mathematical finance - math.uni-kiel.de

Jan Kallsen

Mathematical FinanceAn introduction in discrete time

CAU zu Kiel, WS 16/17, as of January 31, 2017

Contents

0 Some notions from stochastics 40.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

0.1.1 Probability spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 40.1.2 Random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 50.1.3 Conditional probabilities and expectations . . . . . . . . . . . . . . 6

0.2 Absolute continuity and equivalence . . . . . . . . . . . . . . . . . . . . . 70.3 Conditional expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

0.3.1 σ-fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80.3.2 Conditional expectation relative to a σ-field . . . . . . . . . . . . . 10

1 Discrete stochastic calculus 161.1 Stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3 Stochastic integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4 Intuitive summary of some notions from this chapter . . . . . . . . . . . . 27

2 Modelling financial markets 362.1 Assets and trading strategies . . . . . . . . . . . . . . . . . . . . . . . . . 362.2 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.3 Dividend payments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.4 Concrete models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Derivative pricing and hedging 533.1 Contingent claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2 Liquidly traded derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3 Individual perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4.1 Forward contract . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4.2 Futures contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.4.3 European call and put options in the binomial model . . . . . . . . 633.4.4 European call and put options in the standard model . . . . . . . . 66

3.5 American options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2

CONTENTS 3

4 Incomplete markets 824.1 Martingale modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.2 Variance-optimal hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5 Portfolio optimization 895.1 Maximizing expected utility of terminal wealth . . . . . . . . . . . . . . . 895.2 Utility-based pricing and hedging . . . . . . . . . . . . . . . . . . . . . . 95

6 Elements of continuous-time finance 1066.1 Continuous-time theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.2 Black-Scholes model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Chapter 0

Some notions from stochastics

0.1 Basics

We start these notes with a small remark concerning the tables and figures that illustratevarious concepts and results. If numbers appear in green, this indicates that they have or canbeen computed from the other (black) numbers, which are given in the first place. In otherwords: black numbers are model input, green ones follow from them using the definitionsand tools in the text.

0.1.1 Probability spaces

This course requires a decent background in stochastics which cannot be provided here. Weonly recall a few indispensible notions and the corresponding notation. Random experimentsare modelled in terms of probability spaces (Ω,P(Ω), P ). The sample space Ω representsthe set of all all possible outcomes of the experiment, e.g. Ω = 1, 2, 3, 4, 5, 6 if you throw adie. We consider sample spaces with finitely many elements in this course. These outcomeshappen with certain probabilities, which are quantified by a probability measure P . Formathematical reasons, probabilities are not assigned to elements ω ∈ Ω (i.e. the outcomes1, . . . , 6 in our example) but to subsets A ⊂ Ω, which are called events. The power setP(Ω) is the set of all subsets A ⊂ Ω, i.e. the set of all events in stochastic language. In ourexample, we have

P(Ω) =∅, 1, . . . , 6, 1, 1, 1, 2, . . . , 1, 2, 3, 4, 5, 6

.

A probability measure P assigns probabilities to any of these events. More precisely, it isa mapping P : P(Ω)→ [0, 1], which is normalized and additive. This means

1. P (Ω) = 1,

2. P (A ∪B) = P (A) + P (B) for any disjoint sets A,B ⊂ Ω (i.e. A ∩B = ∅).

As a side remark, additivity is replaced by the slightly stronger requirement of σ-additivityfor infinite sample spaces.

4

0.1. BASICS 5

Very often, probability measures are defined in terms of a probability mass function,i.e. a function % : Ω → [0, 1] with

∑ω∈Ω %(ω) = 1. The number %(ω) stands for the

probability of the single outcome ω. The probability measure corresponding to % is given by

P (A) :=∑ω∈A

%(ω).

In our example we would assume all outcomes to be equally likely, i.e. %(ω) = 1/6 forω = 1, . . . , 6. This leads to

P (A) =∑ω∈A

1

6=|A|6

for any A ⊂ Ω.

0.1.2 Random variables

Often we are not so much interested in the particular outcome of the random experiment butof a quantitative aspect of it. This is formalized in terms of a random variable, which isfunction X : Ω → R assigning a real number X(ω) to any particular outcome ω. Insteadof R we consider sometimes Rn, in which case X is a vector-valued random variable. Theexpected value E(X) of a random variable is its mean if the values are weighted by theprobabilities of the outcomes, specifically

E(X) :=∑ω∈Ω

X(ω)P (ω).

This can also be written asE(X) =

∑x∈R

xP (X = x),

where P (X = x) is an abbreviation for P (ω ∈ Ω : X(ω) = x) and the sum actuallyextends only to the finitely many values of X . We also recall the variance

Var(X) := E((X − E(X))2

)= E(X2)− (E(X))2.

If we consider X(ω) = ω in our example of throwing a die, we have of course

E(X) =1 + · · ·+ 6

6= 3.5

and

Var(X) =12 + · · ·+ 62

6− (E(X))2 = 211

12.

An important random variable is the indicator of a set A ⊂ Ω, which is defined as

1A(ω) :=

1 if ω ∈ A,0 if ω ∈ AC .

6 CHAPTER 0. SOME NOTIONS FROM STOCHASTICS

ω P (ω) P (ω|B)

1 1/6 0

2 1/6 1/3

3 1/6 0

4 1/6 1/3

5 1/6 0

6 1/6 1/3

Table 1: Conditional probabilities given the event B := 2, 4, 6

Here, AC := Ω \ A denotes the complement of a set A. The expectation of an indicator isthe probability of the set: E(1A) = P (A) for A ⊂ Ω. Moreover, we sometimes write∫

XdP :=

∫X(ω)P (dω) := E(X) =

∑ω∈Ω

X(ω)P (ω)

and ∫A

XdP :=

∫A

X(ω)P (dω) := E(X1A) =∑ω∈A

X(ω)P (ω),

which is motivated by the fact that the expected value in general – not necessarily finite –probability spaces is defined in terms of a (Lebesgue) integral.

0.1.3 Conditional probabilities and expectations

Suppose that we receive partial information about the outcome of a random experiment,namely that the outcome of our experiment belongs the event B ⊂ Ω. We may e.g. knowthat throwing a die procuced an even number, which means that the event B := 2, 4, 6 hashappened.

This partial knowledge changes our assessment. We know that outcomes in BC areimpossible whereas those in B are more likely than without the additional information.More precisely, the conditional probability of an event A given B is defined as

P (A|B) :=P (A ∩B)

P (B)

unless P (B) = 0. It is easy to see that P (A|B) is again a probability measure if we considerit as a function of A with B being fixed. Cf. Table 1 for an illustration in the above example.

To sets A,B are called (stochatically) independent if P (A|B) = P (A), i.e. knowingthat B has happened does not make A more or less probable. Independence is equivalent toP (A∩B) = P (A)P (B) which makes sense if P (B) = 0 as well. Moreover, one can see thatindependence is a symmetric concept. Random variables X, Y are called (stochatically)independent if P (X ∈ A, Y ∈ B) = P (X ∈ A)P (Y ∈ B) for any two sets A,B. Thisimplies that E(XY ) = E(X)E(Y ).

0.2. ABSOLUTE CONTINUITY AND EQUIVALENCE 7

ω P (ω) Q(ω) dQdP

(ω) dPdQ

(ω)

1 1/6 1/10 3/5 5/3

2 1/6 1/10 3/5 5/3

3 1/6 1/10 3/5 5/3

4 1/6 1/10 3/5 5/3

5 1/6 1/10 3/5 5/3

6 1/6 1/2 3 1/3

Table 2: Equivalent probability measures and the corresponding densities

If probabilities change, expectations of random variables X change as well. The con-ditional expectation of X given B is just the ordinary expected value if we replace theoriginal probabilities by conditional ones, i.e.

E(X|B) :=∑ω∈B

X(ω)P (ω|B) =

∑ω∈BX(ω)P (ω)

P (B)=E(X1B)

P (B).

In our example, the expectation of the outcome X(ω) = ω changes from E(X) = 3.5 to

E(X|B) =2 + 4 + 6

3= 4

if we know the result to be even. 26.10.16

0.2 Absolute continuity and equivalence

In statistics and mathematical finance we often need to consider several probability mea-sures at the same time. E.g. the probability measures P and Q in Table 2 correspond to anordinary fair die resp. a loaded die, where the outcome 6 happens much more often. Thetools of statistics are used if we do not know beforehand whether P or Q (or yet anotherprobability measure) corresponds to our experiment. In mathematical finance, however, weusually assume the “real” probabilities to be known. Here, we consider alternative probabil-ity measures as a means to simplify calculations.

The relation between two probability measures P,Q can be described in terms of thedensity dQ

dP, which is the random variable whose value is simply the ratio of Q- and P -

probabilities:dQ

dP(ω) :=

Q(ω)P (ω)

, (0.1)

In other words, it is the ratio of the corresponding probability mass functions, cf. Table 2 foran example. Of course, (0.1) only makes sense if we do not divide by zero. More generally,densities are defined in the case of absolute continuity.

Definition 0.1 1. A set N ⊂ Ω is called null set or, more precisely, P -null set if itsprobability is zero, i.e. P (N) = 0.


2. A probability measure Q is called absolutely continuous relative to P , written QP , if all P -null sets are Q-null sets as well.

3. Probability measures P,Q are called equivalent, written P ∼ Q, if both Q P andP Q, i.e. if both probability measures have the same null sets.

Equivalent probability measures differ with respect to concrete probabilities but not asto whether a given event may happen at all. If Q is absolutely continuous with respect toP , we define the density dQ

dPas in (0.1). For ω with P (ω) = 0, where (0.1) does not

make sense, we take an arbitrary value, e.g. 0. We can then compute Q-probabilities fromP -probabilities via

Q(A) =

∫A

dQ

dPdP

(= EP

(1AdQ

dP

))and, more generally, Q-expectations via

EQ(X) = EP

(XdQ

dP

).

Indeed, we have

EQ(X) =∑ω∈Ω

X(ω)Q(ω)

=∑ω∈Ω

X(ω)dQ

dP(ω)P (ω)

= EP

(XdQ

dP

)and hence

Q(A) = EQ(1A)

= EP

(1AdQ

dP

)=

∫A

dQ

dPdP.

0.3 Conditional expectation

0.3.1 σ-fields

In this course we often consider subsets of the set P(Ω) of all events, which are calledalgebras. Specifically, F ⊂ P(Ω) is called algebra if

1. Ω ∈ F ,

2. AC ∈ F for any A ∈ F (where AC := Ω \ A),

3. A ∪B ∈ F for any A,B ∈ F .

0.3. CONDITIONAL EXPECTATION 9

F2

6

4B /∈ F

2

F1

5

3

A ∈ F1

ω

Figure 1: The partition generating a σ-field

This means that unions, intersections, complements etc. of sets in F always lead to setsin F . For infinite sample spaces one replaces Axiom 3 by a slightly stronger requirement,which leads to the notion of a σ-algebra or σ-field. In our setup of finite Ω this amountsto the same thing. Nevertheless, we use from now on the terminology σ-field instead ofalgebra because it is more common in the literature. It is not hard to show the

Lemma 0.2 For any σ-field F there is a partition F1, . . . , Fn of Ω (i.e. F1, . . . , Fn ⊂ Ω

with Fi ∩ Fj = ∅ for i 6= j and F1 ∪ · · · ∪ Fn = Ω) such that F contains all unions of theFi, i.e.

F =∅, F1, . . . , Fn, F1 ∪ F2, F1 ∪ F3, . . . , F1 ∪ · · · ∪ Fn

.

We call F1, . . . , Fn the partition that generates F and we write F = σ(F1, . . . , Fn).

Consequently, we can identify a σ-field with a partition of the sample space.A σ-fields stands for partial information. Suppose that we do not know the exact out-

come ω of a random experiment. We only know which of the sets F1, . . . , Fn the outcome ωbelongs to. E.g. we only know whether an even or odd number has appeared on our die. Thispartial information is represented by the σ-field which is generated from F1 := 2, 4, 6 andF2 := 1, 3, 5, cf. Figure 1. So we can tell whether A := 1, 3, 5 has happened but notwhether the outcome is in B := 4, 6 or not.

The smallest σ-field F = ∅,Ω is generated by the partition that consists only of Ω.This trivial σ-field stands for absence of any information. The other extreme is the power setF = P(Ω), which is generated by the finest partition ω1, . . . , ωn of Ω = ω1, . . . , ωn.It stands for complete information.

We also need a notion for the fact that a random variable depends only on the informationgiven by a σ-field F .

Definition 0.3 A random variable X is called F -measurable if X is constant on the setsof the partition that generates F . In other words, it is of the form

X(ω) =n∑i=1

xi1Fi(ω), (0.2)

where F1, . . . Fn is the filtraton that generates F and xi is the value of X on Fi.


F2

6 0 1

4 0 1

2 0 0

F1

5 1 0

3 1 0

1 1 0

ω X(ω) Y (ω)

Figure 2: An F -measurable X and a not F -measurable random variable Y

F1 = X = −1−11

−12

F2 = X = 003

04

F3 = X = 115

16

σ(X)X(ω)ω

Figure 3: The σ-field resp. partition generated by a random variable X

This is illustrated in Figure 2 where F is the σ-field from above. X is F -measurablebecause it is determined by the information whether the outcome is even or odd. This is notthe case for Y which is not constant on F2 and hence not F -measurable.

F -measurability means that a random variable may not be “as random” as an arbitraryrandom variable. In the extreme case of the trivial σ-field F = ∅,Ω, any F -measurablerandom variable is deterministic, i.e. constant. On the other hand, any random variable isF -measurable for the power set F = P(Ω).

We can also turn things around and wonder what information – in the sense of a σ-fieldis delivered by a random variable. This σ-field is denoted as σ(X) and it is generated by thepartition X = x1, . . . , X = xn of X , where x1, . . . , xn are the values of X and we usethe shorthand notation X = x1 := ω ∈ Ω : X(ω) = x1 etc. σ(X) is called the σ-fieldgenerated by X . It is the smallest σ-field such that X is F -measurable. For an illustrationcf. Figure 3.

0.3.2 Conditional expectation relative to a σ-field

Similarly as in Section 0.1.3 our assessment of probabilities and expectations change if wereceive partial information. Here, our information comes from a σ-field F . In other words,we know which of the sets F1, . . . , Fn of the generating partition the outcome ω belongs to.


4

6 6 1/6

1/3

4 4 1/61/3

2 2 1/6

1/3

3

5 5 1/6

1/3

3 3 1/61/3

1 1 1/6

1/3

E(X|F )(ω) X(ω) ω P (ω)P (ω|F )(ω)

Figure 4: Conditional probabilities and conditional expectation

2

1 4

6 1·

1/3

4 1·1/3

2 1

2

·1/3

·

1/2

0 3

5 0·

1/3

3 0·1/3

1 0·1/3

·

1/2

E(ZE(X|F )) Z(ω) E(X|F )(ω) X(ω) Z(ω) E(ZX)··

1/6

1/6

1/6

1/6

1/6

1/6

Figure 5: Illustration of Lemma 0.4


The conditional probability of an event A ⊂ Ω given F is defined as

P (A|F )(ω) := P (A|Fi) =P (A ∩ Fi)P (Fi)

if ω ∈ Fi,

i.e. it is the conditional probability of Section 0.1.3 given the particular event Fi we know tohave happened. Similarly, we define the conditional expectation of a random variable Xgiven F as

E(X|F )(ω) := E(X|Fi) =∑ω∈Fi

X(ω)P (ω|Fi) if ω ∈ Fi, (0.3)

Note that conditional probabilities and expectations given F are random variables becausethey depend on ω. But they are not “as random” as an arbitrary random variable becausethey are F -measurable. I.e. they depend not directly on ω but only on which of the setsF1, . . . , Fn has happened.

Here is an important property of conditional expectation:

Lemma 0.4 The random variable E(X|F ) is F -measurable. Moreover, it satisfies

E(ZE(X|F )

)= E(ZX)

for any F -measurable random variable Z.

The previous lemma is illustrated in Figure 5. In general infinite probability spaces, Def-inition (0.3) does not make sense. Then the properties in Lemma 0.4 are used to defineconditional expectations.

The conditional expectation E(X|F ) can be interpreted as a best predictaion or ap-proximation of X given the partial information F . If F is trivial (no information), thenE(X|F ) = E(X). If, on the other hand, F = P(Ω) (full information), we haveE(X|F ) = X . For general F , the conditional expectation E(X|F ) somehow interpo-lates between X and E(X).

Let us state some useful rules.

Lemma 0.5 1. If X is F -measurable, then E(X|F ) = X . This holds in particular forF = P(Ω).

2. If X and F are independent (which means P (X ∈ B ∩ F ) = P (X ∈ B)P (F )

for any sets B and F ∈ F ), then E(X|F ) = E(X). This holds in particular forF = ∅,Ω.

3. E(E(X|F )) = E(X)

4. E(E(X|F )|G ) = E(X|G ) = E(E(X|G )|F ) if G is a sub-σ-field of F (i.e. G is aσ-field with G ⊂ F ).

5. E(X|F ) is linear in X , i.e. E(X + Y |F ) = E(X|F ) +E(Y |F ) and E(cX|F ) =

cE(X|F ).27.10.16


0

0 6 1/6

1/3

0 4 1/61/3

0 2 1/6

1/3

1

1 5 1/6

1/3

1 3 1/61/3

1 1 1/6

1/3

E(X|F )(ω) X(ω) ω P (ω)P (ω|F )(ω)

Figure 6: F -measurable X

6. E(X|F ) is increasing in X , i.e. E(X|F ) ≤ E(Y |F ) if X ≤ Y .

7. If Xn → X for n→∞, then E(Xn|F )→ E(X|F ).

8. If Z is F -measurable, then E(ZX|F ) = ZE(X|F ).

Let us illustrate some of these rules rather than proving them. The situation of State-ment 1 happens in Figure 6, where X is 0 for even ω and 1 for odd ones. In Figure 7,the random variable X is independent of F as required in Statement 2. Here, X(ω) isω/2 rounded to integer values. Statement 3 is illustrated in Figure 8. Here, X(ω) = ω

and F is the σ-field generated by the sets where X is odd resp. even. Figure 9 illus-trates E(E(X|F )|G ) = E(X|G ) for two nested σ-fields F and G . The similar state-ment E(E(X|G )|F ) = E(X|G ) follows from Statement 1 because Y := E(X|G ) isF -measurable and hence E(Y |F ) = Y . Statements 5–7 parallel analogous rules for ex-pected values. The first and the last statement means that F -measurable random variablesbehave as constants if they appear inside a conditional expectation.


2

3 6 1/6

1/3

2 4 1/61/3

1 2 1/6

1/3

2

3 5 1/6

1/3

2 3 1/61/3

1 1 1/6

1/3

E(X|F )(ω) X(ω) ω P (ω)P (ω|F )(ω)

Figure 7: Independence of X and F

3.5 3.5

4

6 6 1/6

1/3

4 4 1/61/3

2 2 1/6

1/3

1/2

3

5 5 1/6

1/3

3 3 1/61/3

1 1 1/6

1/3

1/2

E(X) E(E(X|F )) E(X|F )(ω) X(ω) ω P (ω)

Figure 8: Expected values of X and E(X|F )


4.5

6.5

7.5

8

1/2

71/21/2

5.5

6

1/2

5

4.5

1/2

1/2

1/2

2.5

3.5

4

1/2

31/21/2

1.5

2

1/2

11/2

1/2

1/2

E(X) E(E(X|F )|G )(ω) E(X|F )(ω) X(ω) E(X)

1/8

1/8

1/8

1/8

1/8

1/8

1/8

1/8

Figure 9: Iterated conditional expectations

Chapter 1

Discrete stochastic calculus

In mathematical finance, we treat asset price movements as stochastic processes, i.e. as ran-dom functions of time. The same is true for the varying number of assets in a portfolioand the resulting wealth. In this chapter we introduce the corresponding mathematical no-tions, which are applicable outside mathematical finance as well. We confine ourselves atthis point to a discrete set 0, 1, 2, 3, . . . of periods (e.g. days, minutes, seconds) because thecontinuous case requires considerably more involved mathematics.

1.1 Stochastic processes

A crucial role is played by the information which is available at any particular point intime. Decisions concerning e.g. buying or selling assets can only be based on the presentknowledge about prices or the market as a whole. The flow of information is expressedmathematically in terms of a so-called filtration.

Definition 1.1 A filtration F0,F1,F2, . . . on a probability space (Ω,P(Ω), P ) is an in-creasing sequence of σ-fields on Ω, i.e. Fm ⊂ Fn for m ≤ n. (Ω,P(Ω), (Fn)n∈N, P ) iscalled filtered probability space.

The σ-field Fn represents the information that is available at time n. An event A belongsto Fn if it is known at time n whether it has happened or not. For an illustration cf. Figures1.1-1.3, where σ-fields are represented by their generating partition.

From now on we fix a filtered probability space (Ω,P(Ω), (Fn)n∈N, P ).

Definition 1.2 A stochastic process X = (X0, X1, X2, . . . ) is a sequence of random vari-ables. It is called adapted if Xn is Fn-measurable for n = 0, 1, . . .

A stochastic process represents the random state of a process through time. n stands forthe corresponding time as above. The range of Xn are usually numbers or, more generally,vectors. Xn could denote e.g. the price(s) of one or several stocks at time t. Unless otherwisenoted, we assume the values of all processes to be numbers rather than vectors.

16

1.1. STOCHASTIC PROCESSES 17

Adaptedness means that we observe or know the present state of the process, at least asfar as it is random, cf. Figures 1.5, 1.6, 1.8. In particular, all deterministic processes areadapted. From now on we consider only such adapted processes.

Notation. Occasionally we write n− := n − 1 for the previous time and set 0− := 0.Moreover, we denote by ∆Xn := Xn − Xn− the increment or jump of X in period n.Finally, we write X− for the time-shifted process X , more specifically (X−)n := Xn−.

The following notion is stronger than adaptedness.

Definition 1.3 A process X is called predictable if Xn is Fn−1-measurable for any n, i.eXn is known already at time n− 1.

We will encounter predictable processes in the context of stochastic integration below. Notethat X− is predictable if X is an adapted process.

Where does the filtration in our general mathematical setup come from? Often there isno need to specify it in detail. But if it is, then it is typically of the following form.

Definition 1.4 A filtration F0,F1,F2, . . . is said to be generated by a processX if Fn =

σ(X0, . . . , Xn) for any n, i.e. Fn is the σ-field generated by X0, . . . , Xn.

The filtration generated by X is the smallest filtration relative to which X is adapted, cf.Figure 1.7. It means that all our information on random events comes from observing X .

E.g. for American-style options we need to consider random times, which are only ran-dom in the sense that they depend on random events in the past. They are called stoppingtimes.

Definition 1.5 A stopping time is a random variable T with values in 0, 1, 2, . . . ,∞which satisfies T = n ∈ Fn (or equivalently T ≤ n ∈ Fn) for any n.

Intuitively this means that the decision whether we say “Stop!” at time n can be based onthe information that is available a time n. The time of a vulcanic eruption is a stopping timeif observing the vulcano belongs to the information encoded in the filtration F0,F1, . . .

Any deterministic (i.e. non-random) time is a stopping time as well. The time exactly threehours before the vulcanic eruption is not a stopping time because we cannot look into thefuture. We cannot say for sure whether this point in time has come or not.

A typical instance of a stopping time is the first time when a process enters a given set,e.g. the first time a stock index exceeds a given threshold. In Figure 1.9, T corresponds tothe first time that process X exceeds the level 10.5.

Lemma 1.6 If X denotes an adapted process and B a set of real numbers, then T :=

infn ∈ N : Xn ∈ B is a stopping time.

T in the previous definition is the first time n such that Xn is in B. If this does nothappen at all, T attains the value∞. Note that the argument ω representing randomness issuppressed in most formulas.

18 CHAPTER 1. DISCRETE STOCHASTIC CALCULUS

We also need a mathematical notion for “freezing” a stochastic process at a particulartime.

Definition 1.7 If X denotes an adpated process, we call the process XT with XTn := XT∧n

the process stopped at time T , where T ∧ n := minT, n denotes the minimum of T andn.

The stopped process XT coincides with X up to time T and stays constant afterwards.

1.2 Martingales

The martingale is a key concept in stochastic analysis. It turns out be pivotal for mathemat-ical finance as well.

Definition 1.8 A martingale (resp. submartingale, supermartingale) is an adaptedstochastic process X such that

E(Xn|Fm) = Xm (resp. ≥ Xm, ≤ Xm) (1.1)

for any m ≤ n.

For a martingale, we expect on average the present value for the future, cf. Figure 1.10. Inthe submartingale (resp. submartingale) case we expect at least (resp. at most) the presentvalue.

