Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann

324 Pages·2002·20.058 MB·English

by  Freddy Delbaen (auth.), Klaus Sandmann, Philipp J. Schönbucher (eds.)

Advances in Finance and Stochastics
Springer-Verlag Berlin Heidelberg GmbH
Klaus Sandmann
Philipp J. Schonbucher (Eds.)
Advances in
Finance and
Stochastics
Essays in Honour
of Dieter Sondermann
With 32 Figures
Springer
Klaus Sandmann
Johannes Gutenberg-Universităt Mainz
Lehrstuhl fiir Bankbetriebslehre
Jakob Welder-Weg 9
55128 Mainz
Germany
e-mail: [email protected]
Philipp J. Schonbucher
Rheinische Friedrich-Wilhelms-Universităt Bonn
Inst. f. Gesellschafts-u. Wirtschaftswissenschaften
Statistische Abteilung
Adenauerallee 24-42
53113 Bonn
Germany
e-mail: [email protected]
Catalog·in·Publication Data applied for
Die Deutsche Bibliothek-CIP-Einheitsaufnahme
Advances in finance and stochastics: essays in honour of Dieter Sondermann/
Klaus Sandmann; Philipp J. Schonbucher (ed.).
ISBN 978-3-642-07792-o ISBN 978-3-662-04790-3 (eBook)
DOI 10.1007/978-3-662-04790-3
Mathematics Subject Classification (2ooo): 6o-6, 91B
JEL Classification: G13
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Preface
Finance and Stochastics and Dieter Sondermann are directly and inextricably
linked to each other. The recognition and the success of this journal would
not have been possible without his untiring commitment, his sensitivity for
scientific quality and originality as well as his trustworthiness when dealing
with the authors. One could almost say: Finance and Stochastics is Dieter
Sondermann, since without him this journal would not be.
In the preface of the first issue of Finance and Stochastics in January
1997, Dieter Sondermann referring to the significance of the thesis of Louis
Bachelier, stated: 'Thus, the year 1900 may be considered as the birth date
of both Finance and Stochastics'. Further on he wrote: 'The journal Finance
and Stochastics is devoted to the fruitful interface of these two disciplines'.
What is there to add? It was important to identify and to articulate such
a goal, yet to translate it into action and to make it possible was crucial. It
is due to Dieter Sondermann's initiative and constant work that the idea of
Finance and Stochastics has turned into a highly reputable and successful
project. His unfailing commitment as founder and chief editor has made this
journal an important publication forum of international renown. A publica
tion in Finance and Stochastics is a guarantee of originality and quality for
scientific papers.
Thus, what could have been more natural than the idea of honouring Di
eter Sondermann on the occasion of his 65th birthday with a collection of
research papers entitled Advances in Finance and Stochastics? Those who
know him would surely agree that especially Dieter Sondermann, in his mod
est and undemonstrative way, would never have approved of such an honour.
Luckily, the person to be honoured does not have a say in the matter. How
ever, if he had had one and had not been able to prevent it happening, it is
likely that he would have warned us emphatically against a conception that
was one-sided and looked back upon his own contributions. He might even
have considered the exercise quite superfluous. Instead, his one and only con
cern would have been for the reader interested in scientific knowledge and
the solution of problems.
'The future has more Futures'. This 'bon mot' of the financial market
also holds good for Dieter Sondermann's scientific work and his involvement
which has always been diverse and with a clear focus on the future. Dieter
VI Preface
Sondermann was among the first scientists in Germany to apply themselves
to the study of Mathematical Finance. Influenced by the seminal work of
Fisher Black, Myron Scholes and Robert C. Merton and by virtue of his own
profound understanding of the theory of general equilibrium, his contribu
tions often mark the starting point for further development. In 1985, with
Hedging of Non-Redundant Contingent Claims, Dieter Sondermann, together
with Hans Follmer, paved the way for the pricing and hedging of options in
an incomplete financial market. At an early stage he recognises the import
ance of the theory of arbitrage for the evaluation of insurance risk, which
he demonstrates in Reinsurance in Arbitrage Free Markets (1991). With a
similar feel for new ground, he proved the market model approach to the
term structure of interest rates in his work Closed Form Solutions for Term
Structure Derivatives with Log-Normal Interest Rates (1997, together with
Kristian R. Miltersen and Klaus Sandmann).
