American-Type Options, Volume 2: Stochastic Approximation Methods

573 Pages·2015·3.71 MB·English

by  Dmitrii S. Silvestrov

Dmitrii S. Silvestrov
American-Type Options
De Gruyter Studies
in Mathematics
Edited by
Carsten Carstensen, Berlin, Germany
Nicola Fusco, Napoli, Italy
Fritz Gesztesy, Columbia, Missouri, USA
Niels Jacob, Swansea, United Kingdom
Karl-Hermann Neeb, Erlangen, Germany
Volume 57
Dmitrii S. Silvestrov
American-Type
Options
Stochastic Approximation Methods
Volume 2
DE GRUYTER
MathematicsSubjectClassification2010
91G20,91G60,60G40,60J05,60J10,60G40,60G15,60J22,60J10,60G15,65C40,62L15
Author
Prof.Dr.DmitriiS.Silvestrov
StockholmUniversity
DepartmentofMathematics
SE-10691Stockholm
Sweden
[email protected]
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ISSN0179-0986
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Preface
Thisbookisthe2ndvolumeofthemonographonstochasticapproximationmeth-
ods for American-type options.
The 1st volume, Silvestrov (2014), was devoted to stochastic approximation
methods for models of American-type options with general pay-off functions, for
discrete time Markov log-price processes.
Inthe2ndvolume,wesystematicallypresentstochasticapproximationmeth-
odsforAmerican-typeoptionswithgeneralpay-offfunctions,forcontinuoustime
Markov log-price processes.
Theprincipalnoveltyofourstudiesisthatwesystematicallyconsidercontin-
uous time multivariate modulated Markov log-price processes and general pay-off
functions,whichcandependnotonlyonlog-pricebutalsoanadditionalstochastic
modulatingindexcomponentwithageneralphasespace.Wealsoimposeminimal
conditions of smoothness on transition probabilities and pay-off functions as well
asminimalmomentcompactnessconditionsonlog-priceprocessesandrestrictions
on the rate of growth for pay-off function in log-price arguments.
The book contains ten chapters.
In Chapters 1 and 2, we present results about reward approximations for
American-type options with general pay-off functions, for autoregressive and au-
toregressive stochastic volatility log-price processes. These chapters are included
in the 2nd volume, in order to balance in size two volumes of the monograph.
In Chapter 3, we introduce models of multivariate modulated Markov log-
price processes and consider main examples of such processes, including log-price
processes with independent increments and diffusion log-price processes; define
basic objects connected with American type options, namely, reward functions,
optimal expected rewards and optimal stopping times; and present some typical
examples of options such as call and put options, exchange of assets options, and
their portfolios.
In Chapter 4, we give upper bounds for reward functions and moments of
random rewards for American type options with general pay-off functions, which
have not more than polynomial rate of growth in price arguments, for multi-
variate modulated Markov log-price processes. We also specify these results for
log-priceprocesseswithindependentincrements,diffusionlog-priceprocessesand
their time-skeleton, martingale, and trinomial-tree approximations as well as for
mean-reverse diffusion log-price processes.
In Chapter 5, we give asymptotically uniform explicit upper bounds for dis-
tance between optimal expected rewards for American-type options with general
pay-offfunctions,formultivariatemodulatedMarkovlog-priceprocessesandopti-
malexpectedrewardsforAmerican-typeoptionswithgeneralpay-offfunctions,for
embedded discrete time modulated Markov log-price processes based on skeleton
vi Preface
partitions of time interval. These upper bounds play a key role in getting results
about convergence of reward approximation algorithms studied in this book.
In Chapter 6, we presents results about convergence of time-space-skeleton
reward approximations, for continuous time multivariate modulated Markov log-
priceprocesses,approximated,inthiscase,byatomicMarkovchains,whichtran-
sition probabilities and initial distributions concentrated on finite sets of skeleton
points. These results are essentially used for getting results about convergence of
rewardsforcontinuoustimemodels.Atthesametime,theyhavetheirownvalue,
since American-type options for approximating embedded discrete time log-price
processes can be interpreted as Bermudian-type options for the corresponding
continuous time log-price processes.
In Chapters 7 and 8, we present our main convergence results for rewards
of American-type options with general perturbed pay-off functions, for perturbed
multivariate modulated Markov log-price processes. What is important that we
impose minimal conditions of smoothness on the limiting transition probabilities
and pay-off functions. For the basic case, where the transition probabilities have
densitieswithrespecttosomepivotalLebesgue-typemeasure,itisusuallyrequired
thatthesetsofweakdiscontinuityforthelimitingtransitionprobabilitiesarezero
sets with respect to the above pivotal measures. As far as pay-off functions are
concerned, we impose on them Lipschitz-type conditions, which are weaker than
conditions involving derivatives of pay-off functions used usually in integrodiffer-
ential approximation methods. Chapter 7 presents convergence results for gen-
eralmultivariatemodulatedMarkovlog-priceprocessesandmultivariatelog-price
processes with independent increments, including Gaussian processes with inde-
pendentincrements.Chapter8presentsconvergenceresultsfordiffusionlog-price
processesandtheirskeleton,martingale,andtrinomial-treeapproximationsaswell
as some results about reward approximations for mean-reverse diffusion log-price
processes.
In Chapter 9, we show, in which way results about reward approximations
for American-type options can be applied to simpler European-type options and
to more complex knockout American-type options; present reward convergence