The wealth in a fair game follows a martingale. As an example consider a roulette game,where the stake doubles in case that red turns up. We play several rounds betting e 1 on redand we denote the wealth process as X . If red turns up with probability 1

2, our wealth stays

on average the same after each round, i.e. X is a martingale. For the (bounded) future weexpect neither a profit nor a loss on average.

Real roulette games are biased in favour of the casino because red appears only withprobability 18

37. Hence we are rather facing a supermartingale. By contrast, stocks and

other securities are commonly believed to behave as submartingales, at least if we adjust fordividends. Economic theory claims that investors would to take the risk involved in suchassets unless they are compensated by a positive return on average.

Lemma 1.9 In the previous definition, it suffices to show

E(Xn|Fn−1) = Xn−1 (resp. ≥,≤)

or, equivalently,E(∆Xn|Fn−1) = 0 (resp. ≥,≤)

for any n ≥ 1 instead of the original conditional condition E(Xn|Fm) = Xm (bzw. ≥,≤),m ≤ n.

1.2. MARTINGALES 19

Proof. This follows by induction from E(Xn|Fn−2) = E(E(Xn|Fn−1)|Fn−2) etc., wherewe used Lemma 0.5(4).

The simplest martingales are obtained by successively adding independent, centeredrandom variables.

Example 1.10 Denote by X1, X2, . . . a sequence of independent random variables withexpected value E(Xn) = 0 for any n. Then Sn :=

∑nm=1Xm defines a martingale relative

to the filtration that is generated by X (or equivalently S).

A random variable generates a martingale in a canonical way.

Lemma 1.11 If Y is a random variable, then

Xn := E(Y |Fn)

defines a martingale X , the martingale generated by Y .

Proof. X is adapted by Lemma 0.4. The martingale property follows fromE(Xn|Fm) = E(E(Y |Fn)|Fm) = E(Y |Fm) = Xm for m ≤ n, where we usedLemma 0.5(4).

One may wonder whether any martingale is generated by a random variable, whichmeans that one can basically identify martingales with their generating random variables.This is indeed the case in our present setup of finite probability spaces. In the more generalcase of infinite spaces it is not the case.

Example 1.12 LetQ ∼ P denote another probability measure. The martingale Z generatedby the density dQ

dPis called density process of Q relative to P . The random variable Zn is

the density of Q relative to P , both restricted to Fn. This means that

Q(A) = E(Zn1A)

holds for any event A ∈ Fn. More generally, we have

EQ(X) = E(ZnX)

for any Fn-measurable random variable X .

An example is provided in Figure 1.11. The density process is useful in order to computeconditional expectations relative to a new probability measure:

Lemma 1.13 (Generalized Bayes formula) Let Q ∼ P denote a probability measure withdensity process Z. Moreover, let X denote an Fn -measurable random variable for somen. Then

EQ(X|Fm) =E(XZn|Fm)

Zmfor any m ≤ n.


1.11.16

Martingales are determined by their value in the future in the following sense:

Lemma 1.14 If X, Y are martingales with XN = YN for some N , then Xn = Yn for anyn ≤ N .

Proof. This follows from Xn = E(XN |Fn) = E(YN |Fn) = Yn.

For a martingale, the expected future value is the present value. A more general processcould exhibit a positive, negative, or varying trend. This is formalized in terms of Doob’sdecomposition. It decomposes the increment ∆Xn into a predictable trend ∆An and arandom deviation ∆Mn from that trend.

Theorem 1.15 (Doob decomposition) An adapted processX can be decomposed uniquelyin the form

X = X0 +M + A

whereM is a martingale withM0 = 0 and A a predictable process with A0 = 0. A is calledcompensator of X .

For an illustration cf. Figure 1.12. The one-period prediction

∆An := E(∆Xn|Fn−1) (1.2)

of the increment of X can be interpreted as the present trend. The cumulative sum An :=∑nm=1 ∆Am of these trends is the compensator A from the previous theorem. Its meaning

is less obvious. The difference of X and its compensator A is a martingale because itincrements ∆Mn := ∆Xn −∆An have conditional expectation 0.

If X is a submartingale (resp. supermartingale), then its compensator A is increasing(resp. decreasing).

1.3 Stochastic integral

As before we consider a filtered probability space (Ω,P(Ω), (Fn)n∈N, P ). We assume thatthat all outcomes happen with non-zero probability. The stochastic integral is a key conceptin stochastic analysis. It is nothing but a sum in our discrete-time setup.

Definition 1.16 Let X denote an Rd-valued adapted process and H an Rd-valued pre-dictable (or at least adapted) process. We call the adapted process H • X of the form

H • Xn :=n∑

m=1

H>m∆Xm. (1.3)

stochastic integral of H relative to X . Here, H>m∆Xm :=∑d

i=1Him∆X i

m denotes thescalar product of the vectors Hm = (H1

m, . . . , Hdm) and ∆Xm = (∆X1

m, . . . ,∆Xdm). In the

univariate case d = 1, we simply have H • Xn =∑n

m=1Hm∆Xm.

1.3. STOCHASTIC INTEGRAL 21

An example is provided in Figure 1.13. Note that the dot is the symbol for the operationon the right-hand side of (1.3) and does not denote a simple product. The name integralis motivated from its continuous-time extension, in which case the sum indeed generalizesto an integral. Its relevance for mathematical finance stems from the fact that it stands forfinancial gains. Suppose that X stands for the stock price evolution and H for the investor’sposition through time. More specifically, let Xn denote the stock price at time n and Hn

the number of shares held in the period from n − 1 to n. Due to stock price changes, theinvestor makes a profit of Hn(Xn − Xn−1) = Hn∆Xn in this period. Consequently, theintegral H • Xn stands for the cumulative gais resp. losses between times 0 und n, as theyare due to price changes (and not from buying resp. selling assets).

If we consider a portfolio of d stocks, both X and H turn vector-valued. In this case, X in

denotes the price at time n of stock No. i. Accordingly, H in denotes the number of shares of

stock No. i in the portfolio in the period from n−1 to n. In order to compute the cumulativegainsH • Xn from 0 to n we now have to sum up over all stocks i = 1, . . . , d and all periodsm = 1, . . . , n.

In order to get the bookkeeping right, we need to be careful about the order in whichthings happen at time n. If we interpret Hn(Xn − Xn−1) as profit at time n, this impliesthat we have acquired Hn shares before the stock price changed from Xn−1 to Xn. Putdifferently, the shares are bought at the end of period n − 1, after the stock price settledat Xn−1. This implies that only the information up to time n − 1 can be used when Hn

is chosen because time n and in particular the stock price Xn still belongs to the unknownfuture. This motivates why we typically assume the process H in the above definition to bepredictable.

Definition 1.17 If X, Y are adapted processes, we denote by

[X, Y ]n :=n∑

m=1

∆Xm∆Ym

the covariation process [X, Y ] ofX and Y . It is called quadratic variation ofX ifX = Y .

The covariation process is primarily needed for calculations as in the integration by partsrule below (cf. Lemma 1.19). It does not have an obvious interpretation in terms of financialmathematics. A large covariation [X, Y ] means that X and Y tend to move in the samedirection. A large quadratic variation [X,X] occurs if the process changes a lot.

Definition 1.18 Let X, Y be adapted processes. The compensator [X, Y ] is called pre-dictable covariation of X and Y and it is denoted as 〈X, Y 〉. For X = Y it is calledpredictable quadratic variation of X .

Note that∆〈X, Y 〉n = E(∆[X, Y ]n|Fn−1) = E(∆Xn∆Yn|Fn−1) (1.4)


and hence

〈X, Y 〉n =n∑

m=1

E(∆Xm∆Ym|Fm−1) (1.5)

by (1.2). The predictable quadratic variation can be interpreted as a generalization of the3.11.16

covariance for stochastic processes. It is used e.g. in Girsanov’s theorem below. It does nothave an obvious interpretation in terms of financial mathematics either. An example for bothquadratic variation and predictable quadratic variation can be found in Figure 1.14.

The following rules are helpful in the context of stochastic integration.

Lemma 1.19 Let X, Y be adpated and H,K predictable processes.

1. H • (K • X) = (HK) • X

2. [H • X, Y ] = H • [X, Y ]

3. Integration by parts:

XY = X0Y0 +X− • Y + Y • X (1.6)

= X0Y0 +X− • Y + Y− • X + [X, Y ] (1.7)

4. 〈H • X, Y 〉 = H • 〈X, Y 〉.

5. If X is a martingale, so is H • X .

6. If X is a supermartingale and H ≥ 0, then H • X is a supermartingale as well.

7. If X is a martingale and T a stopping time, XT is a martingale as well.

8. If X is a supermartingale and T a stopping time, XT is a supermartingale as well.

The last four rules mean that fair resp. unfavourable games cannot be turned into favourableones by clever stopping or trading. The restriction H ≥ 0 in Statement 6 effectively standsfor short sale restrictions. If short sales are allowed, one could profit from deceasing prices.

The integration by parts rule (1.7) can be illustrated by considering a US stock whoseprice X is quoted in US dollars. If Y denotes the dollar exchange rate, more precisely theprice of a US dollar in Euro, then XY represents the stock price quoted in Euro. Its changesmay be due to changes in the dollar stock price or due to changes in the exchange rate. Thetwo effects are expressed by Y− • X resp. X− • Y in (1.7). The less intuitive term [X, Y ]

indicates that another contribution occurs if X and Y change at the same time. We willcome back to this issue after Lemma 1.23.

Proof. 1. H • (K • X)n =∑n

m=1 Hm∆(K • X)m =∑n

m=1 HmKm∆Xm = (HK) • Xn


2.

[H • X, Y ]n =∑n

m=1∆(H • X)m∆Ym

=∑n

m=1Hm∆Xm∆Ym

=∑n

m=1Hm∆[X, Y ]m

= H • [X, Y ]n

3.

XnYn = X0Y0 +∑n

m=1(XmYm −Xm−1Ym−1)

= X0Y0 +∑n

m=1(Xm−1(Ym − Ym−1) + Ym(Xm −Xm−1))

= X0Y0 +X− • Yn + Y • Xn

and

Y • Xn =∑n

m=1Ym∆Xm

=∑n

m=1(Ym−1∆Xm + (Ym − Ym−1)∆Xm)

= Y− • Xn + [X, Y ]n

4. It suffices to show that the increments of both sides coincide.

∆(H • 〈X, Y 〉)n = Hn∆〈X, Y 〉n(1.4)= HnE(∆[X, Y ]n|Fn−1)

Lemma 0.5(9)= E(Hn∆[X, Y ]n|Fn−1)2.= E(∆[H • X, Y ]n|Fn−1)

(1.5)= ∆〈H • X, Y 〉n

5. H • X is again an adapted process. The martingale property follows from

E(H • Xn|Fn−1) = E(H • Xn−1 +Hn∆Xn|Fn−1)Lemma 0.5

= H • Xn−1 +HnE(∆Xn|Fn−1)

= H • Xn−1 +Hn(E(Xn|Fn−1)−Xn−1)

= H • Xn−1.

6. This follows along the same lines as 5.7.,8. The stopped process can be written as a stochastic integral of the

form XT = X0 + H • X with Hn := 1T≥n. H is predictable becauseHn = 1 = T ≥ n = T ≤ n − 1C ∈ Fn−1. This is clear if it is interpretedin terms of financial gains: XT − X0 stands for the profit if the stock is held until time T .Since H equals 1 before T and 0 afterwards, the expression H • X represents the samething. The assertion follows from Statements 5 and 6.


Remark.

1. The stochastic integral H • X is linear in H and X .

2. Both the covariation [X, Y ] and the predictable covariation 〈X, Y 〉 are linear inX undY .

3. The above rules hold for vector-valued processes as well if they make sense at all, e.g.H • (K • X) = (HK) • X if K,X are Rd-valued.

The most important rule in continuous-time stochastic calculus is Itô’s formula. In dis-crete time it reduces to a simple telescopic sum and it does not play an important role. It isstated here only for the sake of completeness.

Theorem 1.20 (Itô’s formula) If X denotes an adapted process and f : R → R a differ-entiable function, then the adapted process f(X) satisfies

f(Xn) = f(X0) +n∑

m=1

(f(Xm)− f(Xm−))

= f(X0) + f ′(X−) • Xn +n∑

m=1

(f(Xm)− f(Xm−)− f ′(Xm−)∆Xm

)Proof. The first equation follows because almost all terms in the telescopic sum cancel. Thesecond equation holds because f ′(X−) • Xn =

∑nm=1 f

′(Xm−)∆Xm.

Remark. If the jumps ∆X are small and f is twice continuously differentiable, we have theapproximation

f(Xn) ≈ f(X0) + f ′(X−) • Xn +1

2f ′′(X−) • [X,X]n (1.8)

Proof. A sufficiently smooth function can be approximated by its second order Taylor poly-nomial f(x + h) ≈ f(x) + f ′(x)h + 1

2f ′′(x)h2 if h is small enough. In our setup this

yields

f(Xm) = f(Xm−1 + ∆Xm)

≈ f(Xm−1) + f ′(Xm−1)∆Xm +1

2f ′′(Xm−1)(∆Xm)2

and hence

n∑m=1

(f(Xm)− f(Xm−)− f ′(Xm−)∆Xm

)≈ 1

2

n∑m=1

f ′′(Xm−)∆[X,X]m

=1

2f ′′(X−) • [X,X]n.


Stochastic exponentials are processes of multiplicative structure, which play an impor-tant role in stochastic calculus. They have an obvious financial interpretation. If ∆Xn de-notes a (random) interest rate in period n (i.e. each Euro at time n− 1 turns into e 1 + ∆Xn

at time n), then the process E (X) describes how an initial capital of e 1 grows by interestand compound interest. Conversely, if the price of an asset can be written as S = E (X),then

∆Xn =∆SnSn−1

is the return of this asset in period n. In that sense, X can be called return process of S. 8.11.16

Definition 1.21 If X is an adapted process, then the stochastic exponential is the uniqueadpated process Z satisfying

Z = 1 + Z− • X

(which is equivalent to Z0 = 1 and ∆Zn = Zn−1∆Xn). We denote it as E (X) = Z.

An example can be found in Figure 1.15. The stochastic exponential has a simple explicitrepresentation.

Lemma 1.22 We have E (X)n =∏n

m=1(1 + ∆Xm).

Proof. If we set Zn :=∏n

m=1(1 + ∆Xm), we conclude

Zn = Zn−1 + Zn−1∆Xn

as desired.

Remark.

1. If the jumps ∆X are small, we have the approximation

E (X)n ≈ exp

(Xn −X0 −

1

2[X,X]n

). (1.9)

2. If X is a martingale, so is E (X). This follows from the fact that the latter is — up toan additional constant — an integral relative to a martingale.

3. If c denotes a constant, then cE (X) is the unique processZ satisfyingZ = c+Z− • X .

The following rule for stochastic exponentials resembles a similar statement for the or-dinary exponential function. But by contrast to eXeY = eX+Y , we have an additonal terminvolving the covariation.

Lemma 1.23 (Yor’s formula) For adapted processes X, Y we have

E (X)E (Y ) = E (X + Y + [X, Y ]).


Proof. By Lemma 1.19 we have

E (X)E (Y ) = E (X)0E (Y )0 + E (X)− • E (Y ) + E (Y )− • E (X) + [E (X),E (Y )]

= 1 + (E (X)−E (Y )−) • Y + (E (X)−E (Y )−) • X + (E (X)−E (Y )−) • [X, Y ]

= 1 + (E (X)E (Y ))− • (Y +X + [X, Y ]),

which yields the claim.

Let us illustrate Yor’s formula by coming back to the example illustrating integrationby parts. Suppose that E (X) denotes the price of a stock, expressed in US dollar. If E (Y )

stands for the price in Euro of a US dollar, then E (X)E (Y ) = E (X+Y +[X, Y ]) representsthe price of the stock in Euro. Yor’s formula means that the return ∆X + ∆Y + ∆[X, Y ]

differs from the sum of the returns of E (X) and E (Y ) by the covariation term ∆[X, Y ].E.g. if the stock price in dollar grows by 3% and the dollar exchange rate increases by 3%as well, then the stock price in Euro grows by 6.09% rather than 6%.

Martingales do not remain martingales if the probability measure is changed. The fol-lowing result shows how the martingale property relative to some new probability measureQ can be expressed in terms of the density process ofQ relative to P . Such measure changesplay an important role im financial mathematics.

Lemma 1.24 Let Q ∼ P be a probability measure with density process Z. An adaptedprocess X is a Q-martingale if and only if XZ is a P -martingale.

Proof. According to the generalized Bayes’ rule (Lemma 1.13) we have

Zn−1EQ(Xn|Fn−1) = EP (ZnXn|Fn−1).

X is a Q-martingale if and only if the left-hand side coincides with Zn−1Xn−1. Similarly,ZX is a P -martingale if and only if the right-hand side coincides with Zn−1Xn−1.

10.11.16

Since a P -martingale does not generally remain a martingale under Q, it exhibits a trendrelative to Q. The Doob decomposition of X relative to Q can be expressed in terms of thedensity process of Q.

Lemma 1.25 (Girsanov’s theorem) Let Q ∼ P be a probability measure with density pro-cess Z. If X is a P -martingale,

X − 1

Z−• 〈Z,X〉

is a Q-martingale, where the predictable covariation is to be interpreted relative to P .

Proof. A := 1Z−

• 〈Z,X〉 is a predictable process. By Lemma 1.19

(X − A)Z = XZ − AZ= X0Z0 +X− • Z + Z− • X + [Z,X]− Z− • A− A • Z

= X0Z0 +X− • Z + Z− • X + ([Z,X]− 〈X,Z〉)− A • Z

1.4. INTUITIVE SUMMARY OF SOME NOTIONS FROM THIS CHAPTER 27

is a P -martingale. The assertion follows now from Lemma 1.24.

The following class of processes turns out to be useful for concrete models.

Definition 1.26 A random walk is a process X with X0 = 0 whose increments∆X1,∆X2, . . . are independent and identically distributed random variables. We call itbinomial random walk if the random variables ∆Xn have only two values.

The following property of binomial random walks turns out to be important in the contextof option pricing. It is shared only by very few other processes.

Theorem 1.27 (Martingale representation) Suppose that X is a binomial random walk,which is a martingale. If the filtration is generated by X , then any martingale M can bewritten as stochastic integral M = M0 +H • X for some predictable process H .

Proof. Binomial random walks correspond to a sequence of random experiments withonly two outcomes like e.g. flipping coins. If the filtration is generated by a binomialrandom walk, each vertex in the corresponding binomial tree has only two children. Sinceboth X and M are martingales, the ratio Hn := ∆Mn

∆Xnmust be the same on both edges

leaving a vertex. In other words, Hn is Fn−1-measurable, which means that we havefound a predictable process H such that ∆Mn = Hn∆Xn for all n or, equivalently,M = M0 +H • X .

Theorem 1.27 is illustrated in Figure 1.16. Since any random variable generates a mar-tingale as in Lemma 1.11, the preceding theorem implies that it can be written as the sum ofits expected value and some stochastic integral.

1.4 Intuitive summary of some notions from this chapter

A filtration on a probability space can be represented as a tree (or several trees if F0 is nottrivial, which means that there is no unique root), cf. Figures 1.1-1.3. A filtered probabil-ity space is such a tree together with probabilities. These can be specified as unconditonalprobabilities of all outcomes or as transition probabilities on all edges of the tree, cf. Figure1.4. Suppose first that the unconditonal probabilities of all outcomes are given. The proba-bilities of the events corresponding to the vertices are obtained by adding the probabilitiesof all outcomes which are descendents of any particular vertex under consideration. Thetransition probabilities on the edges are then obtained as the ratio of the probabilities of theadjacent vertices, more specifically child probability over parent probability. If, on the otherhand, only the transition probabilities on all edges are given, one can compute the probabil-ities of the vertices by multiplying all transition probabilities on the edges leading from theroot to the vertex under consideration.

Specifying an adapted process means to assign numbers resp. vectors to all vertices inthe tree, cf. Figure 1.8. A predictable process has the same value on all children of any


F0 F1 F2

ω1 ω1 ω1

ω2 ω2 ω2

ω3 ω3 ω3

ω4 ω4 ω4

Figure 1.1: σ-fields on Ω = ω1, ω2, ω3, ω4, no filtration

F0 F1 F2

ω1 ω1 ω1

ω2 ω2 ω2

ω3 ω3 ω3

ω4 ω4 ω4

Figure 1.2: Filtration on Ω = ω1, ω2, ω3, ω4

vertex. Sometimes, it is useful to write this value Xn at the corresponding parent vertex attime n−1. A stopping time means to cut the tree at certain vertices, cf. Figure 1.9. Stoppinga process means to change the value at all following vertices to the value where the tree wascut.

A martingale is a process where the value at any parent vertex is the conditional expec-tation of the values at the child vertices, cf. Figure 1.10. For super- resp. submartingales wehave inequalities instead of equality in each vertex.


F0 F1 F2 Ω

×

×× ω1

× ω2

×× ω3

× ω4

Figure 1.3: Alternative representation of Figure 1.2

0 1 2 P (ω)

1

23

49

492/3

29

29

1/32/3

13

29

292/3

19

19

1/3

1/3

Figure 1.4: Filtered probability space, unconditional and transition probabilities


F0 F1 F2

Xn(ω1) 1 2 3

Xn(ω2) 2 3 3

Xn(ω3) 2 2 4

Xn(ω4) 2 2 4

n=0 n=1 n=2

Figure 1.5: Non-adapted process X

F0 F1 F2

Xn(ω1) 1 2 4

Xn(ω2) 1 2 4

Xn(ω3) 1 3 4

Xn(ω4) 1 3 5

n=0 n=1 n=2

Figure 1.6: Adapted process X


F0 F1 F2

Xn(ω1) 1 2 4

Xn(ω2) 1 2 4

Xn(ω3) 1 3 4

Xn(ω4) 1 3 5

n=0 n=1 n=2

Figure 1.7: Filtration generated by X

X0 X1 X2

1

2

4

4

3

4

5

n=0 n=1 n=2

Figure 1.8: Alternative representation of Figure 1.6


(X0

XT0

) (X1

XT1

) (X1

XT1

)

(1010

)(

1111

) (1311

)(

911

)

(99

) (1111

)(

77

)

Figure 1.9: Process X , stopping time T , and stopped process XT

X0 X1 X2

10

11

122/3

9

1/32/3

8

92/3

6

1/3

1/3

Figure 1.10: Martingale X


Z0 Z1 Z2 P (ω) Q(ω) dQdP

(ω)

1

34

916

49

14

916

Q: 1/2

P : 2/3

98

29

14

98

Q: 1/2P : 1/3Q: 1/2

P : 2/3

32

98

29

14

98

Q: 1/2

P : 2/3

94

19

14

94

Q: 1/2P : 1/3

Q: 1/2P : 1/3

Figure 1.11: Probability measures P , Q and density process Z

X0

A0

M0

X1

A1

M1

X2

A2

M2

10

0

0

111323

122343

2/3

1023− 2

3

1/3

2/3

913− 4

3

1143− 1

3

2/3

843− 10

3

1/3

1/3

Figure 1.12: Doob decomposition of X


(X0

H • X0

)H1

(X1

H • X1

)H2

(X2

H • X2

)

(10

0

)2

(11

2

)3

(12

5

)

(10

−1

)

(9

−2

)1

(10

−1

)

(8

−3

)

Figure 1.13: Stochastic integral H • X

X0

[X,X]0〈X,X〉0

X1

[X,X]1〈X,X〉1

X2

[X,X]2〈X,X〉2

10

0

0

12

4

2.5

14

8

5

1/2

11

5

5

1/2

1/2

9

1

2.5

11

5

9

1/2

6

10

9

1/2

1/2

Figure 1.14: Quadratic variation and predictable quadratic variation of X


(X0

E (X)0

) (X1

E (X)1

) (X2

E (X)2

)

(0

1

)

(0.1

1.1

)(

0.2

1.21

)1/2

(0

0.99

)1/2

1/2

(−0.1

0.9

)(

0

0.99

)1/2

(−0.2

0.81

)1/2

1/2

Figure 1.15: Stochastic exponential of X

(X0

M0

)H1

(X1

M1

)H2

(X2

M2

)

(0

3.5

)2.5

(1

6

)−2

(2

4

)1/2

(0

8

)1/21/2

(−1

1

)1

(0

2

)1/2

(−2

0

)1/2

1/2

Figure 1.16: Martingale representation M = M0 +H • X

Chapter 2

Modelling financial markets

This chapter introduces the general mathematical framework which is needed for var-ious concrete issues. Our staring point is as before a fixed filtered probability space(Ω,P(Ω), (Fn)n∈N, P ).

