Continuous Time Finance, Part 1 Lecture Notes, SS 2013math4tune.com/CTF.pdf · 2014. 6. 16. ·...

Continuous Time Finance, Part 1

Lecture Notes, SS 2013

Helmut Strasser

June 16, 2014

Contents

Introduction 3

I Introduction to stochastic analysis 5

1 From Wiener process to jump diffusions 61.1 Basics about Levy processes . . . . . . . . . . . . . . . . . . . . 61.2 Path properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Filtrations and martingales . . . . . . . . . . . . . . . . . . . . . 121.4 Exponential Levy processes . . . . . . . . . . . . . . . . . . . . . 141.5 Compound Poisson processes . . . . . . . . . . . . . . . . . . . . 171.6 Review material . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.6.1 Examples of exam questions . . . . . . . . . . . . . . . . 191.6.2 Examples of exam problems . . . . . . . . . . . . . . . . 201.6.3 Exam proofs . . . . . . . . . . . . . . . . . . . . . . . . 21

2 Variations and integrals 222.1 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 222.1.2 The concept of variation . . . . . . . . . . . . . . . . . . 232.1.3 Finite total variation . . . . . . . . . . . . . . . . . . . . 242.1.4 Quadratic variation . . . . . . . . . . . . . . . . . . . . . 25

2.2 The integral of step functions . . . . . . . . . . . . . . . . . . . . 272.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.2 The integral of step functions . . . . . . . . . . . . . . . 302.2.3 Integral transforms . . . . . . . . . . . . . . . . . . . . . 31

2.3 The Stieltjes extension of the integral . . . . . . . . . . . . . . . . 332.4 Review material . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4.1 Examples of exam questions: . . . . . . . . . . . . . . . . 352.4.2 Examples of exam problems: . . . . . . . . . . . . . . . . 352.4.3 Exam proofs: . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Stochastic integrals 37

1

CONTENTS 2

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 The Wiener integral . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 The Ito integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4 The stochastic integral . . . . . . . . . . . . . . . . . . . . . . . 443.5 Local martingales . . . . . . . . . . . . . . . . . . . . . . . . . . 453.6 Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . . . 473.7 Review material . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.7.1 Examples of exam proofs: . . . . . . . . . . . . . . . . . 493.7.2 Examples of exam questions: . . . . . . . . . . . . . . . . 493.7.3 Examples of exam problems: . . . . . . . . . . . . . . . . 50

4 Quadratic variation and the transformation formula 514.1 Quadratic variation . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Special properties of the quadratic variation . . . . . . . . . . . . 544.3 The transformation theorem (continuous case) . . . . . . . . . . . 554.4 Stochastic exponentials and Levy’s theorem . . . . . . . . . . . . 574.5 Review material . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.5.1 Examples of exam proofs . . . . . . . . . . . . . . . . . . 584.5.2 Examples of exam questions . . . . . . . . . . . . . . . . 584.5.3 Examples of exam problems . . . . . . . . . . . . . . . . 59

II Elements of mathematical finance 61

5 Financial markets and trading strategies 625.1 Market models . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2 Trading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.3 Markovian trading strategies . . . . . . . . . . . . . . . . . . . . 665.4 The market price of risk . . . . . . . . . . . . . . . . . . . . . . . 685.5 Review material . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.5.1 Examples of exam proofs . . . . . . . . . . . . . . . . . . 705.5.2 Examples of exam questions . . . . . . . . . . . . . . . . 705.5.3 Examples of exam problems . . . . . . . . . . . . . . . . 70

6 Risk neutral Ito-models and pricing derivatives 716.1 Risk neutral models . . . . . . . . . . . . . . . . . . . . . . . . . 716.2 Hedging contingent claims . . . . . . . . . . . . . . . . . . . . . 726.3 Review material . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.3.1 Exam questions . . . . . . . . . . . . . . . . . . . . . . . 746.3.2 Exam problems . . . . . . . . . . . . . . . . . . . . . . . 75

7 The fundamental theorems of mathematical finance 767.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767.2 Equivalent models . . . . . . . . . . . . . . . . . . . . . . . . . . 78

CONTENTS 3

7.3 Girsanov’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . 797.4 The no-arbitrage theorem . . . . . . . . . . . . . . . . . . . . . . 827.5 The martingale representation theorem . . . . . . . . . . . . . . . 827.6 Complete and incomplete markets . . . . . . . . . . . . . . . . . 837.7 Review material . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

8 Classroom tests 85

Bibliography 86

Introduction

This is an outline of the lecture contents.

Basic sources of reference are Shreve [?] and Shiryaev [?]. For no-arbitrage the-ory we recommend Foellmer and Schied [?] and Delbaen and Schachermayer [?].Have a look into these books, read the introductions of each chapter.

Financial engineering without martingales has to be common body of knowledgefor every quant. A popular reference for this stuff is Wilmott [?, ?].

The main reference for models with jumps is Rama Cont and Tankov [?]. Mathe-matical texts on the same subject are Protter [?] and Jacod and Shiryaev [?].

4

Part I

Introduction to stochasticanalysis

5

Chapter 1

From Wiener process to jumpdiffusions

1.1 Basics about Levy processes

Recall that a random walk (Xn) is a sequence of partial sums of Xn = U1 +U2 +· · ·+ Un of i.i.d. random variables (Uk). Every random walk has independent andstationary increments. Expectations and variances are proportional to n.

Levy processes are concepts which extend the notion of random walks to continu-ous time.

Let (Xt)t≥0 be a stochastic process with continuous time parameter t ≥ 0 andinitial value X0 = x0 ∈ R. The process (Xt) has independent increments iffor any choice of disjoint time intervals [s, t] and [u, v] the increments Xt − Xs

and Xv −Xu are independent random variables. The process (Xt) has stationaryincrements if the probability distribution of incrementsXt−Xs andXt+a−Xs+a

is equal for any interval [s, t] and any shift a > −s.

1.1 DEFINITION. The process (Xt) is called a Levy process if it has independent

and stationary increments, and if XtP→ x0 as t→ 0.

The important properties are the increment’s properties. The second condition is acontinuity condition in order to exclude cases of an irregular kind.

There are many different Levy processes. It is easy to describe the probabilitydistribution of a particular Levy process, since we need only know the distributionsL(Xt) of the single random variables Xt for every t ≥ 0. Due to the properties ofthe increments the joint distributions of any vectors (Xti)i=1,...,n can be obtainedfrom the single random variables Xt.

6

CHAPTER 1. FROM WIENER PROCESS TO JUMP DIFFUSIONS 7

1.2 DISCUSSION. (Obtaining joint distributions) This can be achieved in thefollowing way. First we obtain the distributions of all increments since L(Xt −Xs) = L(Xt−s − X0). Then by independence of increments we get the jointdistribution of (Xs − X0, Xt − Xs) as the product of the marginals, and usingthe transformation (x, y) 7→ (x0 + x, x0 + x + y) yields the joint distributionof (Xs, Xt). In a similar way we may obtain the joint distribution of any vector(Xti)i=1,...,n. 2

Which probability distributions are possible for Levy processes ? As a consequenceof the very special properties of the increments, the distributions µt := L(Xt −X0) must satisfy a special condition. Recall that for any distributions µ and νof independent random variables X and Y the so-called convolution µ ∗ ν is thedistribution of X + Y .

1.3 DEFINITION. A family (µt)t≥0 is called a convolution semigroup if it satis-fies(1) µs ∗ µt = µs+t (where ∗ denotes the convolution of probability distributions),and(2) limt→0 µt(|x| > ε) = 0, ε > 0.

1.4 LEMMA. For every Levy process (Xt) the family of distributions L(Xt−X0)is a convolution semigroup.

Proof: Note that Xs+t−X0 = (Xs+t−Xt) + (Xt−X0) is a sum of independentrandom variables with distributions µs and µt. Therefore µs+t = µs ∗ µt. This iscondition (1). Condition (2) follows immediately from Xt −X0

P→ 0 as t→ 0. 2

Even the converse is true: For every convolution semigroup (µt) of distributionsthere exists a Levy process such that L(Xt − X0) = µt. This is an advancedmathematical theorem (see Protter). Thus, if we look for special Levy processesthen we have to look for families of distributions (µt) satisfying the convolutionsemigroup property.

1.5 EXAMPLE. (Brownian motion, Wiener process) The family of normal dis-tributions µt := N(bt, σ2t) is a convolution semigroup. The corresponding Levyprocess (Xt) is called a Brownian motion. The parameter b is called drift and theσ is the diffusion parameter. The Brownian motion with b = 0 and σ2 = 1 iscalled a Wiener process and is usually denoted by (Wt).

The Fourier transform isE(eiu(Xt−X0)) = eiubt−σ2tu2/2). The moments areE(Xt) =

x0 + bt and V (Xt) = σ2t. 2

1.6 LABORATORY. The first diagram shows a randomly generated path of theWiener process.

x=rWiener(1); plot(x.date,x);

In the next diagram 50 paths are displayed. We see how randomness increases the


dispersion with time.

x=rWiener(50); plot(x.date,x);

Introducing non-zero drift µ = 0.1 and small diffusion σ = 0.05 changes thepicture.

x=rWiener(10,mu:0.1,sig:0.05); plot(x.date,x);

2

1.7 EXAMPLE. (Poisson process) The family of Poisson distributions µt := π(λt)is a also convolution semigroup. The corresponding Levy process (Nt) is called aPoisson process. A Poisson process typically starts at N0 = 0.

The Fourier transform is E(eiu(Nt) = eλt(eiu−1). The moments are E(Nt) = λt

and V (Nt) = λt. 2

1.8 LABORATORY. The first diagram shows a randomly generated path of a Pois-son process with intensity λ = 20.

x=rPoisson(1,lambda:20); plot(x.date,x);

In the next diagram 10 paths are displayed.

x=rPoisson(10,lambda:20); plot(x.date,x);

2

The preceding examples show very simple expressions for expectations and vari-ances. This holds for any Levy process, provided the moments exist at all.

1.9 THEOREM. Mean and variance of a square integrable Levy process are lin-ear t, i.e. E(Xt) = x0 + tE(X1) and V (Xt) = tV (X1). The covariance isCov(Xs, Xt) = min(s, t)V (X1).

Proof: Let f(t) := E(Xt −X0). Then Xs+t −X0 = (Xs+t −Xt) + (Xt −X0)implies f(s + t) = f(s) + f(t). It follows that f(ns) = nf(s) if n ∈ N. Hencef(1/n) = f(1)/n and f(k/n) = k/nf(1). Thus we have proved that f(t) =tf(1) for all t ∈ Q. A mathematically subtle argument shows that t 7→ f(t) is acontinuous function which implies f(t) = tf(1) for all t ∈ R.

A similar argument applies to the variance. For the covariance we note that s < timplies

Cov(Xs, Xt) = Cov(Xs, Xs + (Xt −Xs))

= Cov(Xs, Xs) + Cov(Xs, Xt −Xs) = V (Xs) = sV (X1).

2


The next assertion is related to the Law of Large Numbers for random walks: Theaverage position of square integrable random walk tends to E(X1).

1.10 THEOREM. Every square integrable Levy process satisfies the law of largenumbers:

Xt

t

P→ E(X1) as t→∞.

Proof: Use Chebyshev’s inequality. 2

From instances of Levy processes it is easy to construct further Levy processes.

1.11 THEOREM. Any linear combination of independent Levy processes is aLevy process.

Proof: It is clear that independence and stationarity of increments is kept underlinear combinations of independent components. 2

Note that any linear function is a Levy process (with variance zero). It followsthat centering an integrable Levy process (subtracting tE(X1)) preserves the Levyproperty.

Moreover, we see that any linear combination of independent Brownian motionsand Poisson processes gives a Levy process. There is a remarkable converse of thisfact: Every Levy process can approximated by linear combinations of independentBrownian motions and Poisson processes. To be honest, even Poisson processesare sufficient to approximate all Levy processes by taking linear combinations.This is a very advanced result which cannot be explained now. But it indicates thatBrownian motion and Poisson process are not only the primary examples of Levyprocess but also the ancestors of any further instances.

Let us indicate a very simple case of such an approximation. Let (Nt) be a Poissonprocess with intensity λ. ThenNt−λt is a compensated (centered) Poisson processwith jumps of size 1. If we multiply this process by a number h then the resultingprocess has jumps of size h. Thus Xt = h(Nt − λt) is a Poisson-like process withjumps of size h, expectation E(Xt) = 0 and variance V (Xt) = λh2t. Betweensubsequent jumps the paths of the process are straight lines with slope −hλ. Thefollowing assertion shows that such processes with many small jumps (λ big and hsmall) look like Brownian motions.

1.12 THEOREM. Let Xnt = hn(Nt − λnt) where (Nt) is a Poisson process withintensity λn. If λn →∞ and hn → 0 in such a way that λnh2n → σ2 > 0, then thedistributions of Xnt tend to the distributions of a Brownian motion.

Proof: The logarithms of the Fourier transforms of Xnt are

logE(eiuXnt) = λnt(eiuhn − 1− iuhn) ≈ λnt ·

(− u2h2n

2

)≈ −σ

2u2

2t.


2

1.13 LABORATORY. The first diagram shows a randomly generated path of a stan-dardized Poisson process with intensity λ = 20.

x=rPoisson(1,lambda:20); y=x.minus(x.date.times(20)).div(sqrt(20)); plot(x.date,y);

In the next diagram we increase the intensity to λ = 200.

x=rPoisson(1,lambda:200); y=x.minus(x.date.times(200)).div(sqrt(200)); plot(x.date,y);

2

1.2 Path properties

Let (Xt) be a stochastic process. A path of the stochastic process (Xt) is a singlerealization t 7→ xt of the random function t 7→ Xt, t ≥ 0. From the distribution ofthe increments we may draw first conclusions about simple path properties.

Let (Xt) be a Brownian motion such that L(Xt) = N(bt, σ2t). The incrementsXt−Xs may attain any real numbers where values between b(t−s)−3σ

√t− s and

b(t−s)+3σ√t− s are typical. If b = 0 the distribution of increments is symmetric

about zero, thus ups and downs occur with equal probability. Increments of size 0have probability zero, which implies that paths are never constant.

Things are completely different for Poisson processes. Let (Nt) be a Poisson pro-cess with intensity λ. The incrementsNt−Ns may attain only nonnegative integervalues. Therefore paths are always non-decreasing. Increments may be zero withpositive probability, thus allowing intervals where paths are constant.

In order to talk more precisely about path properties we need some additional con-cepts.

A stochastic process (Xt) has the cadlag property if all paths are continuous fromright and have limits from left. This is a remarkable property. First, the cadlag-property implies that at every time point t0 a path has both a well-defined limitfrom right and and a well-defined limit from left. It is forbidden that as t ↑ t0 ort ↓ t0 the path xt oszillates in a way that prohibits convergence to a limit. Second,the cadlag-property implies that the value of the path at a particular time point t0 isidentical to the limit from right. Sometimes one says that cadlag paths are regulatedfrom the right.

Thus, for cadlag paths t 7→ xt we have to distinguish between the values xt (whichcoincide with the limits from right) and the limits from left xt−. The difference is


denoted by ∆xt := xt − xt−. If this difference is zero then the path is continuousat t. Otherwise there is a jump with jump height ∆xt := xt − xt−.

In general, for any convolution semigroup one can construct a Levy process suchthat its paths have the cadlag property. For a proof see Protter. Brownian motionsand Poisson processes additionally have very special path properties.

A special path property is continuity (i.e. no jumps at all). We are going to usethe self-explaining notations Xt− and ∆Xt := Xt −Xt− for the random variableversions of the corresponding path concepts.

1.14 DEFINITION. A stochastic process (Xt) with cadlag paths is continuous if(almost) all paths are continuous everywhere, i.e.

P⋂t≥0

(∆Xt = 0) = 1. (1)

1.15 REMARK. (Types of continuity) The preceding definition needs some ex-planation. It is important to understand that there is a big difference between (1)and

P (∆Xt = 0) = 1 for every t ≥ 0. (2)

The latter merely states that at every time point t fixed in advance there is chancezero of observing a jump. But since there uncountable many time points this doesnot exclude jumps. As matter of fact every Levy process satisfies the conditionof (2). (This follows easily from the regularity axiom in Definition 1.1 .) Thus,having no jumps at all is a much stronger condition than being not able to predicta jump for a particular time point. 2

Surprisingly there is only a single distribution family which admits Levy processeswith continuous paths. This is the normal distribution. (It should be clear that thereare many stochastic processes with continuous paths and non-normal distributions.But those processes cannot be Levy processes.)

1.16 THEOREM. A Levy process has continuous paths (with probability 1) iff itis a Brownian motion.

Later we will we able to indicate a proof of one part of this assertion: Given con-tinuous paths, any Levy process must be a Brownian motion. Proving the converse,i.e. Brownian motions always have continuous paths, is beyond the scope of thistext.

Another important path property which uniquely determines a certain Levy processis the counting-process property. A counting process is a stochastic process whosepaths are step functions (piecewise constant functions) with steps of height ∆Nt =+1.

1.17 DEFINITION. A process (Nt) is a counting process if its paths are increas-ing steps functions such that N0 = 0, ∆Nt = 1 and P (Nt < ∞) = 1 for all


t ≥ 0.

There are many interesting counting processes in probability theory and its appli-cation. But again there is only one convolution semigroup which allows a Levycounting-process, namely the Poisson distributions.

1.18 THEOREM. A Levy process is a counting process (with probability 1) iff itis a Poisson process.

Again we will we able to indicate a proof of one part of this assertion, later: Giventhe counting process property, any Levy process must be a Poisson process. Prov-ing the converse, i.e. Levy processes with Poissonian increments are countingprocesses, is beyond the scope of this text.

