Comparison of semimartingales and Lévy processes with applications to financial mathematics [Elektronische Ressource] / vorgelegt von Jan Bergenthum

Informations

• Publié par: albert-ludwigs-universitat_freiburg
• Publié le: 01 janvier 2005
• Nombre de lectures: 4
• Langue: English

Extrait

Comparison of semimartingales
and Levy processes with
applications to nancial
mathematics
Dissertation
zur Erlangung des Doktorgrades
der Fakult at fur Mathematik und Physik
der Albert-Ludwigs-Universitat
Freiburg im Breisgau
vorgelegt von
Jan Bergenthum
Juli 2005Dekan: Prof. Dr. J. Honerkamp
1. Referent: Prof. Dr. L. Rusc hendorf
2. Referent: Prof. Dr. J. Kallsen
Datum der Promotion: 07. Oktober 2005Contents
Introduction i
1 Comparison of semimartingales 1
1.1 Multivariate comparison results . . . . . . . . . . . . . . . . 4
1.1.1 Comparison in terms of local characteristics of the
stochastic logarithms . . . . . . . . . . . . . . . . . . 4
1.1.2 Comparison in terms of local characteristics . . . . . 11
1.2 Propagation of order property . . . . . . . . . . . . . . . . . 16
1.2.1 Monotone convexity for diusions . . . . . . . . . . . 17
1.2.2 Monotone convexity for diusions with jumps . . . . 21
1.2.3 PO(g) for processes with independent increments . . 26
2 Comparison of Levy processes 29
2.1 Compound Poisson processes. . . . . . . . . . . . . . . . . . 31
2.2 Levy processes with in nite Levy measures . . . . . . . . . . 41
2.3 Extension to PII . . . . . . . . . . . . . . . . . . . . . . . . 54
3 Applications 59
3.1 Non-trivial bounds for European option prices . . . . . . . . 60
3.2 Comparison of martingale measures . . . . . . . . . . . . . . 71
3.2.1 Jump models . . . . . . . . . . . . . . . . . . . . . . 73
3.2.2 Stochastic volatility models . . . . . . . . . . . . . . 82
3.3 Comparison of path-dependent options . . . . . . . . . . . . 86
3.3.1 Lookback options . . . . . . . . . . . . . . . . . . . . 86
3.3.2 Asian options . . . . . . . . . . . . . . . . . . . . . . 88
3.3.3 American options . . . . . . . . . . . . . . . . . . . . 93
3.3.4 Barrier options . . . . . . . . . . . . . . . . . . . . . 96
3.4 Ordering results for -stable and NIG processes . . . . . . . 99
A Appendix 105
A.1 Stochastic analysis . . . . . . . . . . . . . . . . . . . . . . . 105
A.2 Stochastic orders . . . . . . . . . . . . . . . . . . . . . . . . 111
iIntroduction
This thesis studies univariate and multivariate stochastic and convex type
orderings of semimartingales and of nite-dimensional distributions of Levy
processeswithseveralapplications,especiallyin nancialmathematics.The
orderings are with respect to one of the following order generating function
classesF
dF :={f :R →R,f is increasing},st
dF :={f :R →R,f is convex},cx
dF :={f :R →R,f is directionally convex}, (1)dcx
dF :={f :R →R,f is supermodular},sm
F :=F ∩F , F :=F ∩F , F :=F ∩F .icx cx st idcx dcx st ism sm st
In the literature, various ordering results for univariate and multivariate
random variables are given; we refer to Marshall and Olkin (1979), Tong
(1980), Shaked and Shanthikumar (1994), Szekli (1995), and Mulle r and
Stoyan (2002) for results and references. There is also an extended theory
for ordering of discrete time processes (like queuing sequences, renewal se-
quences, Markov chains) or related point processes with a wide variety of
applications.Stochasticorderingresultsw.r.t.F havealsobeenestablishedst
under various conditions for di usion type processes (cp. Yamada (1973),
Ikeda and Watanabe (1977), O’Brien (1980), Gal’cuk (1982), Gal’cuk and
Davis (1982)) and for Markov processes (cp. Massey (1987)). These results
are parallel to classical comparison theorems for solutions of di erential
equations. Several convex comparison results for one-dimensional (expo-
nential) stochastic models have been developed in recent papers in nancial
mathematics.Themainaiminthesepapersistoderivesharpupperorlower
bounds for option prices in incomplete markets or to derive comparison of
option prices that result from di erent martingale measures. The methods
used in these papers are based on stochastic calculus (Itˆo formula) and
the propagation of convexity property (see El Karoui, Jeanblanc-Picque,
and Shreve (1998), Frey and Sin (1999), Bellamy and Jeanblanc (2000),
Gushchin and Mordecki (2002), Henderson, Hobson, Howison, and Kluge
iii INTRODUCTION
(2003)), as well as on the coupling method (see Hobson (1998b), Hender-
son and Hobson (2003), M ller (2004), Henderson (2005)), and also on
time change for Brownian motions, (see Eriksson (2004, 2005)). The or-
dering results are of the following type. The price of a European option
with convex payo function g w.r.t. a stochastic volatility model S with
(positive) volatility process is bounded above by the price of this optiont