2.1 Assets and trading strategies

Mathematical Finance considers mostly questions that concern trading securities. Therefore,the mathematical model involves primarily two kinds of stochastic processes: securitiesprice processes representing the up and down of quotations, and trading strategies, whichstand for the investor’s portfolio.

We suppose that a fixed adapted process S = (S0, . . . , Sd) represents the price processof the d + 1 traded assets that are traded in the market. The random variable Sin stands forthe price of security i at time n. We consider concrete models at the end of this chapter. Theassets can be traded. This is expressed in terems of stochastic processes as well.

Definition 2.1 A trading strategy (or portfolio) is a predictable process ϕ = (ϕ0, . . . , ϕd).The value process or wealth process of this portfolio is

V (ϕ) := ϕ>S :=d∑i=0

ϕiSi.

ϕin denotes the number of shares of security i in our portfolio at time n. It is randome.g. in the sense that the investor may choose it depending on the random price changesup to time n. This explains why trading strategies should be adapted. Why do we assumepredictability? This has to do with the chronological order of events. In the period fromn−1 to n prices as well as the portfolio changes. We follow the convention that the portfoliotransition from ϕn−1 to ϕn precedes the securities’ price changes from Sn−1 to Sn. Portfolioϕn is hence bought at the old prices Sn−1. In particular, the information on Sn is not yetavailable; the investor’s decision can only depend on events that have happened up to timen− 1. This is precisely the essence of predictability.

36

2.1. ASSETS AND TRADING STRATEGIES 37

The portfolio value Vn(ϕ) is determined after prices have changed, i.e. relative to pricesSn. Note that its definition involves a scalar product, which means that the total value isthe sum of the positions in the various securities. Cash itself does not appear in the model.Instead, we consider a bank account as traded security, cf. Section 2.4.

In this introductory course we confine ourselves to an idealized market (“dry water”).We assume thta all securities can an arbitrary — even fractional or negative — number ofshares can be held at any time. Even though negative positions do not seem to make sense,such short sales are in fact possible in real markets under some limitations. Moreover, ourmathematical model does not involve any transaction costs and dividend payments; interestrate for debit and credit coincide; prices are not affected by the investor’s trade. In fact, thismeans that we consider neither very large investors, who move prices by their transactions,nor very small ones, who are truly affected by fees and transaction costs. Finally, our frame-work does not fit to illiquid markets, where the difference between bid and ask prices cannotbe neglected.

By contrast to securities prices the portfolio can be chosen according to the investor’spreferences. We restrict ourselves to self-financing strategies in the following sense.

Definition 2.2 A trading strategy ϕ is called self financing, if

(∆ϕn)>Sn−1 :=d∑i=0

∆ϕinSin−1 = 0

for n = 1, 2, . . .

The self-financing condition means that wealth may be redistributed amomg assets, butno funds are added or withdrawn after initiation at time 0, cf. Figure 2.2 for a numericalexample. By expressing cumulative profits and losses in terms of the stochastic integral, thefollowing lemma provides an alternative statement of self-financability: The portfolio valueis the initial value plus profits and losses due to price changes.

Lemma 2.3 A trading strategy ϕ is self financing if and only if

V (ϕ) = V0(ϕ) + ϕ • S.

Proof. The integration by parts formula (1.6) yields

V (ϕ) = ϕ>S = V0(ϕ) + ϕ • S + S− • ϕ. (2.1)

Since the increments of the second integral are S>n−1∆ϕn, it vanishes if and only if ϕ is selffinancing.

Observe that Equation (2.1) means that the portfolio value is affected by two effects.The first integral stands for profits and lossed due to price changes. The second integral, onthe other hand, keeps track of supply and withdrawal of funds after time 0. Self financabilitymeans that this second integral vanishes.

38 CHAPTER 2. MODELLING FINANCIAL MARKETS

Both bookkeeping and mathematical theory simplify considerably if prices are not ex-pressed in currency but in multiples of a given reference asset, called numeriare. Often aparticularly simple asset is chosen for this purpose, e.g. a riskless bond with fixed interestrate (called bond, bank account, or savings account). We denote the price process of thenumeriare by S0, which is considered as strictly positive from now on.

Definition 2.4S :=

1

S0S =

(1,S1

S0, . . . ,

Sd

S0

)is called discounted price process. Moreover,

V (ϕ) :=1

S0V (ϕ) = ϕ>S

is called discounted value process oder discounted wealth process of strategy ϕ.

Example 2.5 If the numeraire is of the form S0n = (1 + r)n with fixed deterministic interest

rate r, the discounted asset price Sin = (1 + r)−nSin coincides with the present value of Sin,i.e. the price of Sin in terms of currency units as of time 0. In other words, the computationof S corresponds to discounting in the usual sense.

The self-financing condition can be expressed in terms of discounted quantities.

Lemma 2.6 A strategy ϕ is self financing if and only if

V (ϕ) = V0(ϕ) + ϕ • S.

Proof. ϕ is self-financing if and only if (∆ϕn)>Sn−1 = 0 for any n. The assertion followsfrom Lemma 2.3 for S instead of S.

Note that the stochastic integral ϕ • S for discounted price processes does not dependon the numeraire part ϕ0.

The self-financing condition limits the choice of the d + 1 assets by a constraint. Thefollowing theorem shows that the investor can chose her investment in d securities arbitrarilyand that the position in the remaining (e.g. numeraire) security is determined uniquely bythis choice.

Lemma 2.7 For any predictable process (ϕ1, . . . , ϕd) and any V0 ∈ R there exists a uniquepredictable process ϕ0 such that ϕ = (ϕ0, . . . , ϕd) is self financing with V0(ϕ) = V0.

Proof. By the previous lemma ϕ is self-financing if and only if

ϕ0nS

0n + (ϕ1, . . . , ϕd)>n (S1, . . . , Sd)n = V0 + (ϕ1, . . . , ϕd) • (S1, . . . , Sd)n,

i.e. if and only if

ϕ0n = V0 + (ϕ1, . . . , ϕd) • (S1, . . . , Sd)n − (ϕ1, . . . , ϕd)>n (S1, . . . , Sd)n

= V0 + (ϕ1, . . . , ϕd) • (S1, . . . , Sd)n−1 − (ϕ1, . . . , ϕd)>n (S1, . . . , Sd)n−1.

2.2. ARBITRAGE 39

This is a predictable process.

Discounting simplifies the bookkeeping in two ways. Firstly, self-financing strategiescan be identified in a canonical way with the last d components (ϕ1, . . . , ϕd), which canbe arbitrarily chosen. In addition, the last component ϕ0 is not needed for the computationof the wealth process V (ϕ) = S0V (ϕ) = S0(V0 + ϕ • S). Should it be of interest, it canbe calculated e.g. by ϕ0 = V (ϕ) − (ϕ1, . . . , ϕd)>(S1, . . . , Sd). From now on, we workprimarily with discounted processes.

Notation. Occasionally, we identify (S1, . . . , Sd) with (S0, S1, . . . , Sd) and predictableprocesses (ϕ1, . . . , ϕd) with the self-financing strategies (ϕ0, ϕ1, . . . , ϕd) from Lemma 2.7with V0 = 0.

The discounted version of Figure 2.2 is to be found in Figure 2.3. 15.11.16

2.2 Arbitrage

In this section we fix a terminal date N , i.e. the time index set is now 0, . . . , N instead ofN. In mathematical finance one generally assumes that riskless profits (arbitrage) cannot bemade without initial capital. This is justified by the argument that such arbitrage opportu-nities are exploited immediately by so-called arbitrage traders whose presence makes themdisappear immediately. Consequently, arbitrage opportunities exist only on small scale andfor short times.

And in fact prices and exchange rates differ only marginally if they are quoted at differentexchanges. Otherwise, buying and selling an asset at the same time in different places wouldconsitute an arbitrage. In our setup, arbitrage do not arise from transactions in differentplaces but from trading dynamically in time.

Definition 2.8 A self-financing strategy ϕ is called arbitrage, if

V0(ϕ) = 0, VN(ϕ) ≥ 0, P (VN(ϕ) > 0) > 0,

i.e. without any initial endwoment one can create a possible profit without venturing anylosses. We say that there are no arbitrage opportunities, if such strategies do not exist.

The seemingly weak assumption of absence of arbitrage is the foundation of many state-ments in mathematical finance. Above we argued that it makes prices of the same asset indifferent places coincide. Here, we need it to conclude that portfolios with the same futurevalue must have the same value today.

Lemma 2.9 (Law of one price) Denote by ϕ, ψ self-financing strategies with VN(ψ) =

VN(ϕ) (resp. VN(ψ) ≤ VN(ϕ)). If there are no arbitrage opportunities, we have Vn(ψ) =

Vn(ϕ) (resp. Vn(ψ) ≤ Vn(ϕ)) for n = 0, . . . , N .


In particular, we have Si = V (ϕ) if ϕ denotes a self-financing strategy with SiN =

VN(ϕ) for some i.

Proof. We proceed by an indirect reasoning. If the prices of the two portfolios do not co-incide for all n ≤ N , then there exist arbitrage opportunities. To this end, suppose thatVn(ϕ) < Vn(ψ) with positive probability for some n < N . The idea now is to go long in(i.e. buy) the relatively cheap portfolio and go short in (i.e. sell) the relatively expensive oneat the same time. More specifically, we buy ϕ and we sell ψ at time n if Vn(ϕ) < Vn(ψ)

occurs. Otherwise we do nothing. The difference Vn(ψ) − Vn(ϕ) > 0 is invested in anysecurity, e.g. S0. At time N , we liquidate our portfolio. The revenues from the long positionin ϕ and the obligations from the short position in ψ cancel each other. The positive wealthfrom the investment in S0 constitute our riskless arbitrage gain.

The second statement follows from considering the portfolio ψ that contains one shareof Si and nothing else.

The following deep thereom links absence of arbitrage to the existence of equivalentmartingale measures, which play a key role in financial mathematics. Intuitively, it statesthat there exist fictitious probabilities Q such that the market can be interpreted as a fairgame, where discounted profits and losses cancel on average. Note that this may not holdunder the physical or “real” probabilities, which are expressed by P . Under real probabilitiesone would rather expect risky assets to have a higher return on average as a compensation forthe higher risk, cf. e.g. the capital asset pricing model (CAPM) in economic theory. Sincetaking risks is not rewarded relative to Q, the corresponding probabilities are often calledrisk neutral.

Theorem 2.10 (First fundamental theorem of asset pricing) There are no arbitrage op-portunities if and only if there exists an equivalent martingale measure (EMM), i.e. aprobability measure Q ∼ P such that the discounted price process S is a martingale rela-tive to Q.

Proof. We only show the simpler implication that the existence of an EMM implies thatthere are no arbitrage opportunities. Let ϕ denote a self-financing strategy with V0(ϕ) = 0

and VN(ϕ) ≥ 0. Strategy ϕ serves as a potential arbitrage because it starts with zero initialcapital and does not take any risk of losses. TheQ-expectation of discounted terminal wealthequals

EQ(VN(ϕ)) = EQ(V0(ϕ) + ϕ • SN) = EQ(0 + ϕ • S0) = 0.

We have used the fact that ϕ • S is a martingale relative to Q, which implies that its expecta-tion stays constant over time. A nonnegative random variable with expected value 0 equals0 with probability 1. This means VN(ϕ) = 0 and hence also VN(ϕ) = 0. Consequently, ϕ isnot an arbitrage opportunity.

17.11.16

Theorem 2.10 is particularly useful for showing that a given market model does notallow for arbitrage. It suffices to find an EMM, which is typically much simpler than to

2.3. DIVIDEND PAYMENTS 41

absence of arbitrage directly. Note that the measure EMM Q may not be unique, cf. e.g.Figure 2.7. Moreover, it depends on the choice of the numeraire security.

2.3 Dividend payments

So far we assumed that no dividends or interest is paid on traded assets. In practice thisassumption is obviously violated in many cases. There are two ways to account for dividendpayments. In the indirect approach one identifies them with price gains. More precisely,one defines a fictitious security wihout dividend payments which yields the same profits andlosses as the original one if dividends are reinvested by buying additional shares of the sameasset. As an example consider a coupon-paying bond whose value of 1 Euro stays constantover time. If it pays 3% interest in any period (i.e. 0.03 Euro per share), reinvesting thesepayments yields a wealth of 1.03n at time n. Generally, the dividend-free price processS0n = (1 + r)n can be used to model a constant bond paying a fixed interest rate r per

period. In case of more complex dividend payments the transition from real to fictitioussecurities becomes somewhat messy. But from a theoretical point of view it provides a wayto generalize results as e.g. the fundamental thereoms of asset pricing to dividend-payingsecurities.

In this section we follow the direct approach, which means that we explicitly keep trackof dividend payments in the bookkeeping. As before we denote the d + 1-dimensionalsecurities’ price process by S = (S0, . . . , Sd). Moreover, we consider a d + 1-dimensionaladapted process D = (D0, . . . , Dd) with D0 = (0, . . . , 0), called cumulative dividendprocess. Di

n stands for the dividends that are paid for security Si up to time n, i.e. at time nthe dividend ∆Di

n is paid for secuity i.Even more than in the dividend-free case we must pay attention to the chronological

order of events. Three things happen in the period from n − 1 to n. The portfolio changesfrom ϕn−1 to ϕn, the prices move from Sn−1 to Sn and dividends ∆Di

n are paid, We supposethat these events happen precisely in this order.

The dividend payments affect the portfolio because the bank account to which the div-idend is transferred to appears in our model as a traded security as well, typically as thenumeraire. We interpret ϕn as the portfolio before dividends are paid, Vn(ϕ), on the otherhand, as the value of the portfolio after dividends at time n have been paid. This motivatesthe following definitions.

Definition 2.11 A traging strategy (or a portfolio) is a d+ 1-dimensional predictable pro-cess ϕ = (ϕ0, . . . , ϕd). Its value process or wealth processs is

V (ϕ) := ϕ>(S + ∆D).

Strategy ϕ is called self financing, if

ϕ>nSn−1 = ϕ>n−1(Sn−1 + ∆Dn−1)


or, equivalently,(∆ϕn)>Sn−1 = ϕ>n−1∆Dn−1

for n = 1, 2, . . .

The analogue of Lemma 2.3 reads as follows.


V (ϕ) = V0(ϕ) + ϕ • (S +D).

Proof. This is left as an exercise.

As before the bookkeeping simplifies by working in discounted terms. We suppose thatthe numeraire asset is positive and we define the discounted price process S := S/S0 asbefore.

Definition 2.13D :=

1

S0• D (2.2)

is called discounted dividend process. The discounted value process or discountedwealth process of strategy ϕ is defined as

V (ϕ) :=1

S0V (ϕ) = ϕ>(S + ∆D).

Note that Equation (2.2) involves a stochastic integral instead of a product. Dividends ∆Dn

are paid at time n, which is why they have to be discounted by S0n. Formula

∆Dn =∆Dn

S0n

for the present payments leads to

Dn =n∑

m=1

∆Dm =n∑

m=1

1

S0m

∆Dm =1

S0• Dn

as in the previous definition. Lemma 2.12 can be expressed in discounted terms as well:


V (ϕ) = V0(ϕ) + ϕ • (S + D).

Proof. ϕ is self financing if and only if (∆ϕn)>Sn−1 = ϕ>n−1∆Dn−1 holds for any n.Hence, the assertion follows from Lemma 2.12, applied to S instead of S.

Lemma 2.7 remains literally true in the case of dividend payments.

2.4. CONCRETE MODELS 43

(S00

S10

) (S01

S11

) (S02

S12

)P (ω)

(1

100

)(

1.01110

) (1.0201

121

)0.960.6

(1.0201

99

)0.24

0.4

0.6

(1.0190

) (1.0201

99

)0.240.6

(1.0201

81

)0.16

0.4

0.4

Figure 2.1: A market with two assets (bond and stocks)

Lemma 2.15 For any predictable process (ϕ1, . . . , ϕd) and any V0 ∈ R there exists aunique predictable process ϕ0 such that ϕ = (ϕ0, . . . , ϕd) is self financing with V0(ϕ) = V0.

Proof. The proof is left as an exercise.

In order to link arbitrage and equivalent martingale measures we consider as before afinite time horizon N ∈ N. Moreover, we assume that no dividends are paid for SecurityS0 (i.e. D0 = 0). Arbitrage is defined as in Section 2.1. Note that the value process of anyself-financing strategy is of the same form as in the case without dividends, but with S + D

instead of S. This implies that the first fundamental theorem holds with this modification.

Corollary 2.16 (First fundamental theorem of asset pricing) The market does not allowfor arbitrage if and only if there exists an equivalent martingale measure for S + D, i.e. aprobability measure Q ∼ P such that S + D is a Q-martingale.

2.4 Concrete models

The statements so far do not depend on the distribution of price processes. In order to calcu-late prices and strategies explicitly in the following chapters, we need to consider concretemodels. Its construction is a delicate task as it has to compromise between different goals.Models should be mathematically tractable without contradicting economic intuition. Butultimately they have to be compatible with real financial data, which must be examined bystatistical means. Here we discuss a simple model for two traded securities.

Cash does not appear in our general setup. Its place is taken by a bond (or money marketaccount, savings account, bank account)

S0n = S0

0 exp(rn) = S00(1 + r)n = S0

0E (rI)n, (2.3)


(ϕ00

ϕ10

)V0(ϕ)

(ϕ01

ϕ11

)V1(ϕ)

(ϕ02

ϕ12

)V2(ϕ)

(2000

)200

(1001

)211

(104.460.9591

) 222.61

201.51

191(

94.561.0611

) 201.51

182.41

Figure 2.2: A self-financing strategy ϕ (namely, half of the wealth invested in stock) and itsvalue process in the market of Figure 2.1

(S00

S10

)V0(ϕ)

(S01

S11

)V1(ϕ)

(S02

S12

)V2(ϕ)

(1

100

)200

(1

108.91

)208.91

(1

118.62

)218.220.6

(1

97.05

)197.53

0.4

0.6

(1

89.11

)189.11

(1

97.05

)197.530.6

(1

79.40

)178.81

0.4

0.4

Figure 2.3: Discounted price process S and discounted wealth process corresponding toFigures 2.1, 2.2


(S00

S10

) (S01

S11

) (S02

S12

)

(1

100

)(

1.01110

) (1.02112

)0.6

(1.02111.1

)0.40.6

(1.0290

) (1.0299

)0.6

(1.0281

)0.4

0.4

Figure 2.4: A market allowing for arbitrage(ϕ00

ϕ10

)V0(ϕ)

(ϕ01

ϕ11

)V1(ϕ)

(ϕ02

ϕ12

)V2(ϕ)

(00

)0(

00

)0(−108.91

1

) 0.91

0.01

0(

00

) 0

0

Figure 2.5: An arbitrage strategy ϕ in the market of Figure 2.4

(S00

S10

) (S01

S11

) (S02

S12

)Q(ω)

(1

100

)(

1108.91

) (1

118.62

)0.30250.55

(1

97.05

)0.2475

0.45

0.55

(1

89.11

) (1

97.05

)0.24750.55

(1

79.40

)0.2025

0.45

0.45

Figure 2.6: Equivalent martingale measure probabilities for the market in Figures 2.1 and2.3


(S00

S10

) (S01

S11

)P (ω) Q(ω) Q(ω)

(1

100

)(

1110

)0.5 0.4 0.3

0.5 (1

100

)0.25 0.2 0.4

0.25

(190

)0.25 0.4 0.3

0.25

Figure 2.7: A market with two different equivalent martingale measures Q, Q

(S00

S10

) (D00

D10

) (S01

S11

) (D01

D11

) (S02

S12

) (D02

D12

)

(1

100

) (00

)(

1.01110

) (01

) (1.0201

121

) (0

2.1

)0.6

(1.0201

99

) (01

)0.4

0.6

(1.0190

) (00

) (1.0201

99

) (0

0.9

)0.6

(1.0201

81

) (00

)0.4

0.4

Figure 2.8: A market with two assets and dividend payments on S1

(ϕ00

ϕ10

)V0(ϕ)

(ϕ01

ϕ11

)V1(ϕ)

(ϕ02

ϕ12

)V2(ϕ)

(2000

)200

(1000

)212

(−104.950.9636

) 224.72

202.46

191(

94.551.0611

) 202.46

182.41

Figure 2.9: A self-financing strategy ϕ and its value process in the market with dividends ofFigure 2.8


(S00

S10

) (D00

D10

)V0(ϕ)

(S01

S11

) (D01

D11

)V1(ϕ)

(S02

S12

) (D02

D12

)V2(ϕ)

(1

100

) (00

)200

(1

108.91

) (0

0.99

)209.90

(1

118.62

) (0

2.07

)220.290.6

(1

97.05

) (0

0.99

)198.47

0.40.6

(1

89.11

) (00

)189.11

(1

97.05

) (0

0.88

)198.470.6

(1

79.40

) (00

)178.81

0.4

0.4

Figure 2.10: Discounted price process S, dividend process D, and value process V (ϕ)

corresponding to Figures 2.8 and 2.9

(S00

S10

) (D00

D10

) (S01

S11

) (D01

D11

) (S02

S12

) (D02

D12

)Q(ω)

(1

100

) (00

)(

1108.91

) (0

0.99

) (1

118.62

) (0

2.07

)0.2740.524

(1

97.05

) (0

0.99

)0.249

0.4760.524

(1

89.11

) (00

) (1

97.05

) (0

0.88

)0.2490.524

(1

79.40

) (00

)0.227

0.476

0.476

Figure 2.11: Equivalent martingale measure probabilities for the market in Figures 2.8 and2.10


where r ∈ R, r := er − 1 and In = n for n ∈ N. It corresponds to an entirely risklessinvestment with fixed interest rate, which is not paid in cash. Instead it is reinvested in thebond and hence compounded. For simplicity we assume that the interest rate r is a fixedconstant.

By its simple structure the bond recommends itself as a natural numeriare. But in prin-ciple any tradable assets could be used for this purpose. Note that the existence of the bonddoes not contradict absence in the sense of Definition 2.8, even though it leads to some sortof riskless gains.

The bond can be held in negative quantities, which corresponds to a loan involving thesame interest rate as a deposit. The term structure of interest rates in real markets is ofcourse much more complicated. There exist long and short term investments with more orless fixed interest rates, which are paid at different times.

The only nontrivial security in our market is a stock or foreign currency whose price isassumed to be of the form

S1n = S1

0 exp(Xn) = S10

n∏m=1

(1 + ∆Xm) = S10E (X)n. (2.4)

Here, X0 = 0 and the increments ∆X1,∆X2, . . . of X are supposed to be independent andidentically distributed. Moreover, we set

Xn :=n∑

m=1

(e∆Xm − 1). (2.5)

In principle, the stock price process has the same structure as the bond. However, the return∆Xn for period n varies randomly. This may be due to unexpected news which affect pricesfavourably or unfavourably.

In the classical standard model the daily logarithmic returns ∆Xn are assumed be Gaus-sian or normally distributed, i.e. the model is determined entirely by two parameters µ, σ2.These parameters can be estimated based on stock price data from the past. Since the loga-rithmic returns

∆Xn = log

(S1n

S1n−1

)in this model are independent and identically distributed with mean µ and variance σ2, onemay use the standard estimates

µ :=1

N

N∑n=1

∆Xn,

σ2 :=1

N − 1

N∑n=1

(∆Xn − µ)2

if the observations S1n, n = 0, . . . , N are available.

In view its very simple structure, the standard model represents real data quite well.However, it does not stand up to a more careful examination. Repeatedly observed stylized


0 500 1000 1500 2000

0

2000

4000

6000

8000

Handelstage

Figure 2.12: DAX from April 1992 to April 2001

0 500 1000 1500 2000

0

2000

4000

6000

8000

Handelstage

Figure 2.13: Simulation according to Equation (2.4)


0 500 1000 1500 2000

−0.05

0.00

0.05

Handelstage

Figure 2.14: Daily logarithmic DAX returns

0 500 1000 1500 2000

−0.05

0.00

0.05

Handelstage

Figure 2.15: Simulation of daily normally distributed returns

0 500 1000 1500 2000

−0.05

0.00

0.05

Handelstage

Figure 2.16: Simulation of daily normal-inverse Gaussian returns


facts have lead to a variety of alternative models. Let us illustrate the issue by having alook at real data. Figure 2.12 depicts the evolution of the German stock index (DAX) fromApril 1991 to April 2001. This should be compared to the simulation of stock index datain Figure 2.13, which is based on Equation (2.4) with Gaussian logarithmic returns. At firstglance these diagrams seem to resemble each other quite closely. However, the representa-tion of the daily logarithmic returns ∆Xn = log(S1

n/S1n−1) in Figure 2.14 is more revealing.

