Xt1t<τ +Xτ1t≥τ.

is also adapted and cadlag.

Definition 3.11 (Hitting Time)

Let X be a stochastic process and let A ∈ B(IRn) be a Borel set in IRn. Define

τA(ω) = inft > 0 : Xt(ω) ∈ A.

Then τ is called a hitting time of A for X.

Proposition 3.11

Let X be a right-continuous process adapted to the filtration Ft.

(i) If Ao is an open set, then the hitting time of Ao is an optional time.

(ii) If Ac is a closed set, then the random variable

τAc(ω) = inft > 0 : Xt(ω) ∈ Ac or Xt−(ω) ∈ Ac.

is a stopping time.

Proof. (i) For simplicity n = 1. We have τAo< t =

⋃s∈Q∩[0,t)Xs ∈ Ao. The

inclusion ⊃ is obvious. Suppose that the inclusion ⊂ does not hold. Then we haveXi ∈ Ao for some i ∈ IR \ Q and i < t, but Xq 6∈ Ao for all q ∈ Q with q < t.Since Ao is open, there exists a δ > 0 such that (Xi − δ,Xi + δ) ⊂ Ao. Therefore,Xq 6∈ (Xi− δ,Xi + δ). However, then there exists a sequence (qn)n∈IN with qn s0

and qn ∈ Q ∩ [0, t), but Xqn6→ Xi which is a contradiction to the right-continuity

of X.

(ii) See Karatzas/Shreve (1991, p. 39) and Protter (2005, p. 5). 2

Remarks.

• If the filtration is right-continuous, then the hitting time τAo is a stopping time.• If X is continuous , then the stopping time τAc is a hitting time of Ac.• Even if X is continuous, in general the hitting time of Ao is not a stopping time.

For example, suppose that X is real-valued, that for some ω, Xt(ω) ≤ 1 for t ≤ 1,and that X1(ω) = 1. Let Ao = (1,∞). Without looking slightly ahead of time 1,we cannot tell wheter or not τAo = 1.

3.3 Prominent Examples of Stochastic Processes

Theorem and Definition 3.12 (Brownian Motion)

(i) A Brownian motion is an adapted, real-valued process Wt,Ftt∈[0,∞) such that

W0 = 0 a.s. and


• Wt −Ws ∼ N (0, t− s) for 0 ≤ s < t “stationary increments”

• Wt −Ws independent of Fs for 0 ≤ s < t “independent increments”

exists and is said to be a one-dimensional Brownian motion.

(ii) An n-dimensional Brownian motion W := (W1, . . . ,Wn) is an n-dimensional

vector of independent one-dimensional Brownian motions.

Proof. Applying Kolmogorov’s extension theorem,6 it is straightforward to provethe existence of a Brownian motion. See, e.g., Karatzas/Shreve (1991, pp. 47ff). 2

Theorem 3.13 (Kolmogorov’s Continuity Theorem)

Suppose that the process X = Xtt∈[0,∞) satisfies the following condition: For all

T > 0 there exist positive constants α, β, C such that

E[|Xt −Xs|α] ≤ C · |t− s|1+β , 0 ≤ s, t ≤ T.

Then there exists a continuous modification of X.

Proof. See Karatzas/Shreve (1991, pp.53ff).

Corollary 3.14 (Continuous Version of the Brownian Motion)

An n-dimensional Brownian motion possesses a continuous modification.

Proof. One can show that

E[|Wt −Ws|4] = n(n+ 2)|t− s|2.

Hence, we can apply Kolmogorov’s continuity theorem with α = 4, C = n(n + 2),and β = 1. 2

Remark. We will always work with this continuous version.

Definition 3.12 (Poisson Process)

A Poisson process with intensity λ > 0 is an adapted, integer-valued cadlag process

N = Nt,Ftt[0,∞) such that N0 = 0, and for 0 ≤ t <∞, the difference Nt −Ns is

independent of Fs and is Poisson distributed with mean λ(t− s).7

3.4 Martingales

Definition 3.13 (Martingale, Supermartingale, Submartingale)

A process X = Xtt∈[0,∞) adapted to the the filtration Ftt∈[0,∞) is called a

martingale (resp. supermartingale, submartingale) with respect to Ftt∈[0,∞) if

(i) Xt ∈ L1(dP ), i.e. E[|Xt|] <∞.

6Kolmogorov’s extension theorem basically says that for given (finite-dimensional) distributions

satisfying a certain consistency condition there exists a probability space and a stochastic process

such that its (finite-dimensional) distributions coincide with the given distributions. See, e.g.,

Karatzas/Shreve (1991, p. 50).7A random variable Y taking values in IN ∪ 0 is Poisson distributed with parameter λ > 0 if

P (Y = n) = e−λλn/(n!), n ∈ IN .


(ii) if s ≤ t, then E[Xt|Fs] = Xs, a.s. (resp. E[Xt|Fs] ≤ Xs, resp. E[Xt|Fs] ≥ Xs).

Examples.

• Let W be a one-dimensional Brownian motion. Then W is a martingale and W 2

is a submartingale.• The arithmetic Brownian motion Xt := µt+ σWt, µ, σ ∈ IR, is a martingale forµ = 0, a submartingale for µ > 0, and a supermartingale for µ < 0.

• Let N be a Poisson process with intensity λ. Then Nt − λtt is a martingale.

Definition 3.14 (Square-Integrable)

Let X = Xt,Ftt∈[0,∞) be a right-continuous martingale. We say that X is square-

integrable if E[X2t ] < ∞ for every t ≥ 0. If, in addition, X0 = 0 a.s. we write

X ∈M2 (or X ∈Mc2 if X is also continuous).

Proposition 3.15 (Functions of Submartingales)

(i) Let X = (X1, . . . , Xn) with real-valued components Xk be a martingale w.r.t.

Ftt∈[0,∞) and let ϕ : IRn → IR be a convex function such that E[|ϕ(Xt)|] < ∞for all t ∈ [0,∞). Then ϕ(X) is a submartingale w.r.t. Ftt∈[0,∞)

(ii) LetX = (X1, . . . , Xn) with real-valued componentsXk be a submartingale w.r.t.

Ftt∈[0,∞) and let ϕ : IRn → IR be a convex, non-decreasing function such that

E[|ϕ(Xt)|] <∞ for all t ∈ [0,∞). Then ϕ(X) is a submartingale w.r.t. Ftt∈[0,∞)

Proof. follows from Jensen’s inequality (Exercise). 2

For any X ∈M2 and 0 ≤ t <∞, we define

||X||t :=√

E[X2t ] and ||X|| :=

∞∑n=1

||X||n ∧ 12n

.

Note that ||X−Y || defines a the pseudo-metric onM2. If we identify indistinguish-able processes, the pseudo-metric ||X − Y || becomes a metric onM2.8

Proposition 3.16

If we identify indistinguishable processes, the pair (M2, ||) is a complete metric

space, and Mc2 is a closed subset ofM2.

Proof. See, e.g., Karatzas/Shreve (1991, pp. 37f).

An important question now arises: What happens to the martingale property of astopped martingale? For bounded stopping times, nothing changes.

Theorem 3.17 (Optional Sampling Theorem, 1st Version)

LetX = Xt,Ftt∈[0,∞) be a right-continuous submartingale (supermartingale) and

let τ1 and τ2 be bounded stopping times of the filtration Ft with τ1 ≤ τ2 ≤ c ∈ IRa.s. Then

E[Xτ2 |Fτ1 ] ≥ (≤)Xτ1

8This means: (i) ||X − Y || ≥ 0 and ||X − Y || = 0 iff X = Y . (ii) ||X − Y || = ||Y −X||. (iii)

||X − Y || = ||X − Z||+ ||Z − Y ||.


If the stopping times are unbounded, in general, this result does only hold if X canbe extended to ∞ such that X := Xt,Ftt∈[0,∞] is still a sub- or supermartingale.

Theorem 3.18 (Sub- and Supermartingale Convergence)

Let X = Xt,Ftt∈[0,∞) be a right-continuous submartingale (supermartingale)

and assume supt≥0 E[X+t ] < ∞ (supt≥0 E[X−

t ] < ∞). Then the limit X∞(ω) =limt→∞Xt(ω) exists for a.e. ω ∈ Ω and E|X∞| <∞.

Proof. See Karatzas/Shreve (1991, pp. 17f).

This proposition ensures that a limit exists, but it is not clear at all if X is a sub- orsupermartingale. To solve this problem, we need to require the following condition:

Definition 3.15 (Uniform Integrability)

A familiy of random variables (U)a∈A is uniformly integrable (UI) if

limn→∞

supa

∫|Uα|≥n

|Uα|dP = 0.

Using the definition, it is sometimes not easy to check if a family of random variablesis uniformly integrable.

Theorem 3.19 (Characterization of UI)

Let (U)a∈A be a subset of L1. The following are equivalent:

(i) (U)a∈A is uniformly integrable.

(ii) There exists a positive, increasing, convex function g defined on [0,∞) such that

limx→∞

g(x)x

= +∞ and supa∈A

E[g |Uα|] <∞.

Remark. A good choice is often g(x) = x1+ε with ε > 0.

Proposition 3.20

The following conditions are equivalent for a nonnegative, right-continuous sub-

martingale X = Xt,Ftt∈[0,∞):

(i) It is a uniformly integrable family of random variables.

(ii) It converges in L1, as t→∞.

(iii) It converges P -a.s. (t → ∞) to an integrable random variable X∞, such that

Xt,Ftt∈[0,∞] is a submartingale.

For the implication “(i)⇒(ii)⇒(iii)”, the assumption of nonnegativity can be dropped.

Theorem 3.21 (UI Martingales can be closed)

The following conditions are equivalent for a right-continuous martingale X =Xt,Ftt∈[0,∞):

(i) It is a uniformly integrable family of random variables.


(ii) It converges in L1, as t→∞.

(iii) It converges P -a.s. (t → ∞) to an integrable random variable X∞, such that

Xt,Ftt∈[0,∞] is a martingale, i.e. the martingale is closed by Y .

(iv) There exists an integrable random variable Y , such that Xt = E[Y |Ft] a.s. for

all t ≥ 0.

If (iv) holds and X∞ is the random variable in (c), then E[Y |F∞] = X∞ a.s.

We now formulate a more general version of Doob’s optional sampling theorem:

Theorem 3.22 (Optional Sampling Theorem, 2nd Version)

Let X = Xt,Ftt∈[0,∞] be a right-continuous submartingale (supermartingale)

with a last element X∞ and let τ1 and τ2 be stopping times of the filtration Ftwith τ1 ≤ τ2 a.s. Then

E[Xτ2 |Fτ1 ] ≥ (≤)Xτ1

Proof. See, e.g., Karatzas/Shreve (1991, pp. 19f).

Proposition 3.23 (Stopped (Sub)-Martingales)

Let X = Xt,Ftt∈[0,∞) be an adapted right-continuous (sub)-martingale and let

τ be a bounded stopping time. The stopped process Xt∧τ ,Ftt∈[0,∞) is also an

adapted right-continuous (sub)-martingale.

Proof. Exercise.

Remarks.

• The difficulty in this proposition is to show that the stopped process is a (sub)-martingale for the filtration Ft. It is obvious that the stopped process is a(sub)-martingale for Ft∧τ.• If Xt,Ftt∈[0,∞] is a UI right-continuous martingale, then the stopping time τ

may be unbounded (See Protter (2005, p. 10)).

Corollary 3.24 (Conditional Expectation and Stopped σ-Fields)

Let τ1 and τ2 be stopping times and Z be an integrable random variable. Then

E[E[Z|Fτ1 ]|Fτ2 ] = E[E[Z|Fτ2 ]|Fτ1 ] = E[Z|Fτ1∧τ2 ], P − a.s.

Proof. Exercise.

For technical reasons, we sometimes need to weaken the martingale concept.

Definition 3.16 (Local Martingale)

Let X = Xt,Ftt∈[0,∞) be a (continuous) process. If there exists a non-decreasing

sequence τnn∈IN of stopping times of Ft, such that X(n)t := Xt∧τn

,Ftt∈[0,∞)

is a martingale for each n ≥ 1 and P (limn→∞ τn =∞) = 1, then we say that X is

a (continuous) local martingale.


Remark.

• Every martingale is a local martingale (Choose τn = n).• However, there exist local martingales which are no martingales. Especially,

E[Xt] need not to exist for local martingales.

Proposition 3.25

A non-negative local martingale is a supermartingale.

Proof. follows from Fatou’s lemma (Exercise). 2

This concept can be applied to other situations (e.g. locally bounded, locally squareintegrable):

Definition 3.17 (Local Property)

Let X be a stochastic process. A property π is said to hold locally if there exists a

sequence of stopping times τnn∈IN with P (limn→∞ τn = ∞) = 1 such that X(n)

has property π for each n ≥ 1.

3.5 Completition, Augmentation, Usual Conditions

The following definition is very important:

Definition 3.18 (Usual Conditions)

A filtered probability space (Ω,G, P, Gt) satisfies the usual conditions if

(i) it is complete,9

(ii) G0 contains all the P -null sets of G, and

(iii) Gt is right-continuous.

In the sequel, we impose the following

Standing assumption: We always consider a filtered probability space (Ω,F , P, Ft)satisfying the usual conditions.

One of the main reasons for this assumption is that the following proposition holds:

Proposition 3.26

Let (Ω,G, P, Gt) satisfy the usual conditions. Then every martingale for Gtpossesses a unique cadlag modification.

Proof. See, e.g. Karatzas/Shreve (1991, pp. 16f).

Unfortunately, we are confronted with the following result.

Proposition 3.27

The filtration FWt is left-continuous, but fails to be right-continuous.

9A probability space is complete if for any N ⊂ Ω such that there exists an N ∈ G with N ⊂ N

and P (N) = 0, we have N ∈ G.


Fortunately, there is a way out. Let F∞ := σ⋃

t≥0 Ft and consider the space(Ω,FW

∞ , P ). Define the collection of P -null sets by

N := A ⊂ Ω : ∃B ∈ FW∞ with A ⊂ B,P (B) = 0.

The P -augmentation of FWt is defined by Ft := σFW

t ∪ N and is called theBrownian filtration.

Theorem 3.28 (Brownian Filtration)

The Brownian filtration, is left- and right-continuous. Besides, W is still a Brownian

motion w.r.t. the Brownian filtration.

This is the reason why we work with the Brownian filtration instead of FWt . For

a Poisson process, a similar result holds.

Proposition 3.29 (Augmented Filtration of a Poisson Process)

The P -augmentation of the natural filtration induced by a Poisson process N is

right-continuous. Besides, N is still a Poisson process w.r.t. this P -augmentation.

The proofs of the previous results can be found in Karatzas/Shreve (1991), pp. 89ff.

Remark. More generally, one can prove the following results:

(i) If a stochastic process is left-continuous, then the filtration FXt is left-continuous.

(ii) Even if X is is continuous, FXt need not to be right-continuous.

(iii) For an n-dimensional strong Markov process X the augmented filtration isright-continuous.

3.6 Ito-Integral

Our goal is to define an integral of the form:∫ t

0

X(s)dW (s),

where X is an appropriate stochastic process. From the Lebesgue-Stieltjes theorywe know that for a measurable function f and a processes A of finite variation on[0, t] one can define∫ t

0

f(s)dA(s) = limn→∞

n−1∑k=0

f(

ktn

)·(A(

(k+1)tn

)−A

(ktn

)).

However, the variation of a Brownian motion is unbounded on any interval. There-fore, naive stochastic integration is impossible (See Protter (2005, pp. 43ff)).

Good News: We can define a stochastic integral in terms of L2-convergence!