(with the same convex payo function g) w.r.t. a generalized Black–Scholes
model S with volatility (t,S ), if the volatilities are ordered in a suit-t
able sense, namely (ω) (t,S (ω)) almost surely. As and (t,S )t t t t
are the roots of the di erential Gaussian characteristics of the stochastic
logarithms X = Log(S), X = Log(S ), the ordering result translates as
follows in terms of semimartingale characteristics and convex type orders:
Ordering of the di erential characteristics of the stochastic logarithms of S,
S in a suitable sense implies convex ordering S S in this case.T cx T
InChapter1weextendthestochasticcalculusapproachofElKaroui,Jean-
blanc-Picque, and Shreve (1998), Frey and Sin (1999), Bellamy and Jean-
blanc (2000), and Gushchin and Mordecki (2002) to obtain orderings of
terminal values of d-dimensional semimartingales S, S , where S is as-
sumed to be Markovian. The ordering is derived in terms of the di erential
characteristicsofthestochasticlogarithmsX =Log(S),X =Log(S )and
also in terms of the di erential characteristics of S and S . The argument
strongly relies on the Markov property of S and on the propagation of
order property of the backward function G(t,s) = E (g(S )|S = s). WeT t
developanewapproachtoestablishthispropertyforseveralunivariateand
multivariate models in Section 1.2.
In Chapter 2 we derive orderings of nite-dimensional distributions of uni-
variateandmultivariateLevyprocessesintermsoftheirLevycharacteristics
by a di erent approach. In the rst step we establish that ordering of all
timemarginalsoftwoMarkovprocessesw.r.t.F,whereF isoneoftheorder
generating function classes in (1), implies ordering of the nite-dimensional
distributions w.r.t.F, if a -monotone separating transition kernel exists.F
In Section 2.1 we derive orderings of marginals of compound Poisson pro-
cessesbyacouplingargument,whichreliesonarandomsumrepresentation.
WeestablishthatthetransitionkernelsofLevyprocessesare -monotoneF
forallF in(1),henceorderingofthe nite-dimensionaldistributionsfollows.
ThenweestablishseveralcutanddominationcriteriafortheLevymeasures
of one-dimensional compound Poisson processes that imply stochastic and
(increasing)convexorderingoftheprocesses.InSection2.2weextendthese
resultstoLevyprocesseswithin niteLevymeasures.WetruncatetheLevy
measures around the origin, establish nite-dimensional ordering of the re-INTRODUCTION iii
sultingcompoundPoissonprocesses,andthenobtainorderingofthelimits.
Again, we derive several cut and domination criteria for univariate Levy
processes in terms of the corresponding Levy measures. In Section 2.3 we
extend these results to processes with independent increments.
In Chapter 3 we give several applications of the ordering results of Chap-
ters 1 and 2, mainly to the eld of nancial mathematics. In Section 3.1
we obtain non-trivial bounds for European option prices in several univari-
ate and multivariate incomplete market models. These include stochastic
volatility models with and without jumps, exponential and stochastic ex-
ponential Levy models and Levy driven di usions. Section 3.2 deals with
comparison of martingale measures in incomplete market models. We con-
sider several well established martingale measures in a di usion with jumps
model of stochastic exponential type, in a compound Poisson model, in a
PII model and in a stochastic volatility model, and obtain orderings of Eu-
ropean option prices w.r.t. these measures. In Section 3.3 we derive several
ordering results for prices of path-dependent options, some of which use
ordering results of Chapters 1 and 2, others extend these results and are
derived in a similar fashion. We consider lookback options, Asian options
with continuous averaging, American options, and single-barrier options.
The ordering results are parallel to the results of Chapters 1 and 2. Or-
dering the di erential characteristics of the underlyings implies ordering of
the options prices of the path-dependent options. Finally, in Section 3.4 we
obtain ordering results for nite-dimensional distributions of -stable pro-
cesses, ∈ (1,2), and of NIG processes in the parameters of the models by
twodi erentapproaches.The rstapproachisanapplicationofthecutand
domination criteria of Chapter 2, the second approach makes use of mixing
type representations of generalized hyperbolic (GH) distributions.
I thank my advisor Ludger Rusc hendorf for encouraging me to this project
and for numerous suggestions and comments. In particular, Chapter 1 is
basedonBergenthumandRusc hendorf(2004a)andthe rstpartofBergen-
thum and Rus chendorf (2004b), and Chapter 2 is partly based on Bergen-
thum and Rusc hendorf (2004b). I thank my colleagues from Abteilung fur
Mathematische Stochastik for the good tim