According to the standard model, they should form a sample of independent, identicallydistributed (i.i.d.) random variables. But they differ in two ways from the correspondingsimulation of i.i.d. data in Figure 2.15. Firstly, large daily price changes happen much moreoften than in standard model (2.4) with comparable variance (heavy tails, leptokurtosis).Repeatedly, price changes of more than 6% happened in the observation period. Accordingto the Gaussian model, this should happen only once in 1300 years. Of course, such suddenmajor price changes involve a considerable risk for investors. Therefore, one should care-fully check to what extent theoretical results based on an unsatisfactory model are reallyapplicable in practice.

The above deficiency is not so much caused by model (2.4) itself, but rather by choosingthe Gaussian law. The latter is typically motivated by the central limit theorem, accordingto which sums of small and approximately independent factors are approximately normallydistributed. However, daily returns could very well be dominated by a few large observa-tions, which means that the Gaussian law lacks foundation. As an example, Figure 2.16depicts a simulation of normal-inverse Gaussian data. Compared to the normal law with thesame variance, this law puts more weight on the tails and the centre.

But a comparison of Figures 2.14, 2.15, 2.16 reveals a further difference, which is indeedlinked to model (2.4). Relatively large resp. small returns are clustered in the real data set(volatility clustering). Processes with independent and identically distributed increments dono show this behaviour. Therefore, they are not compatible with the observed data, even ifwe drop the assumption of Gaussian returns.

A way out is to model the increments ∆Xn in (2.4) in the form

∆Xn = σn∆Zn,

where independent, identically distributed random variables ∆Z1,∆Z2, . . . as above areresponsible for daily price changes. These are multiplied by a slowly varying process σ, whomakes calm periods with small σn take turns with busy periods caused by relatively largeσn. The concrete specification and estimation of a parametric model for ∆Z, σ belongs tothe field of statistics and ecomnometrics. A sensible model should be compatible with realdata, easy to estimate, and tractable for purposes of mathematical finance. According to ageneral rule, simple models should be preferred to complex ones unless they do not matchthe data well enough. It depends on the concrete purpose of the model whether this is thecase.

Note that Gaussian returns contradict our assumption of a finite probability space. Inorder to avoid mathematical inconsistencies one could approximate continuous laws by ap-propriate discrete ones. Alternatively, one could extend the mathematical theory to infinite


sample spaces. Indeed, most results of our setup hold with only minor modifications ingeneral discrete-time models.22.11.16

Chapter 3

Derivative pricing and hedging

Derivatives are securities whose price depends in a more or less complex way of the priceof other, so-called underlying securities or assets. Often these are forward deals. Thismeans that it is negotiated now that a particular transaction will be made in the future at acertain price. These contracts may be signed in order to reduce the own exposure to pricefluctuations. However, for the counterparty they involve a potentially larger exposure torisks. Consequently, a key issue in mathematical finance is how the risk caused e.g. byselling derivatives can be reduced or hedged by trading the underlying securities skillfully.This problem is related to the likewise important question how to assign an appropriate priceto a derivative contract.

More specifically, we distinguish two situations. In Section 3.2, the derivative is itselfliquidly traded at the exchange, which means that the results from the previous section can beapplied. A pivotal role will be played played by the no arbitrage assumption. This seeminglyweak and innocent has in some cases far-reaching consequences. In others, however, it doesnot help much.

Alternatively, a forward deal may be contracted over the counter, (OTC) by two parties.In this case, which is discussed in Section 3.3, we must take an individual rather than amarket perspective. In the end, the two situations lead to similar formulas and results. Thismay explain why the literature does not often distinguish very clearly discrimiate betweenOTC and liquidly traded derivatives.

Our general market model rests as before on a finite filtered probability space(Ω,P(Ω), (Fn)n=0,...,N , P ) with finite time horizon N ∈ N. As before we assume that alloutcomes happen with positive probability and that the initial σ-field is trivial, which meansthat we consider the initial state of the model as deterministic or fixed. As in the previouschapter we suppose that the d + 1-dimensional securities price process S = (S0, . . . , Sd)

with positive numeraire S0 is given. This market of so-called underlying securities is sup-posed not to allow for arbitrage in the sense of Definition 2.8.

53

54 CHAPTER 3. DERIVATIVE PRICING AND HEDGING

3.1 Contingent claims

Many — but not all — forward deals can be represented as a random variable X whichstands for the random payoff at time N . We consider some examples.

1. A European call option gives the owner the right to buy a fixed quantity of a fixedasset at a fixed time (maturity) and for a fixed price (strike). There is no obligation tobuy the asset.

Such an option can be represented as a random variable X . If the market price of theasset at maturity is below the strike, it does not make sense to exercise the option,which is hence worthless. If, on the other hand, the market price settles above thestrike, the value of the option is the difference of the two. Indeed, even if the optionholder is not interested in the asset, he can realize the profit by selling it immediatelyat the market price. Hence, the value of a call option written on one share of securityS1 with maturity N and strike K amounts to

X = (S1N −K)+ = max(S1

N −K, 0) (3.1)

at time N .

Often, the asset is not actually physically delivered. Instead a pure cash settlementtakes place. In practise this may in fact make a difference because of transaction andstrorage cost involved in physical delivery, This difference will be neglected in ouranalysis.

In case of a pure cash settlement, the price of the asset in (3.1) could in fact be replacedby any well-defined quantity, even if it is not a proper price. There are e.g. options onindices.

2. The European put option corresponds to the call but the owner has the right to sellrather than buy the asset. Otherwise, the above comments hold for the put as well.The value of a put option written on one share of security S1 with maturity N andstrike K amounts to

X = (K − S1N)+ = max(K − S1

N , 0) (3.2)

at time N .

3. The owner of an American call resp. put can exercise her option any time before ma-turity. She does not necessarily have to wait till maturity. In constrast to the Europeanoption, this involves a true choice. This complicates the mathematical treatment. In-deed, the seller faces the additional uncertainty at which time the option to buy resp.sell the underlying will be exercised. American options cannot generally be expressedin terms of a single random variable.

3.2. LIQUIDLY TRADED DERIVATIVES 55

4. In a forward contract one agrees to buy a fixed quantity of a fixed asset at a fixed date(maturity) and a fixed price (forward price). In contrast to a call option this involvesan obligation. The forward price is chosen such that no money has to be paid when thecontract is settled, i.e. the contract itself is worthless. Observe the different meaningof the word price compared to the price of call and put options. Whereas the forwardprice resembles the strike and hence the price of the underlying at maturity, the optionprice typically refers to the price of the option contract as an asset in its own right.

In order to express its payoff as a random variable, we consider a forward contract onasset S1 with maturity N . We assume that it is entered at time n ∈ 0, . . . , N at aforward price On. The value of this contract at maturity equals

X = S1N −On

because the holder needs to pay “only” On instead of the market price S1N in order to

buy one share of S1. Note that the forward price may well lie above the market price,in which case X is negative.

5. Forwards are usually contracted directly by two parties (over-the-counter). Theexchange-traded version is called futures contract. It differs from a forward by thebookkeeping, which makes it less transparent. Whereas money is paid only at matu-rity for a forward contract, daily payments are made for a futures contract. Dependingon the market price of the underlying, its price change is credited resp. debited to thecounterparties’ margin accounts (marking to market).

A futures can be viewed as a contract which can be entered and terminated free ofcharge at any time. This contract is based on a time varying quotation, the so-calledfutures price Un, which resembles the above forward price. At maturity N , the fu-tures price is laid down as the market price of a corresponding underlying asset (e.g.UN = S1

N ). During the holding period of the future, payments of the daily futuresprice change Un − Un−1 are made to the holder’s margin account. Neglecting interestpayments, the credit and debit notes from time n toN sum up to UN−Un = S1

N−Un.This corresponds to the payoff of a forward at maturity if the futures price Un is re-placed by the forward price On. In general, however, futures cannot be expressed interms of a random variable expressing a payoff at maturity because interest is earnedor charged for the intermediate payments.

24.11.16

3.2 Liquidly traded derivatives

Even if not all traded derivatives allow for such a representation, we call FN -measurablerandom variables X interchangably (contingent) claim, derivative, or option. The randomvariable X represents the value of the contract at time N and

X :=X

S0N


the discounted payoff, respectively. We consider any contingent claim in this section as aliquidly traded security Sd+1, whose market price at any time n = 0, . . . , N equals Sd+1

n .Based on modest assumptions, we would like to draw conclusions on possible or reasonablemarket prices Sd+1 for n ≤ N , in particular n = 0. Common sense probably suggeststwo answers. Firstly, the claim’s market price is subject to supply and demand and henceunpredictable. Secondly, the conditional expectation EQ(X|Fn) may appear as an at leastnot unreasoanble suggestion.

The following corollary implies that absence of arbitrage restricts the set of possibleoption price processes by a martingale condition. The only feasible derivative prices can berepresented as conditional expectation under some equivalent martingale measure.

Corollary 3.1 Let X denote a contingent claim. An adapted derivative price process Sd+1

satisfying Sd+1N = X leads to an arbitrage-free market (S0, . . . , Sd+1) if and only if there

exists an equivalent martingale measure Q for the market (S0, . . . , Sd) such that

Sd+1n = EQ(X|Fn)

for n = 0, . . . , N .

Proof. ⇐: This implication follows from the fundamental theorem 2.10 because Q is anequivalent martingale measure for (S0, . . . , Sd+1).⇒: By the first fundamental theorem 2.10 there exists an EMM Q for (S0, . . . , Sd+1).

In some cases absence of arbitrage determines the derivative price process uniquely. Thisis the case for replicable claims in the sense of the following definition.

Definition 3.2 A contingent claimX is called replicable or attainable if there exists a self-financing strategy ϕ (trading only in the primary assets S0, . . . , Sd) which satisfies X =

VN(ϕ). In this case ϕ is called (perfect) hedging strategy for X .

Remark. Observe that a contingent claim is attainable if and only if there is some x ∈ Rand some Rd-valued predictable process ϕ = (ϕ1, . . . , ϕd) with

X = x+ ϕ • SN .

Here, S is to be interpreted as the d-dimensional process (S1, . . . , Sd), cf. the notation fol-lowing Lemma 2.7. The question whether such an integral representation exists motivatesTheorem 1.27.

For attainable claims it does not make a difference whether one owns the claim or itsreplicating portfolio ϕ — the value at maturity is the same. Hence, such a derivative isredundant in the sense that it is already available in the market, namely in the shape of thedynamic strategy ϕ. This suggests that the market price of a replicable claim should coincidewith the value of its replicating portfolio.

3.2. LIQUIDLY TRADED DERIVATIVES 57

Theorem 3.3 For any contingent claim X the following statements are equivalent.

1. X is attainable by a self-financing strategy ϕ.

2. There is one and only one derivative price process Sd+1 with X = Sd+1N and such that

the market (S0, . . . , Sd+1) does not allow for arbitrage.

3. For any EMM Q the definition Sd+1n := EQ(X|Fn) leads to the same process Sd+1.

In this case we have Sd+1 = V (ϕ).

Proof. 1⇒2: In order to prove existence set Sd+1 := V (ϕ). Let ψ denote some Rd+1-valuedpredictable process with ψ • (S1, . . . , Sd+1)N ≥ 0. By

ψ • (S1, . . . , Sd+1)N =∑d

i=1 ψi • SiN + ψd+1 • (ϕ • S)N

=∑d

i=1 ψi • SiN + (ψd+1ϕ) • SN

=(ψ1 + ψd+1ϕ1, . . . , ψd + ψd+1ϕd

)• SN

and absence of arbitrage of S we have ψ • (S1, . . . , Sd+1)N = 0. i.e. the extended market(S0, . . . , Sd+1) does not allow for arbitrage.

In order to show uniqueness let Sd+1 denote an arbitrary semimartingale with terminalvalue X = VN(ϕ) and such that (S0, . . . , Sd+1) does not allow for arbitrage. By Lemma 2.9we conclude that Sd+1 = V (ϕ).

2⇒3: Corollary 3.13⇒1: This more difficult implication is skipped here.

In the situation of the previous theorem, Sd+1 is the only price process that is fair in thesense that it does not lead to riskless gains in the market.

Definition 3.4 If a contingent claim X is attainable, we call the process Sd+1 in Theorem3.3 its (unique) fair price process.

Recall that we asked for reasonable market prices of derivatives. For attainable claimswe have found a satisfactory answer. In lucky but somewhat rare cases, any contingent claimis in fact replicable.

Definition 3.5 The market is called complete if any contingent claim X is attainable.

Completeness can be characterized in terms of equivalent martingale measures, similarlyas absence of arbitrage in Theorem 2.10.

Theorem 3.6 (2. fundamental theorem of asset pricing) If the market does not allow forarbitrage, we have equivalence between:

1. The market is complete.

2. There is a unique equivalent martingale measure.


Proof. 1⇒2: For any fixed A ∈ FN define the discounted contingent claim X := 1A. ByTheorem 3.3 (1⇒3) EQ(X|F0) = Q(A) coincides for all EMM’s Q. Hence there is onlyone EMM.

2⇒1: This follows from Theorem 3.3 (3⇒1).

Let us come back to the intuitive discussion in the beginning of Section 3.2. We observethat at least in complete markets both intuitive answers are more or less partly wrong. Firstly,absence of arbitrage leaves only one possible option price. Supply and demand cannothave an effect on the option price — at least not without affecting the underlying as well.Secondly, Sd+1

n = E(X|Fn) or its discounted variant Sd+1n = E(X|Fn) may not yield

acceptable prices unless P happens to be an EMM. As reasonable as they may seem, theyboth typically lead to arbitrage.

In complete markets it suffices to specify the law of the underlying securities; it alreadydetermines the dynamics of any derivative uniquely. We study such a complete marketmodel in Section 3.4 below. In general incomplete markets, absence of arbitrage still im-poses constraints on possible derivative price processes (cf. Corollary 3.1). Sometimes,however, they do not tell us much about option prices as we will see in Section 3.4.4.

The following results show that the number of possible outcomes is rather limited incomplete market models.

Theorem 3.7 Suppose that the market is complete and does not allow for arbitrage. Thenthe number of children of each node in the tree representing the market does not exceed thenumber of traded assets, i.e. d+ 1.

Proof. If we represent the filtered probability measure by a tree as in Figure 1.4, any prob-ability measure Q on Ω can be uniquely characterized by conditional probabilities on theedges of this tree. Let us consider a one-period subtree, where one parent event F in Fn−1

splits into k disjoint child events F1, . . . , Fk in Fn. Denote the conditional probabilities onthe edges of the subtee by qj := Q(Fj|F ) = Q(Fj)/Q(F ), j = 1, . . . , k. Moreover, wewrite Sin−1(F ), i = 0, . . . , d for the discounted prices of the d + 1 assets at time n − 1 ifevent F happens. Similarly, Sin(Fj) for i = 0, . . . , d and j = 1, . . . , k denotes the price ofasset i if event Fk happens at time n. Since F = F1 ∪ · · · ∪ Fn, we have

k∑j=1

qj = 1. (3.3)

In order forQ to be a martingale measure, the coefficients qj must satisfy the d+1 equations

k∑j=1

Sin(Fj)qj = Sin−1(F ), i = 0, . . . , d. (3.4)

Since S0 = 1, the equation for i = 0 coincides with (3.3). Hence altogether d + 1 linearequations have to be met in order for q1, . . . , qk to satisfy the requirements of an EMM. If

3.3. INDIVIDUAL PERSPECTIVE 59

k > d+ 1, then these d+ 1 equations have infinitely many solutions if they have a solutionat all. Note that the conditional probabilities can be chosen independently on each one-period subtree because the martingale property holds if and only (3.4) holds separately onany of these subtrees. Market completeness implies that there is only one EMM. Therefore,k > d+ 1 cannot hold, which proves the claim. 29.11.16

Corollary 3.8 Suppose that the market is complete and does not allow for arbitrage. Thenthe sample space Ω has at most (1 + d)N elements.

Proof. From the previous theorem it follows that the partition generating Fn has at most(1 + d)n elements. Since we assumed P(Ω) = F = FN , FN is generated by the partitionω : ω ∈ Ω, which has as many elements as Ω.

3.3 Individual perspective

In this section we consider derivatives that are contracted between two counterparties andmay not be exchange traded. in particular, it is not obvious whether and at what price thecontract can be sold between inception and maturity. Hence we cannot base our theory on aderivatice price process because this usually refers to a market price at which the claim canbe bought and sold at any time.

We work with the same mathematical setup as in the previous section, i.e. we assumeprice processes S0, . . . , Sd to be given, along with some random variable X which repre-sents the random payoff of a contingent claim at time N . As before, X denotes the corre-sponding discounted payoff.

We consider an asymmetric situation where the potential buyer is interested in the claimfor unknown reasons. The seller — e.g. a bank — acts as a pure service provider who is notinterested in the derivative on its own account. The bank faces two basic questions whichare considered in the sequel.

1. What price does the bank need to charge from the potential buyer?

2. How can the bank hedge against the risk of losses that are involved by the unknownrandom payment which is due at maturity?

Generally, the answer to these questions depends on many factors, in particular on thebank’s attitude towards risk. The minimum acceptable price for the bank cannot be de-termined without additional information of this kind. Therefore, we confine ourselves toreasonable rough bounds which are based on general considerations.

Definition 3.9 Let X denote a contingent claim. We call

πU(X) := infx ∈ R : There is a self-financing strategy ϕ

satisfying V0(ϕ) = x and VN(ϕ) ≥ X


upper price and

πL(X) := supx ∈ R : There is a self-financing strategy ϕ

satisfying V0(ϕ) = x and VN(ϕ) ≤ X

lower price of the option.

In what sense do these represent rational bounds for the price that a bank can or willactually charge for the contingent claim. If it obtains a premium x ≥ πU(X) for the option,this allows the bank to buy a self-financing portfolio whose terminal value VN(ϕ) sufficesto meet its obligations at maturity. Consequently, the bank does not face any risk of losses.Except for administrative costs, there is no reason not to accept any potential buyer’s offerto pay x ≥ πU(X) for the option.

On the other hand, the bank should not accept any premium below the lower price. In-deed, for any initial wealth x < πL(X) it can sell a self-financing portfolio with terminalvalue VN(ϕ) ≤ X . The involved obligations at timeN do not exceed those of a shorted con-tingent claim. Consequently, the bank is better off selling such a portfolio at the exchangethan to sell an option to a potential buyer for less than πL(X). Altogether it follows that anyreasonably charged or offered price for the option should lie between πL(X) and πU(X).

In the following theorem, these bounds are characterized in terms of equivalent martin-gale measures.

Theorem 3.10 Let X denote a contingent claim.

1. For the upper price we have

πU(X) = minx ∈ R : Threre exists some self-financing strategy ϕ

with V0(ϕ) = x and VN(ϕ) ≥ X

= supS0

0EQ(X) : Q EMM.

2. Accordingly, the lower price satisfies

πL(X) := maxx ∈ R : Threre exists some self-financing strategy ϕ

with V0(ϕ) = x and VN(ϕ) ≤ X

= infS0

0EQ(X) : Q EMM.

3. If X is not replicable, then S0

0EQ(X) : Q EMM

(3.5)

is an open interval (and a singleton otherwise).

3.4. EXAMPLES 61

Proof. 1. Let ϕ be a self-financing strategy with VN(ϕ) ≥ X and let Q denote an EMM.Since V (ϕ) is a Q-martingale, it follows that V0(ϕ) = EQ(VN(ϕ)) ≥ EQ(X) and henceπU ≥ sup. . . .

The proof of the inequality πU ≤ sup. . . and the remaining equalities is skipped here.2. This statement follows from the first one by considering −X instead of X .3. S0

0EQ(X) : Q EMM is an interval because the set of EMM’s is convex. Theremaining statements are not proved here.

Above we argued that the premium charged by the bank should belong to the interval 1.12.16

[πL(X), πU(X)] =

[infS0

0EQ(X) : Q EMM, sup

S0

0EQ(X) : Q EMM].

The preceding theorem shows that this interval coincides except for the boundary with theset (3.5). The letter represents the set of initial values of arbitrage-free derivative priceprocesses in the sense of the previous section. Consequently, the two seemingly differentsituations and approaches lead ultimately to similar pricing formulas.

In Theorem 3.10 it was stated that the infimum in the definition of the upper price isactually attained. Put differently, one needs exactly the upper price to be able to afford sucha superhedge, which provides perfect protection against the risk of losses from selling theclaim.

Definition 3.11 Let X be a contingent claim. A self-financing strategy ϕ with V0(ϕ) =

πU(X) and VN(ϕ) ≥ X is called cheapest superhedge for X .

If the interval (3.6) is a singleton, the option can be perfectly hedged by Theorem 3.10.In such cases, the questions from the beginning of this section have a clear answer. The bankwill charge the premium πU = πL (plus administrative costs), and it is perfectly protectedagainst losses by investing in the replicating portfolio. In the general case, the bank couldbe tempted to charge the upper price and invest this premium in the cheapest superhedge.As we will see in Section 3.4.4, however, the upper price may be so large that its cannot beobtained in real markets. We will come back to this issue in Chapter 4.

3.4 Examples

We consider now some concrete contingent claims which can be replicated and hence havea unique fair price.

3.4.1 Forward contract

The forward contract corresponds to a contingent claim with payoff X := S1N − K for

some specific K ∈ R. We assume that the numeraire has deterministic terminal value S0N ,

e.g. because the numeriare asset is altogether deterministic or because it represents a zero


coupon bond with maturity N . Consider now the constant and hence self-financing strategyϕ := (−K/S0

N , 1) in the market S = (S0, S1) . Its value process is Vn(ϕ) = S1n − K

S0n

S0N

.Since VN(ϕ) = S1

N − K = X , the claim X is replicable and V (ϕ) is the only reasonableprice process of the claim, both in the sense of Sections 3.2 und 3.3. Suppose that theforward contract is settled at time n ∈ 0, 1, . . . , N. The forward price On is the value Kthat makes the contract worthless at time n, i.e.

On = S1n

S0N

S0n

= S1nS

0N .

Observe that a forward contract can be hedged perfectly whenever there exists a bondwith deterministic payoff at time N . No assumptions must be made about the dynamics ofthe underlying security S1. Moreover, the hedge is static in the sense that it does not involvefrequent rebalancing. In practice, however, forward contracts may involve a risk that doesnot turn up in our mathematical model, namely the counterparty risk that the other partydoes not neet its obligations.

3.4.2 Futures contract

On first glance, futures contracts do not seem to be compatible with the theory becausethey involve a complex cashflow. But on closer look, they can be interpreted as securitieswith — sometimes negative — dividend payments. The value of the asset itself is always0 because entering and terminating the contract is always free of charge. Since a futurescontract involves a payment of ∆Un = Un − Un−1 at time n, the futures price U can beinterpreted as a dividend process satisfying the constraint UN = S1

N .Let us formally consider the market S0, S1, (S2, D2), where S0 denotes the numeraire,

S1 the underlying of the futures contract, and (S2, D2) = (0, U − U0) the futures contractitself as a dividend-paying asset. We assume that the numeraire asset S0 is predictable.

If this market does not allow for arbitrage, there exists an equivalent martingale measureQ by 2.16, i.e. there is a probability measureQ ∼ P such that S1 and S2 +D2 = 1

S0• U and

Q-martingales. Hence U = U0 +S0 • ( 1S0• U) is a Q-martingale as well. Consequently, we

have

Un = EQ(UN |Fn) = EQ(S1N |Fn)

for n = 0, . . . , N . Note that the futures price process itself is a Q-martingale, not thediscounted price process as usual. This does not contradict absence of arbitrage because thefutures price does not represent the price of a liquidly traded asset.

If the money market account S0 is deterministic, it follows that

Un = EQ(S1NS

0N |Fn) = S0

NEQ(S1N |Fn) = S1

nS0N = S1

n

S0N

S0n

,

i. e. the futures price coincides with the forward price above.

3.4. EXAMPLES 63

In this case the cashflow of the futures can be replicated by trading the numeraire and theunderlying. Indeed, by Section 2.3 the self-financing strategy ϕ = (ϕ0, 0, 1) correspondingto one futures contract has value process

V (ϕ) = 0 +1

S0• D2 =

1

S0• (S0

N S1) =

S0N

S0• S1.

This coincides with the discounted value of the self-financing strategy ψ = (ψ0, ψ1, 0)

which has initial value 0 and holds ψ1n =

S0N

S0n

shares of the underlying S1 at time n.We have derived the equality of futures and forward price processes in the case of de-

terministic interest rates, without any assumptions on the law of S1. Interestingly, however,the corresponding replicating strategies differ. While a sold forward is hedged by buyingexactly one share of the underlying asset, the replicating portfolio of the futures contractcontains S0

N/S0n shares of stock, which is a time-varying number.