Agenda. (disregarding measurability for the moment)

1st Step. Define a stochastic integral for simple processes κ (constant processexcept for finitely many jumps).

It(κ) :=∫ t

0

κsdWs :=?


2nd Step. Every L2[0, T ]-process X can be approximated by a sequence κ(n)nof simple processes such that

||X − κ(n)||2T := E

[∫ T

0

(X(s)− κ(n)(s))2 ds

]→ 0.

3rd Step. For each n the stochastic integral

It(κ(n)) =∫ t

0

κ(n)(s) dW (s)

is a well-defined random variable and the sequence It(κ(n))n converges (in somesense) to a random variable I

4th Step. Define the stochastic integral for a L2[0, T ]-process X as follows:∫ t

0

X(s) dW (s) := limn→∞

It(κ(n)) = limn→∞

∫ t

0

κ(n)(s) dW (s).

5th Step. Extend the definition by localization.

Standing Assumption. Ft is the Brownian filtration of a Brownian motion W .

1st Step.

Definition 3.19 (Simple Process)

A stochastic process κtt∈[0,T ] is a simple process if there exist real numbers 0 =t0 < t1 < . . . < tp = T , p ∈ IN , and bounded random variables Φi : Ω → IR,

i = 0, 1, . . . , p such that Φ0 is F0-measurable, Φi is Fti−1-measurable, and for all

ω ∈ Ω we have

κt(ω) = Φ0(ω)10(t) +p∑

i=1

Φi(ω)1(ti−1,ti](t)

Remarks.

• κt is Fti−1-measurable for t ∈ (ti−1, ti] (measurable w.r.t. the left-hand side ofthe interval).• κ is a left-continuous step function.

Definition 3.20 (Stochastic Integral for Simple Processes)

For a simple process κ and t ∈ [0, T ] the stochastic integral I(κ) is defined by

It(κ) :=∫ t

0

κsdWs :=∑

1≤i≤p

Φi(Wti∧t −Wti−1∧t)

Proposition 3.30 (Stochastic Integral for Simple Processes)

For a simple process κ we have:

(i) The integral I(κ) = It(κ)t∈[0,T ] is a continuous martingale for Ftt∈[0,T ].

(ii) E[(It(κ))2

]= E

[∫ t

0κ2

s ds]

for all t ∈ [0, T ].

(iii) E[(

sup0≤t≤T |It(κ)|)2] ≤ 4E

[∫ T

0κ2

s ds]


Proof. (i) Exercise.

(ii) For simplicity, t = tk+1. Then

E[It(κ)2] =k∑

i=1

k∑j=1

E[ΦiΦj(Wti

−Wti−1)(Wtj−Wtj−1)

].

1st case: i > j.

E[ΦiΦj(Wti

−Wti−1)(Wtj−Wtj−1)

]= E

[E[ΦiΦj(Wti

−Wti−1)(Wtj−Wtj−1)|Fti−1

]]= E[ΦiΦj(Wtj −Wtj−1)(E[Wti |Fti−1 ]︸︷︷︸

=Wti−1

−Wti−1)]

= 0.

2nd case i = j.

E[Φ2

i (Wti −Wti−1)2]

= E[Φ2

i E[(Wti −Wti−1)2|Fti−1 ]

]= E[Φ2

i (ti − ti−1)]

Hence,

E[It(κ)2] = E

[k∑

i=1

Φ2i (ti − ti−1)

]= E

[∫ t

0

κ2s ds

].

(iii) Apply (i), (ii), and Doob’s inequality (see Kartzas/Shreve (1991, p. 14): For aright-continuous martingale Mt with E[M2

t ] <∞ for all t ≥ 0 we have

E

[(sup

0≤≤T|Mt|

)2]≤ 4E[M2

T ].

It remains to check:

E[It(κ)2] = E[ k∑

i=1

Φ2i (ti − ti−1)︸︷︷︸

≤T

]≤ T

k∑i=1

E[Φ2

i

]<∞,

since, by Definition 3.19, Φ is bounded. 2

Remarks.

• Stochastic integrals over [t, T ] are defined by∫ T

t

κs dWs :=∫ T

0

κs dWs −∫ t

0

κs dWs.

• Let κ1 and κ2 be simple processes and a, b ∈ IR. The stochastic integral forsimple processes is linear, i.e.

It(aκ1 + bκ2) = aIt(κ1) + bIt(κ2).

This follows directly from Definition 3.20. Note that linear combinations of simplefunctions are simple.• For κ ≡ 1,

∫ t

01 dWs = Wt. Hence,

E[(∫ t

0

dWs

)2]

= E[W 2t ] = t =

∫ t

0

ds.

This is the reason why people sometimes write dWt =√dt, which is wrong from

a mathematical point of view.


2nd Step. Every L2[0, T ]-process X can be approximated by a sequence κ(n)nof simple processes.

Define the vector space

L2[0, T ] := L2([0, T ],Ω,F , P, Ftt∈[0,T ]

):=

Xt,Ftt∈[0,T ] : progressively measurable, real-valued with E

[ ∫ T

0

X2s ds

]<∞

with (semi-)norm

||X||2T := E

[∫ T

0

X2s ds

],

where we identify processes for which X = Y only Lebesgue⊗ P -a.s.

For simple processes the mapping κ→ I(κ) induces a norm on the vector space ofthe corresponding stochastic integrals via

||I(κ)||2LT:= E

(∫ T

0

κs dWs

)2 Prop. 3.30 (ii)= E

[∫ T

0

κ2s ds

]= ||κ||2T .

Since the mapping I(X) is linear and norm preserving, it is an isometry, called theIto isometry.

Proposition 3.31 (Simple Processes Approximate L2[0, T ]-processes)

The linear subspace of simple processes is dense in X ∈ L2([0, T ]), i.e. for all

X ∈ L2[0, T ] there exists a sequence κ(n)n of simple processes with

||X − κ(n)||T = E

[∫ T

0

(Xs − κ(n)s )2 ds

]→ 0 ( as n→∞)

Proof. Assume that X ∈ L2[0, T ] is continuous and bounded. Choose

κ(n)t (ω) := X0(ω)10(t) +

2n−1∑k=0

XkT/2n(ω)1(kT/2n,(k+1)T/2n](t).

Since X is bounded, κ(n)n is uniformly bounded and we can apply the dominatedconvergence theorem to obtain the desired result.

For the general case, see Karatzas/Shreve (1991, p. 133). 2

3rd and 4th Step.

Define

Mc2([0, T ]) := M : M continuous martingale on [0, T ] w.r.t. Ft,

E[M2t ] <∞ for every t ∈ [0, T ]

Theorem and Definition 3.32 (Ito Integral for X ∈ L2[0, T ])There exists a unique linear mapping J : L2[0, T ] →Mc

2([0, T ]) with the following

properties:


(i) If κ is simple, then P (Jt(κ) = It(X) for all t ∈ [0, T ]) = 1.

(ii) E[Jt(X)2] = E[∫ t

0X2

s ds]

“Ito isometry”

J is unique in the following sense: If there exists a second mapping J ′ satisfying (i)

and (ii) for all X ∈ L2[0, T ], then J(X) and J ′(X) are indistinguishable.

For X ∈ L2[0, T ], we define ∫ t

0

Xs dWs := Jt(X)

for every t ∈ [0, T ] and call it the Ito integral or stochastic integral of X (w.r.t. W ).

Proof. Let X ∈ L2[0, T ].

(a) Let κ(n)n be an approximating sequence, i.e. ||X − κ(n)||T → 0 as n → ∞.Since κ(n)n is also a Cauchy sequence w.r.t. || · ||T , we obtain for every t ∈ [0, T ](as n, m→∞):

E[(It(κ(n))−It(κ(m)))2] I linear= E[It(κ(n)−κ(m))2] Ito Isom.= E[ ∫ t

0

(κ(n)−κ(m))2ds]→ 0.

Hence, It(κ(n))n is a Cauchy sequence in the ordinary L2(Ω,Ft, P ) for randomvariables (t fixed!) which is a complete space and thus

It(κ(n)) L2

−→ Zt,

where Zt is Ft-measurable and E[Z2t ] < ∞. Since It(κ(n)) P−→ Zt as well, there

exists a subsequence nkk such that

It(κ(nk)) a.s.−→ Zt(1)

as k →∞, i.e. Zt is only P -a.s. unique (may depend on sequence!).

(b) Z is a martingale: Note that It(κ(n)) is a martingale for all n ∈ IN . By thedefinition of the conditional expected value, we thus get∫

A

It(κ(n)) dP =∫

A

Is(κ(n)) dP

for s < t and for all A ∈ Fs and n ∈ IN . The L2-convergence (1) leads to (asn→∞)10 ∫

A

ZtdP ←−∫

A

It(κ(n)) dP =∫

A

Is(κ(n)) dP −→∫

A

ZsdP.

Since Z is adapted, (ii) is proved.

(c) By Proposition 3.26, Z possesses a right-continuous modification Jt(X),Ftt∈[0,T ]

still being a square-integrable martingale. Applying Doob’s inequality and theBorel-Cantelli Lemma, one can prove that J(X) is even continuous (Exercise).

(d) Jt(X) is independent of the approximating sequence: Let κ(n)n and

κ(n)n be two approximating sequences forX with It(κ(n)) L2

−→ Zt and It(κ(n)) L2

−→10See Bauer (1992, MI, p. 92 and p. 99).


Zt. Then κ(n)n with κ(2n) = κ(2n) (even) and κ(2n+1) = κ2n+1 (odd) is anapproximating sequence for X with It(κ(n))n converges in L2 to both Zt and Zt.Thus they need to coincide with Jt(X) P -a.s.11

(e) Ito isometry:

E[Jt(X)2

] (∗)= limn→∞

E[It(κ(n))2

]= lim

n→∞E[ ∫ t

0

(κ(n)s )2 ds

](∗∗)= E

[ ∫ t

0

X2s ds

](∗) holds because It(κ(n)) L2

−→ Jt(X) (as n → ∞). This is the L2-convergence inthe ordinaryL2(Ω,Ft, P ).

(∗∗) holds because κ(n) L2

−→ X (as n→∞). This is the L2-convergence in ([0, T ]×Ω,B[0, T ]⊗F , Lebesgue⊗ P ).12

(f) Linearity of J follows from the linearity of I and from the fact that if κ(n)X n

approximates X and κ(n)Y n approximates Y , then aκ(n)

X + bκ(n)Y n approximates

aX + bY (a, b ∈ IR). Therefore,

aJt(X) + bJt(Y ) = a limn→∞

It(κ(n)X ) + b lim

n→∞It(κ

(n)Y ) = lim

n→∞It

(aκ

(n)X + bκ

(n)Y

)= Jt(aX + bY ).

(g) Uniqueness of J : Let J ′ be a linear mapping with properties (i) and (ii).Then

J ′t(X) = J ′t( limn→∞

κ(n)) (∗)= limn→∞

J ′t(κ(n)) (i)= lim

n→∞It(κ(n)) = Jt(X) P − a.s.

for all t ∈ [0, T ]. Since J(X) and J ′(X) are continuous processes, by Proposition3.2 (ii), they are indistinguishable, i.e. P (Jt(X) = J ′t(X) for all t ∈ [0, T ]) = 1. Theequality (∗) holds because linear mappings are continuous. 2

5th Step. The class of admissible integrands can be extended. Define

H2[0, T ] := H2([0, T ],Ω,F , P, Ftt∈[0,T ]

):=

Xt,Ftt∈[0,T ] : progressively measurable, real-valued with

P(∫ T

0

X2s ds <∞

)= 1

In general, for X ∈ H2[0, T ] it can happen that

||X||2T = E

[∫ T

0

X2s ds

]=∞.

Therefore, the Ito isometry needs not to hold and the approximation argumentbreaks down. Fortunately, one can use a localization argument:13

τn(ω) := T ∧ inft ∈ [0, T ] :

∫ t

0

X2s (ω) ds ≥ n

,

X(n)t (ω) := Xt(ω)1τn(ω)≥t.

11See Bauer (1992, MI, p. 130).12Both (∗) and (∗∗) are a consequence of Bauer (1992, MI, p. 92).13Every τn is a stopping time because it is the first-hitting time of [n,∞) for the continuous

process ∫ t

0X2

s dst. See Proposition 3.11.


Hence, X(n)t (ω) ∈ L2[0, T ] and J(X(n)) is defined and a martingale for fixed n ∈ IN .

Now define

Jt(X) := Jt(X(n)) for X ∈ H2[0, T ] and 0 ≤ t ≤ τn.

Note that• Jt(X(n)) = Jt(X(m)) for all 0 ≤ t ≤ τn ∧ τm, i.e. J is well-defined,• limn→∞ τn = T a.s. (since P

( ∫ T

0X2

s ds < ∞)

= 1, i.e. there exists an eventA with P (A) = 1 such that for all ω ∈ A there exists an c(ω) > 0 with∫ T

0X2

s (ω) ds ≤ c(ω).)

Important Results.

• By construction, for X ∈ H2[0, T ] the process Jt(X)t∈[0,T ] is a continuous localmartingale with localizing sequence τnn.• J is still linear, i.e. Jt(aX + bY ) = aJt(X) + bJt(Y ) for a, b ∈ IR and X,Y ∈ H2[0, T ]. (Choose τn := τX

n ∧ τYn as localizing sequence of aX + bY , where

τXn n is the localizing sequence of X.)

• E[Jt(X)] needs not to exist.

Definition 3.21 (Multi-dimensional Ito Integral)

Let W (t),Ft be an m-dimensional Brownian motion and X(t),Ft an IRn,m-

valued progressively measurable process with all components Xij ∈ H2[0, T ]. Then

we define the Ito integral of X w.r.t. W as

∫ t

0

X(s) dW (s) :=

∑m

j=1

∫ t

0X1j(s)dWj(s)

...∑mj=1

∫ t

0Xnj(s)dWj(s)

, t ∈ [0, T ],

where∫ t

0Xij(s)dWj(s) is an one-dimensional Ito integral.

3.7 Ito’s Formula

Recall our

Standing assumption: (Ω,F , P, Ft) is a filtered probability space satisfyingthe usual conditions.

Additional Assumption: Wt,Ft is a (multi-dimensional) Brownian motion ofappropriate dimension.

Definition 3.22 (Ito Process)

Let W be an m-dimensional Brownian motion.

(i) X(t),Ft is said to be a real-valued Ito process if for all t ≥ 0 it possesses the

representation

X(t) = X(0) +∫ t

0

α(s) ds+∫ t

0

β(s)′dW (s)


= X(0) +∫ t

0

α(s) ds+m∑

j=1

∫ t

0

βj(s)′dWj(s)

for an F0- measurable random variable X(0), and progressively measurable pro-

cesses α, β with∫ t

0

|α(s)| ds <∞,m∑

j=1

∫ t

0

β2j (s) ds <∞, P − a.s. for all t ≥ 0.(2)

(ii) An n-dimensional Ito processX = (X(1), . . . , X(n)) is a vector of one-dimensional

Ito processes X(i).

Remarks.

• To shorten notations, one sometimes uses the symbolic differential notation

dX(t) = α(t)dt+ β(t)′dW (t)

• It can be shown that the processes α and β are unique up to indistinguishability.• By (2), we have βj ∈ H2[0, T ].• For a real-valued Ito processX and a real-valued progressively measurable processY , we define ∫ t

0

Ys dXs :=∫ t

0

Ysαs ds+∫ t

0

Ysβ′s dWs

Definition 3.23 (Quadratic Variation and Covariation)

Let X(1) and X(2) two real-valued Ito processes with

X(i)(t) = X(0) +∫ t

0

α(i)(s) ds+∫ t

0

β(i)(s)′dW (s),

where i ∈ 1, 2. Then,

< X(1), X(2) >t:=m∑

j=1

∫ t

0

β(1)j (s)β(2)

j (s) ds

is said to be the quadratic covariation of X(1) and X(2). In particular, < X >t:=<X,X >t is said to be the quadratic variation of an Ito process X.

