Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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xx Contents
11.2.4 Itˆo’s Formulae for a One-dimensional L´evy Process . . . . 612
11.2.5 Itˆo’s Formula for L´evy-Itˆo Processes .................613
11.2.6 Martingales . ......................................615
11.2.7 Harness Property . . . . ..............................620
11.2.8 Representation Theorem of Martingales in a
L´evySetting......................................621
11.3 Absolutely Continuous Changes of Measures . ...............623
11.3.1 Esscher Transform . . . ..............................623
11.3.2 Preserving the L´evy Property with Absolute Continuity 625
11.3.3 General Case . . . ..................................627
11.4 Fluctuation Theory ......................................628
11.4.1 Maximum and Minimum . . . . .......................628
11.4.2 Pecherskii-Rogozin Identity . . .......................631
11.5 Spectrally Negative L´evyProcesses ........................632
11.5.1 Two-sided Exit Times. . . . . . . .......................632
11.5.2 Laplace Exponent of the Ladder Process . . ...........633
11.5.3 D. Kendall’s Identity ..............................633
11.6 Subordinators . ..........................................634
11.6.1 Definition and Examples . . . . .......................634
11.6.2 L´evy Characteristics ofaSubordinated Process .......636
11.7 Exponential L´evyProcesses as Stock PriceProcesses.........636
11.7.1 Option Pricing with Esscher Transform . . . ...........636
11.7.2 A Differential Equation for Option Pricing . ...........637
11.7.3 Put-call Symmetry . . ..............................638
11.7.4 Arbitrage and Completeness . .......................639
11.8 Variance-Gamma Model . . . . ..............................639
11.9 Valuation of Contingent Claims . . . . .......................641
11.9.1 Perpetual American Options . .......................641
A List of Special Features, Probability Laws, and Functions . . 647
A.1 MainFormulae..........................................647
A.1.1 Absolute Continuity Relationships . . . . ...............647
A.1.2 Bessel Processes . ..................................648
A.1.3 Brownian Motion . . . . ..............................649
A.1.4 Diffusions . . ......................................650
A.1.5 Finance ..........................................650
A.1.6 Girsanov’s Theorem . ..............................651
A.1.7 Hitting Times . . . ..................................651
A.1.8 Itˆo’sFormulae ....................................651
A.1.9 L´evyProcesses....................................653
A.1.10 Semi-martingales ..................................654
A.2 Processes...............................................655
A.3 SomeMainModels ......................................655
A.4 Some Important Probability Distributions . . . ...............656
A.4.1 Laws with Density . . . ..............................656
Contents xxi
A.4.2 Some Algebraic Properties for Special r.v.’s . . . . .......656
A.4.3 Poisson Law ......................................657
A.4.4 Gamma and Inverse Gaussian Law . . . ...............658
A.4.5 Generalized Inverse Gaussian and Normal Inverse
Gaussian .........................................659
A.4.6 Variance Gamma VG(σ, ν, θ) .......................661
A.4.7 Tempered Stable TS(Y
±
,C
±
,M
±
) ..................661
A.5 Special Functions........................................662
A.5.1 Gamma and Beta Functions . .......................662
A.5.2 Bessel Functions ..................................662
A.5.3 Hermite Functions . . . ..............................663
A.5.4 Parabolic Cylinder Functions .......................663
A.5.5 Airy Function . . . ..................................663
A.5.6 Kummer Functions . ..............................664
A.5.7 Whittaker Functions . ..............................664
A.5.8 Some Laplace Transforms . . . .......................665
References .....................................................667
B Some Papers and Books on Specific Subjects ..............709
B.1 TheoryofContinuousProcesses ...........................709
B.1.1 Books . . ..........................................709
B.1.2 Stochastic Differential Equations . ...................709
B.1.3 Backward SDE . . ..................................709
B.1.4 Martingale Representation Theorems . ...............710
B.1.5 Enlargement of Filtrations . . . .......................710
B.1.6 Exponential Functionals . . . . . .......................710
B.1.7 Uniform Integrability of Martingales . . ...............710
B.2 ParticularProcesses .....................................710
B.2.1 Ornstein-Uhlenbeck Processes . . . ...................710
B.2.2 CIR Processes . . ..................................710
B.2.3 CEV Processes . . ..................................711
B.2.4 Bessel Processes . ..................................711
B.3 Processes withDiscontinuousPaths........................711
B.3.1 Some Books ......................................711
B.3.2 Survey Papers . . ..................................711
B.4 HittingTimes...........................................711
B.5 L´evyProcesses..........................................712
B.5.1 Books . . ..........................................712
B.5.2 Some Papers . . . . ..................................712
B.6 Some BooksonFinance ..................................712
B.6.1 Discrete Time . . . ..................................712
B.6.2 Continuous Time . . . . ..............................712
B.6.3 Collective Books ..................................712
B.6.4 History ..........................................713
xxii Contents
B.7 Arbitrage...............................................713
B.8 ExoticOptions..........................................713
B.8.1 Books . . ..........................................713
B.8.2 Articles ..........................................713
Index of Authors ..............................................715
Index of Symbols ..............................................723
Subject Index .................................................725
User’s Guide
This book consists of two parts: the first part concerns continuous-path
processes, and the second part concernes jump processes.