3.4.3 European call and put options in the binomial model

In general, European call and put options are not replicable. The are attainable only in verysimple market models as in the bimomial or Cox-Ross-Rubinstein model, which is presentedbelow. The latter is even complete, i.e. any contingent claim allows for a perfect hedge andhence for a unique fair price.

The market consists of a bond and a stock as in Section 2.4. More specifically, let

S0n = S0

0(1 + r)n

with constants S00 > 0, r ≥ 0 and 6.12.16

S1n = S1

0

n∏m=1

(1 + ∆Xm)

with constant S10 > 0 and independent, identically distributed random variables ∆X1,

. . . ,∆XN , where

P (∆Xn = u− 1) = p = 1− P (∆Xn = d− 1)

with 0 < d < 1 + r < u and 0 < p < 1. In particular, the stock price may rise or fall in asingle period only by a fixed factor u or d, respectively. The discounted price process is ofthe form

S1n =

S10

S00

n∏m=1

1 + ∆Xm

1 + r= S1

0

n∏m=1

(1 + ∆Xm),

where (Xn)n=0,...,N is defined by Xn :=∑n

m=1(1+∆Xm

1+r− 1) and satisfies

P (∆Xn = u1+r− 1) = p = 1− P (∆Xn = d

1+r− 1).


We assume that the filtration (Fn)n=0,...,N is generated by S1 and that F = FN as inSection 2.4. Since any of the random vectors (S1

1 , . . . , S1n), (S1

1 , . . . , S1n), (∆X1, . . . ,∆Xn),

and (X1, . . . , Xn) can be expressed in terms of any of the other ones, they all carry thesame information. Put differently, the filtration is also generated by S1, by ∆X , or X ,respectively.

In order to study absence of arbitrage and completeness, we determine the set ofequivalent martingale measures Q. Since S1

n = S10

∏nm=1(1 + ∆Xm) we have that

S1n = S1

n−1(1 + ∆Xn). For S1 to be a Q-martingale, we need to satisfy

EQ(1 + ∆Xn|Fn−1) = 1 (3.6)

for all n. In order to analyse this condition, we focus on any fixed one-period subtree, say inperiod n. Two children descend from the parent node of this subtree, corresponding to thesubevents where S1

n/S1n−1 = 1 + ∆Xn = u and S1

n/S1n−1 = 1 + ∆Xn = d, respectively. In

discounted terms, this means 1 + ∆Xn = u/(1 + r) and 1 + ∆Xn = d/(1 + r), respectively.If we denote the conditional Q-probabilities on the edges by q and 1 − q, respectively, themartingale condition (3.6) boils down to q u

1+r+ (1− q) d

1+r= 1, i.e.

q :=1 + r − du− d

.

Note that this conditional probability depends neither on the period n nor on the particularone-period subtree under consideration. Considering the tree as a whole, we have foundunique transition probabilities that warrant the martingale property of S1. Since there isa one-to-one correspondence between probability measures and their transition probabili-ties on the edges, we conclude that there is a unique equivalent martingale measure Q. Infinancial terms the Cox-Ross-Rubinstein model under consideration is arbitrage free andcomplete by Theorems 2.10 and 3.6. Since the transition probabilities q resp. 1 − q are thesame all over the tree, the model has basically the same probabilistic structure under Q asunder P , except for p being replaced by q.

Market completeness can also be shown directly. To this end, consider an arbitraryrandom variable Y . By the martingale representation theorem 1.27 applied to the Q-randomwalk X , there is a real number y and a predictable process H satisfying

Y = y +H • XN = y + H

S1−• S1

N .

The remark following Definition 3.2 yields that the market is complete.Market completeness entails that European call and put options can be hedged perfectly.

We want to determine their fair price and their replicating strategy. To this end denote byY := (S1

N − K)+ the payoff of a European call with strike K > 0. The discounted stockprice at maturity equals

S1N = S1

0( u1+r

)U( d1+r

)N−U

whereU = |n ∈ 1, . . . , N : ∆Xn = u1+r−1| denotes the number of upward movements

of the stock. Observe that ∆X1, . . .∆Xn is a sequence of independent random variables

3.4. EXAMPLES 65

which attain the value u1+r− 1 with probability q under Q. Therefore the number of upward

movements U is a binomial random variable with parameters N and q relative to the pricingmeasure Q. Moreover S1

N > K/S0N (i.e. the call finishes in the money) if and only if 8.12.16

U > a :=log(K/S1

0)−N log(d)

log(u/d)

(i.e. if there are more than a upward movements of the stock price). By Theorem 3.3 thediscounted fair price of the European call is obtained as

S20 = EQ( Y

S0N

)

= EQ((S1N − K

S0N

)+)

= EQ((S10( u

1+r)U( d

1+r)N−U − K

S0N

)1(a,∞)(U))

=∑N

n=0

(Nn

)qn(1− q)N−n

(S1

0( u1+r

)n( d1+r

)N−n − KS0N

)1(a,∞)(n)

= S10

∑Nn=0

(Nn

)qk(1− q)N−n1(a,∞)(n)− K

S0N

∑Nn=0

(Nn

)qn(1− q)N−n1(a,∞)(n)

= S10(1− bN,q(a))− K

S0N

(1− bN,q(a))

= S10bN,1−q(N − a)− K

S0NbN,1−q(N − a),

where bN,p denotes the cumulative distribution function (cdf) of the biomial distribution withN trials and success probability p and where we set

q :=u

1 + rq =

u(1 + r − d)

(1 + r)(u− d).

Along the same lines we obtain

S2n = EQ( Y

S0N|Fn) = S1

nbN−n,1−q(an)− KS0NbN−n,1−q(an)

for any intermediate time n = 0, . . . , N , where

an :=log(S1

n/K)− (N − n) log(u)

log(u/d).

Hence we obtainS2n = S1

nbN−n,1−q(an)−K S0n

S0NbN−n,1−q(an). (3.7)

for the undiscounted price of the option.In order to compute the hedging strategy we write S2

n = C(S1n, n) with some function

C : R+ × 0, . . . , N → R+, which is determined by Equation (3.7). Denote by ϕ =

(ϕ0, ϕ1) a self-financing strategy which replicates the call by trading (S0, S1), i.e. VN(ϕ) =

(S1N − K)+ = S2

N . For the discounted value process we have V (ϕ) = S2 and hence∆S2

n = ϕ1n∆S1

n. If the stock price rises we obtain

1S0nC(S1

n−1u, n)− 1S0n−1

C(S1n−1, n− 1) = ϕ1

nS1n−1( u

1+r− 1),


if it falls we get

1S0nC(S1

n−1d, n)− 1S0n−1

C(S1n−1, n− 1) = ϕ1

nS1n−1( d

1+r− 1).

Subtracting the two and solving for the number of stocks we have

ϕ1n =

C(S1n−1u, n)− C(S1

n−1d, n)

S1n−1(u− d)

.

The hedge for the European put is obtained along the same lines or using the call-putparity (cf. exercise sheet). At maturity N we have (S1

N −K)+ − (K − S1N)+ = S1

N −K.The fair price at time n of the contingent claim on the right is S1

n −KS0n/S

0N , as has been

shown in Section 3.4.1. The law of one price implies that S2n − S3

n = S1n − KS0

n/S0N ,

n ∈ 0, . . . , N holds for the call price S2n and the put price S3

n. After a straightforwardcalculation one obtains

S3n = K S0

n

S0NbN−n,1−q(bn)− S1

nbN−n,1−q(bn)

for the put price at time n, where

bn :=log(K/S1

n)− (N − n) log(d)

log(u/d).

3.4.4 European call and put options in the standard model

Now we turn to the standard market model from Section 2.4 with Gaussian random variables∆Xn having mean µ and variance σ2. According to Theorem 3.8 this cannot be a completemarket model. Hence we cannot expect to obtain unique option prices. Instead we want todetermine the interval of possible initial prices of a European call, which is limited by thelower and the upper price as in the preceding section. Strictly speaking, we cannot apply theearlier results, which were stated for a finite probability space Ω. However, the results canbe extended to the general case with only minor modifications.

Note that ϕ := (0, 1) is a superhedge for the call because (S1N − K)+ ≤ S1

N . Thisyields πU ≤ S1

0 . Considering ϕ := (0, 0) and (S1N − K)+ ≥ 0 we conclude πL ≤ 0 for

the lower price. Moreover, looking at portfolio ϕ := (−Ke−rN , 1) and (S1N − K)+ ≥

−Ke−rNS0N + S1

N we obtain πL ≥ S10 −Ke−rN . Together, this yields

πL ≥ max0, S10 −Ke−rN = (S1

0 −Ke−rN)+.

These almost trivial price bounds πL, πL hold irrespective of the underlying process S1.We will now construct a one-parametric family of equivalent martingale measures,

which are parametrized by σ > 0. Since both N(µ, σ2) and N(r − σ2

2, σ2) are equivalent to

Lebesgue measure, there exists a density

f :=dN(r − σ2

2, σ2)

dN(µ, σ2).

3.4. EXAMPLES 67

We now define a probability measure Q ∼ P by its density

dQ

dP:=

N∏n=1

f(∆Xn).

Relative to this measure, the price process has the same structure as under P , but withdifferent parameters µ and σ2:

Lemma 3.12 1. Q is a probability measure which is equivalent to P . Relative to Q, therandom variables ∆X1, . . . ,∆XN are independent with law N(r − σ2

2, σ2).

2. Q is an equivalent martingale measure.

Proof. We skip the proof of the first statement. For the second note that 13.12.16

EQ(S1n|Fn−1) = EQ(S1

n−1 exp(∆Xn − r)|Fn−1)

= S1n−1EQ(e∆Xn−r)

= S1n−1

1√2πσ2

∫ex exp

(− 1

2

(x+ σ2

2)2

σ2

)dx

= S1n−1

1√2πσ2

∫exp

(− 1

2

(x− σ2

2)2

σ2

)dx

= S1n−1,

which yields the Q-martingale property of S.

Let us compute the option price that is obtained as expectations under measure Q, i.e.

πQ := S00EQ((S1

N −K)+/S0N) = EQ

((S1

0eXN−rN −Ke−rN)+

).

By the previous lemma XN − rN =∑N

n=1 ∆Xn − rN is Gaussian with mean − σ2

2N and

variance σ2N . Therefore, we have

πQ =1√

2πσ2N

∫ (S1

0ex −Ke−rN

)+exp(− (x+Nσ2/2)2

2σ2N)dx

=1√2π

∫ (S1

0 exp(√σ2Ny − σ2

2N)−Ke−rN

)+

exp(−y2

2)dy

=S1

0√2π

∫ ∞y0

exp(√σ2Ny − σ2

2N)e−

y2

2 dy − Ke−rN√2π

∫ ∞y0

e−y2

2 dy

for

y0 :=log K

S10− rN + σ2

2N

σ√N

.


If Φµ,σ2 denotes the cumulative distribution function of the Gaussian law with mean µ andvariance σ2, we have

πQ =S1

0√2π

∫ ∞y0

e−12

(y−σ√N)2dy − Ke−rN√

2π

∫ ∞y0

e−y2

2 dy

= S10(1− Φσ

√N,1(y0))−Ke−rN(1− Φ0,1(y0))

= S10Φ0,1(−y0 + σ

√N)−Ke−rNΦ0,1(−y0)

= S10Φ0,1(d1)−Ke−rNΦ0,1(d2) (3.8)

with

d1 :=log

S10

K+ rN + σ2

2N

σ√N

and d2 :=log

S10

K+ rN − σ2

2N

σ√N

.

For σ → 0 we obtain

Φ0,1(d1),Φ0,1(d2)→

0 if S1

0 < Ke−rN ,12

if S10 = Ke−rN ,

1 if S10 > Ke−rN

and hence πQ → (S10 −Ke−rN)+. For σ →∞ we get instead Φ0,1(d1)→ 1, Φ0,1(d2)→ 0

and hence πQ → S10 . Since πL ≤ πQ ≤ πU and in view of the estimates above we get

πL = (S10 −Ke−rN)+ and πU = S1

0 .

Therefore the price limits in the standard model coincide with the trivial ones. Put dif-ferently, absence of arbitrage does not provide much information on European call prices.

15.12.16

3.5 American options

The exercise time of an American option can be viewed as a stopping time that shouldbe chosen optimally. This indicates why American options are related to optimal stoppingproblems. As before we work in a filtered probability space (Ω,P(Ω), (Fn)n=0,...,N , P ) withfinite time horizon N ∈ N and trivial initial σ-field.

We first study the problem where an expected payoff E(Xτ ) is to be maximized over allstopping times τ . Here X denotes a given stochastic process and (Xτ )(ω) := Xτ(ω)(ω) thevalue of this random process at the random time τ . This stopping problem is closely relatedto the notion of a Snell envelope in the following sense.

Definition 3.13 The Snell envelope U of an adapted process X is defined recursively viaUN := XN and

Un := maxXn, E(Un+1|Fn).

Its relation to optimal stopping is stated in the following

3.5. AMERICAN OPTIONS 69

Theorem 3.14 The Snell envelope U of an adapted process X has the following properties:

1. U is the smallest supermartingale satisfying U ≥ X .

2.Un = max

τ∈Tn

E(Xτ |Fn) = E(Xτf |Fn) = E(Xτs|Fn)

for n = 0, . . . , N , where Tn denotes the set of stopping times with values in n, n +

1, . . . , N and

τf := minm ≥ n : Um = Xm,τs := minm ≥ n : Xm > E(Um+1|Fm) or m = N.

Proof. 1. By definition we have

Un = maxXn, E(Un+1|Fn) ≥ E(Un+1|Fn)

for n = 0, . . . , N − 1, which implies the supermartingale property of U . Moreover, U ≥ X

is obvious.Consider now an arbitrary supermartingale V ≥ X . We show recursively that Vn ≥ Un

for n = N, . . . , 0. We start with VN ≥ XN = UN . For n = N − 1 the supermartingaleproperty yields

VN−1 ≥ E(VN |FN−1) ≥ E(UN |FN−1).

Since we also have VN−1 ≥ XN−1, we get

VN−1 ≥ maxXN−1, EQ(UN |FN−1) = UN−1.

Proceeding in the same way for n = N − 2, . . . , 0 yields V ≥ U .2. For any stopping time τ ∈ Tn we have

E(Xτ |Fn) ≤ E(Uτ |Fn) = E(U τN |Fn) ≤ U τ

n = Un,

because U τ is a supermartingale by Lemma 1.19. By τf , τs ∈ Tn it remains to be shownthat Un = E(Xτf |Fn) = E(Xτs|Fn).

Recursively we show that

E(Umaxτf ,m|Fm) = Um (3.9)

for m = N, . . . , n. For m = N this is obvious. For m = N − 1 let A := τf ≤ N − 1 ∈FN−1. Observe that E(1AUmaxτf ,N−1|FN−1) = E(1AUN−1|FN−1) = 1AUN−1. Thisyields

E(1ACUmaxτf ,N−1|FN−1) = E(1ACUmaxτf ,N|FN−1)

= 1ACE(E(Umaxτf ,N|FN)|FN−1)

= 1ACE(UN |FN−1)

= 1ACUN−1,


because UN−1 = E(UN |FN) if N − 1 < τf . Together we obtain E(Umaxτf ,N−1|FN−1) =

UN−1. Repeating the argument for m = N − 2, . . . , 0 yields (3.9).Forr m = n we obtain Un = E(Uτf |Fn) = E(Xτf |Fn). The assertion for τs follows

along the same lines.

The following characterization of Snell envelopes will turn out to be useful later.

Lemma 3.15 The following statements are equivalent for adapted processes X .

1. U is Snell envelope of X .

2. U is a supermartingale with U ≥ X , UN = XN and such that 1U− 6=X− • U is amartingale.

Proof. ⇒: According to Theorem 3.14 it remains to be shown that 1U− 6=X− • U is amartingale. By definition of the Snell envelope we have

E(1Un−1 6=Xn−1∆Un|Fn−1) = 1Un−1 6=Xn−1(E(Un|Fn−1)− Un−1) = 0

for n = 1, . . . , N , which implies the desired martingale property.⇐: Again by Theorem 3.14 it suffices to show that V ≥ U holds for any supermartingale

V ≥ X . Obviously, we have VN ≥ UN . If UN−1 = XN−1, then VN−1 ≥ UN−1. Moreover,since V is a supermartingale and 1U− 6=X− • U is a martingale, we have

1UN−1 6=XN−1(VN−1 − UN−1) ≥ 1UN−1 6=XN−1(E(VN |FN−1)− UN−1)

≥ 1UN−1 6=XN−1(E(UN |FN−1)− UN−1)

= E(1UN−1 6=N−1∆UN |FN−1) = 0

Together, this implies VN−1 ≥ UN−1. Repeating the argument for N − 2, . . . , 0, it followsthat Vn ≥ Un for n = N, . . . , 0. 20.12.16

The valuation of European options is based on absence of arbitrage and the first funda-mental theorem of asset pricing. These arguments in turn involve the possibility to trade anyasset at any time in any positive or negative amount, This is only partially true for Americanoptions. No problems are involved with buying American options and selling them later.However, holding negative quantities is less obvious. A short position in an American op-tion can always be terminated by the holder, whose right to execute the option may makethe asset and hence the seller’s short position disappear. On the other hand, we can safelyassume that this will not happen as long as the option’s market value exceeds the exerciseprice. Indeed, in this case selling the option on the market results in a higher profit than exer-cising it. Altogether, any trader of an American option faces certain short selling constraintsthat depend on the option’s market price. We consider now a version of the fundamentaltheorem covering this situation.

As in Section 2.2 we work in a market with d+1 securities, terminal dateN and positivenumeraire S0. Short selling constraints are expressed in terms of predictable processes


γi, i = 1, . . . , d whose only values are 0 and 1. We call the 0, 1d-valued process γ =

(γ1, . . . , γd) the short-sale constraint indicator process. The idea is that asset i cannot beheld in negative amounts as long as γi equals 1. As long as γi equals 0, on the other hand,short-selling of asset i is not restricted. Put differently, we consider the set

Θ :=ϕ self financing : ϕiγi ≥ 0 for i = 1, . . . , d

of trading strategies.

Definition 3.16 Let γ be a short-sale constraint indicator process. We say that the marketdoes not allow for γ-arbitrage if there exists no arbitrage strategy ϕ ∈ Θ, where Θ isdefined as above in terms of γ.

The following result generalizes Theorem 2.2 to such strategies.

Theorem 3.17 (First fundamental theorem under short sale constraints) Ifγ = (γ1, . . . , γd) denotes a short-sale constraint indicator process as above, wehave equivalence between:

1. The market does not allow for γ-arbitrage.

2. There exists a probability measure Q ∼ P such that Si is a Q-supermartingale and(1− γi) • Si is a Q-martingale for i = 1, . . . , d.

We skip the proof, which is similar to the one of Theorem 2.2.As in Section 3.2 we consider American options as liquidly traded assets. The goal is

to determine the derivative price processes that are consistent with absence of arbitrage. Tothis end, we suppose that the numeriare asset 0 is positive and arbitrage opportunities do notexist in our market of d+ 1 assets (S0, . . . , Sd) excluding the American option.

From the mathematical perspective, an American option is defined by a nonnegativeadapted process X . The random variable Xn represents the payoff that the holder receivesif she exercises the option at time n. The corresponding discounted exercise process isdefined as X := X

S0 . We denote the option’s market price process as Sd+1. If the option hasnot been exercised prematurely, its terminal value equals Sd+1

N = XN . Evidently, Sd+1 ≥ X

should always hold. Indeed, if the option’s market price fell below the exercise price, onecould buy it for Sd+1

n and at the same time receive the higher value Xn by exercising itimmediately. This can be viewed as a simple form of arbitrage.

As argued above, trading American options involves certain short sale constraints. Anegative number ϕd+1

n of options in the portfolio may not be upheld while Sd+1n− = Xn−

because the option may be exercised and hence vanish from the market. As motivatedabove, this is not going to happen as long as the market value exceeds the exercise price.Consequently, we are facing trading constraints of the form

γd+1n :=

1 if Sd+1

n− = Xn−,

0 otherwise,


i.e.γd+1 = 1Sd+1

− =X−.

We can now derive an analogue of Corollary 3.1 for American options.

Corollary 3.18 The following statements are equivalent for an American option with exer-cise process X .

1. The market (S0, . . . , Sd+1) obtained by adding derivative price process Sd+1 does notallow for arbitrage. More precisely, Sd+1 is an adpated process with Sd+1 ≥ X ,Sd+1N = XN and such that the market does not allow for (0, . . . , 0, γd+1)-arbitrage.

2. There is an equialent martingale measure Q for the market (S0, . . . , Sd) such thatthe discounted derivative price process Sd+1 is the Snell envelope of X relative toprobability measure Q.

Proof. 2⇒1: Lemma 3.15 yields that Sd+1 is a Q-supermartingale and (1− γd+1) • Sd+1 isa Q-martingale. The assertion follows now from Theorem 3.17.

1⇒2: According to Theorem 3.17 there is a probability measure Q ∼ P such thatS1, . . . , Sd+1 are Q-martingales, Sd+1 is a Q-supermartingale, and 1Sd+1

− 6=X−• Sd+1 is a

Q-martingale. By Lemma 3.15, we conclude that Sd+1 is the Q-Snell envelope of X .

In complete markets, there exists a unique price process for the American option whichdoes not lead to arbitrage because there is only one EMM Q for the underlying market. Wecall it the fair price process of the claim.21.12.16

Let us now consider a market where both an American option with exercise process Xand a European option with terminal payoff XN are traded. If this market does not allowfor arbitrage, the American option must be at least as expensive as the European one atany instance. Indeed, this follows from Corollary 3.18 but it can also be seen directly. Ifthe American option happens to be cheaper than the Europen one, one buys the Americanoption, shorts the European one at the same time, and invests the positive difference in thenumeriare. At maturity N the payoff XN and the liability −XN of the two options cancel.The investment in the numeriare remains as arbitrage gain.

By contrast, it is natural to expect that the American option should have a higher marketvalue than the corresponding European one because of the additional right to choose theexercise time. Surprisingly, this is not the case for call options on stocks that do not paydividends:

Theorem 3.19 Consider a market S = (S0, S1, S2, S3) where S2, S3 denote the price pro-cesses of a European and an American call option on S1 with strike price K and maturityN (i.e. XN = (S1

N − K)+ is the terminal payoff of S2, and X = (S1 − K)+ is the ex-ercise process of S3). If the numeriare S0 is increasing and the market does not allow for(0, 0, 1S3

−=X−)-arbitrage, then S2 = S3.


Proof. By Corollary 3.18 there is some Q ∼ P such that S2 is a Q-martingale and S3 is theQ-Snell envelope of X . Since f(x) = (x −K)+ is a convex function, Jensen’s inequalityfor conditional expectations yields

S2n = EQ

(f(S1

N)/S0N

∣∣∣Fn

)= EQ

(f(S1

N)S0n/S

0N

∣∣∣Fn

)/S0

n

≥ EQ

(f(S0

nS1N)∣∣∣Fn

)/S0

n

≥ f(S0nEQ(S1

N |Fn))/S0

n

= f(S1n)/S0

n

= Xn.

(Jensen’s inequality states that E(f(X)) ≥ f(E(X)) for random variables X and convexfunctions f .) Together with the Q-martingale property of S2 we have

S3n = max

τ∈Tn

EQ(Xτ |Fn)

≤ maxτ∈Tn

EQ(S2τ |Fn)

= S2n.

In view ofS3n = max

τ∈Tn

EQ(Xτ |Fn) ≥ EQ(XN |Fn) = S2n,

the assertion follows. The idea of the proof is that early exercise never pays because themarket price of the European call always exceeds the payoff of the American option.

If the numeriare price process S0 is constant or decreasing (i.e. the numeraire has non-positive return), the statement in Theorem 3.19 holds for put options as well.

American put options typically do not allow for a simple pricing formula, not even in thesimple Cox-Ross-Rubinstein model. However, the fair price can be computed recursively.Note that the situation is simpler for the American call because its price coincides with theEuropean call by the previous theorem.

Example 3.20 We consider the binomial model from Section 3.4.3 with a riskless asset S0

and a risky stock S1. Let S2 denote the fair price of an American put option on S1 with strikepriceK ∈ R, i.e. with payoff processX = (K−S1)+. We want to determine S2

n recursivelyfor n = N,N−1, . . . , 0. To this end, denote byQ the unique equivalent martingale measurein the CRR model. Obviously, we have S2

N = (K−S1N)+. Since S2 is the Q-Snell envelope

of X/S0, we have

S2n = max

(KS0n

− S1n

)+

, EQ(S2n+1|Fn)

. (3.10)

One can show that


1. S2n = gn(S1

n) for some function gn : R→ R, i.e. S2n depends only on the present stock

price S1n but not on the whole past S1

0 , . . . , S1n,

2. gn(S1n) is decreasing (and convex) in S1

n,

3. there is some threshold xn such that

S2n =

EQ(S2

n+1|Fn) for S1n > xn,(

KS0n− S1

n

)+

for S1n ≤ xn.

