Introduction
SinceBlackandScholespublishedtheirarticleonoptionpricingin1973, there
has been an explosion of theoretical and empirical work on the subject. How-
ever, over the last thirty years, a vast number of pricing models have been
proposed as an alternative to the classic Black-Scholes approach, whose as-
sumption of lognormal stock diffusion with constant volatility is considered
always more flawed.
OnemajorreasonisthatsincethestockmarketcrashofOctober19, 1987, de-
viations of stock index option prices from the benchmark Black-Scholes model
havebeenextraordinarilypronounced. Infact,sincethen,toequatetheBlack-
Scholes formula with quoted prices of European calls and puts, it is generally
necessary to use different volatilities, so-called implied volatilities, for differ-
entoptionstrikesandmaturities(theBlack-Scholesmodelrequiredaconstant
volatility based on the subjacent historical volatility). That feature suggests
that the distribution perceived by market participant and incorporated into
optionpricesissubstantiallynegativelyskewed(thatistosayleptokurticwith
a fat tail on the negative side), in contrast to the essentially symmetric and
slightly positively lognormal distribution underlying the Black-Scholes model.
The pattern formed by the implied volatilities across the strikes is then called
ix
x
volatility smile or skew, due to the fact that the implied volatility of in-the-
money call options is pretty much higher than the one of out-of-the-money
options. Typically, the steepness of the skew decreases with increasing op-
tion maturities. The existence of the skew is often attributed to fear of large
downward market movements. The research of a new form of models able to
incorporate the smile has been one of the most active fields of studies in mod-
ern quantitative finance.
There are two assumptions that have to be made in order to price derivatives
with the Black-Scholes model: returns are subject to a single source of uncer-
tainty and asset prices follow continuous sample paths (a Brownian motion).
Then, under these two assumptions, a continuously rebalanced portfolio can
beusedtoperfectlyhedgeanoptionsposition,thusdeterminingauniqueprice
for the option.
Therefore, extensions of the Black-Scholes model that capture the existence of
volatility smile can, broadly speaking, be grouped in two approaches, each one
relaxing one of these two assumptions. Relaxing the assumption of a unique
source of uncertainty leads to the stochastic volatility family of models, where
the volatility parameter follows a separate diffusion, as proposed by Heston
[58]. Relaxingtheassumptionofcontinuoussamplepaths, leadstojumpmod-
els, where stock prices follow an exponential L´ evy process of jump-diffusion
type (where evolution of prices is given by a diffusion process, punctuated by
jumps at random intervals) or pure jumps type. Jump models attribute the
biases in Black-Scholes model to fears of a further stock market crash. They
wouldinterpretthecrashasarevelationthatjumpscaninfactoccur. Looking
to a plot of a stock index time series, there is clear evidence that prices don’t
xi
follow a diffusion process and actually jump.
This thesis deals with the study of L´ evy processes for option pricing. L´ evy
processesareanactivefieldofresearchinfinance, andmanymodelshavebeen
presented during the last decade. This thesis is not an attempt to describe all
theL´ evymodelsdiscussedintheliteratureorexplaintheirmathematicalprop-
erties. We focus on four famous models, two of jump-diffusion type (Merton
normal jump-diffusion and Kou double-exponential jump-diffusion) and two
pure jump models (Variance Gamma and Normal Inverse Gaussian) for which
wedescribetheirforemostmathematicalcharacteristicsandweconcentrateon
providing modeling tools.
In the first chapter, we present the mathematical tools useful for option
pricing. We discuss some characteristics of stochastic processes and financial
mathematics in continuous time. This chapter can be seen as a prerequisite.
In a second chapter, we discuss the limitations of the Black-Scholes model and
describe its weaknesses. We also explain from a statistical point of view how
the hypothesis of lognormal returns defined by the Black-Scholes model goes
wrong in describing market returns. Finally, we roughly present some alterna-
tive models discussed in the literature.
In a third chapter, we introduce L´ evy processes and present their major math-
ematicalproperties,beginningfromPoissonprocesswhichisthestartingpoint
of jump processes.
In the fourth chapter, we present the four L´ evy processes we selected for op-
tion pricing: Merton normal jump-diffusion, Kou double-exponential jump-
diffusion, Variance Gamma and Normal Inverse Gaussian. An important part
ofthechapterisdedicatedtothesimulationofsuchprocesses. Inthelastpart,
xii
we give the neutral characteristic function of each model, needed for option
pricing in the successive chapter.