Remarks. The definition may appear a bit unfounded. However, one can showthe following (see Karatzas/Shreve (1991, pp. 32ff)): For fixed t ≥ 0, let Π =t0, . . . , tn with 0 = t0 < t1 < . . . < tn = t be a partition of [0, t]. Then the p-thvariation of X over the partition Π is defined by (p > 0)

V(p)t (Π) =

n∑k=1

|Xtk−Xtk−1 |p.

The mesh of Π is defined by ||Π|| = max1≤k≤n |tk − tk−1|. For X ∈ Mc2, it holds

thatlim

||Π||→0V

(2)t (Π) =< X >t in probability.

The main tool for working with Ito processes is Ito’s formula:


Theorem 3.33 (Ito’s Formula)

Let X be an n-dimensional Ito process with

Xi(t) = Xi(0) +∫ t

0

αi(s) ds+m∑

j=1

∫ t

0

βij(s) dWj(s).

Let f : [0,∞)× IRn → IR be a C1,2-function. Then

f(t,X1(t), . . . , Xn(t)) = f(0, X1(0), . . . , Xn(0)) +∫ t

0

ft(s,X1(s), . . . , Xn(s)) ds

+n∑

i=1

∫ t

0

fxi(s,X1(s), . . . , Xn(s)) dXi(s)

+0.5n∑

i,j=1

∫ t

0

fxixj(s,X1(s), . . . , Xn(s)) d < Xi, Xj >s,

where all integrals are defined.

Proof. For simplicity, X(0) = const., n = m = 1, and f : IR→ IR.

1st step. By applying a localization argument, we can assume that

Mt :=∫ t

0

βs dWs, Bt :=∫ t

0

αs ds, Bt :=∫ t

0

|αs| ds,∫ t

0β2

s ds, and thus X are uniformly bounded by a constant C. Hence, the relevantsupport of f , f ′, and f ′′ is compact implying that these functions are bounded aswell. Besides, M is a martingale (since β ∈ L2[0, T ]).

2nd step. Let Π = t0, . . . , tp be a partition of [0, t]. Taylor expension leads to(Note: f(Xtk

) = f(Xtk−1) + f ′(Xtk−1)(Xtk−Xtk−1) + . . ..)

f(Xt)− f(X0) =p∑

k=1

(f(Xtk)− f(Xtk−1))

=p∑

k=1

f ′(Xtk−1)(Xtk−Xtk−1)︸︷︷︸

(∗)

+0.5p∑

k=1

f ′′(ηk)(Xtk−Xtk−1)

2

︸︷︷︸(∗∗)

with ηk = Xtk−1 + θk(Xtk−Xtk−1), θk ∈ (0, 1).

3rd step. Convergence of (*). We get

(∗) =p∑

k=1

f ′(Xtk−1)(Btk−Btk−1)︸︷︷︸

A1(Π)

+p∑

k=1

f ′(Xtk−1)(Mtk−Mtk−1)︸︷︷︸

A2(Π)

.

(i) For ||Π|| → 0, we obtain (Lebesgue-Stieltjes theory)

A1(Π)a.s.,L1

−→∫ t

0

f ′(Xs)dBs =∫ t

0

f ′(Xs)αs ds

(ii) Approximate f ′(Xs) by a simple process

Y Πs (ω) := f ′(X0)10(s) +

p∑k=1

f ′(Xtk−1(ω))1(tk−1,tk](s).


Note that A2(Π) =∫ t

0Y Π

s dMs.14 By the Ito isometry,

E[∫ t

0

f ′(Xs) dMs −A2(Π)]2

= E[∫ t

0

(f ′(Xs)− Y Πs )βs dWs

]2= E

[∫ t

0

(f ′(Xs)− Y Πs )2β2

s ds

]L2

−→ 0,

where the limit follows from the dominated convergence theorem (as ||Π|| → 0).Therefore,15 (as ||Π|| → 0)

(∗) = A1(Π) +A2(Π) L1

−→∫ t

0

f ′(Xs)αs ds+∫ t

0

f ′(Xs) dMs

4th step. Convergence of (**). It can be shown that (See Korn/Korn (1999,pp. 53ff))

(∗∗) L1

−→∫ t

0

f ′′(Xs)β2s ds.

5th step. By steps 2-4, there exists a subsequence Πkk, such that the integralsconverges a.s. towards the corresponding limits. Therefore, for fixed t both sidesof Ito’s formula are equal a.s. Since both sides are continuous processes in t, byProposition 3.2 (ii), they are indistinguishable. 2

Remark. Often a short-hand symbolic differential notation is used (f : [0,∞) ×IR→ IR)

df(t,Xt) = ft(t,Xt)dt+ fx(t,Xt)dXt + 0.5fxx(t,Xt)d < X >t

Examples.

• Xt = t and f ∈ C2 leads to the fundamental theorem of integration.• Xt = h(t) and f ∈ C2 leads to the chain rule of ordinary calculus.• Xt = Wt and f(x) = x2:

W 2t = 2

∫ t

0

Ws dWs + t

• Xt = x0 + αt+ σWt, x0, α, β ∈ IR, and f(x) = ex.

df(Xt) = f ′(Xt)dXt + 0.5f ′′(Xt)d < X >t

= f(Xt)[αdt+ σdWt + 0.5σ2dt

]= f(Xt)

[(α+ 0.5σ2)dt+ σdWt

].

Therefore, the solution of the stock price dynamics in the Black-Scholes model,dSt = St[µdt+ σdWt], reads

St = s0e(µ−0.5σ2)t+σWt .(3)

14This is actually a consequence of the more general definition of Ito integrals in Karatzas/Shreve

(1991, pp. 129ff).15L2-convergence implies L1-convergence


• Ito’s product rule: Let X and Y be Ito processes and f(x, y) = x · y Then(omitting dependencies)

d(XY ) = fxdX + fydY + 0.5 fxx︸︷︷︸=0

d < X > +0.5 fyy︸︷︷︸=0

d < Y > + fxy︸︷︷︸=1

d < X, Y >

= XdY + Y dX + d < X, Y >

Definition 3.24 (Stochastic Differential Equation)

If for a given filtered probability space (Ω,F , P, Ft) there exists an adapted pro-

cess X with X(0) = x ∈ IRn fixed and

Xi(t) = xi +∫ t

0

αi(s,X(s)) ds+m∑

j=0

∫ t

0

βij(s,X(s)) dWj(s)

P -a.s. for every t ≥ 0 and i ∈ 1, . . . , n, such that

∫ t

0

|αi(s,X(s))|+m∑

j=1

β2ij(s,X(s))

ds <∞

P -a.s. for every t ≥ 0 and i ∈ 1, . . . , n, thenX is a strong solution of the stochastic

differential equation (SDE)

dX(t) = α(t,X(t))dt+ β(t,X(t))dW (t)(4)

X(0) = x,

where α : [0,∞)× IRn → IRn and β : [0,∞)× IRn → IRn,m are given functions.

Theorem 3.34 (Existence and Uniqueness)

If the coefficients α and β are continuous functions satisfying the following global

Lipschitz and linear growth conditions for fixed K > 0

||α(t, z)− α(t, y)||+ ||β(t, z)− β(t, y)|| ≤ K||z − y||,

||α(t, y)||2 + ||β(t, y)||2 ≤ K2(1 + ||y||2),

z, y ∈ IR, then there exists an adapted continuous process X being a strong solution

of (4) and satisfying

E[||Xt||2] ≤ C(1 + ||x||2)eCT for all t ∈ [0, T ],

where C = C(K,T ) > 0. X is unique up to indistinguishability.

Proof. See Korn/Korn (1999, pp. 128ff).

In general, there do not exist explicit solutions to SDEs. However, the importantfamily of linear SDEs admits explicit solutions:


Proposition 3.35 (Variation of Constants)

Fix x ∈ IR and let A, a, B, b be progressively measurable real-valued processes with

(P -a.s. for every t ≥ 0) ∫ t

0

(|A(s)|+ |a(s)|) ds < ∞,

m∑j=1

∫ t

0

(Bj(s)2 + b2j (s)

)ds < ∞.

Then the linear SDE

dX(t) = (A(t)X(t) + a(t))dt+m∑

j=1

(Bj(t)X(t) + bj(t))dWj(t),

X(0) = x,

possesses an adapted Lebesgue⊗ P -unique solution

X(t) = Z(t)

x+∫ t

0

1Z(u)

a(u)− m∑j=1

Bj(u)bj(u)

du+m∑

j=1

∫ t

0

bj(u)Z(u)

dWj(u)

,where

Z(t) = exp(∫ t

0

(A(u)− 0.5||B(u)||2

)du+

∫ t

0

B(u)′dW (u)).(5)

is the solution of the homogeneous SDE

dZ(t) = Z(t) [A(t)dt+B(t)′dW (t)] ,

Z(0) = 1.

Proof. For simplicity, n = m = 1. As in (3), one can show that Z solves thehomogeneous SDE. To prove uniqueness, note that Ito’s formula leads to

d

(1Z

)=

1Z

[(−A+B2)dt−BdW

].

Let Z be another solution. Then, by Ito’s product rule,

d

(Z

1Z

)= Zd

(1Z

)+

1ZdZ+ <

1Z, Z >

=Z

Z

[(−A+B2)dt−BdW

]+Z

Z[Adt+BdW ]− Z

ZB2dt = 0

implying Z 1Z = const. Due to the initial condition, we obtain Z = Z.

To show that X satisfies the linear SDE, define Y := X/Z. Ito’s product ruleapplied to Y Z gives the desired result. To prove uniqueness, let X and X be twosolutions to the linear SDE. Then X := X −X solves the homogeneous SDE

dX = X[Adt+BdW ]

with X0 = 0. Applying Ito’s product rule to 0 ·Z shows Xt = 0 a.s. for all t ≥ 0. 2

4 CONTINUOUS-TIME MARKET MODEL AND OPTION PRICING 47

Example. In Brownian models the money market account and stocks are modeledvia (i = 1, . . . , I)

dS0(t) = S0(t)r(t)dt, S(0) = 1,(6)

dSi(t) = Si(t)

µi(t)dt+m∑

j=1

σij(t)dWj(t)

, S(0) = si0 > 0.

Applying “variation of constants” leads to

S0(t) = e

∫ t

0r(s) ds

,

Si(t) = si0e

∫ t

0µi(s)−0.5

∑m

j=1σ2

ij(s) ds+∑m

j=1

∫ t

0σij(s)dWj(s).

4 Continuous-time Market Model and Option Pric-

ing

4.1 Description of the Model

We consider a continuous-time security market as given by (6).

Mathematical Assumptions.• Recall the standing assumption: (Ω,F , P, Ft) is a filtrered probability space

satisfying the usual conditions.• Wt,Ft is a (multi-dimensional) Brownian motion of appropriate dimension.• Ft is the Brownian filtration.• The coefficients r, µ, and σ are progressively measurable processes being uni-

formly bounded.

Notation.

• ϕi(t): number of stock i at time t• ϕ0(t)S0(t): money invested in the money market account at time t• Xϕ(t) :=

∑Ii=0 ϕi(t)Si(t): wealth of the investor

• Assumption: x0 := X(0) =∑I

i=0 ϕi(0)Si(0) ≥ 0 (non-negative initial wealth)• πi(t) := ϕi(t)Si(t)/X(t): proportion invested in stock i• 1−

∑i πi: proportion invested in the money market account

Definition 4.1 (Trading Strategy, Self-financing, Admissible)

(i) Let ϕ = (ϕ0, . . . , ϕI)′ be an IRI+1-valued process which is progressively measur-

able w.r.t. Ft and satisfies (P -a.s.)∫ T

0

|ϕ0(t)|dt <∞,I∑

i=1

∫ T

0

(ϕi(t)Si(t))2dt <∞.(7)

Then ϕ is said to be a trading strategy.

(ii) If

dX(t) = ϕ(t)′dS(t)


holds, then ϕ is said to be self-financing.

(iii) A self-financing ϕ is said to be admissible if X(t) ≥ 0 for all t ≥ 0.

Remarks.

• The corresponding portfolio process π is said to be self-financing or admissibleif ϕ is self-financing or admissible (Recall πi = ϕiSi/X). Note that if we workwith π, then we need to require that X(t) > 0 for all t ≥ 0. Otherwise, π is notwell-defined.• (7) ensures that all integrals are defined.• Compare the definition (ii) with the result of Proposition 2.11.• Every trading strategy satisfies

dX(t) = d(ϕ(t)′S(t)).

Hence, in abstract mathematical terms, self-financing means that ϕ can be putbefore the d operator.• In the following, we always consider admissible trading strategies.• Since admissible strategies are progressive, investors cannot see into the future

(no clairvoyants).

Definition 4.2 (Arbitrage, Option, Replication, Fair Value)

(i) An admissible trading strategy ϕ is an arbitrage opportunity if (P -a.s.)

Xϕ(0) = 0, Xϕ(T ) ≥ 0, P (Xϕ(T ) > 0) > 0.

(ii) An FT -measurable random variable C(T ) ≥ 0 is said to be a contingent claim,

T ≥ 0.

(iii) An admissible strategy ϕ is a replication strategy for the claim C(T ) if

Xϕ(T ) = C(T ) P − a.s.

(iv) The fair time-0 value of a claim is defined as

C(0) = infp : D(p) 6= ∅,

where

D(x) := ϕ : ϕ admissible and replicates C(T ) and Xϕ(0) = x

Remark. The strategy in (i) is also known as “free lottery”.

Economical Assumptions.

• All investors model the dynamics of the securities as above.• All investors agree upon σ, but may disagree upon µ.• Investors are price takers.• Investors use admissible strategies.


• Frictionless markets• No arbitrage (see Definition 4.2 (i))

Proposition 4.1 (Wealth Equation)

The wealth dynamics of a self-financing strategy ϕ are described by the SDE

dX(t) = X(t)[(r(t) + π(t)′(µ(t)− r(t)1))dt+ π(t)′σ(t)dW (t)

], X(0) = x0,

the so-called wealth equation.

Proof. By definition,

dX = ϕ′dS

= ϕ0S0rdt+I∑

i=1

ϕiSi

[µidt+

m∑j=1

σijdWj

]

=(1−

I∑i=1

πi

)Xrdt+

I∑i=1

πiX[µidt+

m∑j=1

σijdWj

]

= X[(r +

I∑i=1

πi(µi − r))dt+

m∑j=1

I∑i=1

πiσijdWj

]2

Remarks.

• Due to the boundedness of the coefficients, by “variation of constants”, the con-dition (P − a.s.) ∫ t

0

πi(s)2 ds <∞

is sufficient for the wealth equation to possess a unique solution.• One can directly start with π (instead of ϕ) and require that for the wealth

equation ∫ t

0

X(s)2πi(s)2 ds <∞

4.2 Black-Scholes Approach: Pricing by Replication

We wish to price a European call written on a stock S with strike K, maturity T ,and payoff

C(T ) = maxS(T )−K; 0,

wheredS(t) = S(t)

[µdt+ σdW (t)

]with constants µ and σ. The interest rate r is constant as well.

Assumptions.

• No payout are made to the stocks during the life-time of the call.


• There exists a C1,2-function f = f(t, s) such that the time-t price of the call isgiven by C(t) = f(t, S(t)).