Accepting the counting process nature of a Poisson process we may derive thedistribution of waiting times between subsequent jumps.

1.19 THEOREM. Let (Ti) be the jump times of a Poisson process (Nt) withintensity λ. Then the waiting times Tk+1 − Tk are independent and identicallydistributed random variables with P (Tk+1 − Tk > x) = e−λx, x ≥ 0.

Proof: It is easy to see that

P (Ns+x −Ns = 0|Ns−ε < k ≤ Ns) = e−λx.

Letting ε→ 0 we obtain P (Tk+1 − Tk > x|Tk = s). 2

1.3 Filtrations and martingales

Dealing with mathematical models for time dependent random processes is facedwith two major challenges. One challenge is the separation of informative struc-ture from non-informative noise in empirical data. The second and actually muchbigger challenge is concerned with the different role of past and future which dom-inates our perception of reality.

Mathematical probability theory offers concepts and tools for dealing with thesechallenges going far beyond the mathematical tools of deterministic analysis.

The key notion for dealing with past and future is the concept of probabilisticconditioning based on a filtration.

Let (Xt)t≥0 be a stochastic process. Fix a particular time point t0 and consider theset of all statements about (Xt)t≥0, which can be decided (as true or false) basedon the information on (Xt)t≤t0 , i.e. the information available through observationsup to time t0. This set of statements is a σ-field and is denoted by

FXt0 := σ(Xs : s ≤ t0).


It is called the past of the process (Xt)t≥0 at time t0. As time goes by the availableinformation increases leading to an increasing family of σ-fields (FXt )t≥0. Thisis called the history of the process (Xt)t≥0. The history describes the increase ofinformation in the course of time.

1.20 DEFINITION. Let (Ω,F , P ) be a probability space. Any increasing family(Ft)t≥0 of sigma-fieldsFt ⊆ F is called a filtration ofF . The tripel (Ω, (Ft)t≥0, P )is called a filtered probability space.

Usually, a stochastic model starts with a probability space (Ω,F , P ) with a ba-sic filtration (Ft)t≥0, which typically is the history of some stochastic process(ξt)t≥0. This process (ξt)t≥0 need not be observable directly, it may serve only asworkhorse for the dynamics of the model. The basic filtration (Ft)t≥0 is the modelfor observable information. Other processes (Xt)t≥0 in the model are observableiff there pasts do not exceed the observable pasts, i.e. iff FXt ⊆ Ft for every t ≥ 0.

1.21 DEFINITION. Let (Ft)t≥0 be a filtration. A process (Xt)t≥0 is adapted tothe filtration if Xt is Ft-measurable for every t ≥ 0.

Obviously, each process is adapted to its own history.

Let (Xt) be a Levy process. Then a particular incrementXt−Xs is independent ofany other increment Xv −Xu in the past of s. It is clear that the whole past FXs isgenerated by statements about past increments. Therefore the increment Xt −Xs

is independent of FXs . Summing up: A Levy process has increments which areindependent of its own past.

Sometimes we need a slightly stronger property. If we are starting with a basicfiltration (Ft) and we have an observable (adapted) Levy process we would like tohave independence not only w.r.t. the past of the Levy process itself, but also w.r.t.the (possibly) larger past of our basic filtration. This gives rise to a slightly moregeneral concept of a Levy process.

1.22 DEFINITION. A Levy process (Xt) is a Levy process w.r.t. a given fil-tration (Ft) if it is adapted to the filtration (Ft) and if for s < t the incrementsXt −Xs are independent of Fs.

A Wiener process w.r.t. a filtration (Ft) is thus a Levy process (Wt) w.r.t. (Ft),such that the increments satisfy Wt −Ws ∼ N(0, t− s).

We turn to the problem of separating structure from noise. The first question to beanswered is how to define noise. Noise means a process carrying no trend patternat all. Any forecast of future changes results in forecasting no change. To put itinto mathematical terms a noisy process should satisfy

E(Xt −Xs|Fs) = 0,

since this conditional expectation of Xt − Xs given Fs is the best forecast of theincrement on the basis of all available information. Processes with this property


are called martingales.

1.23 DEFINITION. A process (Xt) is a martingale w.r.t. a filtration (Ft) ifit is adapted to the filtraton, integrable and satisfies E(Xt|Fs) = Xs for all pairss < t.

If (Xt) is an integrable Levy process then Xt = Xs + (Xt −Xs) implies that

E(Xt|Fs) = Xs + (t− s)E(X1).

Therefore a (Xt) is a martingale iff it is centered, i.e. iff E(X1) = 0. To put it inother words: Any integrable Levy process (Xt) can be decomposed as

Xt = At +Mt

where At = tE(X1) is the slowly varying trend component and Mt = Xt −tE(X1) is the noise component, being a martingale.

We have seen that centering a Levy process yields a martingale. Therefore anyWiener process is a martingale. If (Nt) is a Poisson process with intensity λ thenNt − λt (the compensated Poisson process) is a martingale.

Sometimes this procedure also works for processes which are not Levy processes.

1.24 THEOREM. Let (Mt) be a square integrable Levy martingale with varianceV (M1) = σ2. Then M2

t − σ2t is a martingale.

Proof: Note that M2t = M2

s + (Mt −Ms)2 + 2Ms(Mt −Ms). This implies

E(M2t |Fs) = M2

s + σ2(t− s). 2

If (Wt) is a Wiener process then W 2t − t is a martingale. This fact will turn out to

be of great importance, later.

1.4 Exponential Levy processes

The main applications of stochastic processes we have in mind are models of pricesfor financial instruments, like commodities, stocks, bonds or derivatives. A de-tailed discussion of financial mathematics is devoted to part 2 of this text. At thispoint we use some concepts from financial mathematics for preparing the intuitivebackground.

Models of non-negative stock prices are often defined to be exponentials of Levyprocesses, i.e.

St = S0ebt+cMt , t ≥ 0,

where (Mt) is a Levy martingale. This is equivalent to the assumption that thelog-returns

logSt − logSs = b(t− s) + c(Mt −Ms)


are increments of a Levy process. It should be clear that such models can giveonly an idealized picture of stock prices in an stationary environment without ex-traordinary events. They are a mathematical idealization like straight lines areidealizations of country roads.

1.25 DEFINITION. A non-negative stochastic process (St) is called an exponen-tial Levy process if logSt is a Levy process.

IfMt = Wt (where (Wt) is a Wiener process) and St = S0ebt+σWt then (St) is the

familiar lognormal model with volatility σ for Stock prices, known under the labelBlack-Scholes. If Mt = Nt − λt (where (Nt) is a Poisson process with intensityλ) then the stock price model is a simple jump model which jump intensity λ andjump height c.

1.26 LABORATORY. The first diagram shows a randomly generated path of theWiener process.

x=rWiener(1,mu:0.04,sig:0.2); plot(x.date,exp(x));

In the next diagram 50 paths are displayed. We see how randomness increases thedispersion with time.

x=rWiener(20,mu:0.04,sig:0.2); plot(x.date,exp(x));

2

Let us calculate the expected returns. By Jensen’s inequality we have

E(logSt) < logE(St)

since log is a concave function. Therefore expected returns E(St) increase at anexponential rate which is greater than the rate of expected log-returns.

1.27 EXAMPLE. (Returns of log-normal models) Let (St) be a log-normal model.Then

E(St) = S0ebt+σ2t/2, whence E

(St − S0S0

)= et(b+σ

2/2) − 1.

The return during period [0, t] grows exponentially with relative rate µ = b+σ2/2.Thus, expected log-returns are bt = t(µ− σ2/2).

For a general period s < t we have

E(St − Ss

Ss

∣∣∣Fs) = e(t−s)(b+σ2/2) − 1.

For short periods s < t this gives

E(St − Ss

Ss

∣∣∣Fs) ≈ (t− s)(b+ σ2/2).


2

1.28 REMARK. (Returns and log-returns) From the last example we see that theexpected log-returns depend on the variance of the log-returns, usually addressedas volatility. This is not surprising.

Note, that ex ≈ 1 + x+x2

2. Applied to returns this yields

St − SsSs

= elogSt−logSs − 1 ≈ (logSt − logSs) +1

2(logSt − logSs)

2

If the log-returns logSt had smooth paths then the quadratic term would be negligi-ble for small periods s < t. However, as will become clear later, stochastic modelsdo not have smooth paths and the squared increments contribute to an extent whichcannot be neglected. 2

A particular important class of models are martingale models satisfyingE(St|Fs) =Ss, s < t. For financial trading the martingale property has heavy conseqences.Usually, a financial agent attempts to improve his position using certain buy-sellstrategies. At time s the agent has the choice of keeping his capital in a risk-freesafe or investing it into shares of a stock with price Ss. Due to the martingaleproperty E(St|Fs) = Ss, s < t, on average the agent cannot improve his positionby investment. Taking risk in a martingale position does not pay. Therefore, infinancial vocabulary martingale models are called risk neutral models.

Things are even more serious. Assume that the agent holds a share of stocks duringrandom intervals [T1, T2] where T1, T2 are stopping times (recall this notion fromdiscrete time stochastics). Then by the optional stopping theorem, even such anadvanced strategy does not improve the agent’s position if the stock prices followa martingale model.

Turning back to mathematics we have good reasons to check martingale propertiesof exponential Levy processes. The proofs are left as exercises.

1.29 LEMMA. An exponential Brownian motion St = S0ebt+σWt is a martingale

iff b = −σ2/2.

1.30 LEMMA. An exponential Poisson process St = S0ebt+c(Nt−λt) is a martin-

gale iff b = −λ(ec − 1− c).

Note, that both conditions state that the expected log-returns should approximatelycompensate the variance of the log-returns, divided by 2. Later, we will see thatthis a property typical for exponential martingales.


1.5 Compound Poisson processes

Looking at stock prices at an intraday time scale shows that prices change in a step-wise manner. If we try to find Levy process models for such a stepwise behaviourthen we arrive at compound Poisson processes.

Let (Nt) be Poisson process with intensity λ and let (Uk) be an independent se-quence of i.i.d. random variables. We are going to construct a process which haspiecewise constant paths, which jumps whenever (Nt) jumps, and whose jumpsheights are randomly distributed according to the distribution of the random vari-ables Uk.

1.31 DEFINITION. A process of the form Xt =

Nt∑k=1

Uk is called a compound

Poisson process (CPP).

Any compound Poisson process has piecewise constant paths which satisfy thecadlag property. It satisfies

∆Xt = Uk iff ∆Nt = 1 and Nt = k.

1.32 THEOREM. Every compound Poisson process is a Levy process.

Proof: The increments of a CPP are given by

Xt −Xs =

Nt∑k=Ns+1

Uk

The increments are stationary since the number of jumps of Nt − Ns is station-ary and the random variables (Uk) are identically distributed. The increments areindependent since the increments of (Nt) are independent and the random vari-ables (Uk) are independent of each other and of (Nt). The regularity condition isinherited from (Nt). 2

A CPP may have jumps of different size. We might be interested in counting jumpsof a particular size, e.g. jumps with size ∆Xt ∈ B, where B ⊆ R is a given subsetof possible jump heights. This counting task is performed by the process

JBt :=

Nt∑k=1

1(Uk∈B).

This is again a compound Poisson process and therefore a Levy process. But itis even more. It is a counting process since it has jumps of size +1 only. Andtherefore it must be a simple Poisson process. It satisfiesE(JBt ) = tλQ(B). Thus,the intensity of this counting process is ν(B) = λQ(B). This is the intensity of


those jumps of (Xt) which have a jump size ∆Xt ∈ B. Due to its importance theset function B 7→ ν(B) has a special name.

1.33 DEFINITION. Let (Xt) be CPP with jump intensity λ and jump size distri-bution Q. Then ν : B 7→ λQ(B) is the Levy measure of (Xt).

1.34 LEMMA. The Fourier transform of a compound Poisson process is

E(eiuXt

)= eλt

∫(eiux−1)Q(dx) = et

∫(eiux−1) ν(dx)

Proof: Note that

E(eiuXt

)=

∞∑n=0

E(eiu

∑nk=1 Uk

∣∣Nt = n)P (Nt = n)

=∞∑n=0

(∫eiunQ(dx)

)nP (Nt = n)

and evaluate the sum of this series. 2

It follows that the distribution of a compound Poisson process is determined by itsLevy measure ν.

1.35 COROLLARY. Let (Xt) be a CPP with Levy measure ν. (Xt) is integrableiff∫|x| ν(dx) <∞. In this case E(X1) =

∫x ν(dx).

It is square integrable iff∫x2 ν(dx) <∞. In this case V (X1) =

∫x2 ν(dx).

Proof: The most easy way is to derive moments from the derivatives of the F-transfrom. 2

Let (Wt) be a Wiener process and (Yt) an independent compound Poisson processwith Levy measure ν. Then the Levy process

Xt = bt+ σWt + Yt

is called a jump diffusion. Its distribution is determined by the characteristictriple (b, σ2, ν).

The paths of a jump diffusion have a rich structure. On time intervals betweenjumps of (Yt) they look like paths of ordinary log-normal processes. Jumps occuraccording to exponential waiting times with intensity λ. There are jump diffusionswith rare but big jumps, serving as models for regular financial markets with rareextraordinary events. But there are also jump diffusions with frequent small jumpswhose paths look completely different from log-normal processes.

From the empiricial point of view there is a big difference between CPP and jumpdiffusions. The reason is that for a CPP one may localize the jumps. Whenever we


observe a non-zero increment of a CPP then we can be sure that there was a jump.This is not the case for jump diffusions. Any increment of a jump diffusion willbe non-zero and it is difficult to distinghuish between increments containing jumpsand those which do not.

Consider the popular jump diffusion model where the diffusion component hasparameters (µ, σ2) and the jump heights follow a normal distribution with param-eters (γ, τ2). Together with jump intensity λ the increments follow a distributionmodel with five parameters. Jumps disturb the diffusion by causing skew (due toγ) and kurtosis (heavy tails, due to τ2). Jumps cannot be localized, and thereforeparameter estimation for such a model is a challenging statistical problem.

1.6 Review material

Concepts: Levy process (square integrable), cadlag property, Wiener process, gen-eralized Brownian motion, counting process, Poisson process, compensatedPoisson process, filtration (past, history), independence of the past, geomet-ric Brownian motion, geometric Poisson process, compound Poisson process(CPP), jump diffusion.

Facts: mean and variance of Levy processes, distribution of Levy processes (Gaus-sian case, Poisson case),covariance structure of a Levy process, law of largenumbers for Levy processes, F-transform of Wiener process and Poissonprocess, normal approximation of the Poisson process.

independence of the past-property of Levy processes, martingale proper-ties of Levy processes (and of squares), martingale properties of geometricWiener process and geometric Poisson process.

CPP are Levy processes, Fourier transform of CPP, moments of CPP, mar-tingale properties of CPP, linear combinations of Levy processes.

1.6.1 Examples of exam questions

1. Why are mean and variance of Levy processes linear functions of t ?

2. What are the properties of the paths of Levy processes ?

3. How can one characterize the Wiener process and the Poisson process bypath properties ?

4. Can a Wiener process have jumps ?

5. Find the distribution of the waiting time for the first jump of a Poisson pro-cess.


6. Can a Poisson process have infinitely many jumps on a bounded time interval?

7. How fast do the paths of a square integrable Levy process increase on aver-age ? (Hint: Use the law of large numbers.)

8. What is a martingale ?

9. Are Levy processes always martingales ?

10. Is the Poisson process a martingale ?

11. Explain in words the construction of a compound Poisson process. Describethe paths of a compound Poisson process.

12. Explain the notion of a Levy measure.

13. Which compound Poisson processes can be written as a linear combinationof independent Poisson processes ?

14. How can one construct a Leyy process with two kinds of jumps ?

15. When is a compound Poisson process a martingale ?

16. How can one construct a Levy process with jumps, which is not constantbetween jumps ?

1.6.2 Examples of exam problems

1. Let (Wt) be a Wiener process and s < t. Find(a) E(W 2

t |Fs), (b) E(W 3t |Fs), (c) E(eαWt |Fs).

2. Let (Nt) be a Poisson process and s < t. Find(a) E(N2

t |Fs), (b) E(N3t |Fs), (c) E(eαNt |Fs).

3. Is Xt :=∫ t0 Wu du a martingale ? (Calculate E(Xt −Xs|Fs) !)

4. Let (Wt) be a Wiener process. When is eat+Wt a martingale ?

5. Let (Nt) be a Poisson process. When is eat+Nt a martingale ?

6. Let (Xt) be a square integrable Levy process. When is eat+Xt a martingale?

7. Let (Xt) be a Levy process. Show thatZt = eiuXt/E(eiuXt) is a martingale.

8. Find the Fourier transform of a compound Poisson process.

9. Analyze the Levy processes with the following Fourier transforms (expec-tation, variance, path properties, decomposition as a jump diffusion, Levymeasure, jump intensity, jump height distribution, martingale property (yes/no),compensator).


(a) log φt(u) = t(−iu− 5u2 + e2iu + e−iu/2 − 2)(b) log φt(u) = t(iu+ 3/2e2iu + 1/2e−iu/2 − 2)(c) log φt(u) = t(iu− u2)(d) log φt(u) = t(e3iu − 1− 3iu)

10. Find the Fourier transform of a jump diffusion with variance 2, jumping withintensity 3, having jump heights +1 and -1 with equal probability.

11. Find the Fourier transform of a jump diffusion with variance 1, jumping withintensity 1, having jump heights uniformly distributed on [−2, 0].

12. Find the Levy measure of a sum of five independent Poisson processes withintensities 1, 2, . . . , 5.

13. Find the Levy measure of a linear combination of five independent Poissonprocesses with intensity 1 and weights 1, 2, . . . , 5.