The fifth chapter treats about option pricing methods. Jump models intro-
duce forms of risk included in option prices that are not directly priced by any
instrument currently traded in financial markets (unlike bonds for example).
The result is that the Black-Scholes arbitrage-based methodology cannot be
used. However, given the risk-neutral pricing formula and the fact that the
characteristic function is always known for a L´ evy process, even if the prob-
ability density is not, Fourier-based option pricing method are possible. We
first expose the FFT option pricing method given by Carr and Madan [24],
and extend it to the Fractional Fourier Transform (FRFT), a lot faster. We
finally give some results about the goodness of the Fourier approximation of
option prices.
Finally, the sixth chapter deals with the calibration of parameters to market
option prices. We first select four sets of option market prices and show how
bad results the Black-Scholes model gives. We then present the non-linear
least-squares method used to recover the parameters of L´ evy processes and
finally show the improvement in the fitting of market and model prices with
these models.
The code written to perform simulation, pricing and calibration in chapter
4, 5 and 6 is an important part of the thesis. All the code is available in the
appendix at the end of the thesis and downloadable at the following website:
http://ddeville.110mb.com/thesis/
The code is written withMatlab. The reason why we choseMatlab rather
xiii
thanaprogramminglanguagelikeCorC++isthelevelofdifficulty. Matlab
code is easy to write and above all easy to read, even to someone with a very
few knowledge of coding.
Chapter 1
Stochastic Processes and
Mathematical Finance
1.1 Probability, Stochastic Processes, Filtra-
tions
Definition 1.1 (Algebra) Let Ω be a nonempty set, and letF be a collection
of subsets of Ω. We say thatF is an algebra provided that:
(i) Ω∈F and∅∈F,
(ii) A∈F⇒A
c
= Ω\A∈F,
(iii) A,B∈F⇒A∪B∈F.
Definition 1.2 (σ-algebra) AnalgebraF ofsubsetsofΩiscalledaσ-algebra
on Ω if for any sequence (A
n
)
n∈N
∈F, we have
∞
[
n=1
A
n
∈F
2 Stochastic Processes and Mathematical Finance
Such a pair (Ω,F) is called a measurable space.
Thus, aσ-algebra on Ω is a family of subsets of Ω closed under any countable
collection of set operations. The σ-algebra generated by all open subsets is
called the Borel σ-algebra:B(E).
Definition 1.3 (Probability) Let Ω be a nonempty set, and letF be a σ-
algebra of subsets of Ω. A probability measure P is a function that, to every
set A∈F assigns a number in [0,1], called the probability of A and written
P(A). We require:
(i) P(Ω) = 1, and
(ii) (countable additivity) whenever A
1
,A
2
,... is a sequence of disjoint sets
inF, then
P
∞
[
n=1
A
n
!
=
∞
X
n=1
P(A
n
). (1.1)
The triple (Ω,F,P) is called a probability space.
A probability space isP-complete if for each B⊂A∈F such thatP(A) = 0,
we have B∈F.
In a dynamic context, as time goes on, more information is progressively
revealed to the observer. We must thus add some time-dependent ingredient
to the structure of our probability space (Ω,F,P).
Definition 1.4 (Filtration) A filtration (or information flow) on (Ω,F,P)
is an increasing family of σ-algebras (F
t
)
t∈[0,T]
:
F
s
⊂F
t
⊂F
T
⊂F for 0≤s<t≤T.
1.1 Probability, Stochastic Processes, Filtrations 3
F
t
represents the information available at time t, and the filtration (F
t
)
t∈[0,T]
represents the information flow evolving (increasing) with time.
A probability space (Ω,F,P) equipped with a filtration is called a filtered
probability space (Ω,F,P,(F
t
)
t∈[0,T]
).
Definition 1.5 (Usual conditions) We say that a filtered probability space
(Ω,F,P,(F
t
)
t∈[0,T]
) satisfies the “usual conditions” if:
(i)F isP-complete.
(ii)F
0
contains all P-null sets of Ω. This means intuitively that we know
which events are possible and which are not.