Then

dC(t) = ft(t, S(t))dt+ fs(t, S(t))dS(t) + 0.5fss(t, (S(t))d < S >t

= ftdt+ fsS(µdt+ σdW

)+ 0.5fssS

2σ2dt

=(ft + fsSµ+ 0.5fssS

2σ2)dt+ fsSσdW

Consider a self-financing trading strategy ϕ = (ϕM , ϕS , ϕC) and choose ϕC(t) ≡−1. Then

dXϕ(t) = ϕM (t)dM(t) + ϕS(t)dS(t)− dC(t)

= ϕ0rMdt+ ϕSS(µdt+ σdW

)−(ft + fsSµ+ 0.5fssS

2σ2)dt− fsSσdW

=(ϕMrM + ϕSSµ− ft − fsSµ− 0.5fssS

2σ2︸︷︷︸(∗)

)dt+

(ϕ1Sσ − fsSσ︸︷︷︸

(∗∗)

)dW

Choosing ϕS(t) = fs(t) leads to (∗∗) = 0. Hence, the strategy is locally risk-freeimplying (∗) = Xϕ(t)r (Otherwise there is an arbitrage opportunity with the moneymarket account). Rewriting (*):

(∗) = ϕMrM − ft − 0.5fssS2σ2

= r(ϕMM + ϕSS − C︸︷︷︸

=Xϕ

)−rfsS + rC − ft − 0.5fssS

2σ2︸︷︷︸(∗∗∗)

.

Therefore, (∗ ∗ ∗) = 0, i.e.

rfs(t, S(t))S(t)− rf(t, S(t)) + ft(t, S(t)) + 0.5fss(t, S(t))S2(t)σ2 = 0.

It follows:

Theorem 4.2 (PDE of Black-Scholes (1973))

In an arbitrage-free market the price function f(t, s) of a European call satisfies the

Black-Scholes PDE

ft(t, s) + rsfs(t, s) + 0.5s2σ2fss(t, s)− rf(t, s) = 0,

(t, s) ∈ [0, T ]× IR+, with terminal condition f(T, s) = maxs−K, 0.

Remark. We have not used the terminal condition to derive the PDE. There-fore, every European (path-independent) option satisfies this PDE. Of course, theterminal condition will be different.


Question: How should we solve this PDE?

1st approach: Merton (1973). Solve the PDE by transforming it to the heatequation ut = uxx which has a well-known solution.16

2nd approach. Apply the Feynman-Kac theorem (see below) which states thatthe time-0 solution of the Black-Scholes PDE is given by

C(0) = E[maxY (T )−K, 0

∣∣∣Y (0) = S(0)]e−rT ,

where Y satisfies the SDE

dY (t) = Y (t)[rdt+ σdW (t)

], Y (0) = S(0).

Theorem 4.3 (Black-Scholes Formula)

(i) The price of the European call is given by

C(t) = S(t) ·N(d1(t))−K ·N(d2(t)) · e−r(T−t)

with

d1(t) =ln(S(t)/K) + (r + 0.52)(T − t)

σ√T − t

,

d2(t) = d1(t)− σ√T − t.

(ii) The trading strategy ϕ := (ϕM , ϕS) with

ϕM (t) =C(t)− fs(t, S(t)) · S(t)

M(t),

ϕS(t) = fs(t, S(t)),

is self-financing and a replication strategy for the call.

Proof. (i) Compute (exercise)

C(t) = E[maxY (T )−K, 0

∣∣∣Y (t) = S(t)]e−r(T−t).

(ii) We haveXϕ(t) = ϕM (t) ·M(t) + ϕS(t) · S(t) = C(t).(8)

Hence, (ϕM , ϕS) replicates the call and is self-financing because

dXϕ (8)= dCIto=

(ft + fsSµ+ 0.5fssS

2σ2)dt+ fsSσdW

(+)=

(r[C − fsS] + fsSµ

)dt+ fsSσdW

Def. (ϕM , ϕS)=

(rMϕM + ϕSSµ

)dt+ ϕSSσdW

=(ϕMrMdt+ ϕSS

[µdt+ SσdW

])= ϕMdM + ϕSdS.

16See, e.g., Korn/Korn (1999, p. 124).


Note: (+) holds due to the Black-Scholes PDE

ft + 0.5fssS2σ2 = r[C − fsS]

2

Remark. Since (ϕM , ϕS) is self-financing, the strategy (ϕM , ϕS , ϕC) with ϕC = −1is self-financing as well because

d(ϕC(t)C(t)) = d(−C(t)) = −dC(t) = ϕC(t)dC(t).

Theorem 4.4 (Feynman-Kac Representation)

Let β and γ be functions of (t, x) ∈ [0,∞) × IR. Under technical conditions (see

Korn/Korn (1999, p. 136)), there exists a unique C1,2-solution F = F (t, x) of the

PDE

Ft(t, x) + β(t, x)Fx(t, x) + 0.5γ2(t, x)Fxx(t, x)− rF (t, x) = 0(9)

with terminal condition F (T, x) = h(x) possessing the representation

F (t0, x) = e−r(T−t0)E[h(X(T ))|X(t0) = x],

where X satisfies the SDE

dX(s) = β(s,X(s))ds+ γ(s,X(s))dW (s)

with initial condition X(t0) = x.

Sketch of Proof. (i) Firstly assume r = 0. If F is a C1,2-function, we can applyIto’s formula

dF = Ftdt+ FxdX + 0.5Fxxd < X > .

Since d < X >= γ2dt, we get

dF = Ftdt+ Fx

(βdt+ γdW

)+ 0.5γ2Fxxdt

=(Ft + βFx + 0.5γ2Fxx

)dt+ γFxdW.

Suppose that there exists a function F satisfying the PDE, i.e.

Ft + βFx + 0.5γ2Fxx = 0

with F (T, x) = h(x). ThendF = γFxdW

or in integral notation

F (s,X(s)) = F (t0, X(t0)) +∫ s

t0

γ(u,X(u))Fx(u,X(u))dW (u).

If E[ ∫ s

t0γ(u,Xu)2Fx(u,Xu)2 du

]<∞, the stochastic integral has zero expectation

and thusF (t0, x) = E[F (s,X(s)|X(t0) = x].


Choose s = T and note that F (T,X(T )) = h(X(T )).

(ii) For r 6= 0, multiply the PDE (9) by e−rt:

Ft(t, x)e−rt + β(t, x)Fx(t, x)e−rt + 0.5γ2(t, x)Fxx(t, x)e−rt − rF (t, x)e−rt = 0.

Set G(t, x) := F (t, x)e−rt. Since

Gt(t, x) = Ft(t, x)e−rt − rF (t, x)e−rt,

we getGt(t, x) + β(t, x)Gx(t, x) + 0.5γ2(t, x)Gxx(t, x) = 0

with terminal condition G(T, x) = e−rTh(x). Now we can apply (i):

G(t0, x) = E[e−rTh(X(T ))|X(t0) = x].

Substituting G(t0, x) = e−rt0F (t0, x) gives the desired result. 2

Remarks.

• Two crucial points are not proved: Firstly, we have not checked under whichconditions a C1,2-function F exists. Secondly, we have not checked under whichconditions γ(·, X)Fx(·, X) ∈ L2[0, T ].

• To solve the Black-Scholes PDE, choose β(t, x) = rx and γ(t, x) = xσ.• The Feynman-Kac representation can be generalized to a multidimensional set-

ting, i.e. X is a multidimensional Ito process, with stochastic short rate r = r(X).

4.3 Pricing with Deflators

In a single-period model the deflator is given by (See remark to Proposition 2.7)

H =1

1 + r

dQ

dP

This motivates the definition of the deflator in continuous-time:

H(t) :=1

M(t)Z(t),

where

M(t) := exp(∫ t

0

r(s) ds),

Z(t) := exp(−0.5

∫ t

0

||θ(s)||2 ds−∫ t

0

θ(s)′ dW (s)),

The process θ is implicitly defined as the solution to the following system of linearequations:

σ(t)θ(t) = µ(t)− r(t)1.(10)

Applying Ito’s product rule yields the following SDE for H:

dH(t) = d(Z(t)M(t)−1) = −H(t)[r(t)dt+ θ(t)′dW (t)]

Remarks.


• In this section we are going to work under the physical measure. Neverthelesswe will see later on that Z can be interpreted as a density inducing a change ofmeasure (Girsanov’s Theorem).• The system of equations (10) needs not to have a (unique) solution.• The process θ is said to be the market price of risk and can be interpreted as a

risk premium (excess return per units of volatility).

The following theorem shows that H has indeed the desired properties:

Theorem 4.5 (Arbitrage-free Pricing with Deflator and Completeness)

(i) Assume that (10) has a bounded solution for θ. Let ϕ be an admissible portfolio

strategy. Then H(t)Xϕ(t)t∈[0,T ] is both a local martingale and a supermartingale

implying

E[H(t)Xϕ(t)] ≤ Xϕ(0) for all t ∈ [0, T ].(11)

Besides, the market is arbitrage-free.

(ii) Assume that I = m (“# stocks = # sources of risk”) and that σ is uniformly

positive definite.17 Then (10) has a unique solution which is uniformly bounded.

Furthermore, let C(T ) be an FT -measurable contingent claim with

x := E[H(T )C(T )] <∞(12)

Then there exists a P ⊗Lebesgue-unique portfolio process ϕ such that ϕ is a repli-

cation strategy for C(T ), i.e. (P -a.s.)

Xϕ(T ) = C(T ),

and Xϕ(0) = x. Hence, the market is complete. The time-t price of the claim

equals

C(t) =E[H(T )C(T )|Ft]

H(t),

i.e. the process H(t)C(t)t∈[0,T ] is a martingale.

Remarks.

• The boundedness assumption in (i) can be weakened.• The price in (ii) is thus equal to the expected discounted payoff under the phys-

ical measure where we discount with the state-dependent discount factor H, thedeflator.• Compare (ii) with Theorem 2.13.

Proof. For simplicity, I = 1. (i) Applying Ito’s product rule, one gets for anadmissible ϕ = (ϕ0, ϕ1):18

d(HXϕ) = HdXϕ +XϕdH + d < H,Xϕ >

17The process σ is uniformly positive definite if there exists a constant K > 0 such that

x′σ(t)σ(t)′x ≥ Kx′x for all x ∈ IRm and all t ∈ [0, T ], P -a.s. This requirement implies that

σ(t) is regular.18Note θσ = µ− r.


= HXϕrdt+Hϕ1S(µ− r)dt+Hϕ1SσdW −XϕHrdt

−XϕHθdW −Hϕ1Sθσdt

= H(ϕ1Sσ −Xϕθ)dW.(13)

Since ϕ is admissible, we have Xϕ(t) ≥ 0. By definition, H(t) ≥ 0 and thusH(t)X(t) ≥ 0. By Proposition 3.25, HX is a supermartingale which proves (11).Note that an arbitrage strategy would violate this supermartingale property.

(ii) Clearly, θ = (µ − r)/σ is unique and uniformly bounded by some Kθ > 0.19

DefineX(t) :=

E[H(T )C(T )|Ft]H(t)

By definition, X(t) is adapted, X(0) = x, X(t) ≥ 0, and X(T ) = C(T ) (P -a.s.).20

Set M(t) := H(t)X(t) = E[H(T )C(T )|Ft]. Due to (12), M is a martingale withM(0) = x. Applying the martingale representation theorem, there exists an Ft-progressively measurable real-valued process ψ with

∫ T

0ψ(s)2 ds < ∞ such that

(P − a.s.)

H(t)X(t) = M(t) = x+∫ t

0

ψ(s) dW (s).(14)

Since H and∫ ·0ψ(s) dW (s) are continuous, we can choose X to be continuous.

Comparing (13) and (14) yields21

H(ϕ1Sσ −Xθ) = ψ.

Since there exists a Kσ > 0 such that σ(t) ≥ Kσ for all t ∈ [0, T ],22 we get

ϕ1 =ψ/H +Xθ

Sσ.

Since Xϕ = ϕ0M + ϕ1S for any strategy ϕ, we define

ϕ0 :=X − ϕ1S

M

which implies that ϕ is self-financing and X = Xϕ. Since M > 0, H > 0, andX ≥ 0 are continuous, M as well as H are pathwise bounded away from zero and Xis pathwise bounded from above, i.e. M(ω) ≥ KM (ω) > 0, H(ω) ≥ KH(ω) > 0, andX(ω) ≤ KX(ω) < ∞ for a.a. ω ∈ Ω. Therefore, the process is a trading strategybecause23∫ T

0

(ϕ1(s)S(s))2 ds =∫ T

0

(ψ(s)/H(s) +X(s)θ(s)

σ(s)

)2

ds

≤ 2K2

σ

(1K2

H

∫ T

0

ψ(s)2 ds+K2XK

2θT

)<∞,

19Note that all coefficients are bounded and σ is uniformly positive definite.20Since F0 is the trivial σ-algebra augemented by null sets, is every F0-measurable random

variable a.s. constant and the conditioned expected value is equal to the unconditioned expected

value.21Note that the representation of an Ito process is unique.22By assumption, σ is uniformly positive definite.23(a + b)2 ≤ 2a2 + 2b2.


∫ T

0

|ϕ0(s)| ds =∫ T

0

|X(s)− ϕ1(s)S(s)|M(s)

ds

≤ 1KM

(KXT +

∫ T

0

1 + (ϕ1(s)S(s))2 ds

)<∞

for P -a.s. 2

Theorem 4.6 (Martingale Representation Theorem)

Let Mtt∈[0,T ] be a real-valued process being adapted to the Brownian filtration.

(i) If M is a square integrable martingale, i.e. E[M2t ] < ∞ for all t ∈ [0, T ], then

there exists a progressively measurable IRm-valued process ψtt∈[0,T ] with

E

[∫ T

0

||ψs||2 ds

]<∞(15)

and

Mt = M0 +∫ t

0

ψ′sdWs, P -a.s.

(ii) If M is a local martingale, then the result from (i) holds with∫ T

0

||ψs||2 ds <∞

instead of (15).

In both cases, ψ is P ⊗ Lebesgue-unique.

Proof. See Korn/Korn (1999, pp. 81ff).

Remarks.

• All Brownian (local) martingales are stochastic integrals and vice versa!• The quadratic (co)variation is defined for all (local) Brownian martingales.

Proposition and Definition 4.7 (Quadratic (Co)variation)

(i) Let X, Y be Brownian local martingales. The quadratic covariation < X,Y >

is the unique adapted, continuous process of bounded variation with < X,Y >0= 0such that XY− < X,Y > is a Brownian local martingale. If X = Y , this process

is non-decreasing.

(ii) Let (Ω,G, P, Gt) be a filtered probability space satisfying the usual conditions

and let X, Y be continuous local martingales for the filtration Gt. Then the result

of (i) holds with XY− < X,Y > being a local martingale for Gt.

Proof. (i) By the martingale representation theorem, dX = αdW and dY = βdW

with progressively measurable processes α and β. Ito’s product rule yields

d(XY ) = Y dX +XdY + d < X, Y >= Y αdW +XβdW + αβdt.

Since Ito integrals are local Brownian martingales, XY− < X,Y > is a Brownianmartingale. The rest is obvious.


(ii) See Kartzas/Shreve (1991, p. 36). 2

Remark. One can show that if X and Y are square-integrable, then XY− <

X,Y > is a martingale.

4.4 Pricing with Risk-neutral Measures

The deflator is defined asH(t) =

1M(t)

Z(t),

wheredZ(t) = −Z(t)θ(t)′ dW (t)

is a local martingale. If it is even a martingale, we have E[Z(T )] = 1 and we candefine a probability measure on FT by

Q(A) := E[1AZ(T )], A ∈ FT .