The interpretation is that the American option price coincides with the payoff if thestock price is below some threshold.

The expectation in (3.10) can be expressed more explicitly. We have

S2n = max

( KS0n− S1

n)+, EQ(gn+1(S1n+1|Fn)

= max

( KS0n− S1

n)+, EQ(gn+1(S1n(1 + ∆Xn+1))|Fn)

= max

( KS0n− S1

n)+, qgn+1(S1n

u1+r

) + (1− q)gn+1(S1n

d1+r

),

which means that the functions gN , . . . , g0 can be determined recursively by gN(x) =

(K/S0N − x)+ and

gn(x) = max

( KS0n− x)+, qgn+1(x u

1+r) + (1− q)gn+1(x d

1+r).

Finally, let us consider American options from the individual perspective of Section 3.3,where over-the-counter deals are considered. In this case the option is not traded liquidlybut instead offered to the potential buyer by a bank. We assume that the market (S0, . . . , Sd)

is complete, i.e. there exists a unique EMM Q. Above we have seen that if the options wereliquidly traded,

π := S00 max

EQ(Xτ ) : τ stopping time

would be the only initial option price that does not lead to arbitrage. The following resultshows that this price π is reasonable from the point of view of OTC deals as well.

Lemma 3.21 At price π the bank can purchase a self-financing portfolio ϕ that warrantsperfect protection against losses in the sense that V (ϕ) ≥ X . If, however, the bank sells theoption for a premium x < π, there is an arbitrage opportunity for the buyer.

Proof. We have π/S00 = U0, where U denotes the Q-Snell envelope of X . We write

U = U0 + M + A for the Q-Doob decomposition of U , i.e. M is a Q-martingale and Aa predictable decreasing process with M0 = 0 = A0.

By market completeness there exists a self-financing strategy ϕwith VN(ϕ) = U0 +MN .The martingale property ofM implies V (ϕ) = U0+M and in particular V0(ϕ) = U0, i.e. the


(S00

S10

) (S01

S11

) (S02

S12

)X

(1

100

)(

1.01110

) (1.0201

121

)260.6

(1.0201

99

)4

0.4

0.6

(1.0190

) (1.0201

99

)40.6

(1.0201

81

)0

0.4

0.4

Figure 3.1: A contingent claim X in the market of Figure 2.1

V0(ϕ)(ϕ0

1

ϕ11

)V1(ϕ)

(ϕ02

ϕ12

)V2(ϕ) X

9.65(−56.18

0.683

)15.94

(−93.121

) 26 26

4 4

2.18(−17.65

0.22

) 4 4

0 0

Figure 3.2: The replicating strategy ϕ for X and its value process

bank can afford ϕ. If the holder of the option exercises at time n, she obtains the discountedpayoff

Xn ≤ Un ≤ U0 +Mn = Vn(ϕ),

i.e. the value of the hedge portfolio ϕ is large enough to cover this obligation.We turn now to the second claim. Since π = S0

0EQ(Xτf ) for an appropriately chosenstopping time τf , we can consider π as unique fair price of a European contingent claim withdiscounted payoff Xτf . Suppose that the investor shorts the perfect hedge of this option andconsequently obtains π. For the smaller amount x she buys the American option. Thedifference is invested in the numeraire asset. At time τf the investor exercises the Americanoption and invests the payoff Xτf in the numeriare asset. The discounted terminal valueof this investment coincides with and hence covers the liabilities from the shorted hedge ofthe European option. Hence the investor profits from the riskless discounted gain of π − xstemming from the initial trade. 10.1.17


S00

S10

S20

S0

0

S10

S20

S0

1

S11

S21

S0

1

S11

S21

S0

2

S12

S22

S0

2

S12

S22

X X

1

100

9.65

1

100

9.65

1.01

110

15.94

1

108.91

15.78

1.0201

121

26

1

118.62

25.49

26 25.490.55

1.0201

99

4

1

97.05

3.92

4 3.92

0.45

0.55

1.01

90

2.18

1

89.11

2.16

1.0201

99

4

1

97.05

3.92

4 3.920.55

1.0201

81

0

1

79.40

0

0 0

0.45

0.45

Figure 3.3: Computation of the fair price process S2 of contingent claim X in Figures 3.1,3.2 using the martingale measure of Figure 2.6

(S00

S10

)S2

0 S20

(S01

S11

)S2

1 S21 X

(1

100

)7 6.5

(1

110

)15 15 15

0.5 (1

100

)5 5 5

0.25

(190

)0 0 0

0.25

Figure 3.4: A contingent claim X in the market of Figure 2.7 and two alternative priceprocesses S2, S2 for X (corresponding to Q, Q in Figure 2.7) that do not lead to arbitrage


S00

S10

S20

S01

S11

S21

S02

S12

S22

P (ω) Q(ω)

1

100

100

1

110

110

1

121

121

14

19

P : 1/2, Q: 1/3

1

121

88

18

19

P : 1/4, Q: 1/3

1

88

121

18

19

P : 1/4, Q: 1/3

P: 1/2

,Q: 1/3

1

110

80

1

121

88

18

19

P : 1/2, Q: 1/3

1

121

64

116

19

P : 1/4, Q: 1/3

1

88

88

116

19

P : 1/4, Q: 1/3

P : 1/4, Q: 1/3

1

80

110

1

88

121

18

19

P : 1/2, Q: 1/3

1

88

88

116

19

P : 1/4, Q: 1/3

1

64

121

116

19

P : 1/4, Q: 1/3

P : 1/4,Q: 1/3

Figure 3.5: A complete market with three assets and the corresponding equivalent martin-gale measure Q


Sn−1 Sn

Sn−1(F )

Sn(F1)

q1 Sn(F2)q2

...

Sn(Fk)

qk

Figure 3.6: One-period subtree in the proof of Theorem 3.7

(S00

S10

) (S01

S11

) (S02

S12

)

(S00

S10

)(S0

0(1+r)

S10u

)(S0

0(1+r)2

S10u

2

)p

(S00(1+r)2

S10ud

)1− pp

(S00(1+r)

S10d

) p

(S00(1+r)2

S10d

2

)1− p

1− p

Figure 3.7: Cox-Ross-Rubinstein model

(S0n−1

S1n−1

) (S0n

S1n

)(

1S1n−1

)(

1S1n−1

u1+r

)q = 1+r−d

u−d

(1

S1n−1

d1+r

)1− q

Figure 3.8: Subtree of the Cox-Ross-Rubinstein model and transition probabilities of theEMM Q


(S00

S10

)πL(X), πU(X)

(S00

S10

)X Q(ω)

(1

100

)5, 7.5

(1

110

)15 1−q

20.5 (

1100

)5 q ∈ (0, 1)

0.25

(190

)0 1−q

2

0.25

Figure 3.9: Lower and upper price of the claim X in Figure 3.4 as well as possible EMM’sQ

V0(ϕ)(ϕ0

1

ϕ11

)V1(ϕ) X

7.5(−67.5

0.75

)15 15

0.5

7.5 50.25

0 0

0.25

Figure 3.10: The cheapest superhedge of X in Figure 3.9


X0

E(U1|F0)

U0

X1

E(U2|F1)

U1

X2

U2

8

8.5

8.5

16.5

15.5

16.5

26

260.5

5

5

0.5

0.5

2.5

2.5

2.5

5

50.5

0

0

0.5

0.5

Figure 3.11: Snell envelope U of process X and stopping times τf (red) and τs (blue)

(S00

S10

)X0

(S01

S11

)X1

(S02

S12

)X2

(1

100

)5

(1.01110

)0

(1.0201

121

)00.6 (

1.020199

)6

0.4

0.6

(1.0190

)15

(1.0201

99

)60.6 (

1.020181

)24

0.4

0.4

Figure 3.12: An American contingent claim X (namely a put) in the market of Figure 2.1


X0

EQ(S21 |F0)

S20

X1

EQ(S22 |F1)

S21

X2

S22

5

8.14

8.14

0

2.65

2.65

0

0Q: 0.55

5.88

5.88

Q: 0.45Q: 0.55

14.85

13.82

14.85

5.88

5.88Q: 0.55

23.53

23.53

Q: 0.45

Q: 0.45

Figure 3.13: The discounted exercise process X , martingale measure probabilities and theAmerican option’s fair price process S2 in Figure 3.12

S20

V0(ϕ)

(ϕ0

1

ϕ11

)S2

1

V1(ϕ)

(ϕ0

2

ϕ12

)S2

2

V2(ϕ)

8.14

8.14

(69.77

−0.61

)2.67

2.67

(32.35

−0.27

) 0

0

6

6

15

15

(103.96

−1

) 6

7.05

24

25.05

Figure 3.14: The fair price S2 of the American claim in Figures 3.12, 3.13, its perfect hedgeϕ and the value process of ϕ

Chapter 4

Incomplete markets

In the preceding chapter we have seen that the modest assumption of absence of arbitragemay have far-reaching consequences for valuation and hedging of derivatives. In other casesarbitrage theory does not provide much useful information. Recall that the market price ofa European call option in the standard model may be arbitrarily close to the current stockprice, even if the strike is very high. Such extreme prices are rather unrealistic becausethey would turn the option into an exceptionally attractive asset. Similarly, superhedgingstrategies as in Section 3.3 are not worth considering in the standard model. Indeed, theseller will hardly receive any premium close to the upper price, which is needed to affordthe cheapest superhedge but which coincides with the stock price in the standard model. Inthis chapter we consider the question how to model and hedge non-attainable contingentclaims. As before, we distinguish exchange-traded options and OTC contracts.

4.1 Martingale modelling

Recall from Section 3.2 that the only reasonable derivative price processes are obtained viaconditional expectation relative to some equivalent martingale measureQ. But who choosesQ? The market does as a result of supply and demand! Arbitrage theory only limits itschoice byimposing the constraint that the prices need to be consistent with some EMM. Buthow do we obtain a mathematical model for derivative price processes, without knowing themanifold preferences of the agents and their effects on market prices?

We consider here an approach which relies on ideas from statistics and which is relatedto what is done in practice. The idea is to use observed quotes of liquidly traded options —so-called vanilla options — in order to obtain information on the market’s pricing measureQ.

We proceed as follows. Denote by S1 resp. S2, . . . , Sd the discounted price processesof the stock and of d − 1 options on the stock. By the fundamental theorem 2.10 there isa probability measure Q ∼ P such that all discounted price processes — of the stock S1

as well as of all derivatives S2, . . . , Sd — are Q-martingales. Unfortunately, Q cannot bedetermined by statistical methods because observed frequencies are subject to the physical

82

4.1. MARTINGALE MODELLING 83

probability measure P rather than pricing measure Q.But similarly as in statistics we may assume that Q belongs to a certain parametric

(or possibly nonparametric) set, cf. Example 4.1 below). More precisely, we express thedynamics of the stock relative to Q in terms of a parametric model. The parameter vectorϑQ must be chosen such that S1 is a Q-martingale. Since Q is supposed to be a martingalemeasure for the whole market, the price processes S2, . . . , Sd and in particular the initialprices S2

0 , . . . , Sd0 are obtained by the martingale conditions

Sin = EQ(SiN |Fn)

and in particularSi0 = EQ(SiN). (4.1)

If the options are liquidly traded, the parameter vector ϑQ must be chosen such that thetheoretical option prices (4.1) coincide — at least approximately — with observed marketquotes at time 0. In mathematical terms, this parallels computing a moment estimator instatistics. “Estimating” ϑQ by equating theoretical and observed option prices is calledcalibration. If the dimension of the parameter vector is smaller than the number of observedoptions, a perfect fit is typically impossible. Instead, one needs to use an approximation,e.g. by least squares. In the non-parametric case, on the other hand, it makes sense to applymethods from nonparametric statistics.

Strictly speaking, two situations should be distinguished. If one only wants to expressoption prices in terms of the underlying stock, it suffices to consider the pricing measure Q.There is no real need to determine the market’s dynamics relative to the real-world measureP as well. If, on the other hand, one wants to make statements on physical probabilities,quantiles, expectations etc. P must be considered as well.

Often one chooses the same parametric class of models for both the P - and the Q-dynamics of the stock. The parameter vector ϑP is obtained by statistical estimation basedon past data of the stock price S1, whereas the corresponding Q-parameters are obtainedby calibration as stated above. Note, however, that the parameter vectors ϑP and ϑQ arepossibly linked by the fact that measures Q and P need to be equivalent, i.e. the set ofpossible and impossible scenarios must be the same. This constraint implies that statisticalestimation of P -parameters may in fact yield some information on Q-parameters.

How can we assess whether the more or less arbitrarily chosen class of possible Q-dynamics is appropriate? In contrast to statistical estimation of P -models, this decisioncannot be based on statistical tests because observed frequencies are not subject to Q-probabilities. However, some evidence can be obtained from observed option prices. Ifno parameter vector ϑQ leads to theoretical option prices that fit observed market quotesreasonably well, then the chosen parametric class of models obviously fails to explain themarket under consideration. But as in statistics, one should beware of choosing a high-dimensional class which allows to fit almost any conceivable set of market quotes. Such anoverfitting may manifest itself after a few days in terms of a growing discrepancy betweentheoretical and observed option prices. One may be tempted to solve this problem by fre-quent recalibration, i.e. by repeatedly choosing new parameter vectors ϑQ. This common

84 CHAPTER 4. INCOMPLETE MARKETS

practice, however, contradicts our general paradigm. Indeed, the martingale measure Q inthe Fundamental Theorem 2.10 is supposed to be fixed once and for all. In other words, itdepends neither on time nor on the options under consideration.

Example 4.1 (Calibration in the standard model using implied volatility) We considerthe standard market model of Section 3.4.4. Denote by Qσ, σ > 0 the equivalent martingalemeasures which were derived there. We suppose that all market prices are consistent (inthe sense of the FTAP 2.10) with some EMM from the corresponding parametric familyQσ : σ > 0. The market’s Qσ can be inferred from the observed price of a call by solving

πQ = S10Φ(d1)−Ke−rNΦ(d2)

for σ, cf. (3.8). The solution σ is called the option’s implicit volatility. If several options areobserved, all such equations must be solved by the same value σ. If this cannot be achievedin reasonable approximation, the parametric family of martingale measure does not seem tobe well suited for explaining the market.

The calibrated model can now be used to price further, not yet liquidly traded options.Indeed, if we assume that Q is a pricing measure as in the first fundamental theorem ofasset pricing, we obtain any initial option price as usual by computing Q-expectations of thediscounted payoff.

However, one should be careful for at least two reasons. Firstly, applying the calibratedmodel to new options amount to extrapolation. It may happen that two different parametricfamilies that have been calibrated successfully to a set of derivatives, yield very differentprices for new contingent claims. Secondly, it is not clear what the computed option pricesmean to the single investor. Absence of arbitrage does not imply that the option can behedged perfectly or at least reasonably well.12.1.17

4.2 Variance-optimal hedgingskipped

We turn back to the questions raised in Section 3.3. We already discussed superreplication,which extends the idea of perfect hedging to non-replicable claims. However, the necessaryinitial endowment, i.e. the upper price of the claim often turns out to be excessively high inconcrete models. As derived in Section 3.4.4, the cheapest superhedge of an arbitrary callin the standard model is to buy the stock.

The literature provides a number of suggestions how to hedge at least partially againstthe risk of losses. We give a brief introduction to the concept of variance-optimal hedging.The idea is to approximate the contingent claims as well as possible by a replicable payoff.Hedging instruments may be the underlying stock but also liquid derivatives if they areavailable on the market.

The general setup is as in Section 3.3. For simplicity we focus on the case that thediscounted price process S is a martingale. This assumption may appear to be somewhat

4.2. VARIANCE-OPTIMAL HEDGING 85

rough, but probably acceptable for hedging purposes. In the general case one can derivesimilar, but considerably more involved results. As in Section 3 we consider a contingentclaim X whose discounted payoff is denoted as X := X/S0

N . In the present section weconsider only discounted quantities, as usual expressed by the ˆ notation.

The following definition specifies the idea of a best approximation:

Definition 4.2 We call ϕ variance-optimal hedging strategy for X if ϕ = ϑ minimizesthe expected squared hedging error

E(

(VN(ϑ)− X)2)

over all self-financing trading strategies ϑ.

The value process of such a strategy ϕ is unique:

Lemma 4.3 Any two variance-optimal hedging strategies have the same discounted wealthprocess V (ϕ).

Proof. For variance-optimal hedging strategies ϕ define ψ := (ϕ+ϕ)/2. If VN(ϕ) 6= VN(ϕ),we have

E(

(VN(ψ)− X)2)

= E(

(12(VN(ϕ)− X) + 1

2(VN(ϕ)− X))2

)< 1

2E((VN(ϕ)− X)2) + 1

2E((VN(ϕ)− X)2)

= E((VN(ϕ)− X)2)

which contradicts the optimality of ϕ. Hence we have VN(ϕ) = VN(ϕ). By Lemma 2.9 weeven have V (ϕ) = V (ϕ).

In the language of functional analysis, the discounted terminal value VN(ϕ) of avariance-optimal hedge is the orthogonal projection of the discounted payoff X on the sub-space of stochastic integrals relative to S, shifted by the initial endowment. This fact playsa role in the following martingale decomposition, which in turn will be the key to variance-optimal hedging.

Theorem 4.4 (Galchouk-Kunita-Watanabe-decomposition) Any martingale V allowsfor a decomposition

V = V0 + ϕ • S +M, (4.2)

where ϕ is some predictable, Rd-valued process and M a martingale which is orthogonalto S in the sense that MSi is a martingale for i = 1, . . . , d. The martingales ϕ • S and Min this decomposition are unique. The integrand ϕ solves the vector equation

∆〈V , S〉 = ϕ>∆〈S, S〉,

or, more precisely,

∆〈V , Si〉n =d∑j=1

ϕjn∆〈Sj, Si〉n (4.3)

for n = 1, . . . , N und i = 1, . . . , d.


Proof. The set VN(ϕ) : ϕ self-financing strategy is a subspace of the finite-dimensionalspace of random variables. Denote by Y = VN(ϕ) the orthogonal projection of VN on thatspace. Moreover, let M be the martingale generated by Y i.e. Un = E(Y |Fn). In view ofLemma 1.19, we have U = V (ϕ). Define M := V − U = V − V (ϕ). As a difference ofmartingales, M is a martingale as well.

Since Y is the orthogonal projection, E((VN − Y )VN(ϑ)) = 0 holds for any self-financing strategy ϑ. In particular, V0 − V0(ϕ) = E(VN − VN(ϕ)) = 0. Indeed, this followsfrom considering the self-financing strategy ϑ with Vn(ϑ) = 1 for any n. This yields (4.2).

Let n ∈ 1, . . . , N and A ∈ Fn−1. If we define a predictable process

ϑim =

1A if j = i and m ≥ n,

0 otherwise,

we have

0 = E((VN − Y )(ϑ • SiN))

= E(MN1A(SiN − Sin−1))

= E(MN1ASiN)− E(E(MN1AS

in−1|Fn−1))

= E(MN1ASiN)− E(E(MN |Fn−1)1AS

in−1)

= E(1AMN SiN)− E(1AMn−1S

in−1),

i.e. MSi is a martingale.Since [M, Si] = MSi − M0S

i0 − M− • Si − Si− • M is a martingale, 〈M, S〉 = 0.

Equation (4.2) yields 〈V , Si〉 = ϕ • 〈S, Si〉+ 〈M, Si〉 = ϕ • 〈S, Si〉, which implies (4.3).For the uniqueness proof consider another decomposition V = V0 + ϕ • S + M . Then

〈M − M,M − M〉 = 〈(ϕ − ϕ) • S,M − M〉 = (ϕ − ϕ) • 〈S,M − M〉 = 0 and henceM − M = 0 (e.g. by the remark following Definition 1.18).

Corollary 4.5 Denote by

V = V0 + (ϕ1, . . . , ϕd) • S +M

the Galchouk-Kunita-Watanabe decomposition of the martingale Vn := E(X|Fn) relativeto S = (S1, . . . , Sd). Moreover, let ϕ be the self-financing trading strategy corresponding to(ϕ1, . . . , ϕd) and initial discounted wealth V0 (cf. Lemma 2.7). Then ϕ is variance-optimal.It solves

∆〈V , S〉 = ϕ>∆〈S, S〉 (4.4)

in the sense of (4.3). The expected quadratic hedging error amounts to

ε2 := E((VN(ϕ)− X)2)

= E(〈V − ϕ • S, V − ϕ • S〉N

)(4.5)

= E

(〈V , V 〉N −

d∑i,j=1

(ϕiϕj) • 〈Si, Sj〉N

). (4.6)

4.2. VARIANCE-OPTIMAL HEDGING 87

Proof. In the proof of Theorem 4.4 it is shown that VN(ϕ) = V0 + ϕ • SN the orthogonalprojection of X on VN(ϑ) ∈ L2 : ϑ self-financing strategy. Hence ϕ variance-optimal.

Since M is a martingale,

M2 − 〈M,M〉 = 2M− •M + [M,M ]− 〈M,M〉

is a martingale as well. Hence we obtain E(M2N) = E(〈M,M〉N), which together with

(4.2) yields (4.5). (4.6) follows from

〈V , ϕ • S〉N =N∑n=1

∆〈V , ϕ • S〉n

=N∑n=1

d∑i=1

ϕin∆〈V , Si〉n

=N∑n=1

d∑i,j=1

ϕinϕjn∆〈Sj, Si〉n

=d∑

i,j=1

(ϕiϕj) • 〈Si, Sj〉N ,

where we used (4.4) in the second but last equation.

In the univariate case d = 1 the variance-optimal hedge is obtained from

ϕ1n =

∆〈V , S1〉n∆〈S1, S1〉n

. (4.7)

The numeraire component ϕ0 is obtained from the self-financing condition as usual.

Remark. The optimisation problem in Definition 4.2 can be considered subject to the con-straint that, the initial wealth V0(ϑ) is fixed. Since S is a martingale, we have

E(

(VN(ϑ)− X)2)

= E(

(V0(ϑ)− V0 + ϑ • SN − X + V0)2)

= (V0(ϑ)− V0)2 + 2(V0(ϑ)− V0)E(ϑ • SN) + E(

(E(X) + ϑ • SN − X)2)

= (V0(ϑ)− E(X))2 + E(

(E(X) + ϑ • SN − X)2).

This implies that the strategy ϑ minimizing the hedging error E((VN(ϑ)− X)2) depends onthe fixed initial endowment V0(ϑ) only through its numeraire component.

For the variance-optimal strategy in Corollary 4.5 we have

V0(ϕ)− E(X) = 0.


If V0(ϑ) is fixed instead, this non-optimal initial endowment raises the expected squarederror by (V0(ϑ)− E(X))2.

A concrete answer to the questions raised in Section 3.3 could be to charge the initialvalue of the variance-optimal hedge ϕ, raised by some risk premium which depends (e.g.linearly) on the expected squared hedging error. By investing in the variance-optimal hedgethe bank can remove its risk at least partially. A weakness of a symmetric criterion as thequadratic one is that profits and losses are penalized in the same way. In the academicliterature alternative concepts have been suggested as e.g. the utility indifference criteriondiscussed in Section 5.2. In practice heuristic approaches seem to dominate, which are noteasily justified from a formal theoretical point of view.

Chapter 5

Portfolio optimization

ab 26.1.17

5.1 Maximizing expected utility of terminal wealth

Without any background one may think that financial mathematics is primarily concernedwith maximizimg the investor’s wealth. As we have seen this is not the case. Nevertheless,the question how to choose one’s portfolio optimally has been studied thoroughly. In thischapter we conider maximization of expected utility of terminal wealth.

As in the previous chapters we work on a finite filtered probability space(Ω,F , (Fn)n=0,...,N , P ) with terminal date N ∈ N, trivial σ-field F0 = ∅,Ω, and suchthat F = P(Ω) and P (ω) > 0 for all ω. As usual we consider a asset price processS = (S0, . . . , Sd) that does not allow for arbitrage and such that the numeraire S0 is posi-tive. Later we will be more specific about the price process.

We conside an investor who wants to maximize her terminal wealth VN(ϕ) by investingher initial capital V0 optimally. However, VN(ϕ) is a random and hence any strategy’ssuccess or failure depends on whether and which asset prices rise or fall, which we cannotpredict. Hence one needs to think about what really is supposed to be optimized here.