Part I:
Chapter 1 introduces the main results for continuous-path processes and
presents many examples, including in particular Brownian motion.
Chapter 2 presents the main tools in finance: self-financing portfolios,
valuation of contingent claims, hedging strategies.
Chapter 3 contains some useful information about hitting times and their
laws. Closed form expressions are given in the case of (geometric) Brownian
motion.
Chapter 4 discusses finer properties of Brownian motion, e.g., local times,
bridges, excursions and meanders.
Chapter 5 is devoted mainly to the presentation of one-dimensional
diffusions, thus extending the scope of Chapter 4. Filtration problems are
also studied.
Chapter 6 focuses on Bessel processes and applications to finance.
Part II:
Chapter 7 is concerned with models of default risk, which involve stochastic
processes with a single jump.
Chapter 8 introduces Poisson and compound Poisson processes, which are
standard examples of jump processes.
Chapter 9 contains general theory of semi-martingales and aims at unifying
results obtained in Chapters 1 and 8.
Chapter 10 presents some jump-diffusion processes and their applications
to Finance.
Chapter 11 gives basic results about L´evy processes.
Chapter 12 consists of a list of useful formulae found throughout this book.
xxiv User’s Guide
At the end of the book, the reader will find an extended bibliography, and
a list of references, sorted by thema, followed by an index of authors, in which
the page number where each author is quoted is specified.
In the text, some important words are in boldface. These words are also
found in the subject index. Some notation can be found in the notation index.
To complete this guide, we emphasize some particular features of this book,
already mentioned in the Preface:
• in some cases, proofs are sketched and/or omitted, but precise references
are given;
• forward references to topics discussed further in the book are indicated
with the arrow ;
• we proceed by generalization: an important case/process is discussed,
followed (a little later) by a general study.
Throughout this book, the symbol indicates the end of a proof, the symbol
indicates the end of an exercise and the symbol is used to separate a long
proof into different parts.
Section 2.1 refers to Chapter 2, Section 1, and Subsection 4.3.7 refers to
Chapter 4, Section 3, Subsection 7. Theorem (Proposition, Lemma) 3.2.1.4
is the 4th in Chapter 3, Section 2, Subsection 1.
Begin at the beginning, and go on till you come to the end. Then, stop.
Lewis Carroll, Alice’s Adventures in Wonderland.
Notation xxv
Notation and Abbreviations
We shall use the standard notation and abbreviations.
We shall use increasing instead of nondecreasing and positive instead
of non-negative.
u.i. : uniformly integrable (for a family of r.v.’s)
BM : Brownian motion
r.v. : random variable
e.m.m. : equivalent martingale measure
a.s. : almost surely
w.r.t. : with respect to
w.l.g. : without loss of generality
SDE : Stochastic Differential Equation
BSDE : Backward Stochastic Differential Equation
PRP : Predictable Representation Property
MCT : Monotone Class Theorem
x ∨ y =sup(x, y)
x ∧ y =inf(x, y)
x y : scalar product of the vectors x, y ∈ R
d
HX : stochastic integral of the process H with respect to the
semi-martingale X
x
+
= x ∨ 0
x
−
=(−x) ∨ 0
∂
x
f =
∂
∂x
f = f
x
C
n
b
: set of functions with continuous bounded derivatives
up to n-th order
μ(t),μ
t
: A function (or a process) evaluated at time t.
If μ is a deterministic function, μ(t) is preferably used;
if μ is a process, when the subscript is not too large,
μ
t
is prefered
b
a
dsf(s)=
b
a
f(s)ds when it seems convenient
X
law
= Y : the random variables (or the processes) X and Y have
thesamelaw
X
mart
= Y : the process X − Y is a local martingale
X ∈F : X is a F-measurable r.v., i.e., X ∈ L
0
(F)
X ∈ bF : X is a bounded F-measurable random variable
N(x)=
1
√
2π
x
−∞
e
−y
2
/2
dy, the cumulative function for a
standard Gaussian law
Other notation can be found in the glossary at the end of the volume.