Hence, Z(T ) is the density of Q and we can rewrite the result from Theorem 4.5 as

C(0) = E[H(T )C(T )] = E[Z(T )

C(T )M(T )

]= EQ

[C(T )M(T )

],

i.e. the price of a claim equals its expected discounted payoff where we discount withthe money market account (instead of the deflator) and expectation is calculatedw.r.t. the measure Q (instead of the physical measure).

Two important questions arise:

Q1. Under which conditions is Z a martingale? (Novikov condition)

Q2. What happens to the dynamics of the assets if we change to the measure Q?(Girsanov’s theorem)

Proposition 4.8

Let θ be progressive andK > 0. If∫ T

0||θ(s)||2 ds ≤ K, then the process Z(t)t∈[0,T ]

defined by

dZ(t) = −Z(t)θ(t)′dW (t), Z(0) = 1,(16)

is a martingale.

Proof. For simplicity, m = 1. The process Z is a local martingale. By variationsof constants, we get

Z(t) = exp(−0.5

∫ t

0

θ(s)2 ds−∫ t

0

θ(s) dW (s))≥ 0.(17)

By Proposition 3.25, it is thus a supermartingale. Hence, it is sufficient to showthat E[Z(T )] = 1. Define

τn := inft ≥ 0 :

∣∣∣ ∫ t

0

θ(s) dW (s)∣∣∣ ≥ n.


Due to (17), for fixed n ∈ IN , we get Z(T∧τn)2 ≤ e2n and thus∫ T∧τn

0Z(s)2θ(s)2 ds ≤

K e2n. From (16), we obtain E[Z(T ∧ τn)] = 1. Therefore, the claim is proved ifZ(T ∧ τn)n∈IN is UI because in this case

limn→∞

E[Z(T ∧ τn)] = E[Z(T )].

We obtain the estimate

|Z(T ∧ τn)|1+ε ≤ exp(0.5(ε+ ε2)

∫ T

0

θ(s)2 ds)

︸︷︷︸independent of n and bounded by assumption

Y (T ∧ τn),

wheredY (t) = −Y (t)(1 + ε)θ(s)dW (s), Y (0) = 1.

With the same arguments leading to E[Z(T ∧τn)] = 1 we conclude E[Y (T ∧τn)] = 1.Therefore,

supn∈IN

E|Z(T ∧ τn)|1+ε <∞,

which, by Proposition 3.19, proves UI of Z(T ∧ τn)n∈IN . 2

Remark. The requirement in the proposition is a strong one. It can be replacedby the Novikov condition (See Karatzas/Shreve (1991, pp. 198ff)):

E[exp

(0.5∫ T

0

||θ(s)||2 ds)]

<∞.

We turn to the second question and define (Q := QT )

Qt(A) := E[1AZ(t)], A ∈ Ft, t ∈ [0, T ].(18)

The following lemma shows that Qt is well-defined.

Lemma 4.9

For a bounded stopping time τ ∈ [0, T ] and A ∈ Fτ , we get QT (A) = Qτ (A).

Proof. The optional sampling theorem (OS) yields (See Theorem 3.17)

QT (A) = E[1AZ(T )] = E[E[1AZ(T )|Fτ ]] = E[1AE[Z(T )|Fτ ]] OS= E[1AZ(τ)] = Qτ (A).

2

To prove Girsanov’s theorem, we need some additional tools.

Theorem 4.10 (Levy’s Characterization of the Brownian Motion)

An m-dimensional stochastic process X with X(0) = 0 is an m-dimensional Brow-

nian motion if and only if it is a continuous local martingale with24

< Xj , Xk >t= δjkt.

24Note that δij = 1i=j is the Kronecker delta.


Proof. For simplicity m = 1. Clearly, the Brownian motion is a continuous localmartingale with < W >t= t. To prove the converse, fix u ∈ IR and set f(t, x) =exp(iux+ 0.5u2t) and Zt := f(t,Xt), where i =

√−1. Since f ∈ C1,2 is analytic, a

complex-valued version of Ito’s formula yields

dZ = ftdt+ fxdX + 0.5fxxd < X >

= 0.5u2fdt+ iufdX − 0.5u2fd < X ><X>t=t= iufdX.

Since an integral w.r.t. a local martingale is a local martingale as well,25 Z is a(complex-valued) local martingale which is bounded on [0, T ] for every T ∈ IR.Hence, Z is a martingale implying E[exp(iuXt + 0.5u2t|Fs] = exp(iuXs + 0.5u2s),i.e.

E[exp(iu(Xt −Xs))|Fs] = exp(−0.5u2(t− s)).

Since this hold for any u ∈ IR, the difference Xt −Xs ∼ N (0, t− s) and Xt −Xs isindependent of Fs. 2

Proposition 4.11 (Bayes Rule)

Let µ and ν be two probability measures on a measurable space (Ω,G) such that

dν/dµ = f for some f ∈ L1(µ). Let X be a random variable on (Ω,G) such that

Eν [|X|] =∫|X|fdµ <∞.

Let H be a σ-field with H ⊂ G. Then

Eν [X|H] Eµ[f |H] = Eµ[fX|H] a.s.

Proof. Let H ∈ H. We start with the left-hand side:∫H

Eµ[fX|H] dµ =∫

H

Xf dµ =∫

H

Xdν =∫

H

Eν [X|H] dν =∫

H

Eν [X|H]f dµ

= Eµ

[Eν [X|H] · f1H

]= Eµ[Eµ[Eν [X|H] · f1H |H]]

= Eµ[1HEν [X|H] · Eµ[f |H]] =∫

H

Eν [X|H] · Eµ[f |H] dµ.

2

Lemma 4.12

Let Yt,Ftt∈[0,T ] be a right-continuous adapted process. The following statements

are equivalent:

(i) ZtYt,Ftt∈[0,T ] is a local P -martingale.

(ii) Yt,Ftt∈[0,T ] is a local QT -martingale.

25For instants, a Brownian local martingale can be represented as Ito integral.


Proof. Assume (i) to hold and let τnn be a localizing sequence of ZY . Then26

EQT[Yt∧τn

|Fs]Lem 4.9= EQt∧τn

[Yt∧τn|Fs]

Bayes=EP [Zt∧τn

Yt∧τn|Fs]

EP [Zt∧τn |Fs]

=Zs∧τn

Ys∧τn

Zs∧τn

= Ys∧τn.

The converse implication can be proved similarly (exercise). 2

Theorem 4.13 (Girsanov’s Theorem)

We assume that Z is a martingale and define the process WQ(t),Ft by

WQj (t) := Wj(t) +

∫ t

0

θj(s) ds, t ≥ 0,

j ∈ 1, . . . ,m. For fixed T ∈ [0,∞), the process WQ(t),Ftt∈[0,T ] is an m-

dimensional Brownian motion on (Ω,FT , QT ), where QT is defined by (18).

Proof. For simplicity, m = 1. By Levy’s characterization, we need to show that

(i) WQ(t),Ftt∈[0,T ] is a continuous local martingale on (Ω,FT , QT ).

(ii) < WQ >t= t on (Ω,FT , QT ).

ad (i). Ito’s product rule yields

d(ZWQ) = ZdWQ +WQdZ + d < WQ, Z >

= ZdW + Zθdt−WQZθdW − Zθdt

= (Z −WQZθ)dW,

i.e. ZWQ is a local P -martingale. By Lemma 4.12, the process WQ is a localQT -martingale as well.

ad (ii). Ito’s product rule leads to

d(Z · ((WQ)2 − t)

)= d(WQ · (ZWQ)

)− d(Z · t)

= WQd(ZWQ) + ZWQdWQ + d < ZWQ,WQ > −Zdt− tdZ

= WQd(ZWQ) + ZWQ(θdt+ dW ) + (Z −WQZθ)dt− Zdt+ tθZdW

= WQd(ZWQ) + Z(WQ + tθ)dW,

i.e. Zt · ((WQt )2 − t) is a local P -martingale. Applying Lemma 4.12 again yields

that (WQt )2 − t is a local QT -martingale. By Proposition 4.7 (ii), the claim (ii)

follows. 2

Corollary 4.14 (Girsanov II)

If Mtt∈[0,T ] is a Brownian local P -martingale and

Dt :=∫ t

0

1Zsd < Z,M >s,

26Note that Yt∧τn is Ft∧τn -measurable, i.e. the conditional expected value can be computed

w.r.t. Qt∧τn instead of QT . Lemma 4.9 ensures that both measures coincide on Ft∧τn . Besides,

Ft∧τn ⊂ FT , i.e. Bayes rule can be applied. Furthermore, by Proposition 3.23, EP [Zt∧τn |Fs] =

Zs∧τn .


then Mt − Dtt∈[0,T ] is a continuous local QT -martingale and < M − D >Q=<M >, i.e. the quadratic variations of M −D under QT and M under P coincide.

Proof. By the martingale representation theorem, we have dM = αdW for someprogressive α. Hence,

dD =1Zd < Z,M >= − 1

ZθZαdt = −θαdt

and thusd(M −D) = αdW + θαdt = αdWQ.

Therefore, M −D is a local Q-martingale. Besides,

d < M −D >Qt = α2dt = d < M >t,

which completes the proof. 2

Theorem 4.15 (Converse Girsanov)

Let Q be an equivalent probability measure to P on FT . Then the density process

Z(t),Ftt∈[0,T ] defined by

Z(t) :=dQ

dP

∣∣∣∣Ft

is a positive Brownian P -martingale possessing the representation

dZ(t) = −Z(t)θ(t)dW (t), Z(0) = 1,

for a progressively measurable m-dimensional process θ with∫ T

0

||θ(s)||2 ds <∞ P − a.s.(19)

Proof. Since Q and P are equivalent on FT , both measures are equivalent on Ft,t ∈ [0, T ] , i.e. Z(t) > 0 for all t ∈ [0, T ]. Besides, for A ∈ Ft∫

Q

Z(T )dP =∫

A

dQ/dP |FT

Lem 4.9=∫

A

dQ/dP |Ft=∫

Q

Z(t)dP,

i.e. Z(t) = E[Z(T )|Ft] implying that Z is a Brownian martingale. By the martingalerepresentation theorem,

Z(t) = 1 +∫ t

0

ψ(s)′dW (s)

with a progressive process ψ satisfying∫ T

0||ψ(s)||2 ds <∞, P − a.s.. Since Z > 0,

we have Z(ω) ≥ KZ(ω) > 0 for a.a. ω ∈ Ω. Defining

θ(t) = −ψ(t)Z(t)

thus implies (19) which completes the proof. 2


Definition 4.3 (Doleans-Dade Exponential)

For an Ito process X with X0 = 0 the Doleans-Dade exponential (stochastic ex-

ponential) of X, written E(X), is the unique Ito process Z that is the solution

of

Yt = 1 +∫ t

0

YsdXs.

Remarks.

• Therefore, the density Z is sometimes written as E(∫ ·0θsdWs).

• One can show that E(X)E(Y ) = E(X + Y+ < X,Y >) and that (E(X))−1 =E(−X+ < X,X >).

We now come back to our pricing problem. We set Q := QT . Recall our definitions

H(t) :=1

M(t)Z(t),

where

M(t) := exp(∫ t

0

r(s) ds),

Z(t) := exp(−0.5

∫ t

0

||θ(s)||2 ds−∫ t

0

θ(s)′ dW (s)),

and θ is implicitly defined by:

σ(t)θ(t) = µ(t)− r(t)1.(20)

For bounded θ, by Girsanov’s theorem, Z(t) = dQ/dP |Ft is a density process forthe measure Qt(A) = E[Z(t)1A] and WQ(t),Ft defined by

WQj (t) := Wj(t) +

∫ t

0

θj(s) ds, t ≥ 0,

is a Q-Brownian motion.

Definition 4.4 (Equivalent Martingale Measure)

A probability measure Q is a (weak) equivalent martingale measure if

(i) Q ∼ P and

(ii) all discounted price processes Si/M , i = 1, . . . , I, are (local) Q-martingales.

Remark. An equivalent martingale measure (EMM) is also said to be a risk-neutralmeasure.

Proposition 4.16

If Q is a weak EMM and

E

[exp

(0.5∫ T

0

||σi(s)||2 ds)]

<∞(21)

for every i ∈ 1, . . . , I, then Q is already an EMM.


Proof. Define Yi := Si/M . Since Q is a weak martingale measure, Yi is a localQ-martingale. By the martingale representation theorem and the uniqueness of therepresentation of an Ito process,

dYi = Yiσ′idW

Q.

By the Novikov condition (21), this is a Q-martingale. 2

Remark. Since we have assumed that σ is uniformly bounded, we need not todistinguish between EMM and weak EMM.

Theorem 4.17 (Characterization of EMM)

In the Brownian market (6), the following are equivalent:

(i) The probability measure Q is an EMM.

(ii) There exists a solution θ to the system of equations (20) such that the process

Z = E(∫ ·0θsdWs) is a Girsanov density inducing the probability measure Q.

Proof. Define Yi := Si/M . Assume (i) to hold. Since Q is a martingale measure,

dYi = Yiσ′idW

Q.(22)

Besides, since Q is equivalent to P , by the converse Girsanov, there exists a processθ such that dQ/dP = E(

∫ ·0θ′sdWs) and dWQ = dW + θdt is a Brownian increment

under Q. Therefore,

dSi = Si[µidt+ σ′idW ] = Si[(µi − σ′iθ)dt+ σ′idWQ]

implyingdYi = Yi[(µi − σ′iθ − r)dt+ σ′idW

Q](23)

Comparing (22) and (23) gives (ii).

On the other hand, assume (ii) to hold. By definition, Q is equivalent to P anddWQ = dW + θdt is a Q-Brownian increment. Ito’s formula leads to

dYi = Yi[(µi − r)dt+ σ′idW ] = Yi[(µi − r − σ′iθ)dt+ σ′idWQ] = Yiσ

′idW

Q,

which is Q-martingale increment because σ is assumed to be uniformly bounded. 2

Corollary 4.18 (Q-Dynamics of the Assets)

Under an EMM Q, the dynamicss of asset i are given by

dSi(t) = Si(t)[r(t)dt+ σi(t)′dWQ(t)],

i.e. the drift µ does not matter under Q.


Theorem 4.19 (Arbitrage-free Pricing under Q and Completeness)

(i) LetQ be an EMM and let ϕ be an admissible portfolio strategy. Then Xϕ(t)/M(t)t∈[0,T ]

is both a local Q-martingale and a Q-supermartingale implying that

EQ

[Xϕ(t)M(t)

]≤ Xϕ(0) for all t ∈ [0, T ].

Besides, the market is arbitrage-free. If additionally

EQ

[∫ T

0

(ϕi(s)

Si(s)M(s)

)2

ds

]<∞(24)

for every i = 1, . . . , I, then Xϕ(t)/M(t)t∈[0,T ] is a Q-martingale.

(ii) Assume that I = m (“# stocks = # sources of risk”) and that σ is uniformly

positive definite. Then (10) has a unique solution which is uniformly bounded

implying that there exists a unique EMM Q. Besides, the market is complete.

Furthermore, let C(T ) be an FT -measurable contingent claim with

x := EQ[C(T )/M(T )] <∞

Then the time-t price of the claim equals

C(t) = M(t) EQ

[C(T )M(T )

∣∣∣∣Ft

],

i.e. the discounted price process C(t)/M(t)t∈[0,T ] is a Q-martingale.

Proof. (i) Consider

dXϕ = ϕ′dS = Xrdt+I∑

i=1

ϕiSiσ′idW

Q.