1.6.3 Exam proofs

1. Explain how a Wiener process can be approximated by compensated Poissonprocesses.

2. Explain the covariance structure of Levy processes.

3. Find the Fourier transform of compound Poisson processes.

4. Find the moments of compound Poisson processes.

Chapter 2

Variations and integrals

2.1 Variations

2.1.1 Introduction

Paths of stochastic processes are of a much more complicated nature than functiongraphs which are familiar from mathematical analysis tought in schools. Thereforewe have to dive a bit more deeply into the foundations of mathematical functionsof a real variable.

Let us start with a familiar object like a differentiable function f(t), defined fort ≥ 0. Then at every point s ≥ 0 there is a derivative

limt→s

f(t)− f(s)

t− s= f ′(t)

which gives the slope of the function at s. This means that locally (within a smallinterval around s) the function f(t) can be approximated very well by its lineartangent. We can state this property saying: The increments f(t)−f(s) are approx-imately proportional to the length t− s of the underlying intervals.

Now we turn to paths of a Levy process (Xt). Assume that the process is centeredand has finite variance. Then we know that

E((Xt −Xs)

2

t− s

)= σ2. (1)

This means that on average the squared increments are approximately proportionalto t− s. Does this mean that increments Xt −Xs are of magnitude

√t− s ? This

would be a very surprising fact since√t− s is much larger than t − s for small

intervals. Actually

limt→s

√t− st− s

= limt→s

1√t− s

=∞!

22

CHAPTER 2. VARIATIONS AND INTEGRALS 23

Indeed, the phenomenon can be explained only in two alternative ways:

Either: If increments Xt − Xs on small intervals s < t are typically of the samemagnitude as t − s then at least some increments must be much larger than t − s.The compensated Poisson process Xt = Nt − λt is of this kind. Linear decreaseis of magnitude t− s but jumps are increments of infinitely larger magnitude thant− s. This explains property (1) for the Poisson process.

Or: If there are no jumps then increments Xt − Xs on small intervals must be ofmagnitude

√t− s. This means for Brownian motion (which does not jump) that

on a microscopic level Brownian paths are not smooth at all. There is no chance ofapproximating a small piece of a Brownian path by a straight line.

2.1.2 The concept of variation

The magnitude of increments can be studied conveniently by adding incrementsover subdivisions and considering the sum.

Let 0 = t0 < t1 < t2 < . . . < tn = t be a subdivision of the interval [0, t]. A se-quence of such subdivisions (tn0, tn1, . . . , tnn) is called a Riemannian sequence(of subdivisions) if n→∞ and

max1≤i≤n

|tni − tn,i−1| → 0 as n→∞.

As the number of intervals increases to infinity the maximal length of the intervalsgoes to zero.

Let us study the sum of increments over such subdivisions.

2.1 DEFINITION. Let f be a function defined on [0, t] and (tni) any Riemanniansequence of subdivisions. Then for α > 0

V αt (f) := lim

n→∞

n∑i=1

|f(tni)− f(tn,i−1)|α

is called the α-variation of f on [0, t]. Th exponent α is called the order of varia-tion.

It can be shown that any such variation is well-defined in the sense that its value isthe same for every Riemannian sequence of subdivisions. If α = 1 then it calledthe total variation, denoted by Vt(f) := V 1

t (f). If α = 2 then it is called thequadratic variation and denoted by [f ]t := V 2

t (f). It is obvious that for anyfunction f any variation t 7→ V α

t (f) is increasing with the length of the interval[0, t].

Variations simply add all ups and downs of a function graph. For piecewise con-stant (cadlag) functions f(t) whose function values vary by jumps only, it is clear


thatV αt (f) =

∑t

|∆f(t)|α.

If there are only finitely many jumps then any variation is finite, too.

Things are more interesting if we turn to monotone functions. Let us start withconsidering the total variation. If f(t) is increasing then any sum over a subdivisiongives

n∑i=1

|f(ti)− f(ti−1)| =n∑i=1

(f(ti)− f(ti−1)) = f(t)− f(0)

which implies V 1t (f) = f(t) − f(0). If f(t) is descreasing we have V 1

t (f) =−(f(t)− f(0)). Thus, total variation is always finite for monotone functions.

2.2 LABORATORY. We start with the variation of a well-known smooth function.

t=subdiv(0,3*Math.PI,500); x=sin(t); y=variation(x); plot(t,cbind(x,y))

For a monotone function the variation coincides with the function.

t=subdiv(0,3*Math.PI,500); x=sqrt(t); y=variation(x); plot(t,cbind(x,y))

For continuous functions variations of higher order are smaller.

t=subdiv(0,3*Math.PI,500); x=sin(t); y=variation(x,1); y1=variation(x,1.5); y2=variation(x,2);plot(t,cbind(x,y,y1,y2))

2

2.1.3 Finite total variation

Familiar mathematical analysis deals with functions f(t) which have finite totalvariation. The set of all function having finite total variation on [0, t] is denoted byFV t

0 . We have seen that monotone functions belong to this class. Surprisingly, thisis almost an exhaustive description.

2.3 THEOREM. A function f(t) has finite total variation iff it is a linear combina-tion of monotone functions.

Proof: It is clear that any linear combination of FV-function is an FV-function,thus every linear combination of monotone functions. We have to prove the con-verse. For this we note that f(t) = Vt(f)−(Vt(f)−f(t)). It is clear that t 7→ Vt(f)is a monotone function. It remains to show that t 7→ Vt(f) − f(t) is a monotonefunction. For this, we note that the value of |f(t) − f(s)| is certainly below thetotal variation of f between s amd t, which implies

f(t)− f(s) ≤ Vt(f)− Vs(f) whenever s < t.


Hence, t 7→ Vt(f)− f(t) is an increasing function. 2

The description of FV-functions as differences of monotone function seems to indi-cate a very special class of function. From an abstract point of view this is certainlytrue. But at least all functions wihich are familiar from school mathematics belongto FV-functions.

2.4 THEOREM. If f is a continuously differentiable function then it is an FV-function and its total variation is

Vt(f) =

∫ t

0|f ′(s)| ds <∞.

Proof: We are able to give a precise proof only for twice differentiable functions.Increments of f satisfy |f(t) − f(s)| ≈ |f ′(s)|(t − s) up to an error term ofmagnitude (t− s)2. Thus, sums of increments are approximately

∑i

|f(ti)− f(ti−1)| ≈∑i

|f ′(ti−1)|(ti − ti−1) ≈∫ t

0|f ′(s)| ds

2

It follows that any function which is piecewise (on a finite subdivision) continu-ously differentiable is an FV-function.

2.1.4 Quadratic variation

What happens to other variations ? For later applications we are mainly interestedin quadratic variation.

There is a remarkable result which we can prove it in full generality. It tells us thatfor a continuous function f(t) the variation V α

t (f) can be positive and finite onlyfor exactly one particular order α > 0.

2.5 THEOREM. Let f(t) be a continuous function. If 0 < V αt (f) < ∞ for a

particular α > 0 then V γt (f) = 0 for all γ > α and V β

t (f) =∞ for all 0 < β < α.

Proof: Since f(t) is a (uniformly) continuous function we may choose our subdi-vision sufficiently small such that all increments satisfy |f(t)− f(s)| < 1. Then itis clear that for 0 < β < α < γ and any subdivision∑

i

|f(ti)− f(ti−1)|γ ≤∑i

|f(ti)− f(ti−1)|α ≤∑i

|f(ti)− f(ti−1)|β


whence V γt ≤ V α

t ≤ Vβt . Moreover, since α < γ we have∑

i

|f(ti)− f(ti−1)|γ ≤∑i

|f(ti)− f(ti−1)|α ·maxi|f(ti)− f(ti−1)|γ−α.

Passing to the limit in a Riemannian sequence of subdivisions implies V γt (f) = 0.

Since β < α we have∑i

|f(ti)− f(ti−1)|α ≤∑i

|f(ti)− f(ti−1)|β ·maxi|f(ti)− f(ti−1)|α−β.

Therefore V αt (f) > 0 is only possible if V β

t (f) =∞. 2

There are two important consequences which should be isolated for easy reference.

2.6 COROLLARY. (1) Every continuous function with finite total variation hasquadratic variation zero.(2) Every continuous function with positive quadratic variation has infinite totalvariation.

2.7 LABORATORY. A Poisson path is equal to all of its variations.

x=rPoisson(1,lambda:20,ngrid:500); y=variation(x,1); y1=variation(x,2); plot(x.date,cbind(x,y,y1))

If jump heights are smaller than 1 then variations decrease with order.

x=rPoisson(1,lambda:20,ngrid:500); y=x.div(2); y1=variation(y,1); y2=variation(y,2);plot(x.date,cbind(y,y1,y2))

The quadratic variation of a compensated Poisson path shows the pure jumps part.

x=rPoisson(1,lambda:20,ngrid:500); y=x.minus(x.date.times(20)); y1=variation(y,1);y2=variation(y,2); plot(x.date,cbind(y,y1,y2))

2

If you think that continuous functions with positive quadratic variation are notimportant for practical application then consider the following theorem.

2.8 THEOREM. Let (Wt) be a Wiener process. Then for P -almost all paths[W ]t = t.

Proof: Using Chebysev’s inequality it is easy to show that for any Riemanniansequence of subdivisions ∑

i

(Wti −Wti−1)2P→ t.


2

2.9 LABORATORY. Let us plot a Wiener path and ist quadratic variation.

x=rWiener(1,ngrid:1000); plot(x.date,cbind(x,variation(x,2)))

The quadratic variation is the same for all paths of a Wiener process.

x=rWiener(20,ngrid:1000); plot(x.date,cbind(x,variation(x,2)))

Total variation is infinite.

x=rWiener(1,ngrid:1000); plot(x.date,cbind(x,variation(x,1),variation(x,2)))

Quadratic variation depends on the variance.

x=rWiener(1,ngrid:1000); y=rWiener(1,ngrid:1000,sig:2); plot(x.date,cbind(x,y,variation(x,2),variation(y,2)))

2

It follows that the paths of a Wiener process (which are continuous, as we know)have positive quadratic variation and therefore have infinite total variation.

2.10 LABORATORY. Let us study the quadratic variation of functions of Wienerprocesses. At this stage we can only carry out experiments. Later we will be ableto understand mathematically what is going on.

We observe that the quadratic variations are nonlinear smooth functions. First weconsider the square of a Wiener path.

x=rWiener(1,ngrid:1000); y=x.up(2); plot(x.date,cbind(y,variation(y,2)))

Our next example is an exponential Brownian motion.

x=rWiener(1,ngrid:1000); y=exp(x); plot(x.date,cbind(y,variation(y,2)))

2

Paths of CPP are FV-functions. However, paths of jump diffusions are not FV-functions.

2.2 The integral of step functions

2.2.1 Introduction

There are many concepts of an integral in mathematics. This is (not the only) onereason why integration is considered a difficult matter. For applications in financeintegrals come into play through expressions of the following kind.


Let g(t) be any cadlag function. Assume that g(t) denotes the price of some finan-cial instrument and that α is the number of units of this instrument in an agents’sportfolio during period [s, t). If Vs and Vt denote the portfolio values at times sand t then we obviously have

Vt − Vs = α (g(t)− g(s)).

Now, let 0 = t0 < t1 < · · · < tn = t be a subdivision and let the step function

h(u) :=n∑i=1

αi1[ti−1,ti)(u) (2)

be the number of instruments in the agent’s portfolio at time u. Note, that αi =h(ti−). Then we have

Vt − V0 =n∑i=1

αi (g(ti)− g(ti−1)) =

n∑i=1

h(ti−) (g(ti)− g(ti−1)) (3)

If the function h(t) is not a step function but an arbitrary cadlag function thenformula (3) can be used in an approximate sense. The function h(t) describes theagent’s portfolio strategy for any trading time t, but practically the portfolio can beadjusted only at finitely many discrete time points.

Now comes the mathematically important idea: The sums in equation (3) are stable.This means: If the underlying subdivision consists of a great number of smallintervals then the value of the sum is more or less independent of the underlyingsubdivision. Mathematically, the (approximate) common value of these sums canbe identified with their limit under a sequence of Riemannian subdivisions. Thislimit is called integral and is denoted by a stylized sum

limn→∞

n∑i=1

h(tni−) (g(tni)− g(tn,i−1)) =:

∫h(s−) dg(s).

This is a much more handy notation than a sum since we neither need indices norintervals. The symbol informs us about the essential components of the integral:about the function h(t), called integrand, which goes in via its values, and aboutthe function g(t), called integrator, which goes in via its increments.

So far a rough description of the basic idea of integration. Mathematically, thingsare a bit more complicated. This is the reason why one has to perform the construc-tion of the concept of an integral very carefully. This is the duty of professionalmathematics.

2.11 LABORATORY. Let us draw some diagrams of integrals which are calculatedby simple evaluation of cumulative sums over a fine grid. We will see that theresults are exactly the same which are obtained by usual integral calculus.


We start with the Riemannian integral∫ t0 x dx.

t=subdiv(0,1,500); plot(t,cbind(t,integral(t,t)))

Our next example is∫ t0 f(x) dx, where f(x) is the indicator function of an interval.

t=subdiv(0,1,500); x=steps(t,[0,1,0],[0.3,0.7]); plot(t,cbind(x,integral(x,t)))

Now we change to nonlinear integrators. Let us draw∫ t0 x dx

2.

t=subdiv(0,1,500); plot(t,cbind(t,integral(t,sqr(t))))

Another example is∫ t0 x

2 d sin(x).

t=subdiv(0,6,500); plot(t,cbind(sqr(t),integral(t,sin(t))))

2

2.12 REMARK. (Integral extension) The intuitive idea of stable sums in (3) isan optimistic picture. Actually, it is the core problem of mathematical integrationtheory to find out for which integrands and integrators the integral is well-defined,i.e. this stability is valid and the integral has nice properties.

In the present chapter we will explain that if the integrator g(t) has finite totalvariation the integral is well-defined for all integrands which are continuous orpiecewise continuous functions. Integrals with FV-integrators are called Stieltjesintegrals. The Stieltjes integral admits paths of stochastic processes as integrands,and even as integrators provided that they have finite total variation.

The Stieltjes integral does not admit paths of the Wiener process or of Brownianmotions as integrators since such paths have infinite total variation. A suitable con-struction of the integral for integrators with infinite total variation is the stochasticintegral. This is the subject of chapter 3. 2

2.13 REMARK. (Regulation of integrands) Does it matter whether an integrandh(t) is right-continuous or left-continuous ? Why are we so scrupulous in writingintegrands as left-regulated versions h(t−) only ?

This is a reasonable question since for usual functions the difference between right-continuity and left-continuity typically affects function values at a few isolatedpoints only but not on intervals of positive length. Indeed, whenever the integratorg(t) is a continuous function then the value of the integral does not change if wereplace a left-continuous integrand h(t−) by its right-continuous version h(t).

But if the integrator has jumps, then the kind of jump regulation matters. The valueof the integral changes if both h(t) and g(t) jump at some point t0 and if h(t0−) isreplaced by h(t0).


When we are dealing with stochastic integration it will turn out that only left-continuous functions are reasonable integrands. This will be our main applicationand therefore we restrict the construction of the integral to left-continuous inte-grands. 2

2.2.2 The integral of step functions

Let us state the ideas of the preceding section in a more formal way.

2.14 DEFINITION. Let g(t) be any cadlag function and h(t) =∑n

i=1 αi1[ti−1,ti)(t)a cadlag step function. Then∫

h− dg =

∫h(t−) dg(t) =

n∑i=1

h(ti−)(g(ti)− g(ti−1))

is called the g-integral of h.

Note, that h(t−) is not defined as a function value of h but is the limit h(t−) =lims↑t h(s). Thus, altering function values of h at finitely many points does notchange the value of h(t−).

From Definition 2.14 it follows that the integral is both linear w.r.t. the integrandas well as linear w.r.t. to the integrator:∫ (

αh1(t−) + βh2(t−))dg(t) = α

∫h1(t−) dg(t) + β

∫h2(t−) dg(t),∫

h(t−) d(αg1(t) + βg2(t)

)= α

∫h(t−) dg1(t) + β

∫h(t−) dg2(t).

For a smooth integrator the g-integral can be written as an ordinary (Riemannian)integral.

2.15 EXAMPLE. (Riemannian integral) A special case is g(t) = t. Then theg-integral is the same as the familiar Riemannian integral∫

h(t−) dg(t) =

∫h(t−) dt

This is clear for h(t) = 1[a,b)(t) and by linearity it follows for any step functionh(t).

Moreover, we know that for continuously differentiable functions g(t) we haveg(b)− g(a) =

∫ ba g′(u) du. This implies∫

h− dg =

∫h(t−)g′(t) dt.


Again it is clear for h(t) = 1[a,b)(t) and by linearity it follows for any step functionh(t). 2

2.16 LABORATORY. Let us compare∫

sin(x) dx2 and∫

sin(x) 2x dx.

t=subdiv(0,1,500); x=integral(sin(t),sqr(t)); y=integral(sin(t).times(t.times(2)),t); plot(t,cbind(x,y.plus(0.01)))

2

The g-integral admits also step functions as integrators.