1
Continuous-Path Random Processes:
Mathematical Prerequisites
Historically, in mathematical finance, continuous-time processes have been
considered from the very beginning, e.g., Bachelier [39, 41] deals with
Brownian motion, which has continuous paths. This may justify making our
starting point in this book to deal with continuous-path random processes,
for which, in this first chapter, we recall some well-known facts. We try to
give all the definitions and to quote all the important facts for further use. In
particular, we state, without proofs, results on stochastic calculus, change of
probability and stochastic differential equations.
For proofs, the reader can refer to the books of Revuz and Yor [730],
denoted hereafter [RY], Chung and Williams [186], Ikeda and Watanabe
[456], Karatzas and Shreve [513], Lamberton and Lapeyre [559], Rogers and
Williams [741, 742] and Williams, R. [845]. See also the reviews of Varadhan
[826], Watanabe [836]andRao[729]. The books of Øksendal [684] and Wong
and Hajek [850] cover a large part of stochastic calculus.
1.1 Some Definitions
1.1.1 Measurability
Given a space Ω,aσ-algebra on Ω is a class F of subsets of Ω, such that F is
closed under complements and countable intersection (hence under countable
union) and ∅∈F(hence, Ω ∈F). For a given class C of subsets of Ω,we
denote by σ(C) the smallest σ-algebra which contains C (i.e., the intersection
of all the σ-algebras containing G).
A measurable space (Ω,F)isaspaceΩ endowed with a σ-algebra F.
A measurable map X from (Ω,F) to another measurable space (E,E)isa
map from Ω to E such that, for any B ∈E,theset
X
−1
(B):={ω ∈ Ω : X(ω) ∈ B}
belongs to F.
M. Jeanblanc, M. Yor, M. Chesney, Mathematical Methods for Financial
Markets, Springer Finance, DOI 10.1007/978-1-84628-737-4
1,
c
Springer-Verlag London Limited 2009
3
4 1 Continuous-Path Random Processes: Mathematical Prerequisites
A real-valued random variable (r.v.) on (Ω,F) is a measurable map from
(Ω,F)to(R, B)whereB is the Borel σ-algebra, i.e., the smallest σ-algebra
that contains the intervals.
Let X be a real-valued random variable on a measurable space (Ω,F).
The σ-algebra generated by X, denoted σ(X), is σ(X):={X
−1
(B);B ∈B}.
Doob’s theorem asserts that any σ(X)-measurable real-valued r.v. can be
written as h(X)whereh is a Borel function, i.e., a measurable map from
(R, B)to(R, B ) (a function such that h
−1
(B):={x ∈ R : h(x) ∈ B}∈Bfor
any B ∈B). The set of bounded Borel functions on a measurable space (E,E)
(i.e., the measurable maps from (E,E)to(R, B)) will be denoted by b(E). If
H is a σ-algebra on Ω, we shall make the slight abuse of notation by writing
X ∈Hfor: X is an H-measurable r.v. and X ∈ bH for: X is a bounded r.v.
in H.
Let (X
i
,i ∈ I) be a set of random variables. There exists a unique r.v.
with values in
¯
R, denoted esssup
i
X
i
(essential supremum of the family
(X
i
; i ∈ I)) such that, for any r.v. Y ,
X
i
≤ Ya.s.∀i ∈ I ⇐⇒ esssup
i
X
i
≤ Y.
If the family is countable, esssup
i
X
i
=sup
i
X
i
. In the case where the set I is
not countable, the map sup
i
X
i
(pointwise supremum) may not be a random
variable.
1.1.2 Monotone Class Theorem
We will frequently use the monotone class theorem which we state without
proof (see Dellacherie and Meyer [242], Chapter 1). We give two different
versions of that theorem, one dealing with sets, the other with functions.
Theorem 1.1.2.1 Let C be a collection of subsets of Ω such that
• Ω ∈C,
• if A, B ∈Cand A ⊂ B, then B\A = B ∩ A
c
∈C,
• if A
n
is an increasing sequence of elements of C, then ∪
n
A
n
∈C.
Then, if F⊂Cwhere F is closed under finite intersections, then σ(F) ⊂C.
Theorem 1.1.2.2 Let V be a vector space of bounded real-valued functions
on Ω such that
• the constant functions are in V,
• if h
n
is an increasing sequence of positive elements of V such that
h =suph
n
is bounded, then h ∈V.
If G is a subset of V which is stable under pointwise multiplication, then V
contains all the bounded σ(G)-measurable functions.