Since d(1/M) = −(1/M)rdt,

d(XM

)= Xd

( 1M

)+

1MdX =

1M

I∑i=1

Siϕiσ′idW

Q

which is a positive local Q-martingale and thus a Q-supermartingale. If (24) issatisfied, then it is a Q-martingale.

(ii) Since θ is unique, by Theorem 4.17, the EMM Q is unique. The claim pricefollows from Theorem 4.5 because H = Z/M . 2

Remark.

• An EMM exists if, for instants, θ is bounded. This follows from Theorem 4.17.• If ϕ is uniquely bounded, then (24) holds because µ, σ, and r are assumed to be

uniformly bounded.• If the market is incomplete, then there exist more than one market price of riskθ. Let Q and Q be two corresponding EMM. If (24) holds, then for an admissiblestrategy ϕ we get

M(t) EQ

[Xϕ(T )M(T )

∣∣∣∣Ft

]= Xϕ(t) = M(t) EQ

[Xϕ(T )M(T )

∣∣∣∣Ft

].


This implies that any EMM assigns the same value to a replicable claim, namlyits replication costs. However, the prices of non-replicable claims may differunder different EMM. This is the reason why pricing in incomplete markets is socomplicated and in some sense arbitrary.

Corollary 4.20 (Black-Scholes Again)

The price of the European call is given by

C(t) = S(t) ·N(d1(t))−K ·N(d2(t)) · e−r(T−t)

with

d1(t) =ln(S(t)/K) + (r + 0.52)(T − t)

σ√T − t

,

d2(t) = d1(t)− σ√T − t.

Proof. By Theorem 4.19,

C(t) = EQ[maxS(T )−K; 0] e−r(T−t),

where due to Corollary 4.18

dS(t) = S(t)[rdt+ σdWQ(t)].

Thus S(T ) has the same distribution as Y (T ) in the proof of Proposition 4.3 andthe result follows. 2

Corollary 4.21 (Q-Dynamics of Claims)

Under the assumptions of Theorem 4.19, the dynamics of a replicable claim C(T )are given by

dC(t) = C(t)[r(t)dt+ ψ(t)′dWQ(t)](25)

with a suitable progressively measurable process ψ.

Proof. By Theorem 4.19, C/M is a (local) Q-martingale and thus (25) followsfrom the martingale representation theorem. 2

Proposition 4.22 (Black-Scholes PDE Again and Hedging)

Under the assumptions of Theorem 4.19, let C(T ) be a replicable claim with C(t) =f(t, S(t)) and C(T ) = h(S(T )), where f is a C1,2-function.

(i) The pricing function f satisfies the following multi-dimensional Black-Scholes

PDE:

ft +I∑

i=1

fisir + 0.5I∑

i,k=1

m∑j=1

siskσijσkjfik − rf = 0, f(T, s) = h(s), s ∈ IRI+,

where fi denotes the partial derivative w.r.t. the i-th asset price.


(ii) A replication strategy for C(T ) is given by

ϕi(t) = fi(t, S(t)), i = 1, . . . , I,

where the remaining funds C(t)−∑I

i=1 ϕi(t)Si(t) are invested in the money market

account, i.e. ϕM (t) = (C(t)−∑I

i=1 ϕi(t)Si(t))/M(t).

Proof. (i) Ito’s formula yields

dC =(ft +

I∑i=1

fiSir + 0.5I∑

i,k=1

m∑j=1

fikSiSkσijσkj

)dt+

I∑i=1

m∑j=1

fiSiσijdWQj .(26)

Since the representation of an Ito process is unique, comparing the drift with thedrift of (25) yields the Black-Scholes PDE.

(ii) Since C(T ) is attainable, there exists a replication strategy ϕ with Xϕ(T ) =C(T ) and Q-dynamics

dXϕ = Xϕrdt+I∑

i=1

m∑j=1

ϕiSiσijdWQj ,

where due to Theorem 4.19 (ii) the drift equals r.27 Comparing the diffusion partwith the diffusion part of (26) shows that one can choose ϕi = fi. 2

Remarks.

• The strategy in (ii) is said to be a delta hedging strategy.• Under the conditions of Theorem 4.19 (ii), the delta hedging strategy is the

unique replication strategy.• In the Black-Scholes model, we get for the European call (see Exercise 29): ϕS =N(d1). Due to the put-call parity, the delta hedge for the put reads ϕS =−N(−d1).• If I > m, i.e. (“# stocks > # sources of risk”), then in general there exist further

replication strategies, i.e. the choice ϕi = fi in the proof of (ii) is not the onlypossible choice.

In Theorem 4.19, we have proved that the following implications hold:

1) Existence of EMM =⇒ No arbitrage

2) Uniqueness of EMM =⇒ Completeness

The following theorems show to which extend the converse is true.

Theorem 4.23 (Existence of a Market Price of Risk)

Assume that the market (6) is arbitrage-free. Then there exists a market price of

risk, i.e. there exists a progressively measurable process θ solving the system (10).

Proof. Define φi := ϕiSi (amount of money invested in asset i) leading to thefollowing version of the wealth equation:

dX(t) = X(t)r(t)dt+ φ(t)′(µ(t)− r(t)1)dt+ φ(t)′σ(t)dW (t).27The representation of the diffusion part follows from rewriting Xπ′σdW Q in terms of ϕ =

πX/S.


Assume that there exists a self-financing strategy with

φ(t)′σ(t) = 0 and φ(t)′(µ(t)− r(t)1) 6= 0.

This strategy would be an arbitrage since it does not involve risk, but its wealthequation has a drift different from r. Thus in an arbitrage-free market every vectorin the kernel Ker(σ(t)′) should be orthogonal to the vector µ(t) − r(t)1. But theorthogonal complement of Ker(σ(t)′) is Range(σ(t)). Hence, µ(t)− r(t)1 should bein the range of σ(t), i.e. there exists a θ(t) such that

µ(t)− r(t)1 = σ(t)θ(t)

The progressive measurability of θ is proved in Karatzas/Shreve (1999, pp. 12ff).2

Remark. In this theorem, nothing is said about whether the corresponding processZ is a martingale. Although a market price of risk exists, it is not clear if an EMMexists.

Theorem 4.24 (Uniqueness of EMM)

Assume that the market is complete and letQ be an EMM such that Xϕ(t)/M(t)t∈[0,T ]

is a Q-martingale for all admissible strategies ϕ with E[Xϕ(T )/M(T )] < ∞. If Q

is another EMM, then Q = Q, i.e. Q is the unique EMM.

Proof. Fix j ∈ 1, . . . ,m. Since the market is complete, there exists an admissibletrading strategy ϕj such that Xϕj (T ) = Yj(T ), where

Yj(T ) = exp(∫ T

0

r(s) ds− 0.5T +WQj (T )

).

From Theorem 4.19, we know that the process Xϕj/M is a local martingale underQ and Q. By assumption, Xϕj/M is even a Q-martingale implying

Xϕj (t) = M(t)EQ

[Xϕj (T )M(T )

∣∣∣∣Ft

]= M(t)EQ

[Yj(T )M(T )

∣∣∣∣Ft

]= Yj(t) P − a.s.,

i.e. Yj is a modification of Xϕj . By Proposition 3.2, Xϕj and Yj are indistinguish-able. Hence, Yj/M is a local martingale under Q and Q. By Theorem 4.17, bothmeasures are characterized by market prices of risk θ and θ such that

dWQj (t) = dWj(t) + θ(t)dt,

dW Qj (t) = dWj(t) + θ(t)dt.

Therefore,

Yj(t)M(t)

= 1 +∫ t

0

Yj(s)M(s)

dWQj (s) = 1 +

∫ t

0

Yj(s)M(s)

dW Qj (s) +

∫ t

0

Yj(s)M(s)

(θj(s)− θj(s)) ds.

Since Yj/M is a Brownian local Q-martingale, the process Yj(θj − θj)/M needs tobe indistinguishable from zero.28 Since Yj/M > 0, the market prices of risk θj andθj are indistinguishable. Since this holds for every j, we get Q = Q. 2

Remarks.28Recall that the representation of Ito processes is unique up to indistinguishability.

5 CONTINUOUS-TIME PORTFOLIO OPTIMIZATION 68

• Since Xϕ(t)/M(t)t∈[0,T ] is a Q-supermartingale, this process is a Q-martingaleif, for instants,

EQ

[Xϕ(T )M(T )

]= Xϕ(0).

• Obviously it is sufficient to check if Xϕj (t)/M(t)t∈[0,T ] is a Q-martingale forevery j = 1, . . . ,m.

5 Continuous-time Portfolio Optimization

5.1 Motivation

In the theory of option pricing the following problem is tackled:

Given: Payoff C(T )

Wanted: Today’s price C(0) and hedging strategy ϕ

In the theory of portfolio optimization the converse problem is considered:

Given: Initial wealth X(0)

Wanted: Optimal final wealth X∗(T ) and optimal strategy ϕ∗

Question: What is “optimal”?

1st Idea. Maximze expected final wealth, i.e. maxϕ E[Xϕ(T )].

Problem: Petersburg Paradox. Consider a game where a coin is flipped. If inthe n-th round, the first time head occurs, then the player gets 2n−1 Euros, i.e. theexpected value is

12· 1 +

14· 2 +

18· 4 + . . .+

12n· 2n−1 + . . . =

∞∑n=0

12

=∞.

Hence, investors who maximize expected final wealth should be willing to pay ∞Euros to play this game! Therefore, maximization of expected final wealth is usuallynot a reasonable criterion.

Question: What is the reason for this result?

Answer: People are risk averse, i.e. if they can choose between• 50 Euros or• 100 Euros with probability 0.5 and 0 Euros with probability 0.5,

they choose the 50 Euros.

Bernoulli’s idea. Measure payoffs in terms of a utility function U(x) = ln(x), i.e.

12· ln(1) +

14· ln(2) +

18· ln(4) + . . .+

12n· ln(2n−1) + . . .

=∞∑

n=1

12n· ln(2n−1) = ln(2)

∞∑n=1

n− 12n

= ln(2).


Hence, an investor following this criterion would be willing to pay at most 2 Eurosto play this game.

This idea can be generalized: For a given initial wealth x0 > 0, the investormaximizes the following utility functional

supπ∈A

E[U(Xπ(T ))],(27)

where A := π : π admissible and E[U(Xπ(T ))−] <∞ and

dXπ(t) = Xπ(t)[(r(t) + π(t)′(µ(t)− r(t)1))dt+ π(t)′σdW (t)

], Xπ(0) = x0,

and the utility function U has the following properties:

Definition 5.1 (Utility Function)

A strictly concave, strictly increasing, and continuously differentiable function U :(0,∞)→ IR is said to be a utility function if

U ′(0) := limx0

U ′(x) =∞, U ′(∞) := limx∞

U ′(x) = 0.

Remarks.

• The investor’s greed is reflected by U ′ > 0, i.e. loosely speaking: “More is betterthan less”.• The investor’s risk aversion is reflected by the concavity of U which implies

that for any non-constant random variable Z, by Jensen’s inequality,

E[U(Z)] < U(E[Z]).

i.e. loosely speaking: “A bird in the hand is worth two in the bush”.

There exist two approaches to solve the dynamic optimization problem (27):

1. Martingale approach: Theorem 4.5 ensures that in a complete market anyclaim can be replicated.=⇒ Determine the optimal wealth via a static optimization problem and interpretthis wealth as a contingent claim

2. Optimal control approac: We know that expectations can be represented assolutions to PDEs (Feynman-Kac theorem)=⇒ Represent the functional (27) as PDE and solve the PDE

Standing Assumption. We consider a Brownian market (6) with bounded coef-ficients.

5.2 Martingale Approach

Assumption in the Martingale Approach: The market is complete.

The martingale approach consists of a three-step procedure:


1st step. Compute the optimal wealth X∗(T ) by solving a static optimizationproblem.

2nd step. Prove that the static optimization problem is equivalent to the dynamicoptimization problem (27)

3rd step. Determine the replication strategy for X∗(T ) which, in the context ofportfolio optimization, is said to be the optimal portfolio strategy π∗.

1st step. Consider the constrained static optimization problem

supX∈C

E[U(X )],

subject to the so-called budget constraint (BC)

E[H(T )X ] ≤ x0

and C := C(T ) ≥ 0 : C(T ) is FT -measurable and E[U(C(T ))−] <∞.

The BC states that the present value of terminal wealth cannot exceed the initialwealth x0 (see Theorem 4.5 (i)).

Heuristic solution. Pointwise Lagrangian

L(X , λ) := U(X ) + λ(x0 −H(T )X ).

First-order condition (FOC)

U ′(X )− λH(T ) = 0,

i.e. X ∗ = V (λH(T )), where V := (U ′)−1 : (0,∞)→ (0,∞) is strictly decreasing.

Since the investor is greedy, the BC is satisfied as equality and we can choose λ∗

such thatΓ(λ) := E[H(T )V (λH(T ))] = x0(28)

leading to λ∗ = Λ(x0) := Γ−1(x0).

2nd step. Verify that the heuristic solution is indeed the optimal solution.

Lemma 5.1 (Properties of Γ)

Assume Γ(λ) <∞ for all λ > 0. Then Γ is

(i) strictly decreasing,

(ii) continuous on (0,∞),

(iii) Γ(0) := limλ0 Γ(λ) =∞, and Γ(∞) := limλ∞ Γ(λ) = 0.

Proof. (i) V is strictly decreasing on (0,∞). Since H(t) > 0 for every t ∈ [0, T ],the function Γ is also strictly decreasing.

(ii) follows from the continuity of H and V as well as the dominated convergencetheorem.29

29To check continuity at λ0, choose some λ1 < λ0. Then due to (i) we can use Γ(λ1) < ∞ as a

majorant.


(iii) Sincelimy0

V (y) =∞, limy∞

V (y) = 0,

and V is strictly decreasing, the first result follows from the monotonous convergencetheorem and the second from the dominated convergence theorem. 2

Remark. Hence, Λ(x) = Γ−1(x) exists on (0,∞) with Λ ≥ 0 and

Λ(0) := limx0

Λ(x) =∞, Λ(∞) := limx∞

Λ(x) = 0.

Therefore, one can always find a λ such that (28) is satisfied as equality.

Lemma 5.2

For a utility function U with U−1 = V we get

U(V (y)) ≥ U(x) + y(V (y)− x) for 0 < x, y <∞.

Proof. By concavity of U , Taylor approximation leads to U(x2) ≤ U(x1) +U ′(x1)(x2 − x1) and thus U(x1) ≥ U(x2) + U ′(x1)(x1 − x2) implying

U(V (y)) ≥ U(x) + U ′(V (y))(V (y)− x) = U(x) + y(V (y)− x).

2

Theorem 5.3 (Optimal Final Wealth)

Assume x0 > 0 and Γ(λ) <∞ for all λ > 0. Then the optimal final wealth reads

X ∗ = V (Λ(x0) ·H(T ))

and there exists an admissible optimal strategy π∗ ∈ A such that Xπ∗(T ) = x0 and

Xπ∗(T ) = X ∗, P -a.s.

Proof. By definition of Λ(x0), we get E[H(T )X ∗] = x0. Besides, X ∗ > 0 becauseV > 0. Hence, X ∗ satisfies all constraints and, by Theorem 4.5 (ii), there exists astrategy π∗ replicating X ∗.

We now verify E[U(X ∗)−] <∞, i.e. π∗ ∈ A. Due to Lemma 5.2,

U(X ∗) ≥ U(1) + Λ(x0)H(T )(X ∗ − 1).