Dieter Sondermann's academic career might be considered surprising and
unusual, especially in its initial phase. Yet, if one looks at it from today's
perspective, one can see easily how each step and each stage form an integ
ral part of a consistent whole. He was born on May 10th, 1937 in Duisburg,
Germany. His early years do not directly point to an academic career: his em
ployment as a forwarding agent in the 'Rhenania Allgemeine Speditions AG'
in 1953, his examination in 1956 as a business assistant and finally his activ
ity until 1958 as an expedient in the shipping company 'Vereinigte Stinnes
Reederei GmbH' in Duisburg Ruhrort. During those years, two notions must
have become rooted in his mind, his love of the Rhine and of shipping and
his love of pursuing promising ideas. After his Abitur in 1960, Dieter Sonder
mann embarked on his studies of Mathematics, Physics and Economics at the
University of Bonn. Little did he know (or even hope), when leaving Bonn in
1962 with a Vordiplom and heading for Hamburg, that he was to return as a
professor of Economics and Statistics not quite 17 years later. Many stages
and formative encounters awaited him still. After his Diplom in Mathemat
ics in 1966, Prof. Dr. Heinz Bauer who had noticed this promising young
mathematician from Hamburg, invited him to the University of Erlangen.
Here, after only two years, Dieter Sondermann obtained his Ph.D. in 1968.
There was no respite for the young academic with such diverse interests. No
sooner had he obtained his doctoral degree than his interest in Economics
was kindled - and this with lasting effect, not least because of Theory of
Value, the 'magic' book by Gerard Debreu. It fascinated him and filled him
with enthusiasm. With his innate capacity for sound judgement he clearly
grasped the opportunities and perspectives contained in the work - and he
used them. In 1970 Dieter Sondermann was appointed lecturer in Mathemat
ical Economics at the University of Saarbriicken. Yet, curiosity deriving from
fascination requires scientific discussion. Thus his path led to the Center of
Operations Research and Econometrics, CORE, in Louvain, Belgium, where
from 1970 to 1972 he was a visiting research professor. Here at CORE a
Preface VII
great number of committed young scientists met up, and it was here that the
foundations were laid for many scientific and personal friendships which were
to last until the present day. In 1972 Dieter Sondermann returned to Bonn,
this time as Visiting Associate Professor, in 1973 he was in Berkeley, USA,
and a year later he accepted a full professorship in Economics at the Uni
versity of Hamburg, Germany. The same year he joined the editorial board
of the Journal of Mathematical Economics, founded by Werner Hildenbrand,
on which he served until 1985. At the same time, from 1973 until 1980, he
was a member of the editorial board of the Journal of Economic Theory,
and from 1983 until 1992 of that of the Applicandae Mathematicae (Acta). In
1979 he became Fellow of the lAS at the Hebrew University, Jerusalem. This
is also the year in which he accepted a chair in Economics and Statistics at
the University of Bonn.
Dieter Sondermann, Bonn, the Rheinische Friedrich-Wilhelms University
and the Rhine are intertwined in so many different ways. His house by the
Rhine serves as a refuge for him, his wife, his family and their friends. Even
the perennial threat of high water cannot mar his lifelong attachment to the
Rhine and Rhine shipping. Instead, with a calmness that is so typical of
him, he will contemplate such a phenomenon of nature in statistical terms.
With the same calmness, full of determination, and most successfully, Dieter
Sondermann manages, from 1985 until 1999, Stochastics of Financial Mar
kets, the subproject B 3 of the Sonderforschungsbereich 303. During these
15 years, this research team, under his leadership, gains recognition at home
and abroad and makes a lasting contribution towards the development and
importance of Mathematical Finance. His open and problem-oriented style
of discussion deeply influences work methods and fosters an atmosphere of
curiosity. To bring into accord both research and teaching has always been
for him - and still is - a constant matter of concern. In a personal and human
manner that is so characteristic of him, Dieter Sondermann has, through
out the years, supported and influenced the career of his numerous members
of staff. Many of his students, themselves now in responsible positions at
universities or in industry, remain deeply indebted to him.