2.17 EXAMPLE. (Integrator step functions) It is obvious that the integrator g(t)contributes to the integral only where its increments do not vanish. Thus, if theintegrator g(t) is a cadlag step function with jumps at a1 < a2 < · · · an then itsincrements at ai contribute on intervals si < ai < ti which implies∫

h(s−) dg(s) =n∑i=1

h(ai−)(g(ti)− g(si))

=n∑i=1

h(ai−)(g(ai)− g(ai−)) =∑t

h(t−)∆g(t)

where the latter is a convenient abuse of notation. 2

2.18 LABORATORY. Let us draw∫ t0 x dNx.

t=subdiv(0,1,500); x=rPoisson(1,lambda:10,ngrid:500); plot(t,cbind(x,integral(t,x)))

2

2.2.3 Integral transforms

For convenience the following notation is used∫ ba h− dg :=

∫1(a,b]h− dg which

is familiar from Riemannian integrals. Another familiar notion are indefinite in-tegrals H(t) :=

∫ t0 h(s) ds which are primitives in the sense that H ′(t) = h(t)

(where h(t) is continuous). For g-integrals the corresponding notion

t 7→∫ t

0h− dg

is called the integral transform of h (w.r.t. g).

2.19 REMARK. (Integral transform) Note the convenient representation

h =n∑i=1

αi1[ti−1,ti) ⇒∫ t

0h− dg =

n∑i=1

αi(g(ti ∩ t)− g(ti−1 ∩ t))


2

There are three basic rules for the integral transform which remain valid for any ofthe subsequent extensions of the integral. We begin with the jump rule.

2.20 THEOREM. The integral transform f(t) =∫ t0 h− dg is a cadlag function and

satisfies ∆f(t) = h(t)∆g(t).

Proof: Whenever t ∈ (ti−1, ti] we have f(t) = . . .+ αi(g(t)− g(ti−1). 2

It follows that the integral transform jumps at most at those points where the inte-grator jumps. It is continuous where the integrator is continuous.

Recall, that stopping a function f at a point a means defining a new function fa by

fa(t) := f(a ∩ t)

Our next rule is the stopping rule. The stopping rule tells us that an integraltransform can be stopped in two ways: Either the integrand is truncated, or theintegrator is stopped.

2.21 THEOREM. ∫ a∩t

0h− dg =

∫ t

01(0,a]h− dg =

∫ t

0h− dg

a.

Proof: The first equation follows from∫ a∩t

0h− dg =

∫1(0,a∩t]h− dg =

∫1(0,a]1(0,t]h− dg =

∫ t

01(0,a]h− dg

For the second equation let h = 1[u,v). Then∫ a∩t

0h− dg = g(v ∩ t ∩ a)− g(u ∩ t ∩ a) =

∫ t

0h− dg

a.

2

The last of those basic rules is the associativity rule. This rule has a nice formalstructure. If we denote (like some authors) the integral transform by (h g)(t) :=∫ f0 h− dg then the rule states that

h (q g) = (h · q) g.

2.22 THEOREM. Let f(t) =∫ t0 q− dg. Then for any caglad step-function function

h(s) ∫ t

0h− df =

∫ t

0h−q− dg


Proof: Let h = 1[u,v). Then∫ t

0h− df = f(v ∩ t)− f(u ∩ t) =

∫ v∩t

u∩tq− dg =

∫ t

01(u,v]q− dg

2

This rule is usually abbreviated by the shorthand df = q− dg. However, it shouldbe kept in mind that such „differential abbreviations” do make sense only if theyare interpreted as assertions about integral transforms.

2.3 The Stieltjes extension of the integral

In order to extend the notion of the integral to a larger class of cadlag functionsthan only step-functions we could proceed as follows.

Let h be an arbitrary cadlag function. Then it easy to see that for any Riemanniansequence of subdivisions of [0, t] we have

hn(s−) :=n∑i=1

h(tn,i−1)1(tn,i−1,tni](s)→ h(s−).

Since (hn) is a sequence of cadlag step-function the integrals∫ t0 hn,− dg are well-

defined. It is a natural idea to define∫ t

0h− dg := lim

n→∞

∫ t

0hn,− dg. (4)

But how can we be sure that this limit exists and shares all the nice properties wehave proved for the integral of step-functions so far ?

If there exists a reasonable limit in equation (4) then at least the special case

hn → 0 ⇒∫ t

0hn,− dg → 0 (5)

is necessary to have any hope for the general case. Advanced mathematical the-ory tells us that the „continuity condition” (5) is actually even sufficient for (4).Moreover, condition (5) is true iff the integrator g(t) has finite total variation.

2.23 THEOREM. If the integrator g(t) has finite total variation then the Stieltjes-integral (4) is well-defined, linear in the integrand and the integrator, and satisfiesthe jump rule, the associativity rule and the stopping rule.

It is nice to observe how the familiar integral formulas from Riemannian inte-gration can be recovered in a very elegant manner. We consider only continuous


functions since the general situation will be covered later by the rules of stochasticintegration.

2.24 REMARK. (Integration by parts) Let f(s) and g(s) be continuous FV-functions.Then the integration-by-parts rule says that

f(t)g(t)− f(s)g(s) =

∫ t

sf(u) dg(u) +

∫ t

sg(u) df(u)

The familiar version is obtained for differentiable functions putting dg(u) = g′(u) duand df(u) = f ′(u) du.

For the easy proof note that

f(t)g(t)−f(s)g(s) = f(s)(g(t)−g(s))+g(s)(f(t)−f(s))+(f(t)−f(s))(g(t)−g(s)).

Use this for a Riemannian sequence of subdivisions and show that the third termtends to zero by the Cauchy-Schwarz inequality. 2

2.25 LABORATORY. Let us check sin(t) t2 =∫ t0 sin(x) dx2 +

∫ t0 x

2 d sin(x).

t=subdiv(0,1,500); f=sin(t); g=sqr(t); x=f.times(g); temp1=integral(f,g); y=temp1.plus(integral(g,f));plot(t,cbind(x,y.plus(0.01)))

2

2.26 REMARK. (Integration by substitution) Let f(s) be a continuous FV-Functionand φ(x) any continuously differentiable function. Then the substitution rule saysthat

φ(f(t))− φ(f(s)) =

∫ t

sφ′(f(u)) df(u)

Again the familiar version is obtained for differentiable functions putting df(u) =f ′(u) du.

For the proof use induction to show that the assertion is valid for φ(x) = xk,k = 2, 3, . . .. For the induction step use integration by parts. 2

2.4 Review material

Concepts: variation of a function, total variation, quadration variation, finite vari-ation property.

integral of step functions, integral transform, extension of the integral forFV-integrators.


Facts: magnitude of increments on small time intervals, total variation of smoothfunctions, relation between total variation and quadratic variation, variationof monotone functions, characterization of FV-functions, quadratic variationof the Wiener process, variations of the Poisson process.

financial motivation of integrals by trading, problem of continuous trading.

Integral of step functions: bilinearity, upper bounds, integral transform as astopped sum, increasing integrators, jump rule, stopping rule, associativityrule, variation of an integral transform.

Continuity of the integral with FV-integrators, how the extension works (Rie-mannian sums).

Evaluation of integrals for smooth integrators, for pure jump integrators,stochastic integral w.r.t the Poisson process.

2.4.1 Examples of exam questions:

1. Are the increments of a Wiener process on small time intervals bigger orsmaller than the increments of a linear function ?

2. How do the variations of a continuous function depend on the order of vari-ation ?

3. Are there functions whose variations are equal for every order ? (exampleand explanation)

4. Is it always true (or always wrong) that the variation of a function increases(decreases) with the order of variation ? (explain by counterexamples)

5. Which increasing functions are finite variation functions ?

6. Explain why V 1t (W ) =∞, where (Wt) is a Wiener process.

7. What is the total (quadratic) variation of CPP, of a jump diffusion ?

8. Describe the difference between left-continuous and right-continous stepfunctions. Which are used for integrands, which for integrators ?

9. Why does the extension of the integral work only for FV-integrators ?

10. Discuss the total variation of an integral transform ! (consider smooth inte-grators, pure jump integrators, try to grasp the case of FV-integrators)

2.4.2 Examples of exam problems:

1. Find: (a)∫ t0 s d(s2), (b)

∫ t0 s d(es

2).


2. Assume that the Poisson process (Nt) has three jumps in [0, t] at T1, T2 andT3. Calculate the integrals∫ t

0s dNs,

∫ t

0Ns− ds,

∫ t

0Ns− dNs

3. Calculate∫ t0 e

Ns− dNs, where (Nt) is a Poisson process.

2.4.3 Exam proofs:

1. Show that FV-functions are exactly the linear combinations of monotonefunctions.

2. Show that the Wiener process has finite quadratic variation.

3. State and explain the jump rule for integral transforms of step function.

4. State and explain the associativity rule for integral transforms of step func-tion.

5. State and explain the stopping rule for integral transforms of step function.

Chapter 3

Stochastic integrals

3.1 Introduction

In the preceding chapter we saw that for financial applications mathematical instru-ments like integrals are useful. The integrators used in such applications typicallyare the prices of financial instruments. It is therefore important to have a concept ofintegrals admitting as integrators a broad class of mathematical models for financialinstruments. Stochastic processes are presently the most important mathematicalparadigm for prices of financial instruments.

Shortly, what we need is a stochastic integral. So let’s see whether and how we canpaths of stochastic processes plug into the integrator slot of our integral.

Recalling the definitions of a g-integral, it is clear that there is no problem as longas integrands are step functions. Integrals of step function simply are sums offinitely many terms, requiring no further advanced mathematical operations (likelimits of complicated sequences of mathematical objects). Actually, such step-function integrands already cover plenty of applications. In real world any financialtrading strategy is based on finitely many trading periods only and is thereforea step function. Why is there any need at all of going beyond integrals of stepfunctions when we are interested in financial applications ?

Although practical financial trading takes place at finitely time periods only, thenumber of those periods is large. From the mere calculation point of view thiswould be a minor problem, since todays computing machinery is able to calculateresults of gigantic numbers of numerical operations within milliseconds. But cal-culating numerical results is not the end of the story. In the preceding chapter wealready explained that sums over subdivisions tend to exhibit (mathematical) pat-terns and properties which are independent of the particular subdivision used forcalculation. Such patterns cannot be discovered by machine calculations, but theyare the indispensable basis of intellectual perception. Integrals with general inte-

37

CHAPTER 3. STOCHASTIC INTEGRALS 38

grands realize this perception and thus are the appropriate intellectual abstraction,enabling us to study patterns and properties of sums with a large number of terms.

Returning from philosophical reasoning back to earth, let us discuss how to realizeour goal of a stochastic integral. How can we extend the trivial integral of stepfunctions into a reasonable integral admitting integrands useful for applications ?

In the preceding chapter we were able to define a reasonable integral, whenever theintegrator function has finite total variation. The resulting Stieltjes integral admitsany piecewise continuous integrands. How far can this Stieltjes integral be usedfor stochastic integrators ? The answer is straightforward: Whenever a stochasticprocess has paths with finite total variation, then it may be used as an integrator of aStieltjes integral. Applying this recipe to Levy processes it follows that every Pois-son or compound Poisson process may be used as integrator of Stieltjes integralswithout any troubles. However, difficulties arise with the Wiener process and withBrownian motions. We have seen that the paths of a Wiener process have positivequadratic variation, which implies that their total variation is infinite. Thereforethe mathematical approach to integration via Stieltjes integrals is no answer to theproblem of stochastic integration as soon as Wiener processes are involved as inte-grators.

A first solution to the problem of defining a more powerful stochastic integral hasbeen suggested by the Polish mathematician Norbert Wiener in the 1st half of the20th century. The so-called Wiener-integral admits deterministic functions as in-tegrands and paths of a Wiener process as integrators. It is important to note: Itremains true that for a given (non-trivial) integrand the integral cannot be definedfor an arbitrary integrator with infinite total variation. But by the Wiener integralconstruction one can be sure (with probability 1) that a Wiener process producesonly paths where the integral is well-defined (i.e. Riemannian sums converge to alimit with nice properties).

However, it took another thirty years until the Japanese mathematician Kyoshi Itowas able to construct a stochastic integral where also the integrands may be pathsof a stochastic process. Before that was possible the concept of martingales hadto be developed, which is due to Joseph L. Doob. Combining the ideas of NorbertWiener with those of Doob, Ito showed that the Wiener integral can be extendedto very special stochastic integrands. In short, the integrands must be such that theRiemannian sums, used for defining the integral, are martingales. In other words:Only if the integrands are of a nature which turns the integrator martingale (theWiener process) into a Riemannian sum which is a martingale again, the Rieman-nian sums converge to a limit with nice properties, called the Ito-integral.

The immense power of the Ito-integral, both for mathematical theory and for appli-cations, comes from the astonishing fact (confirming the strong coherence betweenbeauty and applicability in mathematics) that the above mentioned speciality of in-tegrands, necessary for Ito-integrals, is a very natural one: The integrands must be


stochastic processes which cannot anticipate the future, i.e. they must be adaptedto the past of the integrating Wiener process.

Let us conclude this introduction by a remark on the significance of martingales.Usually martingales are introduced as models for non-informative time series. Thedefining property - being motivated by the idea of lacking any inherent dyamicalstructure - is located somewhere between having uncorrelated increments and hav-ing independent increments. At first sight there is no obvious reason for superiorityof the martingale property over intuitively smoother running alternatives. The realpower of the martingale concept comes from another bulk of properties, namelyfrom the law of large numbers. The martingale concept is the final answer to theproblem of stating the probabilistic law of large numbers. Probabilistic assertionsbeing called law of large numbers are concerned with the idea that in the increas-ing light of historic information deterministic patterns emerge from the shadow ofstochastic noise. Doob was the first one to show that it is the martingale propertyof (cumulated) noise which is responsible for this phenomenon. The constructionof the Ito-integral makes heavy use of this fundamental fact. Thus the martingaleconcept was the starting point for an almost completely new mathematical theoryof probability in the 2nd half of the 20th century.

3.2 The Wiener integral

The paths of the Wiener process are not of finite variation. Therefore the Stieltjesintegral does not allow using Wiener paths as integrators. However, it is possibleto extend the integral of step functions in such a way that Wiener paths can be usedas integrators. If the integrand is a deterministic function then the integral is calleda Wiener integral.

Let (Ft) be a filtration and let (Wt) be a Wiener process w.r.t. this filtration. More-over, let

f(s) =n∑i=1

ai1[ti−1,ti)(s)

be a cadlag step function. For such step functions we may define the integral in anelementary way as

Xt =

∫ t

0f(s−) dWs =

n∑i=1

ai(Wti∩t −Wti−1∩t). (1)

This is well-defined and shares all properties of the integral of step functions (seesection 2.2). It is called the Wiener integral of a step function.

New features are due to the fact that the integral depends on a random path of aWiener process and is therefore a random variable.

3.1 LEMMA. The integral Xt =∫ t0 f− dW defined by (1) is a random vari-


able with a normal distribution, expectation E(Xt) = 0 and variance V (Xt) =∫ t0 f

2(s−) ds.

Proof: The proof is straightforward and left as an exercise. 2

If we consider the integral (1) as a function of the upper limit t, then (Xt) is astochastic process. This Wiener integral process has interesting properties. Let usdenote

At := V (Xt) =

∫ t

0f2(s) ds =

n∑i=1

a2i (ti ∩ t− ti−1 ∩ t).

3.2 THEOREM. The process (Xt) defined by (1) has independent increments andis therefore a (square integrable) martingale. Moreover, the process X2

t − At is amartingale.

Proof: The first part of the assertion is easy to see and left as an exercise.

For the second part of the assertion we have to show that E(X2t − X2

s |Fs) =V (Xt)− V (Xs). By the martingale property of (Xt) this is equivalent to

E((Xt −Xs)2|Fs) = V (Xt)− V (Xs).

Since the increments of the Wiener integral are independent of the past we have

E((Xt −Xs)2|Fs) = E((Xt −Xs)

2) =

∫ t

sf2(u−) du = V (Xt)− V (Xs).

2

As a consequence we get the important isometric equality

E((∫ t

0fs− dWs

)2)=

∫ t

0f2s ds (2)

This equality implies that small integrands f lead to integrals with small variance.We can state this fact as a kind of „continuity condition”

fn(s−)→ 0 ⇒∫ t

0fn(s−) dWs

P→ 0

in terms of second moments. A similar continuity condition was the basis of theStieltjes extension of the g-integral if g is an FV-function. Now we have estab-lished the continuity condition for integrals of adapted step processes w.r.t. pathsof the Wiener process. This is the mathematical reason why the following exten-sion works.

Let f(s) be a cadlag function such that∫ t

0f2(s) ds <∞.


It is not difficult to see that for any Riemannian sequence of subdivisions of [0, t]the corresponding step functions fn approximate the function f , i.e. we have

fn(s−) :=

n∑i=1

ai1(tn,i−1,tni](s)→ f(s−).

Since the functions fn(s) are cadlag step-functions the integrals∫ t0 fn(s−) dWs

are well-defined. As a consequence of the continuity condition one can show byadvanced mathematical arguments that those integrals converge to a limit whichdoes not depend on the particular Riemannian sequence of subdivisions chosen.Therefore it is logically consistent to define

Xt =

∫ t

0f(s−) dWs := lim

n→∞

∫ t

0fn(s−) dWs. (3)

This is actually the mathematical definition of the Wiener integral. All importantproperties of the Wiener integral for step functions remain valid for the generalWiener integral. Let us summarize these properties.

3.3 THEOREM. The Wiener integral Xt defined in (3) is well-defined for anysquare integrable cadlag function f , is linear in the integrand and the integrator,and satisfies the jump rule, the associativity rule and the stopping rule.

The process (Xt) is a Gauss process (all joint distributions are normal distributions)with E(Xt) = 0 and At = V (Xt) =

∫ t0 f(s) ds. Moreover, the process (Xt) is

a martingale with continuous paths, independent increments, and (X2t − At) is a

martingale, too.

It can be shown that the stopping rule is valid even for random stopping times(adapted to the underlying filtration).