Therefore,30

E[U(X ∗)−] ≤ |U(1)|+Λ(x0) E[H(T )(X ∗+1)] ≤ |U(1)|+Λ(x0) (x0+E[H(T )]) <∞.

Finally, we show that π∗ and X ∗ are indeed optimal. Consider some π ∈ A. Dueto Lemma 5.2,

U(X ∗) ≥ U(Xπ(T )) + Λ(x0)H(T )(X ∗ −Xπ(T ))

30a ≥ b implies a− ≤ b− ≤ |b|.


and thus

E[U(X ∗)] ≥ E[U(Xπ(T ))] + Λ(x0)E[H(T )(X ∗ −Xπ(T ))]

= E[U(Xπ(T ))] + Λ(x0) (x0 − E[H(T )(Xπ(T ))])︸︷︷︸BC

≥ 0

which gives the desired result. 2

3rd step. To determine the optimal strategy π∗, one can apply a similar methodas in Proposition 4.22 (ii). This is demonstrated in the following example.

Example “Power Utility with Constant Coefficients”.We make the following assumptions:• Power utility function U(x) = 1

γxγ

• I = m = 1 (“one stock, one Brownian motion”)• The coefficients r, µ, σ are constant.• σ > 0, i.e. the market is complete.

Since V (y) = (U ′)−1(y) = y1

γ−1 , the optimal final wealth reads:

X ∗ = V (Λ(x0)H(T )) = (Λ(x0)H(T ))1

γ−1 ,

whereH(T )

1γ−1 = exp

(1

1−γ rT + 0.5 11−γ θ

2T + 11−γ θW (T )

)(29)

and θ = (µ− r)/σ. On the other hand, the final wealth for a strategy π is given by

Xπ(T ) = x0 exp(rT +

∫ T

0

(µ− r)πs − 0.5σ2π2s ds+

∫ t

0

σπs dWs

).(30)

By comparing the diffusion parts of (29) and (30) we guess the optimal strategy:

π∗(t) =1

1− γµ− rσ2

implying

Xπ∗(T ) = exp(rT + 1

1−γ θ2T − 0.5 1

(1−γ)2 θ2T + 1

1−γ θW (T )).

Since Γ(λ) = E[H(T )V (λH(T ))] = λ1

γ−1 E[H(T )γ

γ−1 ] = x0, we get

Λ(x0)1

γ−1 = Γ−1(x0) = x0 exp(− γ

1−γ rT − 0.5 γ1−γ θ

2T − 0.5(

γ1−γ

)2θ2T

).

Therefore,

X ∗ = (Λ(x0)H(T ))1

γ−1 = x0 exp(rT + 0.5θ2T − 0.5

(γ

1−γ

)2θ2T + 1

1−γ θW (T )).

It is easy to check that

11−γ − 0.5 1

(1−γ)2 = 0.5− 0.5(

γ1−γ

)2.

Hence, X ∗ = Xπ∗(T ) and due to completeness π∗ is the unique optimal portfoliostrategy.


5.3 Stochastic Optimal Control Approach

Assumption. The coefficients r, µ, σ are constant and I = m = 1.

Define the value function

G(t, x) := supπ∈A

Et,x[U(Xπ(T ))],

where Et,x[U(Xπ(T ))] := E[U(Xπ(T ))|X(t) = x]. Note that G(T, x) = U(x).

5.3.1 Heuristic Derivation of the HJB

We start with a heuristic derivation of the so-called Hamiltion-Jacobi-Bellmanequation (HJB) which is a PDE for G.

Assume that there exists an optimal portfolio strategy π∗. We now carry out thefollowing procedure:

(i) For given (t, x) ∈ [0, T ] × (0,∞), we consider the following strategies over[t, T ]:• Strategy I: Use the optimal strategy π∗.• Strategy II: For an arbitrary π use the strategy

π =

π on [t, t+ ∆]

π∗ on (t+ ∆, T ]

(ii) Compute Et,x[U(Xπ∗(T ))] and Et,x[U(X π(T ))].

(iii) Using the fact that π∗ is at least as good as π and taking the limit ∆ → 0yields the HJB.

We set X∗ := Xπ∗ and X := X π.

ad (ii).

Strategy I. By assumption, Et,x[U(X∗(T ))] = G(t, x).

Strategy II.

Et,x[U(X(T ))] = Et,x[Et+∆,X(t+∆)[U(X(T ))]] = Et,x[Et+∆,X(t+∆)[U(X∗(T ))]]

= Et,x[G(t+ ∆, X(t+ ∆))] = Et,x[G(t+ ∆, Xπ(t+ ∆))],

for any admissible π.

ad (iii). By definition of the strategies,

Et,x[G(t+ ∆, Xπ(t+ ∆))] ≤ G(t, x),(31)

where we have equality if π = π∗. Given that G ∈ C1,2, Ito’s formula yields

G(t+ ∆, Xπ(t+ ∆)) = G(t, x) +∫ t+∆

t

Gt(s,Xπs )ds+

∫ t+∆

t

Gx(s,Xπs )dXπ

s


+0.5∫ t+∆

t

Gxx(s,Xπs )d < Xπ >s

= G(t, x) +∫ t+∆

t

(Gt(s,Xπ

s ) +Gx(s,Xπs )Xπ

s (r + πs(µ− r))

+0.5Gxx(s,Xπs )(Xπ

s )2π2sσ

2)ds

+∫ t+∆

t

Gx(s,Xπs )Xπ

s πsσ dWs

Given Et,x[∫ t+∆

tGx(s,Xπ

s )Xπs πsσ dWs] = 0, we can rewrite (31):

Et,x

[ ∫ t+∆

t

(Gt(s,Xπ

s ) +Gx(s,Xπs )Xπ

s (r + πs(µ− r))


s )2π2sσ

2)ds

]≤ 0.

Dividing by ∆ and taking the limit ∆→ 0 we get (Note: Xπ(t) = x)

Gt(t, x) +Gx(t, x)x(r + πt(µ− r)) + 0.5Gxx(t, x)x2π2t σ

2 ≤ 0

for any admissible π. As above, we have equality for π = π∗. Therefore, we get theHJB

supπ

Gt(t, x) +Gx(t, x)x(r + π(µ− r)) + 0.5Gxx(t, x)x2π2σ2

= 0.

with final condition G(T, x) = U(x).

Remarks.

• Since (t, x) was arbitrarily chosen, the HJB holds for all (t, x) ∈ [0, T ]× (0,∞).• Note that we take the supremum over a number π and not over a process.

5.3.2 Solving the HJB

We consider the HJB for an investor with power utility U(x) = 1γx

γ :

supπ

Gt(t, x) +Gx(t, x)x(r + π(µ− r)) + 0.5Gxx(t, x)x2π2σ2

= 0.

with final condition G(T, x) = 1γx

γ . Pointwise maximization over π yields thefollowing FOC

Gx(t, x)x(µ− r) +Gxx(t, x)x2π(t)σ2 = 0(32)

implying

π∗(t) = −µ− rσ2

Gx(t, x)xGxx(t, x)

.

Since the derivative of (32) w.r.t. π is Gxx(t, x)x2σ2, the candidate π∗ is the globalmaximum if Gxx < 0.

Substituting π∗ into the HJB yields a PDE for G:

Gt(t, x) + rxGx(t, x)− 0.5θ2G2

x(t, x)Gxx(t, x)

= 0


with G(T, x) = 1γx

γ . We try the separation G(t, x) = 1γx

γf(t)1−γ for some functionf with f(T ) = 1 and obtain

f ′ = − γ1−γ (r + 0.5 1

1−γ θ2)︸︷︷︸

=:K

f.

This is an ODE for f with the solution f(t) = exp(−K(T − t)). Hence, we get thefollowing candidates for the value function and the optimal strategy:

G(t, x) =1γxγ exp

(− (1− γ)K(T − t)

),(33)

π∗(t) =1

1− γµ− rσ2

.

Remark. Our candidate G is indeed a C1,2-function. Since γ < 1, we also getGxx(t, x) = (γ − 1)xγ−1f(t) < 0.

5.3.3 Verification

In this section, we show that our heuristic derivation of the HJB has lead to thecorrect result.

Proposition 5.4 (Verification Result)

Assume that the coefficients are constant and that the investor has a power utility

function. Then candidates (33) are the value function and the optimal strategy of

the portfolio problem (27), where for γ < 0 only bounded admissible strategies are

considered.

Proof. Let π ∈ A. Since G ∈ C1,2, Ito’s formula yields

G(T,Xπ(T )) = G(t, x) +∫ T

t

Gt(s,Xπs )ds+

∫ T

t

Gx(s,Xπs )dXπ

s

+0.5∫ T

t

Gxx(s,Xπs )d < Xπ >s

= G(t, x) +∫ T

t

(Gt(s,Xπ

s ) +Gx(s,Xπs )Xπ

s (r + πs(µ− r))


s )2π2sσ

2)ds+

∫ T

t

Gx(s,Xπs )Xπ

s πsσ dWs

(∗)≤ G(t, x) +

∫ T

t

Gx(s,Xπs )Xπ

s πsσ dWs,

where (∗) holds because G satisfies the HJB.

The right-hand side is a local martingale in T . For γ > 0 we have G > 0 and thusthe right-hand side is even a supermartingale implying

Et,x[ 1γ (Xπ(T ))γ ] ≤ G(t, x),(34)

where we use G(T, x) = 1γx

γ . Besides, since Et,x[∫ T

tGx(s,X∗

s )X∗sπ

∗sσ dWs] = 0

(exercise) and for π∗

Gt(s,X∗s ) +Gx(s,X∗

s )X∗s (r + π∗s (µ− r)) + 0.5Gxx(s,X∗

s )(X∗s )2(π∗s )2σ2 = 0,


we obtainEt,x[ 1

γ (X∗(T ))γ ] = G(t, x),

i.e. G is the value function and π∗ is the optimal solution.

Since for a bounded strategy π we get E[∫ T

tGx(s,Xπ

s )Xπs πsσ dWs] = 0, the relation

(34) also holds for γ < 0 and bounded admissible strategies. 2

Remarks.

• For γ < 0 the boundedness assumption of π can be weakened.• Note that we have not explicitly used the fact that the market is complete.

5.4 Intermediate Consumption

So far the investor has maximized expected utility from terminal wealth only. Wenow consider a situation where the investor maximizes

• expected utility from terminal wealth as well as

• expected utility from intermediate consumption.

Assumption. At time t, the investor consumes at a rate c(t) ≥ 0, i.e. over theinterval [0, T ] he consumes

∫ T

0c(t) dt.

Remark. Hence, in our model consumption can be interpreted as a continuousstream of dividends (See Exercise 40).

To formalize the problem we need to generalize the concept of admissibility:

Definition 5.2 (Consumption Process, Self-financing, Admissible)

(i) A progressively measurable, real-valued process ctt∈[0,T ] with c ≥ 0 and∫ T

0c(t) dt <∞ P − a.s. is said to be a consumption process.

(ii) Let ϕ be a trading strategy (see Definition 4.1) and c be a consumption process.

If

dX(t) = ϕ(t)′dS(t)− c(t)dt

holds, then the pair (ϕ, c) is said to be self-financing.

(iii) A self-financing (ϕ, c) is said to be admissible if X(t) ≥ 0 for all t ≥ 0.

Hence, the investor’s goal function reads

sup(π,c)∈A

E[ ∫ T

0

U1(t, c(t)) dt+ U2(Xπ,c(T ))],(35)

where U1 is a utility function for fixed t ∈ [0, T ],

A :=

(π, c) : (π, c) admissible and E[ ∫ T

0

U1(t, c(t))− dt+ U2(Xπ,c(T ))−]<∞


and

dXπ,c(t) = Xπ,c(t)[(r(t) + π(t)′(µ(t)− r(t)1))dt+ π(t)′σdW (t)

]− c(t)dt,

Xπ,c(0) = x0.

We set V1 := (U ′1)−1 and V2 := (U ′2)

−1 and define

Γ(λ) := E[ ∫ T

0

H(t)V1(t, λH(t)) dt+H(T )V2(λH(T ))]

It is straightforward to generalize the result of Section 5.2.

Theorem 5.5 (Optimal Consumption and Optimal Final Wealth)

Assume x0 > 0 and Γ(λ) < ∞ for all λ > 0. Then the optimal consumption and

the optimal final wealth reads

c∗(t) = V1(t,Λ(x0) ·H(t))

X ∗ = V2(Λ(x0) ·H(T ))

and there exists a portfolio strategy π∗ such that (π∗, c∗) ∈ A and Xπ∗,c∗(T ) = x0

and Xπ∗,c∗(T ) = X ∗, P -a.s.

We will present the solution by using the stochastic control approach. Thevalue function now reads

G(t, x) := sup(π,c)∈A

Et,x[ ∫ T

t

U1(s, c(s)) ds+ U2(Xπ,c(T ))].

As in Section 5.3 we make the following

Assumption. The coefficients r, µ, σ are constant and I = m = 1.

Assume that there exists an optimal strategy (π∗, c∗). We now carry out the pro-

cedure of Section 5.3:

(i) For given (t, x) ∈ [0, T ] × (0,∞), we consider the following strategies over[t, T ]:• Strategy I: Use the optimal strategy (π∗, c∗).• Strategy II: For an arbitrary (π, c) use the strategy

(π, c) =

(π, c) on [t, t+ ∆]

(π∗, c∗) on (t+ ∆, T ]

(ii) Compute Et,x[∫ T

tU1(s, c∗(s)) ds + U2(Xπ∗,c∗(T ))] and Et,x[

∫ T

tU1(s, c(s)) ds +

U2(X π,c(T ))].

(iii) Using the fact that (π∗, c∗) is at least as good as (π, c) and taking the limit∆→ 0 yields the HJB.

We set X∗ := Xπ∗,c∗ and X := X π,c.


ad (ii).

Strategy I. By assumption, Et,x[∫ T

tU1(s, c∗(s)) ds+ U2(X∗(T ))] = G(t, x).

Strategy II.

Et,x[ ∫ T

t

U1(s, c(s)) ds+ U2(X(T ))]

= Et,x[Et+∆,X(t+∆)

[ ∫ T

t

U1(s, c(s)) ds+ U2(X(T ))]]

= Et,x[ ∫ t+∆

t

U1(s, c(s)) ds+ Et+∆,X(t+∆)[ ∫ T

t+∆

U1(s, c(s)) ds+ U2(X(T ))]]

= Et,x[ ∫ t+∆

t

U1(s, c(s)) ds+ Et+∆,X(t+∆)[ ∫ T

t+∆

U1(s, c∗(s)) ds+ U2(X∗(T ))]]

= Et,x[ ∫ t+∆

t

U1(s, c(s)) ds+G(t+ ∆, Xπ,c(t+ ∆))]

for any admissible (π, c).

ad (iii). By definition of the strategies,

Et,x[ ∫ t+∆

t

U1(s, c(s)) ds+G(t+ ∆, Xπ,c(t+ ∆))]≤ G(t, x),(36)

where we have equality if (π, c) = (π∗, c∗). To shorten notations, we set X := Xπ,c.Given that G ∈ C1,2, Ito’s formula yields

G(t+ ∆, Xt+∆) = G(t, x) +∫ t+∆

t

Gt(s,Xs)ds+∫ t+∆

t

Gx(s,Xs)dXs

+0.5∫ t+∆

t

Gxx(s,Xs)d < X >s

= G(t, x) +∫ t+∆

t

(Gt(s,Xs) +Gx(s,Xs)Xs(r + πs(µ− r))−Gx(s,Xs)cs

+0.5Gxx(s,Xs)(Xs)2π2sσ

2)ds

+∫ t+∆

t

Gx(s,Xs)Xsπsσ dWs

Given Et,x[∫ t+∆

tGx(s,Xs)Xsπsσ dWs] = 0, we can rewrite (36):

Et,x

[ ∫ t+∆

t

U1(s, cs) ds+∫ t+∆

t

(Gt(s,Xs) +Gx(s,Xs)Xs(r + πs(µ− r))

−Gx(s,Xs)cs + 0.5Gxx(s,Xs)(Xs)2π2sσ

2)ds

]≤ 0.