Naïvely, one may consider maximizing the expectation E(VN(ϕ)). This criterion, how-ever, does not appear to be attractive on closer inspection. It neglects that most investorsare risk averse. Focusing on the expectation makes a highly speculative investment seemsuperior to a riskless savings account, even if the average return is only slightly higher.

As an alternative we consider a classical criterion that reflects both the desire to achievea high average return as well as the aversion to possibly threatening losses. The idea isto replace mean terminal wealth by the mean E(u(VN(ϕ))) of its utility. This utility isexpressed in terms of a utility function u : R→ R∪−∞, which maps an amount of moneyto the degree of happiness it creates. For obvious reasons one assumes this function to beincreasing. As a further important property one demands convexity. Intuitively, this meansthat an additional Euro creates less additional utility than a lost Euro kills. Put differently,a lottery paying one Euro on average is perceived as less attractive than a safe Euro. Thisproperty of weighting losses more than gains makes the criterion consider both competinggoals. The choice of the utility function allows to incorporate one’s personal preferences

89

90 CHAPTER 5. PORTFOLIO OPTIMIZATION

and risk aversion. Popular utility functions are the power and logarithmic utility consideredbelow.

Definition 5.1 Any increasing, strictly concave function u : R→ R∪−∞ is called util-ity function. A self-financing strategy ϕ with discounted initial wealth V0 is called optimalin the sense of expected utility from expected terminal wealth if ϕ maximizes E(u(VN(ϑ)))

as a function of such strategies ϑ.

The most popular utility functions are of power type u(x) = x1−p

1−p for x > 0 and fixedrisk aversion parameter p ∈ (0,∞) with p 6= 1, the logarithm u(x) = log(x) for x > 0, andexponential functions u(x) = 1− exp(−λx) with fixed risk aversion parameter λ > 0. Thedenominator in the power case only warrants that u is increasing for any choice of p.

Exploiting convextity, it is easy to show that optimal strategies are essentially unique.More precisely, their value process is uniquely determined. The corresponding trading strat-egy itself may be ambiguous if there are e.g. two assets with identical price process in themarket.

Lemma 5.2 All optimal strategies ϕ corresponding the same utility function u and the sameinitial capital have the same (discounted) wealth process V (ϕ).

Proof. Suppose that there are optimal strategies ϕ, ϕ with different terminal wealth, i.e.VN(ϕ) 6= VN(ϕ). For the average portfolio ψ := 1

2(ϕ+ ϕ) we have

E(u(VN(ψ))) = E(u(1

2VN(ϕ) + 1

2VN(ϕ))

)> E

(12u(VN(ϕ)) + 1

2u(VN(ϕ))

)= E(u(VN(ϕ))),

which contradicts the optimality of ϕ.Consequently, the discounted wealth of all optimal strategies ϕ coincides. By Lemma

2.9 this implies that the whole value process V (ϕ) is the same for all optimal strategies.

In general, it is not easy to compute optimal portfolios because the set of trading strate-gies is of very high dimension. A classical approach is called dynamic programming, wherethe optimal strategy is obtained recursively, similarly as we determined the price of an Amer-ican put in the Cox-Ross-Rubinstein model. Here we apply instead martingale methods toobtain optimal portfolios in concrete models. They are based on the following suffcientcondition for optimality. It does not provide the solution but it helps to prove that a givencandidate strategy is in fact optimal.

Theorem 5.3 Suppose that utility function u is differentiable (on the set where it is finite).Moreover, let ϕ be a self-financing strategy with discounted initial capital V0. Define aprobability measure Q ∼ P by its density

dQ

dP=

u′(VN(ϕ))

E(u′(VN(ϕ))). (5.1)

5.1. MAXIMIZING EXPECTED UTILITY OF TERMINAL WEALTH 91

If Q happens to be an equivalent martingale measure, then ϕ is the optimal strategy for u.26.1.17

Proof. Suppose that ψ is another self-financing strategy with the same initial value. Byconcavity of u we have

u(VN(ψ))− u(VN(ϕ)) ≤ u′(VN(ϕ))(VN(ψ)− VN(ϕ))

= E(u′(VN(ϕ)))dQdP

((ψ − ϕ) • SN). (5.2)

Since (ψ − ϕ) • S is a Q-martingale by Lemma 1.19, we have EQ((ψ − ϕ) • SN) = 0 andhence E(u(VN(ψ))) ≤ E(u(VN(ϕ))) by (5.2).

The previous theorem underlines once more the key role played by equivalent martingalemeasures in Mathematical Finance. Their existence and uniqueness characterize absence ofarbitrage resp. completeness. And now they occur again in the context of portfolio opti-mization. In the present case of finite probability spaces, the above sufficient criterion isalso necessary for most utility functions. Moreover, the above sufficient condition holds ininfinite probability spaces as well. The above EMM Q can be shown to solve some dualminimization problem, i.e. it is the martingale measure which is closest to P with respectto some distance that depends on u. However, these results are beyond of the scope of thepresent introduction.

In arbitrary market models it is not easy to determine the optimal strategy unless oneconsiders the logarithmic utility function. For the latter, the problem can be solved explicitlyin general. Here, we focus on a time-homogeneous market model, which generalizes the twoasset model of Section 2.4 to multiple risky assets. The money market account or bond isassumed to be of the form (2.3) as before. Parallel to (2.4), the price processes of d stocksis supposed to be given by

Sin = Si0 exp(X in) = Si0

n∏m=1

(1 + ∆X im) = Si0E (X i)n

for i = 1, . . . , d and n = 0, . . . , N . Here, X (resp. X) denotes a Rd-valued adapted processwith X0 = X0 = 0, whose increments ∆Xn (resp. ∆Xn), n = 1, . . . , N are independent,identically distributed random variables. X and X are related to each other via ∆X i

n =

e∆Xin − 1, which parallels (2.5). For the discounted price process we have

Sin = Si0

n∏m=1

(1 + ∆X im) = Si0E (X i)n

with X i0 = 0 and ∆X i

n = 1+∆Xin

1+r− 1. Note that the relative price changes of different assets

may be stochastically dependent, but not relative time changes in different periods.We want to apply the above martingale criterion in order to determine optimal strategies

for utility functions of power and logarithmic type. The difficult part is to guess a reason-able candidate portfolio whose optimality can then be proved by Theorem 5.3. To this end


we make a parametric ansatz, hoping that the true solution is of the assumed form. Theparametres are determined in second step such that the proof works.

In the above homogeneous market the following ansatz turns out to be successful. Weconsider portfolios investing a constant fraction of wealth in each of the risky assets. Theproportions are denoted by the vector γ below. The actual number of shares varies randomlyover time because both the investor’s wealth and the asset prices move. The discountedwealth and the numeraire part ϕ0 are determined by self financability.

Another obstacle must be overcome to apply the above criterion. Knowing the densitydQ/dP generally does not suffice to decide whether Q is a martingale measure. The wholeprice process Z is needed for that purpose. This problem is solved by once again making anansatz. We assume that proportionality of Zn and u′(Vn(ϕ)) holds not only at N , but also inearlier periods with a possibly changing factor.

Theorem 5.4 Consider the utility function u with

u(x) :=

x1−p

1−p for x > 0

−∞ for x ≤ 0

for fixed p ∈ (0,∞) . In the case p = 1 the above undefined expression is to be replacedby u(x) = log(x) for x > 0. Define a vector γ ∈ Rd of portfolio weights such thatγ>∆X1 > −1 and

E

(∆X1

(1 + γ>∆X1)p

)= 0. (5.3)

Moreover, setVn := V0E (γ>X)n,

ϕin :=γi

Sin−1

Vn−1 for i = 1, . . . , d,

ϕ0n := Vn−1 − (ϕ1, . . . , ϕd)>n (S1, . . . , Sd)n−1.

Then ϕ is the optimal strategy for u and discounted initial wealth V0. Its discounted wealthprocess equals V .

Proof. Since

V0 + ϕ • S = V0

(1 +

∑di=1(E (γ>X)−

γi

Si−Si−) • X i

)= V0

(1 + E (γ>X)− • (γ>X)

)= V0E (γ>X) = V ,

we have that V is the discounted wealth process of the self-financing strategy ϕ correspond-ing to (ϕ1, . . . , ϕd) and discounted initial wealth V0. Moreover, the numeraire componentϕ0 is given by the above expression.

5.1. MAXIMIZING EXPECTED UTILITY OF TERMINAL WEALTH 93

Setα := (E((1 + γ>∆X1)−p))1/p

andZn := (αnE (γ>X)n)−p

for n = 0, . . . , N . From

Zn = Zn−1α−p(

E (γ>X)n

E (γ>X)n−1

)−p= Zn−1α

−p(1 + γ>∆Xn)−p

we conclude Z = E (M) with Mn :=∑n

m=1((α(1 + γ>∆Xm))−p − 1). Since

E(α−p(1 + γ>∆Xn)−p|Fn−1) = E((1+γ>∆Xn)−p|Fn−1)

E((1+γ>∆Xn)−p)= 1,

M and hence also Z are martingales. Therefore Z is the density process of a probabilitymeasure Q ∼ P with density ZN . According to Bayes’ formula (Lemma 1.13) and (5.3) wehave 31.1.17

EQ(∆Xn|Fn−1) = E(∆XnZn

Zn−1|Fn−1)

= E(∆Xnα−p(1 + γ>∆Xn)−p|Fn−1)

= α−pE(∆Xn(1 + γ>∆Xn)−p) = 0

for n = 1, . . . , N , i.e. X is a Q-martingale. Consequently Si = Si0E (X i) is a Q-martingalefor n = 1, . . . , d. Theorem 5.3 yields that ϕ is optimal for u.

Equation (5.3) is a system of d equations with d unknowns γ1, . . . , γd. In most concretemodels it allows for a unique solution.

Note that the logarithm corresponds to p = 1 in the above proof because the latter usesonly the derivative of u. How do the solutions depend on the risk aversion parameter p? Thisis easiest to explain by considering a linear approximation. If the relative price changes ∆Xn

are small, we can approximate the denominator in (5.3) by (1+γ>∆X1)−p ≈ 1−pγ>∆X1.Then (5.3) turns into the quadratic equation E(∆X1) ≈ pE(∆X1∆X>1 )γ. Consequently,we have

γ ≈ 1

p

(E(∆X1∆X>1 )

)−1

E(∆X1), (5.4)

if the matrix

E(∆X1∆X>1 ) = Kov(∆X1) + E(∆X1)(E(∆X1))> ≈ Kov(∆X1)

is invertible.(5.4) can be interpreted as follows. Any investor with power or logarithmic utility tries

to keep the fraction of wealth held in any of the assets constant. The proportion investedin the numeriare S0 on the one hand and in the risky assets S1, . . . , Sd on the other handdepends on p. However, the relative contribution of S1, . . . , Sd is approximately the samefor all p if we assume price changes in single periods to be small. Put differently, the risk


aversion p only determines how wealth is split between risky and riskless assets. In the caseof a single risky stock, we see that the optimal investment in this stock is proportional to theexcess return E(∆X1) (compared to the money market account) and inversely proportionalto the variance Var(∆X1). Since the variance can be interpreted as riskyness of the stock,(5.4) seems quite plausible.

The case of exponential utility u(x) = 1− exp(−λx) can be treated similarly. However,in this case the optimal strategy does not assign a constant fraction of wealth to any of theassets. Instead, a fixed amount of money is invested in any of the risky assets. This amountdoes not depend on initial wealth, which does not seem very reasonable from an economicperspective.

In the case of logarithmic utility the ansatz of Theorem 5.4 works also for arbitrary priceprocesses whose relative increments are not necessarily independent. In this case the optimalfractions γn differ from period to period. Similarly to (5.3), they are given as solution to theequation

E

(∆Xn

1 + γ>n ∆Xn

∣∣∣∣∣Fn−1

)= 0. (5.5)

Let us have a look at the general structure of the optimal portfolio in Thereom 5.4.Surprisingly, it does not depend on the time horizonN . This contradicts the common adviceto invest a larger fraction of weath in risky assets if the time horizon is large. This advice istypically justified by arguing that random fluctuations average out in the long run, while thehigher average return persists.

The optimal portfolio for logarithmic utility has another interesting property. As notedin the beginning of this chapter we cannot expect there to be a portfolio which certainlyyields a higher return than any other. Surprisingly, this changes if we consider the return inthe long run. To this end, note that the wealth of a bank account with fixed continuouslycompounded interest rate r grows expontially according to S0

N = S00erN . The rate of return

r is obtained from S0 viar =

1

Nlog(S0

N/S00). (5.6)

Now, let us consider an arbitrary portfolio ϕ with value process V (ϕ). Inspired by (5.6),

limN→∞1N

log(VN(ϕ)/V0)

can be interpreted as the long run growth rate of wealth of portfolio ϕ. Somewhat surpris-ingly, there typically exists a portfolio which maximizes this long run growth rate of wealthwith probability one. This happens to be the strategy which maximizes expected utlity ofterminal wealth for logarithmic utility and which — as we observed before — does not de-pend on the time horizon. Put differently, in the very long run this growth optimal portfoliooutperforms any other strategy with probability one! However, it may take a rather longtime to do so. The proof of this result is beyond the scope of this course.

At the end of this chapter we want to address a problem that concerns the applicability ofthe results in practice. In order to determine optimal portfolios, we need a reliable estimateof the common law of asset returns.

5.2. UTILITY-BASED PRICING AND HEDGING 95

We consider the particularly simple case of a single stock which has been observed forten years. Let us assume for simplicity that daily logarithmic returns

log(S1n/S

1n−1) = log(1 + ∆X1) = r + log(1 + ∆X1)

are normally distributed with mean r + µ/250 and variance σ2/250. The factor 250 standsfor converting yearly into daily parameters (1 year ≈ 250 trading days). By (5.4) andE(∆X1) ≈ E(log(1 + ∆X1)) the parameter µ enters the optimal portfolio linearly.

Suppose that the volatility is known to be 25%, i.e. σ = 0.25. If the average loga-rithmic return in the previous 10 years amounts to 5% above the riskless interest rate, thestandard estimate for µ is 0.05. To be more precise, standard statistical theory yields a 95%-confidence interval [−0.10, 0.20] for the unkown drift parameter µ. This precision does notincrease if we dispose of high frequency or even tick-by-tick data. Only if longer periodshave been observed, we obtain more reliable estimates. On the other hand, it is less obvi-ous whether the assumption of constant parameters can be trusted if the time horizon spansseveral decades.

Consequently, with ten years of data it is hard even to make a reliable statement whetherthe stock’s excess rate of return compared to the riskless asset is positive at all. Since ϕ isapproximately linear in µ, the same is true for the fraction of wealth invested in stock. Andthis holds even if we make the simpifying assumption of constant parameters in a Gaussianframework.

Nevertheless the above statements provide interesting insights, even if real-world pa-rameters are hard to come by. Indeed, we have seen that it makes sense to hold assets inconstant fractions of wealth that — at least according to the above criteria — do not dependon the time horizon. In particular, it seems preferable to distribute wealth over the all avail-able assets rather than to invest only in the stock with highest rate of return. Finally, for thelong-term investor thinking in centuries rather than years (foundations, the church, . . . ) itmay make sense to invest in the growth optimal portfolio. 2.2.17

5.2 Utility-based pricing and hedgingskipped

We consider the situation is Sections 3.3 und 4.2, where a client asks the bank for a derivativewith discounted payoff X = X/S0

N . If the bank is offered the premium π, it can choose be-tween two alternatives: either to enter the trade or to decline it. In this section we discuss theutility indifference principle, which provides a threshold premium dividing the favourablefrom the unfavourable deals. Moreover, it indicates how to sensibly invest the premium.

We assume that the bank’s objective is to maximize its expected utility based on someutility function u : R → R ∪ −∞. Its initial discounted wealth is denoted as V0. If nooption trade is involved, the maximal expected utility amounts to

U0 := supϕE(u(VN(ϕ))), (5.7)


V0(ϕ) V1(ϕ) V2(ϕ) X = cdPdQ

c

1.0909c

1.1903c 1.1903cQ: 0.55

0.9697c 0.9697c

Q: 0.45

Q: 0.55

0.8888c

0.9697c 0.9697cQ: 0.55

0.7901c 0.7901c

Q: 0.45

Q: 0.45

Figure 5.1: The discounted fair price V (ϕ) of the claim with discounted payoff X =

c(u′)−1(dQdP

) = cdPdQ

for the market in Figures 2.1, 2.6 and utility function u(x) = log(x)

V0(ϕ)(ϕ0

1

ϕ11

)V1(ϕ)

(ϕ02

ϕ12

)V2(ϕ)

c( −0.02c

1.02c/100

)1.1018c

( −0.02c1.02c/100

) 1.214c0.6

0.9892c

0.40.6

0.8978c( −0.02c

1.02c/100

) 0.9892c0.6

0.8060c

0.4

0.4

Figure 5.2: The optimal portfolio ϕ for expected logarithmic utility and its wealth processV (ϕ)


where ϕ runs through all self-financing trading strategies with discounted initial value V0. If,on the other hand, the bank decides to enter the OTC option trade at a discounted premiumπ = π/S0

0 , it can obtain the maximal expected utility

UX(π) := supϕE(u(π − X + VN(ϕ))) (5.8)

because the option’s premium and obligation change the discounted terminal wealth by π−X . The bank will enter the trade only if it is not unfavourable, i.e. if UX(π) ≥ U0. Thishappens if and only if π ≥ π? for some threshold premium π?. This utility indifferenceprice is characterised by the condition

UX(π?) = U0.

π? is unique because the utility function is supposed to be strictly increasing. We denotethe optimal portfolios in (5.7) und (5.8) as ϕ? resp. ϕX,π. Their difference ϕ := ϕX,π − ϕ?(or, more precisely, the corresponding self-financing trading strategy with initial value π)is called utility-based hedging strategy. It represents the correction of the bank’s optimalportfolio which is due to the option trade.

How can we compute π?, ϕ?, ϕX,π? The plain utility maximisation problem (5.7) isdiscussed in the previous section. The modified problem (5.8) can be treated along the samelines. In contrast to (5.1), the density of an equivalent martingale measure Q must be of theform

dQ

dP=

u′(π − X + VN(ϕ))

E(u′(π − X + VN(ϕ)))(5.9)

in order for ϕ to solve the optimisation problem (5.8). The utility indifference price can bedetermined in a second step. However, closed-form expression are available only in rarecases.

Remark. In general it is not obvious whether the optimal values in (5.7) and (5.8) areattained. We simply assume the existence of optimal strategies in the following. For themost common utility functions, absence of arbitrage implies that it suffices in (5.7, 5.8) toconsider portfolio ϕ from some compact set. By continuity of expected utility this impliesthat the maximal value is in fact obtained by some ϕ. In the sequel we also assume with-out proof that the optimisers in (5.7, 5.8) are linked to some EMM’s via Equations (5.1, 5.9).

More specific statements can be made in the case of exponential utility. To this end,we henceforth suppose that u(x) = 1 − exp(−λx) for some λ > 0. Observe that thediscounted initial wealth V0 affects the shifted utility of terminal wealth u(VN(ϕ)) − 1)

only by a constant factor e−λV0 . Consequently, it essentially does not affect optimality ofstrategies, indifference prices etc. For ease of notation, we therefore focus without loss ofgenerality on the case V0 = 0.

In order to solve (5.7, 5.8) and for computing the indifference price, the notion of entropyturns out useful.


Definition 5.5 For probability measures P ∼ Q

H(Q,P ) := EQ(log dQdP

) = EP (dQdP

log dQdP

)

is called relative entropy of Q with respect to P .

We generally haveH(Q,P ) ≥ H(P, P ) = 0,

because Jensen’s inequality for f(x) = − log x yields

H(Q,P ) = EQ(log dQdP

)

= EQ(− log dPdQ

)

≥ − logEQ(dPdQ

)

= − log 1

= 0 = H(P, P ).

The entropy plays a key role in the context of optimization for exponential utility.

Theorem 5.6 Define the probability measure PX ∼ P by its density

dP λ

dP:=

eλX

E(eλX).

Moreover, denote by Q0 and Qx the EMM’s that correspond via (5.1,5.9) to the utility max-imisation problems (5.7) resp. (5.8).

1. Q0 maximizes the relative entropy H(Q,P ) among all EMM’s Q. In particular, itdoes not depend on λ. The manual expected utility in (5.7) amounts to

E(u(VN(ϕ?))) = 1− exp(−H(Q0, P )).

2. QX maximizes the relative entropy

H(Q,PX) = H(Q,P )− λEQ(X) + log(cX)

among all EMM’s Q, where cX := E(eλX). The maximal expected utility in (5.8)amounts to

E(u(π − X + VN(ϕX,π))) = 1− cX exp(−H(QX , PX)− λπ)

= 1− exp(−H(QX , P ) + λEQX(X)− λπ).

3. The utility-based hedging strategy ϕ = ϕX,π−ϕ? and the EMM Qx do not depend onthe negotiated option premium π (except for the numeraire part ϕ0) .


4. The utility indifference price π? = π?S00 is given by

π? =1

λ

(logE(eλX) +H(Q0, P )−H(QX , PX)

)= EQX

(X) +1

λ(H(Q0, P )−H(QX , P )) (5.10)

=1

λlogEQ0

(exp(−λ(ϕ • SN − X))

), (5.11)

where ϕ denotes the utility-based hedging strategy.

Proof. 1. For x > 0 define v(x) = x log x. This is a strictly convex function by v′′(x) =

1/x > 0. For Q0 as in 5.3 we have

v′(dQ0

dP) = 1 + log(dQ0

dP) = c− λVN(ϕ?)

with some constant c ∈ R. Convexity of v implies v(y) ≥ v(x) + v′(x)(y − x) for anyx, y > 0. This yields

H(Q,P ) = E(v(dQdP

))

≤ E(v(dQ0

dP)) + E(v′(dQ0

dP)(dQdP− dQ0

dP))

= H(Q0, P ) + EQ(λVN(ϕ?))− EQ0(λVN(ϕ?))

= H(Q0, P )

for any EMM Q. By strict convexity of v equality holds only for Q = Q0, whence thisminimal entropy martingale measure is unique.

From dQ0

dP= c−1 exp(−λVN(ϕ?)) with c = E(exp(−λVN(ϕ?))) we conclude

H(Q0, P ) = EQ0(log(dQ0

dP))

= log(c−1)− λEQ0(VN(ϕ?))

= − log c

and hence

E(u(VN(ϕ?))) = 1− E(exp(−λVN(ϕ?)))

= 1− c= 1− exp(−H(Q0, P )).

2. Firstly we have

H(Q,PX) = EQ(log dQdPX

)

= EQ(log dQdP− log dPX

dP)

= H(Q,P )− λEQ(X) + log(cX).


Note that

dQX

dPX=

dQX

dP

dP

dPX

=exp(−λ(π − X + VN(ϕX,π)))

E(exp(−λ(π − X + VN(ϕX,π))))

E(eλX)

eλX

=E(eλX)

E(exp(−λ(−X + VN(ϕX,π))))exp(−λVN(ϕX,π))

= cu′(VN(ϕX,π))

for some constant c > 0. This density is of the form in Theorem 5.3 if we consider theplain investment problem without option, but subject to probabilities PX rather than P .Moreover, ϕX,π is the corresponding optimal portfolio. By Statement 1, QX minimises therelative entropy with respect to PX among all EMM’s Q. Statement 1 for QX , PX instead ofQ0, P also yields

E(u(π − X + VN(ϕX,π))) = 1− E(exp(−λ(π − X + VN(ϕX,π))))

= 1− e−λπE(eλX)EPX(exp(−λVN(ϕX,π)))

= 1 + e−λπcX

(EPX

(u(VN(ϕX,π)))− 1)

= 1− e−λπcX exp(−H(QX , PX))

as claimed.3. As observed for V0, the discounted option premium π enters shifted utility in (5.8)

only as a multiplicative constant. Therefore, it does not affect the optimality of a strategyand the corresponding EMM QX in (5.9).

4. By Statements 1,3 the utility indifference price π∗ satisfies

1− exp(−H(Q0, P )) = 1− cX exp(−H(QX , PX))e−λπ,

i.e.eλπ = cX exp(−H(QX , PX) +H(Q0, P ))

or

π =1

λ

(log(E(eλX))−H(QX , PX) +H(Q0, P )

)=

1

λ

(λEQX

(X)−H(QX , P ) +H(Q0, P )).

On the other hand, U0 = UX(π?) means

E(exp(−λϕ? • SN))) = E(exp(−λ(π? − X + ϕX,π • SN)))

and hence1 = e−λπ

?