3.3 The Ito integral

The Ito integral extends the construction of the preceding section to integrandswhich may be random, but have to be adapted. The integrator is still the Wienerprocess.

Let (Ft) be a filtration and let (Wt) be a Wiener process w.r.t. this filtration. More-over, let

Hs =n∑i=1

Zi1[ti−1,ti)(s)

be a cadlag step process (a process whose paths are right-continuous step func-tions). This step process is adapted (does not anticipate the future) if Hs ∈ Fs


which implies Zi = Hti−1 ∈ Fti−1 , i = 0, 1, . . . , n. For step processes as integ-rands the integral

Xt =

∫ t

0Hs−dWs =

n∑i=1

Zi(Wti∩t −Wti−1∩t)

is well-defined and shares all properties of the integral of step functions (see section2.2). Note, that the integral is a random variable.

A major role is played by the process

At =

∫ t

0H2s ds =

n∑i=1

Z2i (ti ∩ t− ti−1 ∩ t).

This process corresponds to the variance of the Wiener integral, but now is astochastic process. It is a continuous increasing process (process with increas-ing paths) and therefore an FV-process (a process with paths of finite variation). Itis integrable iff each Zi is square integrable.

The following theorem is the key for the extension of the integral. It generalizesthe fact that W 2

t − t is a martingale.

3.4 THEOREM. Let (Ht) be an adapted step process. If (At) is integrable then(Xt) is a square integrable martingale and (X2

t −At) is a martingale, too.

Proof: Let s < t and assume that the subdivision is chosen such that s = tk andt = tn. Note that

E(Z2i (Wti −Wti−1)2|Fti−1) = Z2

i E((Wti −Wti−1)2) = Z2i (ti − ti−1) (4)

which implies

E(Z2i (Wti −Wti−1)2) = E(Z2

i )(ti − ti−1).

Therefore (Xt) is square integrable. From

E(Zi(Wti −Wti−1)|Fti−1) = ZiE(Wti −Wti−1) = 0

it follows thatE(Xt −Xs|Fs) = 0.

Hence, (Xt) is a square integrable martingale.

For the second part of the assertion we have to show that E(X2t − X2

s |Fs) =E(At −As|Fs). By the martingale property of (Xt) this is equivalent to

E((Xt −Xs)2|Fs) = E(At −As|Fs).


Note that

(Xt −Xs)2 =

( n∑i=k+1

Zi(Wti −Wti−1))2

=

n∑i=k+1

Z2i (Wti −Wti−1)2 + 2

∑k+1≤i<j

Zi−1(Wti −Wti−1)Zj−1(Wtj −Wtj−1)

For any pair i < j we have

E(Zi(Wti −Wti−1)Zj(Wtj −Wtj−1)

∣∣∣Ftj−1

)= Zi(Wti −Wti−1)ZjE(Wtj −Wtj−1 |Ftj−1) = 0

which together with (4) implies

E((Xt −Xs)2|Fs) = E

( n∑i=k+1

Z2i (ti − ti−1)|Fs

)= E(At −As|Fs).

2

As a consequence we get the important isometric equality

E((∫ t

0Hs− dWs

)2)= E(At) = E

(∫ t

0H2s ds

)(5)

It should be noted that this equality depends on the martingale property of theintegral and thus on the adaptedness of the integrands. The isometric equalityimplies that small integrands Hs lead to integrals with small variance. Again wecan state this fact as a kind of „continuity condition”

Hns → 0 ⇒∫ t

0Hns− dWs

P→ 0

in terms of second moments. This is the reason why the following extension works.

Let (Hs) be an adapted cadlag process such that

E(∫ t

0H2s ds

)<∞.

It is easy to see that for any Riemannian sequence of subdivisions of [0, t] we have

Hns− :=

n∑i=1

Htni−1(tn,i−1,tni](s)→ Hs−.

Since (Hns) is a sequence of adapted caglad step-processes the integrals∫ t0 Hns− dWs

are well-defined. As a consequence of the continuity condition (and some advanced


mathematical arguments) those integrals converge to a limit which does not dependon the particular Riemannian sequence of subdivisions chosen. Therefore we de-fine

Xt =

∫ t

0Hs− dWs := lim

n→∞

∫ t

0Hns− dWs. (6)

This is actually the definition of the Ito-integral.

3.5 THEOREM. Assume that At is integrable. Then the Ito-integral Xt defined in(6) is well-defined for any adapted cadlag process (Hs), is linear in the integrandand the integrator, and satisfies the jump rule, the associativity rule and the stoppingrule.

Moreover, the process (Xt) is a square integrable martingale with continuous paths,and (X2

t −At) is a martingale, too.

It can be shown that the stopping rule is valid even for random stopping times(adapted to the underlying filtration).

3.4 The stochastic integral

The extension process which has been applied in the preceding sections to theWiener process as an integrator, can be extended to any square integrable martin-gale (Mt). The essential step is the existence of an adapted continuous increasingprocess (At) such that (M2

t −At) is a martingale.

The preceding section is concerned with Mt = Wt then At = t. But there aremany other examples of square integrable martingales which are candidates forintegrators of stochastic integrals. If Mt =

∫ t0 Hs− dWs is square integrable then

then At =∫ t0 H

2s ds. If Mt is any square integrable Levy martingale then At =

σ2t.

General results of martingale theory (the so-called Doob-Meyer decomposition)show that for any square integrable martingale (Mt) there is an adapted continuousincreasing process (At) such that (M2

t − At) is a martingale. To be honest, theadapted process (At) is in general not necessarily continuous, but has a slightlyweaker property called predictability. However, in cases typical for applicationsthe process is even continuous.

Let (Ft) be a filtration and for a cadlag step process

Hs =

n∑i=1

Zi1[ti−1,ti)(s)


define the integral by

Xt =

∫ t

0Hs− dMs =

n∑i=1

Zi(Mti∩t −Mti−1∩t).

The process

Bt =

∫ t

0H2s dAs =

n∑i=1

Z2i (Ati∩t −Ati−1∩t)

is a continuous increasing process. The assertion of Lemma 3.4 is valid also inthis case and leads to an isometric equality

E((∫ t

0H− dM

)2)= E(Bt) = E

(∫ t

0H2s dAs

)Based on this equality the extension is possible defining as usual

Xt =

∫ t

0H− dM := lim

n→∞

∫ t

0Hn− dM. (7)

The resulting integral is called the stochastic integral.

3.6 THEOREM. Let (Mt) be a square integrable martingale and (At) a continuousincreasing process such that M2

t −At is a martingale. Then the stochastic integraldefined in (7) is well-defined, linear in the integrand and the integrator, and satisfiesthe jump rule, the associativity rule and the stopping rule.

Moreover, the process (Xt) is a square integrable martingale, and (X2t − Bt) is a

martingale, too.

3.5 Local martingales

This section contains a topic which is generally considered as too advanced for anapplied audience. Let’s give it a try.

The construction of the Ito-integral relies on an integrability condition for the inte-grands, namely

E(∫ t

0H2s ds

)<∞.

This condition means that the Riemannian integrals∫ t0 H

2s ds should have a finite

statistical average (finite expectation). Note, that for every realized path of (Hs) theRiemannian integral

∫ t0 H

2s ds is always finite since every single path (Hs)0≤s≤t

is bounded (with probability 1). It is therefore strange that the definition of thestochastic integral

∫ t0 Hs− dWs for particular paths of (Hs) and (Ws) should de-

pend on a condition where also other paths are involved which are not realized.


Feeling uncomfortable with this has a sound basis. It is actually very easy to definethe Ito-integral for any adapted integrand (Hs), no matter whether the integrabilitycondition is satisfied or not.

Let τa := infs : |Hs−| > a be the random time (stopping time) where (Hs)exceeds a given bound a <∞. Consider the process

1[0,τa](s) :=

1 if s ≤ τa,0 if s > τa.

Note, that the process (1[0,τa])(s) is adapted and left-continuous. The process(Hs−1[0,τa](s)) is a truncated version of (Hs−) and satisfies the integrability con-dition since

E(∫ t

0(Hs−)21[0,τa](s) ds

)≤ a2t

Therefore the Ito-integral∫ t

0Hs−1[0,τa](s) dWs =

∫ t∩τa

0Hs− dWs

is well-defined. For paths of (Hs) not exceeding the bound a this procedure shouldbe a suitable definition of the integral

∫ t0 Hs− dWs.

3.7 DEFINITION. Let τa := infs : |Hs| > a and Ma = (Hs) : maxs |Hs| ≤a. Then∫ t

0Hs− dWs :=

∫ t

0Hs−1[0,τa](s) dWs whenever (Hs) ∈Ma.

From the stopping rule it follows that this definition is not ambiguous. SinceP (⋃aMa) = 1 this is a complete definition of the integral.

3.8 THEOREM. For every adapted process (Hs) the Ito-integral defined by Def-inition 3.7 is well-defined, linear in the integrand, and satisfies the jump rule, theassociativity rule and the stopping rule.

However, the martingale properties stated in Lemma 3.4 are violated ifE(∫ t0 H

2sds) =

∞. In this case the Ito-integral is not a martingale. But it is „almost” a martingale.

3.9 DEFINITION. An adapted cadlag process (Xt) is a local (square integrable)martingale if there is a sequence of stopping times (τn) such that τn ↑ ∞ and thestopped processes (Xτn

t ) are (square integrable) martingales for every n.

Now we may put terms together.

3.10 THEOREM. The Ito-integral∫ t0 Hs−dWs is defined for every adapted cadlag

process (Hs) and is a local (square integrable) martingale. It is a square integrable


martingale iff E(∫ t0 H

2sds) <∞. In this case the assertion of Lemma 3.4 is valid

and the isometric equality holds.

The stochastic integral for square integrable martingales explained in section 3.4can also be extended to general integrands using the concept of local martingales.Even the integrators themselves need not be square integrable martingales but onlylocal martingales.

3.11 THEOREM. Assume that (Mt) is a local (square integrable) martingaleand let (At) be a continuous increasing process such that (M2

t − At) is a localmartingale.

The stochastic integral∫ t0 Hs− dMs is well-defined for every adapted cadlag pro-

cess (Hs) and is a local (square integrable) martingale.

The stochastic integral∫ t0 Hs− dMs is a square integrable martingale iffE(

∫ t0 H

2s− dAs) <

∞. In this case the assertion of Lemma 3.4 is valid and the isometric equality (7)holds.

3.6 Semimartingales

Now we have at our disposal a concept of an integral where we may use as integra-tors both cadlag FV-processes and local martingales. Integrands have to be cadlagand adapted.

It is convenient to have a name for the class of processes which may serve asintegrators of stochastic integrals.

3.12 DEFINITION. A semimartingale is an adapted cadlag process which is thesum of an adapted FV-process and a local martingale.

If Xt is a semimartingale with a decomposition Xt = Mt + Vt where (Mt) is alocal martingale and (Vt) is an adapted FV process, then the stochastic integral isdefined by ∫ t

0Hs−dXs :=

∫ t

0Hs−dMs +

∫ t

0Hs−dVs.

It can be shown that this definition does not depend on the particular decompositionXt = Mt + Vt.

The class of semimartingales is a large and useful class of stochastic processes.

3.13 EXAMPLE. (Semimartingales) (1) Every right continuous deterministic func-tion of finite variation is a semimartingale.

(2) A process of the form

Xt = x0 +

∫ t

0as ds+

∫ t

0bs− dWs, shorthand: dXt = at dt+ bt− dWt


is called an Ito-process. Every Ito-process is a semimartingale.

(3) Every Levy process is a semimartingale. (With our means we can prove: everysquare integrable Levy process is a semimartingale).

(4) The stochastic integral with respect to a semimartingale is a semimartingale. 2

It is a useful and important fact that for any semimartingale the stochastic integralcan be obtained as a limit of Riemannian sums. The proof is based on the steps ofthe extension process.

3.14 THEOREM. Let (Xt) be a semimartingale and (Ht) an adapted cadlag pro-cess. Then for every Riemannian sequence of subdivisions∫ t

0Hs− dXs = lim

n→∞

n∑i=1

Htni−(Xtni −Xtn,i−1) in P -probability.

Are there semimartingales which are both FV-processes and local martingales ?Clearly, there are: Take Nt − λt. But this is possible only with jumps.

3.15 THEOREM. A continuous local martingale can be FV only if it is constant.

Proof: We give a proof for square integrable martingales. The extension to localmartingales follows the usual pattern.

Let (Mt) be a square integrable martingale which is continuous and FV. W.l.g. letM0 = 0. Then the integration by parts formula of Remark 2.24 gives

M2t = 2

∫ t

0Ms dMs, shorthand: dM2

t = 2Mt dMt.

If (Mt) is bounded then E(M2t ) = 0. Otherwise, use localization. 2

3.7 Review material

Concepts: extension problem, Ito integral, local martingale, stochastic integralw.r.t. a (square integrable) martingale, Wiener integral, semimartingales (asadmissible integrators).

Facts: extension process (Riemannian sums), martingale properties of the Ito in-tegral.

Wiener integral: distribution, independent increments property, martingaleproperties, covariance structure.


3.7.1 Examples of exam proofs:

1. Prove Theorem 3.4 .

2. Show that Wiener integrals have a normal distribution. (Hint: Prove it forintegrands, which are step functions.)

3. Show that the joint distribution of a vector (∫fk dW )k=1,...,n of Wiener in-

tegrals with the same driving Wiener process (Ws) is a multivariate normaldistribution. (Hint: It is sufficient to show that every linear combination ofthe components has a normal distribution.)

4. Find the covariance of two Wiener integrals∫ s0 f dW and

∫ t0 f dW . (Hint:

Apply the martingale property.)

5. Show that Wiener integrals have independent increments. (Hint: Show thatthe increments are uncorrelated and apply your knowledge about the jointdistribution.)

6. Find the covariance of two Wiener integrals∫ t0 f dW and

∫ t0 g dW . (Hint:

Express the variance of the sum in two different ways.)

7. Find the covariance of two Ito integrals∫ s0 H dW and

∫ t0 H dW . (Hint:

Apply the martingale property.)

8. Show that Ito integrals have uncorrelated increments.

9. Find the covariance of two Ito integrals∫ t0 H dW and

∫ t0 GdW . (Hint: Ex-

press the variance of the sum in two different ways.)

10. Show that a square integrable martingale which is continuous and FV mustbe constant.

11. Show that the representation of an Ito process is uniquely determined.

12. Let (Mt) be a square integrable martingale. Show that there is at most onecontinuous FV-process such that M2

t −At is a square integrable martingale.

3.7.2 Examples of exam questions:

1. Explain the steps for defining the Ito integral.

2. When is an Ito integral a (square integrable) martingale ?

3. Compare the martingale properties of the Ito integral and the integral w.r.t. aPoisson process.

4. When is the Wiener integral a Levy process ?

5. What can we say about the increments of Ito-integrals ?


6. What can we say about the paths of Wiener-integrals ?

7. What can we say about the paths of Ito-integrals ?

8. Explain why a square integrable Levy process is a semimartingale.

9. Give examples of continuous martingales and semimartingales which aredifferent from the Wiener process.

3.7.3 Examples of exam problems:

1. Find expectation and variance and check the martingale property.(a)∫ t0 e

2Ws dWs, (b)∫ t0 2Ws dWs, (c)

∫ t0 W

2s dWs,

(d)∫ t0 e

2s dWs, (e)∫ t0 2s dWs, (f)

∫ t0 s

2 dWs,

2. Find the covariance of(a) Wt and

∫ t0 s dWs, (b) Wt and

∫ t0 Ws dWs,

(c)∫ t0 s dWs and

∫ t0 (t− s) dWs, (d)

∫ t0 e

2s−Ws dWs and∫ t0 e

Ws−1 dWs.

3. Let Xt =∫ t0 s dWs. Find expectation and variance and check the martingale

property of(a)∫ t0 Xs dWs, (b)

∫ t0 Ws dXs.

Chapter 4

Quadratic variation and thetransformation formula

4.1 Quadratic variation

Let (Wt) be a Wiener process. Then we know that W 2t − t is a martingale. Now

we can make a more precise statement.

4.1 LEMMA. Let (Wt) be a Wiener process. Then

W 2t = 2

∫ t

0Ws dWs + t, shorthand: dW 2

t = 2Wt dWt + dt

Proof: Apply

W 2t −W 2

s = 2Ws(Wt −Ws) + (Wt −Ws)2

to a Riemannian sequence of subdivisions and apply Theorem 3.14 . 2

This result is remarkable. From the elementary integration by parts formula ofTheorem 2.24 it follows that for continuous FV-processes (Xt) the formula

X2t −X2

0 = 2

∫ t

0Xs dXs, shorthand: dX2

t = 2Xt dXt

holds. Therefore the preceding lemma indicates that the integration by parts for-mula has to be modified for stochastic integrals.

4.2 LABORATORY. Let f(t) = sin(t). Then we know from integration by partsthat f2(t) = 2

∫ t0 f(s) df(s).

t=subdiv(0,1,500); x=sin(t); y=integral(x,x).times(2); plot(t,cbind(sqr(x),y.plus(0.01)))

51

CHAPTER 4. QUADRATIC VARIATION AND THE TRANSFORMATION FORMULA52

Now we replace f(t) by some path Wt of the Wiener process. Then the usualintegration by parts formula is no longer true:

W 2t 6= 2

∫ t

0Ws dWs.

x=rWiener(1); y=integral(x,x).times(2); plot(x.date,cbind(sqr(x),y))

Instead we have

W 2t = 2

∫ t

0Ws dWs + t.

x=rWiener(1); y=integral(x,x).times(2); plot(x.date,cbind(sqr(x),y.plus(x.date).plus(0.01)))

2

As the following lemma shows this is even necessary for non-continuous FV-processes.