Dividing by ∆ and taking the limit ∆→ 0 we get (Note: X(t) = x)

U1(t, ct) +Gt(t, x) +Gx(t, x)x(r+ πt(µ− r))−Gx(t, x)ct + 0.5Gxx(t, x)x2π2t σ

2 ≤ 0

for any admissible (π, c). As above, we have equality for (π, c) = (π∗, c∗). Therefore,we get the HJB

supπ,c

U1(t, c) +Gt(t, x) +Gx(t, x)x(r + π(µ− r))−Gx(t, x)c(37)

+0.5Gxx(t, x)x2π2σ2

= 0


with final condition G(T, x) = U2(x).

Remarks.

• Since (t, x) was arbitrarily chosen, the HJB holds for all (t, x) ∈ [0, T ]× (0,∞).• Note that we take the supremum over a number (π, c) and not over a process.

Example “Power Utility”. We consider the following power utility functions

U1(t, c) = e−αt 1γcγ ,

U2(x) = e−αT 1γxγ ,

where α ≥ 0 resembles the investor’s time preferences. The HJB reads

supπ,c

e−αt 1

γ cγ +Gt(t, x) +Gx(t, x)x(r + π(µ− r))−Gx(t, x)c


= 0.

with final condition G(T, x) = e−αT 1γx

γ . The FOCs read

Gx(t, x)x(µ− r) +Gxx(t, x)x2π(t)σ2 = 0

e−αtcγ−1 −Gx(t, x) = 0

implying

π∗(t) = −µ− rσ2

Gx(t, x)xGxx(t, x)

c∗(t) = eα

γ−1 tG1

γ−1x (t, x)

Substituting (π∗, c∗) into the HJB yields

1−γγ e

αγ−1 tG

γγ−1x (t, x) +Gt(t, x) + rxGx(t, x)− 0.5θ2

G2x(t, x)

Gxx(t, x)= 0.

We try the separation G(t, x) = 1γx

γfβ(t) for some function f with f(T ) = 1 andsome constant β. Simplifying yields

1−γγ e−

α1−γ tf

β+γ−1γ−1 + β

γ f′ +(r + 0.5 1

1−γ θ2)f = 0

Choosing β = 1− γ leads to

1−γγ e−

α1−γ t + 1−γ

γ f ′ +(r + 0.5 1

1−γ θ2)f = 0,

which can be rewritten asf ′ = Kf + g,

with K := − γ1−γ (r+0.5 1

1−γ θ2) and g(t) := −e

αγ−1 t. This is an inhomgeneous ODE

for f . Variations of Constants yields

f(t) = fH(t)(f(T )−

∫ T

t

g(s)fH(s)

ds),


where fH(t) = e−K(T−t) is the solution of the corresponding homogeneous ODE(see Section 5.3). Hence,

f(t) = e−K(T−t)(1 +

∫ T

t

e−α

1−γ s+K(T−s) ds︸︷︷︸≥0

)

= e−K(T−t)

1− eKT (1−γ)α+K(1−γ)

(e−( α

1−γ +K)T − e−( α1−γ +K)t

).

Especially, f > 0. Hence, we get the following candidates for the value functionand the optimal strategy:

G(t, x) =1γxγf1−γ(t),(38)

π∗(t) =1

1− γµ− rσ2

,

c∗(t) =X∗(t)f(t)

e−α

1−γ t.

Remark.

• Our candidate G is indeed a C1,2-function. Since γ < 1, we also get Gxx(t, x) =(γ − 1)xγ−1f(t) < 0.• Clearly, it needs to be shown that these candidates are indeed the value function

and the optimal strategy! This can be done analogously to Proposition 5.4. Forthe convenience of the reader, the proof of this verification result is also presented.



function. Then the candidates (38) are the value function and the optimal strategy

of the portfolio problem (35), where for γ < 0 only bounded portfolio strategies π

are considered.

Proof. Let (π, c) ∈ A. Since G ∈ C1,2, Ito’s formula yields

G(T,Xπ,c(T )) +∫ T

t

e−αs 1γ c

γs ds

= G(t, x) +∫ T

t

Gt(s,Xπ,cs )ds+

∫ T

t

Gx(s,Xπ,cs )dXπ,c

s

+0.5∫ T

t

Gxx(s,Xπ,cs )d < Xπ,c >s +

∫ T

t

1γ e−αscγs ds

= G(t, x) +∫ T

t

(Gt(s,Xπ,c

s ) +Gx(s,Xπ,cs )Xπ,c

s (r + πs(µ− r))−Gx(s,Xπ,cs )cs

+0.5Gxx(s,Xπ,cs )(Xπ,c

s )2π2sσ

2 + 1γ e−αscγs

)ds

+∫ T

t

Gx(s,Xπ,cs )Xπ,c

s πsσ dWs(39)

It is straightforward to show that Et,x[∫ T

tGx(s,X∗

s )X∗sπ

∗sσ dWs] = 0 (exercise).

Besides, G satisfies the HJB, i.e. for (π, c) = (π∗, c∗)

Gt(s,X∗s ) +Gx(s,X∗

s )X∗s (r + π∗s (µ− r))−Gx(s,X∗

s )c∗s

+0.5Gxx(s,X∗s )(X∗

s )2(π∗s )2σ2 + 1γ e−αs(c∗s)

γ = 0,


we obtain

Et,x[G(T,X∗(T ))] + Et,x[ ∫ T

t

e−αs 1γ (c∗s)

γ ds]

= G(t, x).(40)

implying

Et,x[e−αT 1γX

∗(T )γ ] + Et,x[ ∫ T

t

e−αs 1γ (c∗s)

γ ds]

= G(t, x)

because G(T, x) = e−αT 1γx

γ .

Finally, we need to show that for all (π, c) ∈ A

Et,x[e−αT 1γX

π,c(T )γ ] + Et,x[ ∫ T

t

e−αs 1γ c

γs ds

]≤ G(t, x).

Since G satisfies the HJB, for all (π, c) ∈ A we get

Gt(s,Xπ,cs ) +Gx(s,Xπ,c

s )Xπ,cs (r + πs(µ− r))−Gx(s,Xπ,c

s )cs

+0.5Gxx(s,Xπ,cs )(Xπ,c

s )2(πs)2σ2 + 1γ e−αs(cs)γ ≤ 0.

Therefore, (39) yields

G(T,Xπ,c(T )) +∫ T

t

e−αs 1γ c

γs ds ≤ G(t, x) +

∫ T

t

Gx(s,Xπ,cs )Xπ,c

s πsσ dWs(41)

and now two cases are distinguished.

1st case: γ > 0. The right-hand side of (41) is a local martingale in T . Sincefor γ > 0 the left-hand side is greater than zero, this is even a supermartingaleimplying

Et,x[G(T,Xπ,c(T ))] + Et,x[ ∫ T

t

e−αs 1γ c

γs ds

]≤ G(t, x)(42)

and thus

Et,x[e−αT 1γX

π,c(T )γ ] + Et,x[ ∫ T

t

e−αs 1γ c

γs ds

]≤ G(t, x).

2nd case: γ < 0. Since for a bounded strategy π we obtain

E[∫ T

t

Gx(s,Xπs )Xπ

s πsσ dWs] = 0,

the result of the 1st case also holds for γ < 0 and bounded admissible strategies.2

Remark. If we set c ≡ 0, the previous proof is identical to the proof of Proposition5.4.

5.5 Infinite Horizon

If the optimal portfolio decision over the life-time of an investor is to be deter-mined, then the investment horizon lies very far into the future. For this reason, itreasonable to approximate this situation by a portfolio problem with T =∞.


Hence, the investor’s goal function reads

sup(π,c)∈A

E[ ∫ ∞

0

U1(t, c(t)) dt],(43)

where

A :=

(π, c) : (π, c) admissible and E[ ∫ ∞

0

U1(t, c(t))− dt]<∞

and

dXπ,c(t) = Xπ,c(t)[(r(t) + π(t)′(µ(t)− r(t)1))dt+ π(t)′σdW (t)

]− c(t)dt,

Xπ,c(0) = x0.

As in the previous section we make the following

Assumption 1. The coefficients r, µ, σ are constant and I = m = 1.

Additionally, we make the

Assumption 2. U1(t, c(t)) = e−αtU(c(t)), where α ≥ 0 and U is a utility function.

The value function is given by

G(t, x) := sup(π,c)∈A

Et,x[ ∫ ∞

t

e−αsU(c(s)) ds].

Lemma 5.7

The value function has the representation

G(t, x) = e−αtH(x),

where H is a function depending on wealth only.

Proof. We obtain

G(t, x)eαt = sup(π,c)∈A

Et,x[ ∫ ∞

t

e−α(s−t)U(c(s)) ds]

= sup(π,c)∈A

E0,x[ ∫ ∞

0

e−αsU(c(s)) ds]

=: H(x)

2

Hence, the HJB (37) can be rewritten as follows:

supπ,c

e−αtU(c) +Gt(t, x) +Gx(t, x)x(r + π(µ− r))−Gx(t, x)c


=

supπ,c

U(c)− αH(x) +Hx(x)x(r + π(µ− r))−Hx(x)c+ 0.5Hxx(x)x2π2σ2

= 0

Example “Power Utility”. Let U(c) = 1γ c

γ . Then the FOCs read

Hx(x)x(µ− r) +Hxx(x)x2π(t)σ2 = 0

cγ−1 −Hx(x) = 0


implying

π∗(t) = −µ− rσ2

Hx(x)xHxx(x)

c∗(t) = H1

γ−1x (x).

Substituting (π∗, c∗) into the HJB leads to

1−γγ H

γγ−1x (x)− αH(x) + rxHx(x)− 0.5θ2

H2x(x)

Hxx(x)= 0.

We try the ansatz H(x) = 1γx

γkβ for some constants k and β. Simplifying yields

1−γγ k

βγ−1 − α

γ + r + 0.5 11−γ θ

2 = 0

Choosing β = γ − 1 leads to

k = γ1−γ

(αγ − r − 0.5 1

1−γ θ2).

Our ansatz is only reasonable if k ≥ 0. For γ < 0 this is valid in any case. Forγ > 0 we need to assume that

α ≥ γ(r + 0.5 11−γ θ

2).(44)

Hence, we get the following candidates for the value function and the optimal strat-egy:

G(t, x) =1γe−αtxγkγ−1,(45)

π∗(t) =1

1− γµ− rσ2

,

c∗(t) = kX∗(t).

Remark. In contrast to the previous section, the investor now consumes a time-

independent constant fraction of wealth.



function where for γ < 0 only bounded portfolio strategies π are considered. If (44)

holds as strict inequality, i.e.

α > γ(r + 0.5 11−γ θ

2),(46)

then the candidates (45) are the value function and the optimal strategy of the

portfolio problem (43).

Proof. Let (π, c) ∈ A. Analogously to the proof of Proposition 5.6, we get therelations (42) and (40) for the candidate G for the value function of this subsectiongiven by (45)

Et,x[G(T,Xπ,c(T ))] + Et,x[ ∫ T

t

e−αs 1γ c

γs ds

]≤ G(t, x)

Et,x[G(T,X∗(T ))] + Et,x[ ∫ T

t

e−αs 1γ (c∗s)

γ ds]

= G(t, x).


Firstly note that, by the monotonous convergence theorem,31 for any consumptionprocess c

limT→∞

Et,x[ ∫ T

t

e−αscγs ds]

= Et,x[ ∫ ∞

t

e−αscγs ds].(47)

Secondly, since G(T,X∗(T )) = 1γ e−αTX∗(T )γkγ−1, we get

Et,x[G(T,X∗(T ))] = 1γ k

γ−1 e−αT Et,x[X∗(T )γ ]

= 1γ k

γ−1 e−αTxγEt,x[eγr+ 1

1−γ θ2−0.5 1(1−γ)2

θ2−kT+ γ1−γ θW (T−t)

]= 1

γ kγ−1xγe

−αT+γr+ 11−γ θ2−0.5 1

(1−γ)2θ2−k+0.5 γ2

(1−γ)2θ2T

= 1γ k

γ−1xγe−kT −→ 0 as T →∞,(48)

since (46) ensures that k > 0. Therefore,

Et,x[ ∫ ∞

t

e−αs 1γ (c∗s)

γ ds]

= G(t, x).

To prove

Et,x[ ∫ ∞

t

e−αs 1γ c

γs ds

]≤ G(t, x)(49)

two cases are distinguished.

1st case: γ > 0. Since in this case G(T,Xπ,c(T )) ≥ 0, the relation (49) followsimmediately from (42).

2nd case: γ < 0. Since in this case G < 0, the proof is more involved. From (42)it follows that

supπ,c

Et,x[G(T,Xπ,c(T ))]

+ Et,x

[ ∫ T

t

e−αs 1γ c

γs ds

]≤ G(t, x)

and we obtain the following estimate

0 ≥ supπ,c

Et,x[G(T,Xπ,c(T ))]

= sup

π,c

Et,x

[1γ e

−αTXπ,c(T )γ]

kγ−1

(∗)≥ sup

π,c

Et,x

[1γ e

−αTXπ,c(T )γ +∫ T

t

e−αs 1γ c

γs ds

]kγ−1

≥ supπ

Et,x

[1γ e

−αTXπ,0(T )γ]

kγ−1 = supπ

Et,x

[1γX

π,0(T )γ]

e−αT kγ−1

(+)= 1

γxγe

γ(r+0.51

1−γ θ2)(T−t)e−αT kγ−1 (46)−→ 0 as T →∞,

where (∗) is valid because∫ T

te−αs 1

γ cγs ds ≤ 0 and (+) follows from Proposition 5.4.

This establishes (49). 2

Remarks.

• The relationlim

T→∞Et,x[G(T,Xπ,c(T ))] = 0

is the so-called transversality condition.31We emphasize that 1/γ is deliberately omitted because without this factor the argument of

the expected value is positive.


• For γ > 0 the property (48) can be proved by using a different argument:

0 ≤ Et,x[G(T,X∗(T ))] = Et,x[ 1γX

∗(T )γ ] e−αT kγ−1

≤ Et,x[ 1γX

π∗,0(T )γ ] e−αT kγ−1

≤ supπ

Et,x[ 1γX

π,0(T )γ ] e−αT kγ−1

(+)= 1

γxγe

γ(r+0.51

1−γ θ2)(T−t)e−αT kγ−1 (46)−→ 0 as T →∞

where (+) follows from Proposition 5.4. Applying this argument one can avoidsome tedious calculations.• As in Proposition 5.4, the boundedness assumption for γ < 0 can be weakened.

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Duffie, D. (2001): Dynamic asset pricing theory, 3rd ed., Princeton University Press,Princeton.

Karatzas, I.; S. E. Shreve (1991): Brownian motion and stochastic calculus, 2nd ed.,Springer, New York.

Karatzas, I.; S. E. Shreve (1999): Methods of mathematical finance, Springer, NewYork.

Korn, R. (1997): Optimal portfolios, World Scientific, Singapore.

Korn, R.; E. Korn (1999): Optionsbewertung und Portfolio-Optimierung, Vieweg,Braunschweig/Wiesbaden.

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