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

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

Downloading requires you to have access to the YouScribe library
Learn all about the services we offer

Description

Comparison of semimartingalesand Levy processes withapplications to nancialmathematicsDissertationzur Erlangung des Doktorgradesder Fakult at fur Mathematik und Physikder Albert-Ludwigs-UniversitatFreiburg im Breisgauvorgelegt vonJan BergenthumJuli 2005Dekan: Prof. Dr. J. Honerkamp1. Referent: Prof. Dr. L. Rusc hendorf2. Referent: Prof. Dr. J. KallsenDatum der Promotion: 07. Oktober 2005ContentsIntroduction i1 Comparison of semimartingales 11.1 Multivariate comparison results . . . . . . . . . . . . . . . . 41.1.1 Comparison in terms of local characteristics of thestochastic logarithms . . . . . . . . . . . . . . . . . . 41.1.2 Comparison in terms of local characteristics . . . . . 111.2 Propagation of order property . . . . . . . . . . . . . . . . . 161.2.1 Monotone convexity for diusions . . . . . . . . . . . 171.2.2 Monotone convexity for diusions with jumps . . . . 211.2.3 PO(g) for processes with independent increments . . 262 Comparison of Levy processes 292.1 Compound Poisson processes. . . . . . . . . . . . . . . . . . 312.2 Levy processes with in nite Levy measures . . . . . . . . . . 412.3 Extension to PII . . . . . . . . . . . . . . . . . . . . . . . . 543 Applications 593.1 Non-trivial bounds for European option prices . . . . . . . . 603.2 Comparison of martingale measures . . . . . . . . . . . . . . 713.2.1 Jump models . . . . . . . . . . . . . . . . . . . . . . 733.2.2 Stochastic volatility models . . . . . . . . . .

Subjects

Informations

Published by
Published 01 January 2005
Reads 4
Language English

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 time I had during my PhD studies,
and especially I thank Monika Hattenbach for helping with the Latex prob-
lems and checking through the manuscript. Finally, I thank my family and
friends, and, last but not least, very special thanks go to Charlotte Bergen.Chapter 1
Comparison of terminal values
of semimartingales
In this chapter we derive several comparison results for terminal values of
multivariate special semimartingales in terms of di erential characteristics
(oftheirstochasticlogarithms).Thisisanextensiontothemultivariatecase
of the stochastic calculus approach given in El Karoui, Jeanblanc-Picque,
and Shreve (1998), Frey and Sin (1999), Bellamy and Jeanblanc (2000) and
Gushchin and Mordecki (2002). El Karoui, Jeanblanc-Picque, and Shreve
(1998)establishthatEuropeanoptionpricesw.r.t.stochasticvolatilitymod-
elsS are smaller than the prices that come from a di usion model S , if the
di usion coe cient dominates the stochastic volatility in a suitable
sense, see also the example in the introduction. The reversed comparison
result also holds true. Frey and Sin (1999) consider the special case where
the bounding di usion S is given by a geometric Brownian motion with
constant volatility. Bellamy and Jeanblanc (2000) establish that European
option prices w.r.t. a di usion with jumps model S are bounded below by
the European option price w.r.t. a di usion S that has the same di usion
coe cient as S. Gushchin and Mordecki (2002) extend these results to the
comparisonofEuropeanoptionpricesw.r.t.aspecialsemimartingalemodel
S with the corresponding price w.r.t. a Markovian special semimartingale
S .
Inallthesecomparisonresultsthecomparison processS isMarkovian.This
assumption seems to be necessary, as the following comparison example for
two stochastic volatility models shows. This example was communicated to
us by J. Kallsen.
Example 1 (Larger volatility does not imply a larger price in SV
models). Let ∈R and W be a one-dimensional Brownian motion on a+
12 CHAPTER 1. COMPARISON OF SEMIMARTINGALES
stochastic basis ( ,A,(A ) ,P). Let S be the solution oft t∈[0,T]
dS =S dW , S = 1.t t t 0
For K > 1 we de ne an ( A )-adapted process ast t
(
, max S K,ut u :=t
0, else.
Let S be a solution of dS = S dW , S = 1. Let 0 < e < andtt t t 0
T ∈ [0,T] be xed. De ne an ( A )-adapted process as0 t t

e, max S K, 0tT , ut u 0
:= , max S K, T <tT,t ut u 0

0, otherwise,
and let S be a solution of dS = S dW ,S = 1. Then and S,St t t t 0 t t
are not Markovian as their volatilities are path-dependent.
The price of a European call option with strike K is zero under the S
model.ButasthereisapositiveprobabilityforpathswithS >S ,henceT0 T0
P(S >K)> 0, it follows that the price of that call option is positive withT
respect to the S model, although the volatility of S is smaller than the
volatility of S .
We derive stochastic and convex type comparison results for terminal val-
ues of d-dimensional special semimartingales S,S with nite time horizon
T that are de ned on stochastic bases ( ,A,(A ) ,P) and ( ,A ,t t∈[0,T]
(A ) ,P ), respectively. We assume that the characteristics B(h), C,t∈[0,T]t
of S are absolutely continuous w.r.t. the Lebesgue measure. In this case,
B(h), C, have representation
Z Zt t
i i ij ijB (h) = b (h)ds, C = c ds, (ω;dt,dx) =dtK (dx),ω,tt s t s
0 0

iwith di erential characteristics b(h), c and K, where b(h) = b (h) is
id
a d-dimensional predictable process with associated truncation function h,
ijc = (c ) is a predictable process with values in M (d,R), the set ofi,jd +
all symmetric, positive semide nite dd-matrices with entries in R, and
d dK (dx) is a transition kernel from ( [0,T],P) into (R ,B ) that sat-ω,t
is es some conditions, cp. Jacod and Shiryaev (2003, Proposition II.2.9).
Similarly, we assume that the characteristics of the comparison process S