4.3 LEMMA. Let (Nt) be a Poisson process. Then

N2t = 2

∫ t

0Ns− dNs +Nt, shorthand: dN2

t = 2Nt− dNt + dNt

Proof: This formula is well-known to us. 2

4.4 LABORATORY. Consider some path Nt of the Poisson process. We have

N2t = 2

∫ t

0Ns− dNs +Nt, shorthand: dN2

t = 2Nt− dNt + dNt

x=rPoisson(1,lambda:20); y=integral(x,x).times(2); plot(x.date,cbind(sqr(x),y.plus(x).plus(5)))

2

Let (Xt) and (Yt) semimartingales. We introduce a modification term [X,Y ]t inthe integration by parts formula:

XtYt −X0Y0 =

∫ t

0Xs− dYs +

∫ t

0Ys− dXs + [X,Y ]t, (1)

shorthand: d(XtYt) = Xt− dYt + Yt− dXt + d[X,Y ]t


This is not a theorem but only a notation ! At this point it is clear that [X,Y ] islinear in both components. Now comes the theorem:

4.5 THEOREM. The expression [X,Y ]t is an FV-process satisfying

limn→∞

n∑i=1

(Xtni −Xtn,i−1)(Ytni − Ytn,i−1)P→ [X,Y ]t

for every Riemannian sequence of subdivisions of [0, t].

Proof: For the convergence assertion use approximation by Riemannian sums andapply Theorem 3.14 .

Let us show that [X,Y ]t is an FV-process. Since [X,Y ]t is linear in both com-ponents the quadratic covariation can be recovered from the quadratic variationby

[X,Y ] =[X + Y ]− [X]− [Y ]

2

The quadratic variation is clearly an increasing process. Hence [X,Y ]t is a linearcombination of increasing processes and therefore an FV-process. 2

In the light of the preceding theorem [X,Y ]t is called the quadratic covariationof the semimartingales (Xt) and (Yt).

The special case [X]t := [X,X]t is obviously the quadratic variation of (Xt). It isuseful to consider the special case of (1) concerning squares

X2t −X2

0 = 2

∫ t

0Xs− dXs + [X]t, shorthand: dX2

t = 2Xt− dXt + [X]t

which explains the assertions of Lemmas 4.1 and 4.3 .

4.6 LEMMA. Let (Xt) and (Yt) be semimartingales and assume that either (Xt)or (Yt) is both continuous and FV. Then [X]t = 0 and [X,Y ]t = 0.

Proof: Use the Cauchy-Schwarz inequality. 2

The preceding lemma gives the special case considered in Theorem 2.24 . It is,however, only a preliminary assertion. Later (Corollary 4.11 ) we will see that forthe same conclusion it is sufficient that continuity and FV need not be fulfilled bythe same process.

It is more or less obvious that the quadratic variation satisfies [Xa]t = [X]at (stop-ping stops variation). This property is helpful in the proof of the next theorem.

4.7 THEOREM. Let (Xt) and (Yt) be semimartingales and (Ht) and (Gt) adaptedcadlag processes. Then[ ∫

H− dX,

∫G− dY

]t

=

∫ t

0Hs−Gs− d[X,Y ]s.


Proof: It is sufficient to prove the assertion for Gs = 1 and Hs− = Ba1(a,b](s).For this we have using the stopping rule[ ∫

Ba1(a,b] dX, Y]t

= [Ba(Xb −Xa), Y ]t

= Ba[Xb −Xa, Y ]t = Ba([X

b, Y ]t − [Xa, Y ]t)

= Ba([X,Y ]bt − [X,Y ]at ) =

∫ t

0Ba1(a,b] d[X,Y ]

2

Now we have got sufficient information to consider interesting examples.

4.8 EXAMPLE. (Applications of integration by parts) (i) The quadratic variationof Ito-integrals is

[ ∫H dW

]t

=∫ t0 H

2s ds.

(ii) The integration by parts formula gives dW 3t = 3W 2

t dWt + 3Wtdt.

(iii) The preceding example can be extended to all powers and all continuous semi-martingales:

dXkt = kXk−1

t dXt +k(k − 1)

2Xk−2t d[X]t

(iv) Let (W(1)t ) and (W

(2)t ) be independent Wiener processes. Then their quadratic

covariation is zero and their product is a martingale. (Hint: Consider the quadraticvariation of the sum of the processes.) 2

4.2 Special properties of the quadratic variation

The quadratic variation shows the jumps of the process.

4.9 LEMMA. For any semimartingale (Xt) we have ∆[X]t = (∆Xt)2.

Proof: Integration by parts gives X2t = X2

0 + 2∫ t0 Xs− dXs + [X]t. Passing to

the jumps we obtain (applying the jump rule)

X2t −X2

t− = 2Xt−∆Xt + ∆[X]t

This is the assertion. 2

Hence, the quadratic variation [X]t is continuous iff the semimartingale (Xt) iscontinuous. Moreover, it follows that for every semimartingale the sum of squaredjumps is finite.

4.10 THEOREM. Let (Xt) and (Yt) be semimartingales and let any of them be an


FV semimartingale. Then

[X,Y ]t =∑s≤t

∆Xs∆Ys

Proof: Any FV semimartingale can be written as the sum of a continuous FVsemimartingale and a pure jump process. The rest is straightforward. 2

4.11 COROLLARY. If any of the semimartingales is continuous and any is FV then[X,Y ]t = 0.

4.12 EXAMPLE. (Degenerate covariation) Let (Wt) be a Wiener process and (Nt)a Poisson process. Then

WtNt =

∫ t

0Ws dNs +

∫ t

0Ns− dWs

2

4.13 EXAMPLE. (Linear stochastic differential equations) Consider the equation

Xt = x0 +

∫ t

0αXs ds+Wt, shorthand: dXt = αXt dt+ dWt.

This is a linear stochastic differential equation with a deterministic homogeneouspart. It can be solved by multiplying by the integrating factor e−αt and applyingintegration by parts.

See the problem section for more examples. 2

4.3 The transformation theorem (continuous case)

The following assertion is the the most powerfull tool of stochastic integral calcu-lus. In this lecture we present only the version for continuous semimartingales. Ifthe underlying process is a Wiener process, then it is called Ito’s formula.

4.14 THEOREM. Let (Xt) be a continuous semimartingale and let φ(x) be a twicecontinuously differentiable function. Then

φ(Xt) = φ(X0) +

∫ t

0φ′(Xs) dXs +

1

2

∫ t

0φ′′(Xs) d[X]s

or in shorthanddφ(Xt) = φ′(Xt) dXt +

1

2φ′′(Xt) d[X]t

Proof: There is a short and elegant proof: Use integration by parts and inductionfor showing that the assertion is true for each power function and therefore for each


polynomial. The rest is (almost) straightforward. A subtle point: The argument isimmediate if the range of (Xt) is contained in a compact subset of the domain ofφ. Otherwise, one has to argue a bit more careful (e.g. in order to cover functionslike φ(x) = 1/x).

For understanding the structure of the formula its connection with Taylor’s formulais helpful. For this we have to know that

n∑i=1

Htn,i−1(Xtni −Xtn,i−1)2P→∫ t

0Hs d[X]s

This is a subtle fact following from

[X]t − [X]s = (Xt −Xs)2 + 2Xs(Xt −Xs)− 2

∫ t

sX dX

2

For examples see the problems section.

From the transformation theorem it follows that every smooth function of a contin-uous semimartingale is a semimartingale. Actually every continuous function of asemimartingale is a semimartingale.

There are multidimensional versions of the transformation theorem. They can eas-ily recovered and memorized by its similarity to Taylor’s formula.

4.15 THEOREM. Let (Xt) and Yt) be continuous semimartingales and let φ(x, y)be a twice continuously differentiable function. Then

φ(Xt, Yt) = φ(X0, Y0)

+

∫ t

0φx(Xs, Ys) dXs +

∫ t

0φy(Xs, Ys) dYs

+1

2

∫ t

0φxx(Xs, Ys) d[X]s +

∫ t

0φxy(Xs, Ys) d[X,Y ]s +

1

2

∫ t

0φyy(Xs, Ys) d[Y ]s

or in differential shorthand

dφ(Xt, Yt) = φx(Xt, Yt) dXt + φy(Xt, Yt) dYt

+1

2φxx(Xt, Yt) d[X]t + φxy(Xt, Yt) d[X,Y ]t +

1

2φyy(Xt, Yt) d[Y ]t

For later applications we need a special case of the bivariate transformation theo-rem.

4.16 COROLLARY. Let (Xt) be a continuous semimartingale and let φ(t, x) be a


twice continuously differentiable function. Then

φ(t,Xt) = φ(0, X0)

+

∫ t

0φt(s,Xs) ds+

∫ t

0φx(s,Xs) dXs

+1

2

∫ t

0φxx(s,Xs) d[X]s

or in differential shorthand

dφ(Xt, Yt) = φt(t,Xt) dt+ φx(t,Xt) dXt +1

2φxx(t,Xt) d[X]t

4.4 Stochastic exponentials and Levy’s theorem

4.17 DEFINITION. Let (Xt) be a continuous semimartingale. Then

E(X)t := exp(Xt − [X]t/2)

is called the stochastic exponential of (Xt).

4.18 THEOREM. The stochastic exponential St = S0E(X)t is the solution of thelinear stochastic differential equation

St = S0 +

∫ t

0Su dXu, shorthand: dSt = St dXt

Proof: Let Yt = Xt − [X]t/2 and apply the transformation theorem to St =S0 exp(Yt). 2

It follows that the stochastic exponential of a (local) martingale is a local martin-gale.

The stochastic exponential is a positive process. Also the converse is true.

4.19 THEOREM. Every positive continuous semimartingale is the stochastic ex-ponential of another semimartingale.

Proof: Let (St) be a positive continuous semimartingale. Define dLt = 1/StdStand show that dSt = StdLt. Is it trivial or do we need some non-trivial rule ? 2

The process (Lt) is called the stochastic logarithm of (Xt).


The power of these seemingly formal calculus is illustrated by the following fa-mous theorem.

4.20 THEOREM. The only continuous local martingale (Mt) such that [M ]t = tis the Wiener process.

Proof: From the stochastic differential equation of the stochastic exponential itfollows that the process Zt = exp(iaMt+a

2t/2) is a square integrable martingale.The martingale property of (Zt) leads to the evaluation of the Fourier transform of(Mt). 2

4.5 Review material

Concepts: integration by parts for semimartingales, quadratic variation and co-variation, linear stochastic differential equation, differential notation.

Transformation theorem for continuous semimartingales, stochastic expo-nential of continuous semimartingales, stochastic logarithm.

Two-dimensional Ito-formula.

Facts: explanation of the integration by parts formula, properties of quadratic vari-ation and covariation (bilinear, homogeneous, FV-property), quadratic co-variation of integral transforms.

Powers of semimartingales (of Wiener process), covariation of independentWiener processes, covariation of jump processes, covariation of FV-processes.

Solution of the Vasicek differential equation, properties of the solution (mean,variance, covariance).

Proofs of the transformation formula, solution of a homogeneous stochasticdifferential equation, uniqueness of Ito representation..

4.5.1 Examples of exam proofs

1. Prove [X,Y ] = [X+Y ]−[X]−[Y ]2 .

2. Prove ∆[X]t = (∆Xt)2

3. Prove ∆[X,Y ]t = ∆Xt ∆Yt


1. Explain the idea of differential notation. (How to pass from one notation tothe other ?)


2. When does the quadratic variation or the quadratic covariation vanish ?

3. Explain why each polynomial of semimartingales is a semimartingale.


1. What is the quadratic variation of a Wiener integral ?

2. What is the quadratic covariation of two Wiener integrals(a) for the same integrator, (b) for independent integrators ?

3. Expand by integration by parts:(a) dW 2

t , (b) dW 3t , (c) dN2

t , (d) d(µt+ σWt)2.

4. Find the quadratic variation of(a) W 2

t , (b) W 3t , (c) N2

t , (d) (µt+ σWt)2.

5. Write as Ito-process and identify the components:(a)∫ t0 W dW 2, (b) W 2

t , (c) W 3t , (d) tWt, (e) etW 2

t .

6. Find expectation and variance and check the martingale property.(a)∫ t0 Ws dW

2s , (b)

∫ t0 W

2s dW

2s .

7. Find the covariance of W 2t and

∫ t0 Ws dWs.

8. Let Xt =∫ t0 Ws ds.

(a) Show that Xt =∫ t0 (t− s) dWs.

(b) Show that (Xt) is a Gaussian process (variables are normally distributed).(c) Show that cov(Xs, Xt) = s3

3 + (t− s) s22 , s ≤ t.

9. Find the expectations of(a) [W,

∫W dW ], (b) [

∫s dW,

∫W dW ], (c) [

∫s dW,

∫W ds],

10. Let (Xt) be any semimartingale. Prove by induction on k:

dXk = kXk−1dX +k(k − 1)

2Xk−2d[X].

11. Solve the stochastic differential equation dSt = rStdt+ σdWt.

12. Solve the stochastic differential equation dSt = (a− St)dt+ σdWt.

13. Assume that (St) satisfies dSt = µtStdt + σtdWt. Find the stochastic dif-ferential equation of St := Ste

−rt.

14. Solve the Vasicek equation (Ornstein-Uhlenbeck process):

dXt = (ν − µXt) dt+ σdWt


(a) Calculate expectations and variances as well as their limits for t→∞.(b) Calculate covariances as well as their limits for t→∞.(c) Show that the solution is stationary if the initial value is a Gaussian ran-dom variable with asymptotic values for expectation and variance.

15. Let (Wt) be a Wiener process. Represent as an Ito-process:(a) W 2

t , (b) W 3t , (c) W k

t , (d) (µt+ σWt)k,

(e) eσWt , (f) e(σWt)2 , , (g) eµt+σWt

16. Let Xt =∫ t0 W dW , where (Wt) is a Wiener process. Represent as an Ito-

process:(a) X2

t , (b) X3t , (c) Xk

t , (d) (µt+ σXt)k,

(e) eσXt , (f) e(σXt)2 , , (g) eµt+σXt .Hint: First of all simplify (Xt) !

17. Solve the stochastic differential equations(a) dSt = µStdt, (b) dSt = σStdWt, (c) dSt = µStdt+ σStdWt

Part II

Elements of mathematical finance

61

Chapter 5

Financial markets and tradingstrategies

5.1 Market models

Let us start with a very general definition of market models.

5.1 DEFINITION. A (finite) financial market model is a finite family (S0t , S

1t , . . . , S

kt )

of semimartingales on a filtered probability space (Ω, (Ft), P ). The componentsof the family are called basic assets.

In the present text we consider mainly market models consisting of two assets(Bt, St) where (Bt) is a bank account with a fixed interest rate.

In general, asset models (with positive prices) are defined in terms of the cumula-tive returns

Rt =

∫ t

0

1

SudSu

which means dRt = dSt/St. The asset prices (St) are then defined by the lineardifferential equation

dSt = StdRt

For a bank account the cumulative return process is Rt = rt. Therefore the priceprocess (Bt) satisfies dBt = rBtdt leading to a deterministic exponential evolu-tion

Bt = B0ert.

For general assets with continuous cumulative returns (no jumps) the differentialequation has the solution

St = S0eRt−[R]t/2

62

CHAPTER 5. FINANCIAL MARKETS AND TRADING STRATEGIES 63

5.2 EXAMPLE. (Black-Scholes model) Let (Wt) be a Wiener process. A simplecase are cumulative returns which follow a Brownian motion

dStSt

= dRt = µdt+ σ dWt ⇔ dSt = St dRt = µSt dt+ σSt dWt

The asset prices are then given by the stochastic exponential

St = S0e(µ−σ2/2)t+σWt

This is the Black-Scholes model. The parameter σ is called volatility.

Note, how the parameters are µ and σ2 are related to paths. We have [logS]t = σ2tfor every single path, but µt = logE(St/S0). Hence, the two parameters µ and σare of a completely different nature. The parameter σ (the volatility) is a propertyof a single path. It can be measured by analyzing the quadratic variation of thesingle path alone. It is not a statistical property (in the sense that it were an averagemeasure of a property varying over a given set of paths). On the other hand, theaverage log-return µ is not a path property but a statistical property which dependson how frequently particular paths (with varying returns) are observed. 2

5.3 EXAMPLE. (Types of models) (1) A slightly more general model arises if theparameters are still deterministic, but time dependent, i.e.

dRt = µ(t) dt+ σ(t) dWt

Such models are sometimes labelled as Merton models.

(2) An exponential Ito model is a model where the log-returns are given by anIto-process

dRt = µt dt+ σt dWt

where (µt) and (σt) are adapted caglad processes.

(3) For practical purposes the most important model for equity prices is an Ito-model which satisfies a diffusion equation

dSt = µ(t, St)St dt+ σ(t, St)St dWt.

In this case the volatility σ(t, x) depends both on time and on the spot price. Thesemodels are called local volatility models. 2

Later we will call models to be equivalent models if they are defined on the sameset of paths. Being equivalent implies that the models have the same volatilitystructure. Equivalent models differ only with respect to the probabilities of theirpaths, but they attribute probability one to the same set of paths.

5.4 REMARK. (Role of drift and volatility) At this point it is necessary to say afew words about the role of market models.


It should be kept in mind that the continuous time models we are considering inthis lecture are approximations of a reality which is only appropriate if we do notlook at the microstructure with respect to time.

A simple model of the microstructure could be the binomial model. The binomialmodel consists of two components, the size of the up and downs, and the probabil-ities of the up and downs. For pricing and hedging in the binomial model we onlyneed to know the sizes, but not the probabilities.