EQ0(exp(−λ(−X + ϕ • SN)))


by ϕX,π = ϕ? + ϕ and dQ0

dP= exp(−λϕ?•SN ))

E(exp(−λϕ?•SN ))). This yields the second representation.

The preceding theorem clarifies the structure of the problem but it does not providea concrete solution, which is generally not easy to obtain. In the following we focus onvery small resp. very large risk aversion λ, i.e. we study the asymptotic for λ → 0 andλ → ∞. Since the utility function u, the maximal expected utilities U0 resp. UX(π), theoptimal portfolios ϕ?, ϕX,π, ϕ := ϕX,π − ϕ?, and the indifference price π? depend on λ wesubsequently use the notation uλ, U0(λ), UX(π, λ), ϕ?(λ), ϕX,π(λ), ϕ(λ), π?(λ).

Lemma 5.7 We have ϕ?(λ) = 1λϕ?(1) (or it can be choosen in this way in case of ambigu-

ity). Moreover, U0(λ) = U0(1) for any λ > 0.

Proof. We have

dQ0

dP= cu′1(VN(ϕ?(1))) = c exp(−ϕ?(1) • SN) = c

λu′λ(VN( 1

λϕ?(1)))

for some normalizing constant c > 0. Since Q0 is an EMM, 1λϕ?(1) is an optimal portfolio

for uλ by Th. 5.3. The second statement follows from Theorem 5.6(1).

It is not obvious how the utility indifference price changes with λ. At least, we can showmonotonicity.

Lemma 5.8 π?(λ) is increasing in λ.

Proof. Theorem 5.6 yields

π?(λ) = EQX(X) +

1

λ(H(Q0, P )−H(QX , P ))

= infQ ÄMM

(EQ(X) +

1

λ(H(Q0, P )−H(Q,P ))

).

Since Q0 is the minimal entropy martingale measure, we have H(Q0, P ) − H(Q,P ) ≤ 0

for any EMM Q. Consequently, the above expression is increasing in λ.

We now turn to first-order approximations for small λ, on the basis of a purely heuristicreasoning. A rigorous proof is beyond of the score of this introductory text. We start fromthe natural assumption that the utility-based hedge converges for vanishing λ in the sensethat

ϕ(λ) = η +O(λ)

for λ → 0, where η denotes the desired limiting hedge and O(λ) stands for some strategysuch that O(λ)/λ is bounded by some constant that does not depend on λ. For the aggregateportfolio ϕX,π?

(λ) we obtain

ϕX,π?

(λ) = ϕ?(λ) + ϕ(λ) =1

λϕ?(1) + η +O(λ). (5.12)


For the utility indifference price, we expect a linear approximation

π?(λ) = π?(0) + λδ +O(λ2) (5.13)

with real numbers π?(0), δ ∈ R and some O(λ2) such that O(λ)/λ2 is bounded by someconstant that does not depend on λ. In the following we want to determine the unknownquantities η, π?(0), δ.

To this end, we consider the maximization problem (5.8) for strategies of the form ϕ =1λϕ?(1)+η+O(λ). In view of the Taylor approximation ex = 1+x+ x2

2+O(x3) we obtain

E(uλ(π − X + VN(ϕ)))

= 1− E(

exp(−ϕ?(1) • SN − λ(π − X + (η +O(λ)) • SN)))

= 1− cEQ0

(exp(−λ(π − X + (η +O(λ)) • SN))

)= 1− c− λcEQ0

(π − X + (η +O(λ2)) • SN

)+λ2

2cEQ0

((π − X + η • SN)2

)+O(λ3)

= 1− c+ λc(π − EQ0(X)

)− λ2

2cEQ0

((π − X + η • SN)2

)+O(λ3)

with c = E(exp(−ϕ?(1) • SN), where the last equality holds because that S is a martingalerelative to the EMM Q0. If we neglect the O(λ3) term, we have to minimise

EQ0

((π − X + η • SN)2

)(5.14)

as a function of strategy η. This can be viewed as a quadratic hedging problem as in Section4.2. Corollary. 4.5 and the subsequent remark yield that

∆〈V , S〉Q0 = η>∆〈S, S〉Q0

holds for the optimal strategy η, where V denotes the Q0-martingale generated by the dis-counted payoff X , i. e.

Vn := EQ0(X|Fn)

and the predictable covariation refers to probability measure Q0. In particular, we have

ηn =∆〈V , S1〉Q0

n

∆〈S1, S1〉Q0n

for d = 1. By Corollary 4.5 and the subsequent remark, the minimal value in (5.14) is givenby (π − V0)2 + ε2 for

ε2 = E

(〈V , V 〉Q0

N −d∑

i,j=1

(ϕiϕj) • 〈Si, Sj〉Q0

N

).


We obtain

UX(π, λ) = 1− c+ λc(π − V0)− λ2

2c(

(π − V0)2 + ε2)

+O(λ3)

if the linear expansion (5.12) of the optimal hedge holds. The maximal utility for the plaininvestment problem amounts to

U0(λ) = U0(1) = 1− E(exp(−ϕ?(1) • SN)) = 1− c.

The utility indifference price π?(λ) solves UX(π?(λ), λ) = U0(λ). If the linear approxima-tion (5.13) holds, we obtain

0 = (π?(0)− V0)− λ

2c(

2δ + (π?(0)− V0)2 + ε2)

+O(λ2).

This impliesπ?(0) = V0 = EQ0(X)

and, in view of the linear term in λ,

δ =ε2

2.

Consequently, the discounted utility indifference price converges for λ → 0 to the ex-pectation of the discounted payoff relative to the minimal entropy martingale measure Q0.On top of this zeroth order approximation, we have in first order a risk premium dependinglinearly on λ. It depends on how well the option can be approximated by a self-financingportfolio. Moreover, the utility-based hedge equals to the leading order the variance-optimalhedge of the claim relative to the MEMM.

As another extreme, let us consider instead the hint λ→∞ of large risk aversion. Here,we can provide a rigorous proof in our setup of finite underlying sample space.

Theorem 5.9 1. We have π?(λ)→ πU for λ→∞, where πU denotes the upper price inSection 3.3.

2. ϕ(λ) is an asymptotic superhedge in the sense that

lim infλ→∞

VN(ϕ(λ)) ≥ X, (5.15)

where VN(ϕ(λ)) = π?(λ) + ϕ(λ) • SN denotes the discounted terminal value of theutility-based hedge.

3. If all cheapest superhedges ϕU in Section 3.3 have the same final value VN(ϕU) (e.g.since the cheapest superhedge is unique), we have

Vn(ϕ(λ))→ Vn(ϕU)

fur λ→∞ and n = 0, . . . , N .


Proof. 1. The inequality H(Q0, P )−H(QX , P ) ≤ 0 and (5.10) imply π?(λ) ≤ EQX(X) ≤

πU for λ > 0.Wrongly suppose that supλ>0 π

?(λ) ≤ πU − 2ε for some ε > 0. For fixed λ we haveωλ ∈ Ω with

VN(ϕ(λ))(ωλ) := π?(λ) + ϕ(λ) • SN(ωλ) ≤ X(ωλ)− ε.

Indeed, otherwise π?(λ) + ε < πU would be the discounted initial value of a superhedge.We conclude

U0(λ) = UX(π?(λ), λ)

= E(uλ(VN(ϕ(λ))− X))

= 1− E(exp(−λ(π?(λ) + ϕ(λ) • SN − X)))

≤ 1− P (ωλ) exp(λε)

→ −∞ für λ→∞,

because the finiteness of Ω implies minω∈Ω P (ω) > 0. This, however, contradictsU0(λ) ≥ 1− e0 = 0.

2. Wrongly suppose that (5.15) does not hold. Then there are ω ∈ Ω and ε > 0, suchthat VN(ϕ(λ))(ω) ≤ X(ω)−ε holds for some sequence of arbitrarily large λ. This, howeverleads to the contradiction in the proof of Statement 1.

3. Choose a sequence (λk)k∈N in R+ such that λk → ∞. We have to provelimk→∞ Vn(ϕ(λk)) = Vn(ϕU) for n = 0, . . . , N . We start with n = N .

By (5.15) the sequence (VN(ϕ(λk))(ω))k∈N is bounded from below for any ω ∈ Ω. Since

EQ0(VN(ϕ(λk))) = V0(ϕ(λk)) = π?(λk) ≤ πU

and minω∈Ω P (ω) > 0, these sequences are bounded from above as well. Since(VN(ϕ(λk)))k∈N is bounded in RΩ, there is a convergent subsequence. We must show thatany of these convergent subsequences converge to VN(ϕU). W.l.o.g., we denote such anarbitrary subsequence again by (VN(ϕ(λk)))k∈N. Since VN(ϕ) : ϕ self-financing strategyis a subspace of RΩ and in particular bounded, we have VN(ϕ(λk)) → VN(ϕ) for someself-financing strategy ϕ. (5.15) yields VN(ϕ) = limk→∞ VN(ϕ(λk)) ≥ X , i.e., ϕ is asuperhedge. On the other hand, the finiteness of Ω implies

V0(ϕ) = EQ0(VN(ϕ)) = limk→∞

EQ0(VN(ϕ(λk))) = limk→∞

π?(λ) ≤ πU ,

which means that ϕ is a cheapest superhege and hence VN(ϕ) = VN(ϕU).It remains to be shown that convergence holds also for n < N . Pointwise convergence

for k →∞ and once more finiteness of Ω yield

Vn(ϕ(λk)) = EQ0(VN(ϕ(λk))|Fn)→ EQ0(VN(ϕU)|Fn) = Vn(ϕU)

as desired.


The above results provide a link between the different approaches to valuing and hedg-ing OTC-derivates. For exponential utility, one obtains a continuum of prices and hedgingstrategies. Superreplication is located at one and of this spectrum, requiring typically – asdiscussed in Section 3.3 – an exceedingly high option premium. At the other end we haveobtained expressions which are closely linked to variance-optimal hedging as presented inSection 4.2. Hence we observe in hindsight that this approach is of interest even if onerejects the symmetric treatment of profits and losses inherent in quadratic loss functions.

Chapter 6

Elements of continuous-time financeab 17.1.17

In this course we consider mainly discrete-time models with time set N resp. 0, . . . , N.Alternatively, mathematical finance can be based on continuous-time models with time setR+ or [0, t], respectively. Many concepts and results of the preceding chapters can be carriedover to this case. The mathematical theory, however, is too involved to be covered in thisintroductory course. Nevertheless, we want to give a short overview. In particular, we brieflyaddress the Black-Scholes model as a main cornerstone of mathematical finance.

6.1 Continuous-time theory

The majority of definitions and results from Chapter 1 allows for a continuous-time counter-part. This includes filtrations, filtered probability spaces, stochastic processes, adaptedness,predictability, generated filtrations, stopping times, stopped processes, martingales, sub- andsupermartingales, generated martingales, density processes, the generalized Bayes’ fomula,the Doob decomposition, compensators, stochastic integrals, covariation and predictable co-variation, integration by parts and other rules, Itô’s formula, stochastic exponential, Yor’sformula, Girsanov’s theorem, martingale representation, the Snell envelope. Some notionsrequire a more refined theory as e.g. predictability and stochastic integration. In this chapterwe liberally use notions and results from Chapter 1 in continuous time without precise defi-nitions, in particular related to stochastic integration. The reader should regard them simplyas counterparts or limits of the familiar discrete objects.

Sums of independent, identically distributed rendom variables play an important role indiscrete time. We applied them e.g. in the market models of Section 2.4. Their continous-time counterpart is called Lévy process.

Definition 6.1 A Lévy process (process with stationary and independent increments) isan adapted process X = (Xt)t≥0 such that

1. X0 = 0

2. Xt −Xs is stochastically independent of Fs for s ≤ t.

106

6.1. CONTINUOUS-TIME THEORY 107

3. The law of Xt −Xs depends only on t− s.

4. As a function of t, the process (Xt)t≥0 is continuous from the right and has left-handlimits.

In some sense, Lévy processes play a similar role as linear functions in analysis. Firstly,they represent constant growth, where constant here is to be interpreted in a stochastic sense.Secondly, they can be characterized by relatively few parameters. Finally, a large class ofmore general processes resembles Lévy processes on a local scale, similarly as differentiablefunctions in analysis locally look like linear functions.

For simplicity we focus on univariate processes. The most important Lévy process nextto deterministic linear functions is standard Brownian motion.

Definition 6.2 A Lévy process is called standard Brownian motion if X1 is a standardnormal random variable, i.e. with mean 0 and variance 1.

Remark. The law of standard Brownian motion is uniquely determined. We have that Xt isnormally distributed with mean 0 and variance t. Moreover, the definition easily yields thatstandard Brownian motion is a martingale.

From now on we consider only continuous processes, i.e. (Xt)t≥0 is supposed to be acontinuous in time t. The follwing theorem states the surprising and deep fact that anycontinuous process exhibiting constant growth in the sense of Definition 6.2 is a linear com-bination of a linear function and standard Brownian motion. This underlines the importanceof Brownian motion for stochastic calculus. Moreover, we see that — as in deterministiccalculus — objects of constant growth are determined by relatively few parameters, namelyµ and σ compared to the slope µ in the deterministic case.

Theorem 6.3 A real-valued Lévy process X is continuous if and only if it can be written as

Xt = µt+ σWt

for some standard Brownian motion W and constants µ ∈ R, σ ∈ R+.

Definition 6.4 In view of the previous throrem we call continuous Lévy processes Brown-ian motion with drift.

Notation. By I we denote the identity process It := t, i.e. a simple linear function.

In calculus, differentiable functions can be viewed as approximately linear on a localscale. E.g. they can be written as f(t) =

∫ t0µ(s)ds, where the derivative µ(s) equals the

slope of the linear function whose growth resembles that of f around s. Similarly, one canconsider processes that resemble Brownian motion with parameters µ, σ on a local scale.They are called Itô processes.

108 CHAPTER 6. ELEMENTS OF CONTINUOUS-TIME FINANCE

Definition 6.5 Let W be a standard Brownian motion. Processes of the form

X = X0 + µ • I + σ • W (6.1)

are called Itô process where µ, σ denote predictable processes µ, σ which are integrablewith respect to I resp. W .

Remark. If (6.1) referred to discrete-time integrals, we could write it as

Xt = X0 +t∑

s=1

µs∆Is +t∑

s=1

σs∆Ws

or∆Xt = µt∆It + σt∆Wt.

If we replace∑

by the integral sign∫

(which is a stylized “S” for sum) and also ∆ by d toindicate a limit, we obtain the two more common notations for (6.1) in the literature, namely

Xt = X0 +

∫ t

0

µsds+

∫ t

0

σsdWs

anddXt = µtdt+ σtdWt.

Covariation is defined in continuous time as well. For Itô processes X = X0 + µ •

I + σ • W and X = X0 + µ • I + σ • W it is given by

[X, X] = (σσ) • I. (6.2)

It is needed in Itô’s formula, which plays a key role in stochastic calculus.

Theorem 6.6 (Itô’s formula) If X = X0 + µ • I + σ • W is an Itô process and f : R→ Ra twice continuously differentiable function, we have

f(Xt) = f(X0) + f ′(X) • Xt +1

2f ′′(X) • [X,X]t (6.3)

= f(X0) +

(f ′(X)µ+

1

2f ′′(X)σ2

)• It + (f ′(X)σ) • Wt.

More generally, if f : R+ × R→ R is twice continuously differentiable, then

f(t,Xt) = f(0, X0) +

(f(I,X) + f ′(I,X)µ+

1

2f ′′(I,X)σ2

)• It + (f ′(I,X)σ) • Wt,

(6.4)where dot and prime represent the derivatives with respect to the first and second argumentof f , i.e. f = D1f , f ′ = D2f etc.

6.1. CONTINUOUS-TIME THEORY 109

Interestingly, the approximation (1.8) holds exactly for Itô processes. On closer look, how-ever, a difference can be detected. Instead of X− we have written X in (6.3). But theprevious value Xn− = Xn−1 is naturally replaced by a left-hand limit Xt− := lims↑tXs incontinuous time. Since we consider only continuous processes here, we have X = X−, i.e.the two formulas coincide when interpreted properly.

The integration by parts rule reads as

XY = X0Y0 +X • Y + Y • X + [X, Y ],

in continuous time. Note that this corresponds to (1.7), but not to (1.6). 17.1.17

As in discrete time, the stochastic exponential Z = E (X) of a process X is defined asunique solution to the equation Z = 1+Z− • X . In the present case of continuous processeswe can write again Z for Z−.

Theorem 6.7 (Stochastic exponential) For Itô processes X we have

E (X)t = exp

(Xt −X0 −

1

2[X,X]t

). (6.5)

Proof. The Itô process Y := X − X0 − 12[X,X] satisfies [Y, Y ] = [X,X] because the

constant X(0) and integrals relative to I do not change the covariation by (6.2). Using Itô’sformula (6.3 we obtain for Z := exp(Y ):

Z = exp(Y )

= exp(Y0) + eY • Y +1

2eY • [Y, Y ]

= 1 + eY • X − 1

2eY • [X,X] +

1

2eY • [X,X]

= 1 + Z • X

as desired.

Hence the approximation (1.9) is exact as well in continuous time.Transferring the results from discrete-time mathematical finance to continuous time is

less obvious than for the theory of stochastic processes. Continuous-time counterparts existfor price processes, trading strategies, value processes, self-fiancing strategies, discountedprocesses, arbitrage, the first fundamental theorem of asset pricing, equivalent martingalemeasures, dividend processes, market valuation, attainability, completeness, the second fun-damental theorem of asset pricing, OTC valuation, upper and lower price, superhedging,American options and their relation to stopping problems and the Snell envelope, martin-gale modelling, variance-optimal hedging, expected utility optimization. The validity of thecorresponding results, however, depends sensitively on the precise definitions of the set ofadmissible trading strategies etc. These definitions differ in the literature depending on thesetup and on the problem under consideration. The continuous-time analogues have not yetalways been stated in a clear and convincing fashion, at least compared to the situation inthe theory of stochastic processes.


6.2 Black-Scholes model

Having finished the general overview we turn now to a concrete market model with twoassets. This so-called Black-Scholes model goes back to Osborne and Samuelson and cor-responds directly to the its discrete-time analogue in Section 2.4. Again we assume thatrelative price movements evolve homogeneously in time and independently of the past. Inaddition, we suppose that the price process is continuous. These assumptions already deter-mine the model uniquely up to few parameters, in contrast to discrete time, where the law ofdaily relative time changes could be freely chosen. This follows from the surprising obser-vation from Theorem 6.3, namely that any continuous process with stationary, independentincrements is a Brownian motion with drift. In this sense, the process below can indeedbe viewed as a standard market model even if the reservations of Section 2.4 apply here aswell.

The money market account or bond is modelled as

S0t = S0

0 exp(rt) = S00E (rI)t (6.6)

with constant interest rate r ∈ R. Moreover, we consider a stock or foreign currency whoseprice evolves in the following form:

S1t = S1

0 exp(µt+ σWt) = S10E (µI + σW )t, (6.7)

with µ ∈ R, σ > 0, µ = µ + σ2

2and some standard Brownian motion W . A process as in

(6.7) is called geometric Brownian motion. The Itô-process representations of S0 and S1

are

S0t = S0

0 + (S0r) • It,

S1t = S1

0 + (S1µ) • It + (S1σ) • Wt.

One can show that this Black-Scholes model is complete. Relative to the correspondingequivalent martingale measure Q the process Wt := Wt + µ−r

σt is a standard Brownian

motion. Obviously we have

S1t = S1

0 exp

(r − σ2

2t+ σWt

)= S1

0E (r + σW )t.

For the return process X := logS1 we have that the increments Xt − Xt−1, t = 1, 2, . . .

are independent and normally distributed with mean r − σ2

2and variance σ2. Restricted to

integer times, the process X has the same law as in Lemma 3.12 for σ = σ. This impliesthat we obatin the option prices from (3.8).19.1.17

In the remainder of this section we take a different approach. We show that the Europeancall (S1

T −K)+ is replicable and we determine the corresponding perfect hedging strategy.Put differently, we look for a self-financing strategy ϕ satisfying VT (ϕ) = (S1

T −K)+. Wemake the natural ansatz that the value of this portfolio is a deterministic function of the stockprice at any time, i.e.

Vt(ϕ) = f(t, S1t )

6.2. BLACK-SCHOLES MODEL 111

for some deterministic, twice continuously differentiable function f : [0, T ] × R+ → R.Itô’s fomula implies that

Vt(ϕ) = f(t, S1t )

= V0(ϕ) +

(f(I, S1) + f ′(I, S1)S1µ+

1

2f ′′(I, S1)(S1σ)2

)• It

+(f ′(I, S1)S1σ

)• Wt. (6.8)

On the other hand, we obtain from the self-financing condition

Vt(ϕ) = V0(ϕ) + ϕ0 • S0t + ϕ1 • S1

t (6.9)

= V0(ϕ) +(ϕ0S0r + ϕ1S1µ

)• It + (ϕ1S1σ) • Wt (6.10)

Since the decomposition of an Itô process in drift and diffusion part, i.e. in integrals withrespect to time and W is essentially unique, we expect from comparing (6.8) and (6.10) that

ϕ1 = f ′(I, S1).

Since the value of a portfolio is

V (ϕ) = ϕ0S0 + ϕ1S1,

the previous equation implies

ϕ0S0 = f(I, S1)− f ′(I, S1)S1.

Equating the drift terms in (6.8,6.10) we get

f(I, S1) + f ′(I, S1)S1µ+1

2f ′′(I, S1)(S1σ)2

= ϕ0S0r + ϕ1S1µ

= f(I, S1)r − f ′(I, S1)S1r + f ′(I, S1)S1µ.

Since this is supposed to hold for arbitrary values of It = t and S1t , we obtain the following

partial differential equation (PDE) for f :

f(t, x) = −1

2f ′′(t, x)(xσ)2 − f ′(t, x)xr + f(t, x)r

together with the terminal condition

f(T, x) = (x−K)+,

which is obtained from VT (ϕ) = (S1T −K)+.

Since the setup is related to Section 3.4.4, we can try and use the pricing formula fromthere in order to guess a solution. Indeed, one easily verfies that the above PDE is solved byfunction

f(t, x) = xΦ(d1(t, x))−Ke−r(T−t)Φ(d2(t, x))


with

d1(t, x) :=log x

K+ r(T − t) + σ2

2(T − t)

σ√T − t

d2(t, x) :=log x

K+ r(T − t)− σ2

2(T − t)

σ√T − t

,

which is known from (3.8). The corresponding portfolio equals

ϕ1t = f ′(t, S1

t ) = Φ(d1(t, S1t ))

and

ϕ0t =

f(t, S1t )− f ′(t, S1

t )S1t

S0t

= −Ke−rTΦ(d2(t, S1t )).

Reversing the above computations yields that this strategy really has value

Vt(ϕ) = f(t, S1t ) = S1

t Φ(d1(t, S1t ))−Ke−r(T−t)Φ(d2(t, S1

t )) (6.11)

and that its satisfies the self-financing condition (6.9). Since in addition VT (ϕ) = (S1T−K)+,

we have found a replicating strategy for the European call.Note that the fair call price in this Black Scholes formula (6.11) depends only on the

volatility parameter σ but non on the drift parameter µ, which is notoriously hard to estimate.It may appear as very surprising or even paradox that the very parameter is missing whichcrucially affects the probability of receiving a non-zero payoff at all.

Moreover, one may observe that the number of shares in the hedge is obtained by differ-entiating the pricing function relative to the stock price. This derivative is called Delta ofthe option.

Finally we mention that the Black-Scholes model along with pricing formula (6.11) canbe obtained as a limit of Cox-Ross-Rubinstein models. From the point of view of the stan-dard model in Section 3.4.4, however, the completeness of the Black-Scholes model mayseem rather surprising. Indeed, no matter how fine we choose the mesh size of a grid dis-cretizing proceses (6.6,6.7) in time, we obtain the trivial price bounds for the European call.In the limiting continuous-time model, however, only one price is cosistent with absence ofarbitrage.

The Black-Scholes formula and the arbitrage reasoning are of utmost importance inpractice. Indeed, tremendous sums are turned over in the golbal derivatives market. Thiswas acknowleged by awarding the so-called Nobel price in economics to M. Scholes undR. Merton; F. Black had already passed away earlier. On the other hand, the theory hasbeen critized to have contributed to the global financial crisis because it gives the impres-sion of perfect control of financial risks which in fact one cannot handle. As can be seenfrom the previous sections, unique pricing formuals and perfect hedging strategies are theexception rather than the rule, even if we assume the absence of transaction costs, illiquidity,model risks etc. In any case, it is crucial to understand the limitations and weaknesses ofmathematical models when decisions are based on them in market practice.24.1.17

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