(iii) (F
t
)
t∈[0,T]
is right-continuous, i.e.F
t
=F
t+
:=
T
s>t
F
s
.
Definition 1.6 (Stochastic processes) A stochastic process (X
t
)
t∈[0,T]
is a
family of random variables indexed by time, defined on a filtered probability
space (Ω,F,P,(F
t
)
t∈[0,T]
).
Thetimeparametertmaybeeitherdiscreteorcontinuous. Foreachrealization
of the randomness ω, the trajectory X(ω) : t→ X
t
(ω) defines a function of
time called the sample path of the process. Thus stochastic processes can also
be viewed as random functions.
Definition 1.7 (C` adl` ag function) A function f : [0,T]→ R
d
is said to
be c` adl` ag (from French “continu ` a droite, limite ` a gauche”) if it is right-
continuouswithleftlimits. Iftheprocessisc` agl` ad(left-continuous), oneshould
be able to “predict” the value at t -“see it coming”- knowing the values before
t.
4 Stochastic Processes and Mathematical Finance
Definition 1.8 (Adapted processes) Astochasticprocess(X
t
)
t∈[0,T]
issaid
to beF
t
-adapted (or nonanticipating with respect to the information structure
(F
t
)
t∈[0,T]
) if, for each t∈ [0,T], the value of X
t
is revealed at time t: the
random variable X
t
isF
t
-measurable.
Definition 1.9 (Stopping times) A random time is a positive random vari-
ableT≥ 0whichrepresentsthetimeatwhichsomeeventisgoingtotakeplace.
If, given an information flowF
t
, someone can determine whether the event has
happened (τ≤t) or not (τ >t), the random time τ is called a stopping time
(or nonanticipating random time). In other words, τ is a non-anticipating
random time ((F
t
)-stopping time) if
∀t≥ 0, {τ≤t}∈F
t
.
1.2 Classes of Processes
1.2.1 Markov Processes
A Markov process is a particular type of stochastic process where only the
present value of a variable is relevant for predicting the future. The past
history of the variable and the way that the present has emerged from the
past are irrelevant (the past history is, say, integrated in the present value).
Definition 1.10 (Markov process) Let (Ω,F,P) be a probability space, let
T be a fixed positive number, and let (F
t
)
t∈[0,T]
be a filtration. Consider
an adapted stochastic process (X
t
)
t∈[0,T]
. If, for a well-behaved (i.e. Borel-
measurable) function f:
E[f(X
t
)|F
s
] =E[f(X
t
)|X
s
] (1.2)
1.2 Classes of Processes 5
the process (X
t
)
t∈[0,T]
is a Markov process.
1.2.2 Martingales
Definition 1.11 (Martingale) A c` adl` ag stochastic process X = (X
t
)
t∈[0,T]
is a martingale relative to (P,F
t
) if
(i) X isF
t
-adapted,
(ii) E[|X
t
|]<∞ for any t∈ [0,T],
(iii)∀s<t
E[X
t
|F
s
] =X
s
(1.3)
X is a supermartingale if in place of (iii)
E[X
t
|F
s
]≤X
s
∀s<t (1.4)
X is a submartingale if in place of (iii)
E[X
t
|F
s
]≥X
s
∀s<t (1.5)
In other words, the best prediction of a martingale’s future value is its present
value. Martingale have a useful interpretation in terms of dynamic games: a
martingaleis“constantonaverage”,andmodelsafairgame;asupermartingale
is “decreasing on average”, and models an unfavorable game; a submartingale
is “increasing on average”, and models a favorable game.
Martingalesrepresentsituationsinwhichthereisnodrift, ortendency, though
there may be a lot of randomness. In the typical statistical situation where we
have data = signal + noise, martingales are used to model the noise compo-
nent.
A familiar example of martingale is the Wiener process W
t
.
6 Stochastic Processes and Mathematical Finance
1.3 Characteristic Functions
The characteristic function of a random variable is the Fourier transform of
its distribution. Many probabilistic properties of random variables correspond
to analytical properties of their characteristic functions, making this concept
very useful for studying random variables.