Recall the well-known approximation of the Black-Scholes model by the binomialmodel (running under the label of Cox-Ross-Rubinstein). In the limit of this ap-proximation the size of the up and downs shows up as the volatility of the model,and the probabilities of the up and downs show up only in the average return of themodel. This gives rise to the suggestion that for pricing and hedging in continuoustime models only the volatility of the model is essential, but not the average return.

We will see in the following that by the mathematics of continuous time theory thissuggestion actually turns out to being true: For pricing and hedging the structureof the paths is important (in terms of volatility), but not the frequencies of the pathsobserved. 2

5.5 REMARK. (Statistical issues) There is another important aspect of this subject,related to statistics.

It is rather easy to get a consistent statistical picture of the volatility structure ofasset prices. This is true both for historical volatility by means of historical assetprices, and for expected future volatility by means of liquid option prices. How-ever, it is almost impossible to make any valid statistical assertions concerning thefuture of average returns.

Let us give a quantitative picture of the problem.

Typical values of volatility (per year) are about 0.2 to 0.3. Valid confidence in-tervals for average log-returns therefore have length 0.8 to 1.2. But for averagelog-returns it is necessary to know the second digit, e.g. to distinguish between0.03 and 0.05, say. There is obviously no chance to do this. Even if some angeltells us the value of the average log-return, it is not at all reliable to bet on thisvalue, since the volatility may completely destroy any such trend in a particularobserved path.

Summing up, betting on future returns is mainly a matter of subjective confidence,and not at all of aleatoric models . It may even be the case that asset prices andfinancial markets do not follow any mechanical aleatoric laws (in the sense thatthe frequencies of particular parts are distributed according to „laws”). The only(partly) stable property of the observed paths is (hopefully) the volatility (measur-ing the size of up and down). Any weighting of those paths (by additional modelparameters) is either for pricing purposes or for explaining subjective preferences,but has „probably” nothing to do with frequencies. 2


5.2 Trading

Let us start with a review of discrete trading.

5.6 DISCUSSION. (Integrals in discrete trading) Let (Xt) be a single asset. As-sume that at time t1 we buy a units of the asset and we sell them at time t2 > t1.Thus, our asset position is Hs = a1[t1,t2)(s), i.e. Hs− = a1(t1,t2](s). The valuechange caused by this trading can be expressed as

a(Xt2 −Xt1) =

∫Hs− dXs

If we perform trading at n time points t0 < t1 < . . . tn then our asset position is

Hs =

n∑i=1

ai1[ti−1,ti)(s)

and the value change amounts to∫Hs− dXs =

n∑i=1

ai(Xti −Xti−1)

The investments ai may be random and depend on the past but not on future infor-mation. This means that the stock position (Hs) must be an adapted step process.2

Consider a market with two assets (S0t , S

1t ). Then a trading strategy consists of a

pair (H0s , H

1s ) of adapted processes and the value of the resulting portfolio at time

t isVt = H0

t S0t +H1

t S1t

A trading strategy is called self-financing if any change of the portfolio does neverrequire any change of capital. At time t the portfolio is changed from (H0

t−, H1t−)

to (H0t , H

1t ). This change is capital neutral iff

0 = (H0t −H0

t−)S0t +(H1

t −H1t−)S1

t = S0t−∆H0

t +S1t−∆H1

t +∆S0t ∆H0

t +∆S1t ∆H1

t

We would like to admit continuous trading. Therefore we admit general adaptedprocesses (Hs) as trading strategies. For continuous trading this is equivalent to

S0t−dH

0t + S1

t−dH1t + d[S0, H0]t + d[S1, H1]t = 0 (1)

5.7 LEMMA. A trading strategy with value process Vt = H0t S

0t + H1

t S1t is self-

financing iff dVt = H0t dS

0 +H1t dS

1t .


Proof: Integration by parts gives

dVt = H0t−dS

0t +H1

t−dS1t + S0

t−dH0t + S1

t−dH1t + d[S0, H0]t + d[S1, H1]t

Apply equation (1). 2

Note, that being self-financing is a path property. It does not depend on prob-abilities. If a trading strategy is self-financing for a particular model then it isself-financing for any equivalent model.

Let (Nt) be a positive continuous FV process. Denote

St :=StNt, V t :=

VtNt

If (S0t , S

1t ) is a market then (S

0t , S

1t ) is called the market obtained by a change of

numeraire.

5.8 LEMMA. The self-financing property of a trading strategy is not affected by achange of numeraire.

Proof: Use integration by parts. 2

5.9 EXAMPLE. (Normalized markets) Let (Bt, St) be a market model with a bankaccount (Bt). Choose (Bt) as numeraire. Then the market model (1, St) is calleda normalized market model.

A trading strategy (H0t , H

1t ) is self-financing for the normalized market (and as

well for the original market) if

V t = V 0 +

∫ t

0H1u− dSu, shorthand: dV t = H1

t− dSt.

This is remarkable since after the normalization any self-financing trading strategycan be expressed as stochastic integrals where only the stock position of the tradingstrategy shows up. In order to construct a self-financing trading strategy we maytherefore proceed as follows:

• Choose (H1t ) arbitrarily.

• Calculate V t according to dV t = H1t− dSt.

• Choose (H0t ) appropriately according to V t = H0

t +H1t St.

2

5.3 Markovian trading strategies

Let (Bt, St) be an Ito market model, i.e.

Bt = ert,dStSt

= µt dt+ σt dWt.


5.10 DEFINITION. An adapted process (Yt) is called Markovian (w.r.t the model)if Yt = φ(t, St), i.e. if it depends on the past only through the present value of (St)(the spot price).

Let (H0t , H

1t ) be a Markovian trading strategy. This means that we may write

H0t = φ(t, St), H1

t = ψ(t, St)

and also the portfolio value of the trading strategy

Vt = φ(t, St)Bt + ψ(t, St)St =: g(t, St)

is Markovian. For simplicity we assume that g(t, x) is twice continuously differ-entiable. Later we will see that this condition is satisfied in typical cases.

Let us apply Ito’s formula to the portfolio value:

dVt = dg(t, St) = gt dt+ gx dSt +1

2gxx d[S]t

= gt dt+ gx(µtSt dt+ σtSt dWt) +1

2gxxσ

2t S

2t dt

= (gt + gxµtSt +1

2gxxσ

2t S

2t ) dt+ gxσtSt dWt

This is a respresentation of the portfolio value as an Ito-process.

If the portfolio is self-financing then we have another representation as an Ito-process:

dVt = φ(t, St) dBt + ψ(t, St) dSt

= (φ(t, St)rBt + ψ(t, St)µtSt) dt+ ψ(t, St)σtSt dWt

From the uniqueness of components of Ito-processes we obtain two fundamentalassertions.

The first assertion is that the stock position of the trading strategy is given by theso-called Delta of the portfolio value, i.e. its partial derivative w.r.t. the stock price.

5.11 THEOREM. Let g(t, x) be the portfolio value of a self-financing Markoviantrading strategy. Then the trading strategy can be recovered from the portfoliovalue by

ψ(t, x) = gx(t, x), φ(t, x) =g(t, x)− gx(t, x)x

Bt

The second assertion is that the self-financing property is equivalent to a partialdifferential equation.

5.12 THEOREM. Let g(t, x) be the portfolio value of a trading strategy. Thistrading strategy is self-financing iff the portfolio g(t, x) value satisfies the partial


differential equation

rg = gt + rgxx+1

2gxxσ

2t x

2

Proof: Comparing the dt-part of the Ito-expansions gives

gt + gxµtx+1

2gxxσ

2t x

2 = r(g(t, x)− gx(t, x)x)) + gxµtx

2

Historically, it was noted as highly remarkable fact that the pdf, which character-izes the self-financing property, does not contain the parameter µt. But this is notat all surprising since the self-financing property is a path property. It does notdepend on probabilities and therefore cannot depend on µt.

For the special case of the Black-Scholes model (constant parameters) the partialdifferential equation is called the Black-Scholes partial differential equation.

5.4 The market price of risk

Later it woill turn out that the hedge portfolio of a positive claim has always apositive value. In such a case the Ito-expansion of the stochastic logarithm dVt/Vtcan be written as

dVt/Vt = (gt + gxµtSt +1

2gxxσ

2t S

2t )/g dt+ gxσtSt/g dWt

= (r − rStgx/g + µtStgx/g) dt+ σtStgx/g dWt

=: µVt dt+ σVt dWt

This yields the following remarkable result which displays the core structure of thepdf.

5.13 THEOREM. Let (Vt) be a positive value process of a Markovian tradingstrategy. This trading strategy is self-financing iff it has the same market price ofrisk as the stock (St):

µVt − rσVt

=µt − rσt

This result shows that trading can improve mean returns only at the cost of takingrisk. For constant parameters this means that the risk-return plot is a straight line

µVt = r + pσVt , where p = (µ− r)/σ.


5.14 EXAMPLES. Let us show how risks and returns can be modified by tak-ing portfolios. We assume that the market model is a Black-Scholes model withparameters µ and σ.

(1) What can we get, if we double the risk ?

We would like to have σV = 2σ. This implies

µV − r =µ− rσ

2σ = 2µ− 2r

which gives µV = 2µ− r. We cannot double the return.

Let us find the portfolio for doubling the risk. The value process satisfies

dVt = Vt(µV dt+ σV dWt) = Vt((2µ− r)dt+ 2σdWt).

The solution is

Vt = V0e(2µ−r−2σ2)t−2σWt = V0e

−rt−σ2te2(µ−σ2/2)t−2σWt = V0e

−(r+σ2)tS2t .

The portfolio components can be obtained in the usual way.

(2) What is the risk, if we double the return ? 2

If µt = r then even taking risk does not help for improving trading returns. Thereturns are then neutral to taking risk, or shortly risk neutral.

5.15 DEFINITION. An Ito-model is called risk-neutral if the market price of riskis zero, ie. if µt = r.

For a risk neutral model the average log-return of a self-financing portfolio cannotbe modified by changing the risk. The average log-return is always equal to thelog-return of the bank account.

A portfolio is called riskless if its value is an FV process (σV = 0).

5.16 COROLLARY. Every riskless self-financing portfolio is proportional to thebank account.

Proof: A riskless portfolio has σV = 0 which implies µV = r. 2

5.5 Review material

Concepts: financial market model, fixed interest rate, log-return process, expo-nential model, Ito-model.

market price of risk, riskless portfolio, risk-neutral model


Facts: examples of continuous exponential financial models (Black-Scholes, Mer-ton, local volatility), trading strategy, self-financing property, change of nu-meraire, stability of the self-financing property under change of numeraire,Delta, Black-Scholes partial differential equation.

Markovian trading strategies, Black-Scholes pdf for Ito-models, role of driftand volatility in the Black-Scholes pdf.

uniqueness of the market price of risk in Ito-models, uniqueness of the risk-less portfolio in Ito-models, meaning of risk neutrality in terms of marketprice of risk.

5.5.1 Examples of exam proofs

1. Derive the Black-Scholes partial differential equation.

2. Show that in a Black-Scholes market any riskless self-financing portfolio isproportional to the bank account.


1. Consider a Black-Scholes market with fixed interest rate r = 0, 08. Thestock has log-return of 10 percent with a volatility of 0,2. What is the marketprice of risk ?

2. Consider a market model (Bt, St) such that Bt = 1 and St = σWt. Is this arisk neutral model ?

3. How can self-financing trading strategies be characterized ? (Definition, par-tial differential equation, integral representation, market price of risk.)

4. How can we construct a self-financing trading strategy ?

5. How can we obtain the components of a self-financing trading strategy fromthe value process ?


1. Find a self-financing portfolio whose volatility is ασ.

Chapter 6

Risk neutral Ito-models andpricing derivatives

6.1 Risk neutral models

In this section we will have a more detailed look ar risk-neutral models. The rea-son is that risk-neutral models are the basic tool for pricing derivatives. We haveintroduced the concept of risk-neutrality at hand of Ito-models where the marketprice of risk is an intuitive appealing parameter. However, risk-neutrality goes farbeyond Ito-models but the definition has to be modified in terms of certain martin-gale properties.

Consider an Ito-model (Bt, St) where Bt = e−rt and dSt/St = µt dt+ σt dWt.

6.1 LEMMA. The Ito-model (Bt, St) is risk-neutral iff (Ste−rt) is a martingale.

Proof: Let St := Ste−rt. Then

dSt = d(Ste−rt) = −rSte−rt dt+ e−rtdSt

= −rSt dt+ e−rtSt(µtdt+ σt dWt)

= St((µt − r) dt+ σt dWt)

2

Therefore risk-neutral models are also called martingale models. The normalizedmarket (1, St) consists of martingales.

6.2 DEFINITION. A market model (Bt, St) on a probability space (Ω, (Ft), P ) isa martingale model if the discounted stock price process (St) is a martingale underP .

71

CHAPTER 6. RISK NEUTRAL ITO-MODELS AND PRICING DERIVATIVES72

For martingale models we may express the stock prices St as discounted condi-tional expectations of the terminal stock prices ST , i.e.

St = E(ST |Ft) ⇔ St = E(ST |Ft)er(t−T ).

This property extends immediately to self-financing portfolios.

Let (H0t , H

1t ) be a trading strategy. The trading strategy is self-financing iff the

normalized portfolio value satisfies

V t = V 0 +

∫ t

0H1u− dSu

Therefore it is a local martingale. Under appropriate integrability conditions (to bechecked for the particular situation) this is even a martingale and satisfies

E(V t|Fs) = V s, s < t.

In particular, we have

V t = E(V T |Ft) ⇔ Vt = E(VT |Ft)er(t−T ).

6.2 Hedging contingent claims

Let CT = f(ST ) be the payoff of a contingent claim with maturity T . Hedg-ing the claim means constructing (and managing) a self-financing portfolio Vt =HBt Bt +HS

t St with given terminal value VT = CT (hedge portfolio).

How can we find such a portfolio ?

For Ito-models the Black-Scholes pde is a natural approach. Whenever a solutiong(t, x) of the pde with terminal condition g(T, x) = f(x) is available it returnsa value process Vt = g(t, St) of a self-financing portfolio with the correct finalpayoff.

Although this pde-method is well established it has several drawbacks which willbecome clear later. An alternative approach is based on risk-neutrality.

6.3 DISCUSSION. (Finding hedge portfolios) Assume that our market model isrisk neutral. Consider a claim CT . If there exists a hedge-portfolio (with appropri-ate integrability properties) then its value process (Vt) satisfies VT = CT and itsdiscounted version must be a martingale, i.e.

V t = E(CT e−rT |Ft). (1)

In order to find the value process (V t) (if it exists at all) we need only study theconditional expectation E(CT e

−rT |Ft).


Typically, E(CT e−rT |Ft) = h(t, St) where h(t, x) is some smooth function. This

is our candidate for the discounted value process (V t). It is certainly a martingale,since every conditional expectation gives rise to a martingale. The question iswhether h(t, St) can be written as a stochastic integral. Only then can we take itas a value process of a self-financing portfolio.

Now Ito’s formula comes in. Expanding h(t, St) by Ito’s formula we obtain

h(t, St) = h(0, S0) +

∫ t

0hx(u, Su) dSu,

since the FV terms vanish in view of the martingale property of h(t, St). Thus, weneed not worry about the existence of a hedge portfolio as long as the conditionalexpectations h(t, St)are sufficiently smooth functions of t and St. 2

The preceding discussion shows that in finding hedge portfolios an important stepis to find closed forms of conditional expectations. Let us consider some examples.

6.4 EXAMPLE. (Power options) Let us consider the claim with payoff CT = S2T .

In order to find a hedge portfolio we study

V t = E(S2T e−rT |Ft) = e−rTE(S2

T |Ft).

We have

E(S2T |Ft) = E

(S2t

(STSt

)2∣∣∣Ft) = S2tE((ST

St

)2)= S2

tE(e2(r−σ

2/2)(T−t)+2σ(WT−Wt))

= S2t e

2(r−σ2/2)(T−t)+4σ2(T−t)/2

= S2t e

(2r+σ2)(T−t).

Thus we obtain

V t = e−rTS2t e

(2r+σ2)(T−t) = e(r+σ2)TS

2t e−σ2t =: h(t, St).

The function h(t, x) is a smooth function and Ito’s formula implies

h(t, St) = e(r+σ2)T + 2e(r+σ

2)T

∫ t

0Sue

−σ2u dSu

which shows that the discounted value process has the form of a self-financingportfolio. For value process we get

Vt = e−r(T−t)S2t e

(2r+σ2)(T−t) = e(r+σ2)(T−t)S2

t =: g(t, St).

That means

g(t, x) = e(r+σ2)(T−t)x2, gx(t, x) = 2xe(r+σ

2)(T−t)


Therefore the price of the claim ist V0 = e(r+σ2)TS2

0 .

Now we can calculate the hedge portfolio. The stock position is HSt = gx(t, St)

and the cash position is

HBt = (g(t, St)−gx(t, St)St)e

−rt = e(r+σ2)(T−t)(S2

t−2S2t )e−rt = −g(t, St)e

−rt.

This gives

Vt = HBt Bt +HS

t St = −g(t, St)e−rt ·Bt + 2Ste

(r+σ2)(T−t) · St.

2

The following example illustrates this procedure.

6.5 EXAMPLE. (Black-Scholes model) Assume that (Bt, St) is a risk-neutralBlack-Scholes model under P . Let CT = (ST − K)+ be the payoff of a calloption with strike K. Calculating

Vt = E(CT |Ft)e−r(T−t) =: g(t, St)

leads to

g(t, x) := xN( log(x/K) + (r + σ2/2)(T − t)

σ√T − t

)−e−rT−tKN

( log(x/K) + (r − σ2/2)(T − t)σ√T − t

)Therefore, it is the value of a self-financing hedge portfolio with price g(0, S0) andstock position

gx(t, x) = N( log(x/K) + (r + σ2/2)(T − t)

σ√T − t

)2

6.3 Review material

Concepts: martingale model.