results for American-type options, which appear in the model of reselling of
European options; and introduce and present results about convergence of re-
wards for American-type options with random pay-off.
In Chapter 10, we present some results of experimental numerical studies
for reward stochastic approximation algorithms based on theoretical results for
discrete and continuous time models of American-type options presented in the
1st and 2nd volumes of the book.
We should also point out that the plan of the 2nd volume, announced in the
1st volume, was slightly changed. It was decided to concentrate presentation on
resultsrelatedtostochasticapproximationmethodsforrewardsofAmerican-type
options, which play a central role in our research studies. Some interesting them-
Preface vii
selvesresultsaboutoptimalstoppingtimes,inparticular,aboutmulti-threthhold
structure of optimal stopping domains, and Monte Carlo reward approximation
algorithms were excluded, in order to satisfy requirements concerned the size of
the book. However, we give references to papers containing the corresponding
results.
The bibliography, which contains more than 300 references, is also preceded
bybriefremarks.Itshouldbenotedthatthebibliographyforthe2ndvolumejust
supplements the bibliography for the 1st volume containing about 600 references.
The presentation of material in the present book is organized in a way that
will hopefully be appreciated by readers. Each chapter has a preamble, in which
the main results are outlined and the content of chapter by sections is presented.
Each section is broken up into titled subsections.
I would like also to comment on the notation system used in the book.
Throughout the text I make use of several basic classes of conditions. Condi-
tions that belong to a specific class are denoted by the same letter. For example,
the letter B is used for conditions imposed on pay-off functions, the letter C
for moment compactness conditions imposed on log-price processes, and so forth.
Conditions belonging to a specific class have subscripts numbering conditions in
the class. A list of all conditions is given in an index.
Chaptersandsectionshave,respectively,asingularandadoublenumeration.
Formulasalsohaveadoublenumeration.Forexample,label(1.2)referstoformula
2 in Chapter 1. Subsections, theorems, lemmas, definitions and remarks have a
triple numeration. For example, Theorem 1.2.3 means Theorem 3 in Section 1.2.
References to chapters, sections, lemmas, theorems, formulas or conditions
from the 1st volume are indexed by symbol ∗, for example, the symbol B refer
∗
to condition B in the 1st volume. Reference Theorem 1.2.3 means Theorem 3 in
∗
Section 1.2, in the 1st volume of the book.
I hope that the publication of this 2nd volume of the monograph and the
additional bibliography of works related to problems of stochastic approximation
methods for American-type options and related problems will be, together with
the 1st volume, a useful contribution to the continuing intensive studies in this
actual area of financial mathematics. In addition to its employment for research
andreferencepurposes,thebookcanbeusedinspecialcoursesrelatedtostochas-
tic approximation methods and option pricing and as a complementary reading
in general courses on stochastic processes. In this respect, it may be useful for
specialists as well as doctoral and advanced undergraduate students.
I am much indebted to Dr. Evelina Silvestrova for her continuous encourage-
ment and support of various aspects of my work on this book.
Iwouldliketothankmycollaboratorsatdifferentstagesofmyresearchstud-
iesinthearea,ProfessorAlexanderKukush,ProfessorAnatoliyMalyarenko,Pro-
fessor Raimondo Manca, Dr. Henrik Jönson, Dr. Robin Lundgren, Dr. Evelina
Silvestrova, and Dr. Fredrik Srenberg for the fruitful cooperation.
viii Preface
IwouldalsoliketothankallmycolleaguesattheDepartmentofMathematics,
at Stockholm University, and the School of Education, Culture and Communica-
tion,attheMälardalenUniversity,forcreatingtheinspiringresearchenvironment
and friendly atmosphere which so stimulated my work.
I also would like to thank the Riksbankens Jubileumsfond for the important
and stimulating support of my research in the area and work on this book.
Stockholm, Dmitrii Silvestrov
September 2014
Contents
Preface v
1 Reward approximations for autoregressive log-price processes
(LPP) 1
1.1 Markov Gaussian LPP 2
1.2 Autoregressive LPP 11
1.3 Autoregressive moving average LPP 22
1.4 Modulated Markov Gaussian LPP 33
1.5 Modulated autoregressive LPP 43
1.6 Modulated autoregressive moving average LPP 58
2 Reward approximations for autoregressive stochastic volatility
LPP 74
2.1 Nonlinear autoregressive stochastic volatility LPP 75
2.2 Autoregressive conditional heteroskedastic LPP 87
2.3 Generalized autoregressive conditional heteroskedastic LPP 100
2.4 Modulated nonlinear autoregressive stochastic volatility LPP 113
2.5 Modulated autoregressive conditional heteroskedastic LPP 127
2.6 Modulated generalized autoregressive conditional heteroskedastic
LPP 142
3 American-type options for continuous time Markov LPP 157
3.1 Markov LPP 157
3.2 LPP with independent increments 162
3.3 Diffusion LPP 167
3.4 American-type options for Markov LPP 171
4 Upper bounds for option rewards for Markov LPP 180
4.1 Upper bounds for rewards for Markov LPP 180
4.2 Asymptotically uniform conditions of compactness for Markov
LPP 194
4.3 UpperboundsforrewardsforLPPwithindependentincrements 206
4.4 Upper bounds for rewards for diffusion LPP 215
4.5 Upper bounds for rewards for mean-reverse diffusion LPP 239
5 Time-skeleton reward approximations for Markov LPP 256
5.1 Lipschitz-type conditions for pay-off functions 257
5.2 Time-skeleton approximations for optimal expected rewards 270
5.3 Time-skeleton approximations for reward functions 287