Definition 1.12 (Characteristic function) The characteristic function of
theR
d
-valued random variable X is the function Φ
X
:R
d
→R defined by
Φ
X
(t) =E(e
itX
) =Ecos(tX)+iEsin(tX) (1.6)
Let F
X
be the distribution function of X. Then
Φ
X
(t) =E(e
itX
) =
Z
+∞
−∞
e
itx
dF(x) (1.7)
so that Φ is the Fourier transform ofF, but without a constant multiplier such
as (2π)
−1/2
which is used in much of Fourier analysis.
The characteristic function of a random variable determines the probability
distribution: twovariableswiththesamecharacteristicfunctionareidentically
distributed. A characteristic function is always continuous and verifies
Φ
X
(0) = 1 |Φ
X
(t)|≤ 1 Φ
aX+b
(t) =e
itb
Φ
X
(at)
Theorem 1.13 If Φ
X
is integrable, then X has a density which is given by
f
X
(x) =
1
2π
Z
∞
−∞
e
−iux
Φ
X
(u)du.
1.4 Brownian Motion 7
Example 1.14 (Gaussian CF) For a normal distribution N(μ,σ
2
), we can
define the density and characteristic function as:
f(x) =
1
σ
√
2π
e
−
1
2
(x−μ)
2
σ
2
Φ
X
(z) =e
iμz−
1
2
σ
2
z
2
(1.8)
Example 1.15 (Poisson CF) For a Poisson distribution P(λ), we can de-
fine the probability mass and characteristic function as:
f(k) :=P(X =k) =
e
−λ
λ
k
k!
Φ
X
(z) =e
−λ(1−e
iz
)
(1.9)
1.4 Brownian Motion
1.4.1 Normal Distribution
Thenormaldistribution,N(μ,σ
2
)is(oneof)themostimportantdistributions.
As seen before, its characteristic function is given by:
Φ
Normal
(z;μ,σ
2
) =e
iμz−
1
2
σ
2
z
2
(1.10)
and the density function is:
f
Normal
(x;μ,σ) =
1
σ
√
2π
e
−
1
2
(x−μ)
2
σ
2
(1.11)
Thenormal, bydefinition, issymmetricarounditsmean, hasaskewnessequal
to 0 and a kurtosis equal to 3.
1.4.2 Brownian Motion
Brownian motion is the dynamic counterpart - where we work with evolution
in time - of the Normal distribution. Brownian motion originates in work of
8 Stochastic Processes and Mathematical Finance
the botanist Robert Brown in 1828. It was first introduced into finance by
Louis Bachelier in 1900, and developed in physics by Albert Einstein in 1905.
Brownian motion was first proved mathematically by Norbert Wiener in 1923.
In honor of this, Brownian motion is also known as the Wiener process.
Definition 1.16 (Brownian motion) A stochastic process X = (X
t
)
t≥0
is
a standard (one-dimensional) Brownian motion,W, on some probability space
(Ω,F,P), if
(i) X(0) = 0, almost surely,
(ii) X hasindependentincrements: X(t+u)−X(t)isindependentofσ(X(s) :
s≤t) for u≥ 0,
(iii ) X has stationary increments: the law of X(t+u)−X(t) depends only
on u,
(iv ) X has Gaussian increments: X(t+u)−X(t) is normally distributed
with mean 0 and variance u, i.e. X(t+u)−X(t)∼N(0,u),
(v) X has continuous paths: X(t) is a continuous function of t, i.e. t→
X(t,ω) is continuous in t for all ω∈ Ω.
Filtration for Brownian motion
Definition 1.17 Let (Ω,F,P) be a probability space on which is defined a
Brownian motion W
t
, t≥ 0. A filtration for the Brownian motion is a collec-
tion of σ-algebrasF
t
, t≥ 0, satisfying:
1.4 Brownian Motion 9
Figure 1.1: Sample path of a standard Brownian motion
(i) (Information accumulates) For 0≤ s < t, every set inF
s
is also in
F
t
. In other words, there is at least as much information available at the later
timeF
t
as there is at the earlier timeF
s
.
(ii) (Adaptivity) For each t≥ 0, the Brownian motion W
t
at time t isF
t
measurable. In other words, the information available at time t is sufficient to
evaluate the Brownian motion W
t
at that time.
(iii) (Independence of future increments) For 0≤t<u, the increment
W
u
−W
t
is independent ofF
t
. In other words, any increment of the Brownian
motion after time t is independent of the information available at time t.
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