Facts: martingale property of risk-neutral models, martingale representation byIto’ formula, construction of replicating portfolios in risk-neutral Ito-models.

6.3.1 Exam questions

1. Consider a market model (Bt, St) such that Bt = ert and St = µt + σWt.When is this a martingale model ?


6.3.2 Exam problems

1. Let (Wt) be a Wiener process. Write as a+∫ t0 Hs dWs:

(a) W 2T , (b) W 3

T , (c) eαWT , (d) 3W 2T − 2eWT , ,

2. Consider a market model (Bt, St) such that Bt = ert and dSt = rStdt +σdWt. Find (St).

3. Consider the market model (Bt, St) such that Bt = 1 and St = σWt. Findreplicating portfolios and prices for the following claims:(a) S2

T , (b) S3T , (c) eST , (d)K1(ST>K), (e) ST 1(ST>k), (f) (ST−

K)+,

4. Consider a risk neutral Black-Scholes market model (Bt, St). Find replicat-ing portfolios and prices for the following claims:(a) S2

T , (b) logST , (c) K1(ST>K), (d) ST 1(ST>k), (e) (ST −K)+,

Chapter 7

The fundamental theorems ofmathematical finance

7.1 Introduction

In the preceding chapter we have seen that for martingale models there is an elegantand efficient method of finding hedge portfolios. However, probabilistic marketmodels which quantify statistical frequencies of paths practically do never satisfythe martingale property. To put it in other words: Models of real markets are notrisk neutral. Typically, there is a positive market price of risk.

How can we utililze the convenient properties of martingale models for statisticalmodels which do not satisfy the martingale property ? The answer is that for pricingwe may - actually we must - use different models than for statistical purposes.

Being a self-financing trading strategy with a particular terminal value is a pathproperty. It does not depend on frequencies, i.e. on statistical probabilities. Forthe construction of such a trading strategy we may use any model which is con-centrated on the same path family as our statistical model; in mathematical terms:which is equivalent to our statistical model.

Carrying out this idea relies on the following conditions:

(1) Existence of an equivalent martingale model: Let P be the statistical prob-ability of the market model (Bt, St). Then we need an equivalent model Q underwhich (Bt, St) is a martingale model (and EQ(C2

T ) <∞).

(2) Possibility of martingale representation: In order to construct an hedge port-folio for a payoff CT it is necessary that the discounted payoff CT can be writtenas a stochastic integral

CT = a+

∫ T

0Hu dSu.

76

CHAPTER 7. THE FUNDAMENTAL THEOREMS OF MATHEMATICAL FINANCE77

Is there any hope that there exist equivalent martingale models ? There is verygeneral positive answer which is called the (1st) Fundamental Theorem of Math-ematical Finance. Roughly speaking, it says that for every arbitrage-free marketmodel there exists at least one equivalent market model.

Since the NA-property is indispensable for any reasonable modelling of financialmarkets we need not worry about the existence of an equivalent martingale model.However, things are different with the problem of finding such a model. We willdiscuss this issue in sections ...

Assume that we have decided to use a particular martingale model for pricingderivatives. Taking conditional expectations of the payoff we obtain a process (Vt)which is a candidate for the value of the derivative.

If condition (2) is satisfied then

V t := a+

∫ t

0Hu dSu

is the value process of a self-financing portfolio such that V T = CT (a replicat-ing portfolio) and a is the discounted cost of this portfolio. Since the market isarbitrage-free the cost of the replicating portfolio is a arbitrage-free price of thederivative.

For carrying out the second step of our pricing and hedging procedure we have torepresent the discounted value process V t as a stochastic integral w.r.t. the price(St) of the underlying. When is this possible ?

Recall that the discounted value process is a martingale under the martingale model.The problem of representing martingales as stochastic integrals is a mathematicalproblem of general martingale theory.

7.1 REMARK. (Martingale representation) The martingale representation is oftenavailable by Ito’s formula.

Let (Bt, St) be a risk-neutral market model (under Q). Consider the martingaleMt = EQ(CT |Ft) and assume that it can be written as Mt = h(t,Wt) for somesmooth function h(t, x) and aQ-Wiener process (Wt). Then Ito’s formula togetherwith the martingale property implies that

CT = h(0,W0) +

∫ T

0hx(u,Wu) dWu (1)

Therefore in this special case we obtain an explicit formula for martingale repre-sentation. 2


7.2 Equivalent models

Let (Ω,F , P ) be a probability measure where F = F(X) is the σ-field consistingof all events (X ∈ B) which can be expressed in terms of X . The probabilitymodel of X is simply the distribution of X , i.e. the collection of all probabilitiesP (X ∈ B) of events in F .

Recall the causality theorem which says that a random variable Y is F(X)-measurable iff it can be written as Y = f(X).

7.2 DEFINITION. Let P and Q be two probability models defined on F(X). Themodels are called equivalent (on F(X)) if

P (X ∈ B) > 0 iff Q(X ∈ B) > 0, B ∈ F(X).

By the theorem of Radon-Nikodym the models P and Q are equivalent on F(X)(i.e. P |F ∼ Q|F) iff there is an F(X)-measurable random variable L = f(X) >0 such that

Q(X ∈ B) = EP (L1(X∈B)) and P (X ∈ B) = EQ

( 1

L1(X∈B)

)The random variable L is called the Radon-Nikodym derivative of Q w.r.t. P andis denoted by

L =dQ

dP,

1

L=dP

dQ.

7.3 EXAMPLE. (Measure change for normal distributions) Assume that X P∼N(0, 1) and X

Q∼ N(α, 1). It is clear that the models P and Q are equivalent (atleast for intervals B ⊆ R). Let us show that

L =dQ

dP= eαX−α

2/2.

Let φ(x) be the density of N(0, 1). Then

EP (L1(X∈B)) =

∫Beαx−α

2/2φ(x) dx =1√2π

∫Beαx−α

2/2e−x2/2 dx

=1√2π

∫Be(x−α)

2/2 dx = Q(X ∈ B)

The shift difference α between P and Q is identical to the covariance between Xand logL. This is a general pattern which in the case of semimartingales is calledGirsanov’s theorem (see Theorem 7.7 ). In mathematical statistics it is called the3rd Lemma of LeCam. 2


It is often necessary to express conditional expectations w.r.t Q in terms of condi-tional expectation w.r.t P . The following assertion settles this problem.

7.4 THEOREM. Let P |F ∼ Q|F , L = dQ/dP and A ⊆ F . Then for everyF-measurable random variable X (Q-integrable or non-negative)

EQ(X|A) =EP (XL|A)

EP (L|A).

Proof: It is sufficient to prove the formula for elementary conditional expecta-tions. Then it is valid for finite σ-fields and covers the general case by the usualarguments.

Let A ∈ F . Then

EQ(X|A) =1

Q(A)EQ(X1A) =

EP (LX1A)

EP (L1A)=EP (LX|A)

EP (L|A)

2

7.3 Girsanov’s theorem

Let (Ω, (Ft), P ) be a filtered probability space.

Consider a finite horizon T > 0 and let Q|FT ∼ P |FT . It is clear that this impliesQ|Ft ∼ P |Ft for every t ∈ [0, T ]. How are the RN-derivatives

Lt :=dQ|FtdP |Ft

related to each other ?

7.5 LEMMA. The process (Lt)t≤T is a positive P -martingale.

Proof: Show that

E(dQ|FTdP |FT

∣∣∣Ft) =dQ|FtdP |Ft

2

It is easy to check the Q-martingale property of a process in terms of conditionalP -expectations.

7.6 LEMMA. A semimartingale (Xt) is a local Q-martingale iff (LtXt) is a localP -martingale.

Proof: Let us prove the assertion for martingales. The Bayes formula implies

EQ(Xt|Fs) =EP (XtLt|Fs)EP (Lt|Fs)

=EP (XtLt|Fs)

Ls


which is equivalent to

EQ(Xt|Fs)Ls = EP (XtLt|Fs)

HenceEQ(Xt|Fs) = Xs ⇔ EP (XtLt|Fs) = XsLs

2

Assume now that (Lt) is even a continuous martingale. Then it can be written as astochastic exponential

Lt = E(Z)t = eZt−[Z]t/2 ⇔ dLt = Lt dZt

where (Zt) is some local martingale.

7.7 THEOREM. Let Q|FT ∼ P |FT with continuous likelihood process Lt =E(Zt), t ≤ T . If (Xt)t≤T is a continuous localP -martingale thenXt−[X,Z]t, t ≤T, is a local Q-martingale.

Proof: In order to show that Xt− [X,Z]t is a local Q-martingale we have to showthat Lt(Xt − [X,Z]t) is a local P -martingale. This follows from

d(Lt(Xt − [X,Z]t) = Lt d(Xt − [X,Z]t) + (Xt − [X,Z]t) dLt + d[L,X]t

= Lt dXt + (Xt − [X,Z]t) dLt.

2

There are several important consequences of Girsanov’s theorem. The first asser-tion says that the semimartingale property is not affected by passing to an equiva-lent model.

7.8 COROLLARY. Assume that (Xt) is a continuous semimartingale under P andthat (Lt) is a continuous process. Then (Xt) is a semimartingale under Q, too.

How to obtain equivalent models ? How to obtain equivalent risk-neutral models ?

7.9 REMARK. (Statistical and risk-neutral probabilities) Assume that we are deal-ing with an Ito-model model dSt/St = µt dt+ σt dWt. The model attributes pos-itive probabilities to paths with volatility σt. The probabilities depend on µt. Ifµt 6= r then the model is not risk neutral.

In order to obtain a risk neutral model we have to change the parameter µt. Wemust not change the volatility parameter σt since for obtaining an equivalent modelwe must not change the set of possible paths of the model (which all have to havevolatility σt). We have to keep the set of possible paths and may only change theprobabilities.


What do the original model assumptions say about the probabilities ? We may putthe assumption as

1

σt

(dStSt− µt dt

)is a Wiener process.

The process on the left is a continuous process with quadratic variation t. ByLevy’s theorem, being a Wiener process is equivalent to being a local martingale.Thus, the present state is

1

σt

(dStSt− µt dt

)is a local martingale,

and what we need is that the shifted process

1

σt

(dStSt− r dt

)=

1

σt

(dStSt− µt dt

)+ (µt − r) is a local martingale.

Hence, the problem is: Can we change the probabilities of the original model insuch a way that the shift by µt − r restores the martingale property ? The answergiven in Corollary 7.11 below. As a result, we may move from the original modelto an equivalent risk-neutral model. 2

Any shift of a Wiener process is a Wiener process under some equivalent model.

7.10 COROLLARY. Let (Wt) be a P -Wiener process (w.r.t. (Ft)) and let Lt =eαWt−α2t/2. Then (Wt − αt) is a Q-Wiener process.

There is a direct application to Black-Scholes models.

7.11 COROLLARY. Assume that (Bt, St)t≤T is a Black-Scholes market under Pwhere

dSt = St(µdt+ σ dWt).

Let

LT = exp(− (µ− r)2

2σ2T − µ− r

σWT

)and define Q = LTP . Then

Wt := Wt +µ− rσ

t

is a Wiener process under Q and (Bt, St) is a risk-neutral Black-Scholes marketunder Q.

Proof: Show that dSt = Stσ dWt. 2

The result can easily be extended to Ito-markets.

It follows that for every Ito-market there exists an equivalent model where it is arisk neutral Ito-market.


7.4 The no-arbitrage theorem

7.12 DEFINITION. A market model is arbitrage-free if for every value processof a self-financing portfolio which is bounded from below the assertion

V0 ≤ 0 ≤ VT ⇒ VT = 0

is valid.

Note, that in an arbitrage-free market prices are uniquely determined: For any twovalue processes (V

(1)t ) and (V

(2)t ) of self-financing portfolios we have

V(1)T = V

(2)T ⇒ V

(1)0 = V

(2)0 .

7.13 THEOREM. Let (Bt, St) be a market model. Assume that there exists anequivalent model Q such that (Bt, St) are local Q-martingales. Then the market isarbitrage-free.

Proof: Under Q each value process (V t) of a self-financing portfolio is a localmartingale. Let (τn) be a localizing sequence of stopping times. Then we have

E(V T ) = E( limn→∞

V T∩τn) ≤ lim infn→∞

E(V T∩τn) = V 0∩τn = V0 ≤ 0.

(where we used Fatou’s lemma). 2

As a result it follows that every Ito-market is arbitrage-free. Market models withjumps (e.g. based on Poisson processes) are not always arbitrage-free.

The no-arbitrage theorem is easily extended to arbitary finite markets. It can beshown that even a converse assertion is true (with slight but essential modifica-tions).

7.5 The martingale representation theorem

The following theorem is a general version of formula (1). It is a purely existenceassertion and it does not claim that the integrand process is left-continuous. How-ever, in most examples there will be a left-continuous adapted integrand process.

Let (Ω, (Ft), P ) be a filtered probability space where the filtration is the history ofa Wiener process (Wt).

7.14 THEOREM. Let CT ∈ FT be square integrable. Then there exists a (pre-dictable) process (Ht) such that

CT = E(CT ) +

∫ T

0Hu dWu


Proof: Our proof shows that the assertion is true for all multivariable polynomi-als of eWt , t ≤ T . Since CT can be approximated by such polynomials this issufficient.

Let CT = (eWs)k(eWt)`, s < t. Then we have

CT = exp(∫ T

0hu dWu

)where hu = (k + `)1(0,s](u) + `1(s,t](u).

Let

Xt = exp(∫ t

0hu dWu −

1

2

∫ t

0h2u du

)Then

dXt = Xtht dWt

which implies

Xt = 1 +

∫ t

0Xuhu dWu

Abbreviating a := exp(12

∫ t0 h

2u du

)we obtain

CT = aXT = a+

∫ T

0aXuhu dWu

2

The assertion says that for filtration which are generated by a Wiener process,martingale representation is possible. This implies for financial theory that forIto-models every contingent claim can be replicated by a self-financing portfolio.

7.6 Complete and incomplete markets

7.15 DEFINITION. A market model is complete if every contingent claim can bereplicated by a self-financing portfolio.

In other words, Ito-models are complete market models. Models based on the Pois-son process are complete. More complicated models with jumps are not complete.

7.16 COROLLARY. For a complete market model there exists at most one equiv-alent martingale model.

Proof: Let CT = 1A where A ∈ FT . Let Q1 and Q2 be two equivalent martingalemodels. By completeness we have

CT = a+

∫ T

0Hu dSu


Assuming integrability it follows that Q1(A) = Q2(A). 2

The interesting point is the converse: Whenever there is more than only one equiva-lent martingale model then the market is incomplete. In such cases it is not possibleto replicate every contingent claim by a self-financing portfolio.

It is an important point to ask how to obtain arbitrage-free prices for claims whichcannot be replicated.

7.17 THEOREM. Let (Bt, St) be an arbitrage-free market and let Q be anyequivalent martingale model. Then for every contingent claim CT the expectationEQ(CT )e−rT is a consistent price which excludes arbitrage.

Proof: Consider the market (Bt, St, EQ(CT |Ft)er(t−T )). Then Q is a martingalemodel for this market which therefore is free of arbitrage. 2

If a market is incomplete with several (infinitely many) equivalent martingale mod-els then every claim has inifinitely many consistent prices. It can be shown that theset of these prices is an open interval.

7.7 Review material

Concepts: Equivalent probability distributions, Radon-Nikodym derivative.

No-arbitrage property, complete markets.

Facts: Equivalent models for shifting Gaussian variables (form of the RN-derivative),Bayes formula for conditional expectations.

Girsanov’s theorem, equivalent models for shifting Wiener processes (formof the RN-derivative), equivalent risk neutral models for Black-Scholes mod-els (form of the RN-derivative).

No-arbitrage theorem, arbitrage-free pricing in incomplete markets.

Martingale representation theorem.

Chapter 8

Classroom tests

Test 1:

(1) Why are mean and variance of Levy processes linear functions of t ?

(2) Let (Wt) be a Wiener process and s < t. Find E(eαWt |Fs).

(3) Find the Fourier transform of a jump diffusion with variance 1, jumping with intensity1, having jump heights uniformly distributed on [−2, 0].

Test 2:

(1) Find the first and second moment of a compound Poisson process.

(2) Explain why V 1t (W ) =∞, where (Wt) is a Wiener process.

(3) Calculate∫ t0

2Ns− dNs, where (Nt) is a Poisson process.

Test 3:

(1) Explain the steps for defining the Ito-integral.

(2) Show that Wiener integrals have independent increments.

(3) Find the covariance of tWt and∫ t0s dWs.

Test 4:

(1) Explain the idea of differential notation. (How to pass from one notation to the other ?)

(2) Write as Ito-process and identify the components: tWt

(3) Prove ∆[X]t = (∆Xt)2

Test 5:

(1) How can we obtain the components of a self-financing trading strategy from the valueprocess ? (Consider smooth Markovian trading strategies for Ito market models.)

(2) Derive the Black-Scholes partial differential equation.

(3) Prove that the NA-property implies uniqueness of prices for self-financing portfolios.

85

CHAPTER 8. CLASSROOM TESTS 86

(4) Consider a market model (Bt, St) such that Bt = ert and dSt = rStdt+ σdWt. Find(St).

(5) Let (Wt) be a Wiener process. Write as a+∫ T0Hs dWs: eαWT .

(6) Consider a risk neutral Black-Scholes market model (Bt, St). Find replicating portfo-lios and prices for the claim CT = logST .

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