1.1 Some Definitions 5
1.1.3 Probability Measures
A probability measure P on a measurable space (Ω,F)isamapfromF
to [0, 1] such that:
• P(Ω)=1,
• P(∪
∞
i=1
A
i
)=
∞
i=1
P(A
i
) for any countable family of disjoint sets A
i
∈F,
i.e., such that A
i
∩ A
j
= ∅ for i = j.
Note that, for A ∈F, P(A)=1− P(A
c
)whereA
c
is the complement set of
A, hence P (∅)=0.
We shall often write, for J a countable set, P(A
j
,j ∈ J)forP(∩
j∈J
A
j
).
Warning 1.1.3.1 The property P(∪
∞
i=1
A
i
)=
∞
i=1
P(A
i
) does not extend to
a non-countable family.
A measurable space (Ω,F) endowed with a probability measure P is called
a probability space.
The “elementary” negligible sets are the sets A ∈Fsuch that P(A)=0.
Sets Γ ⊂ Γ
with Γ
∈Fand P(Γ
) = 0 are said to be (P, F)-negligible.
If (Ω, F) is a measurable space and P a probability measure on F,the
completion of F with respect to P is the σ-algebra of subsets A of Ω such
that there exist A
1
and A
2
in F with A
1
⊂ A ⊂ A
2
and P(A
1
)=P(A
2
)(or,
equivalently, P(A
2
∩A
c
1
) = 0). In particular, the completion of F contains all
the P-negligible sets.
1.1.4 Filtration
A filtration F =(F
t
,t ≥ 0) is a family of σ-algebras F
t
on the same
probability space (Ω,F, P), which is increasing, i.e., such that F
s
⊂F
t
for
s<t(that is: if A ∈F
s
,thenA ∈F
t
for s<t). We note F
∞
= ∨
t∈R
F
t
.
It is generally assumed that the filtration satisfies the so-called “usual
hypotheses,”thatis,
(i) the filtration is right-continuous, i.e., F
t
= ∩
u>t
F
u
,
(ii) the σ-algebra F
0
contains the (P, F)-negligible sets of F
∞
.
Usually, (but not always) the σ-algebra F
0
is the trivial σ-algebra, up to
completion.
A probability space endowed with a filtration which satisfies the usual
hypotheses is called a filtered probability space.
We shall say that a filtration G is larger than F, and write F ⊂ G,if
F
t
⊂G
t
, ∀t.
Comment 1.1.4.1 It is important that the usual hypotheses are satisfied in
order to be able to apply general results on stochastic processes, especially
when studying processes with jumps.
6 1 Continuous-Path Random Processes: Mathematical Prerequisites
1.1.5 Law of a Random Variable, Expectation
The law of a real-valued r.v. X defined on the space (Ω,F, P) is the probability
measure P
X
on (R, B) defined by
∀A ∈B, P
X
(A)=P(X ∈ A) .
It is the image on (R, B)ofP by the map ω → X(ω). This definition extends
to an R
n
-valued random variable, and, more generally, to an E-valued random
variable (a measurable map from (Ω,F)to(E,E)). If X and Y have the same
law, we shall write X
law
= Y .
The cumulative distribution function of a real valued r.v. X is the
right-continuous function F defined as F (x)=P(X ≤ x).
The expectation of a positive random variable Z is defined as
E(Z)=
ZdP =
R
+
xdP
Z
(x) ,
and, if E(|X|) < ∞,thenE(X)=E(X
+
) − E(X
−
). In case of ambiguity, we
shall denote by E
P
the expectation with respect to the probability measure P.
The r.v. X is said to be P-integrable (or integrable if there is no ambiguity)
if E(|X|) < ∞.
There are a few important transforms T of probabilities (on R,say)which
characterize a given probability μ, i.e., such that the map μ → T (μ)isone-
to-one.
• The Fourier transform F
μ
(t)=
R
e
itx
μ(dx) (where t ∈ R).
• The Laplace transform L
μ
(λ)=
R
e
−λx
μ(dx) defined on the interval
{λ ∈ R : E(e
−λX
) < ∞}. Note that the Laplace transform is well defined
on R
+
if X is positive. We shall also use, when it is defined, the Laplace
transform E(e
λX
), λ ∈ R.
1.1.6 Independence
A family of random variables (X
i
,i∈ I), defined on the space (Ω, F, P), is said
to be independent if, for any n distinct indices (i
1
,i
2
,...,i
n
) with i
k
∈ I
and for any (A
1
,...,A
n
)whereA
k
∈B,
P(∩
n
k=1
(X
i
k
∈ A
k
)) =
n
k=1
P(X
i
k
∈ A
k
) .
A classical application of the monotone class theorem is that, if the r.vs
(X
i
,i ∈ I) are independent, then, with the same notation as above, for any
bounded Borel functions f
k
,