There are as many reasons for showing our gratitude to Dieter Sonder
mann as there are possibilities for expressing this. With Advances in Finance
and Stochastics we simply want to say: Thank you!
The future has more Futures, Dieter!
February 2002, Bonn
Klaus Sandmann Philipp Schonbucher
Introduction
In many areas of finance and stochastics, significant advances have been made
since this field of research was opened by Black, Scholes and Merton in 1973.
The collection of contributions in Advances in Finance and Stochastics re
flects this variety. Necessarily, a selection of topics had to be made, and we
endeavoured to choose those that are currently in the focus of active research
and will remain so in future. This selection spans risk management, port
folio theory and multi-asset derivatives, market imperfections, interest-rate
modelling and exotic options.
Since Follmer and Sondermann (1986) published one of the first mathem
atical finance papers on risk management in incomplete markets, quantitat
ive research has developed rapidly in this area. The first three papers of this
volume represent the recent developments in this area.
In the first paper on risk management, Delbaen extends the fundamental
notion of a coherent risk measure in two directions from the original definition
in Artzner et.al. (1999): the underlying probability space is now be a general
probability space (and not finite) and the class of risks that are measured
is extended to encompass all random variables on this space. Using methods
from the theory of convex games he is able to prove the analogies of the results
of Artzner et.al. (1999) in this much more general setup. But not everything
carries through identically from the discrete setup: Delbaen shows that now a
coherent risk measure has to be allowed to assume infinite values, representing
completely unacceptable risks. The following contribution by Follmer and
Schied also treats coherent risk measures, but only as a special case of a more
general class of risk measures: the convex risk measures. The authors show
that convex risk measures can be represented as a supremum of expectations
under different measures, corrected by a penalty function that depends on the
probability measure alone. They also connect these risk measures to utility
based risk measurement. The third article on risk management is authored
by Embrechts and Novak who give a survey of recent developments in the
modelling and measurement of extremal events. While the first two articles
are concerned with the question of a consistent allocation of risk capital to a
given set of risks, this article gives asymptotic answers to the question of the
probability with which this level of risk capital will be exceeded.
X Introduction
The part on portfolio theory opens with a paper by Werner in which he
develops a multi-period extension to the CAPM, the APT and similar factor
pricing models. By measuring the risk of the assets in terms of the risk of the
underlying dividend streams (instead of the one-period returns), the author
is able to give conditions under which exact factor pricing relationships hold.
In contrast to this portfolio-selection problem, Duan and Pliska consider the
pricing of options on multiple co-integrated assets. Apart from providing ne
cessary conditions for cointegration of a set of assets with GARCH-stochastic
volatilities, they also study the effect that cointegrating relationships under
the physical measure have on the dynamics of the assets under the equilib
rium pricing measure and on the dynamics of risk premia. In the following
paper, Madan, Milne and Elliott study the effects that arise when several in
vestors use different, individual factor pricing models, and these models are
aggregated. While Werner took the factor structure as given in his model,
Madan et.al. want to understand where economy-wide risk factors and risk
premia arise from, they shift the focus from asset-returns to identifying and
explaining investor-specific risk exposures.
Market imperfections are the theme of the next three contributions.
Kabanov and Stricker consider super-hedging strategies under transaction
costs. They characterise the hedging-set (the set of initial endowments that
allow a self-financing super-replication) of a contingent claim in a general
setup with non-constant transaction costs. In the following paper, Frey and
Patie address the problem of hedging options in illiquid markets. In a simula
tion study they show that a hedging strategy based upon a nonlinear partial
differential equation that includes liquidity effects can significantly improve
the performance of the hedge. In Frey and Patie's contribution illiquidity
takes the form of market impact, Le. the transactions of a large trader move
prices, but he is able to trade at any time he chooses. Rogers and Zane con
sider a different kind of illiquidity in the third paper of this group: Here,
traders are only allowed to trade at Poisson arrival times which they cannot
influence. The traders' objective is a consumption/investment problem sim
ilar to Merton (1969). Rogers and Zane establish that Merton's investment
rule (investing a fixed proportion of wealth in the risky asset) is still optimal,
and characterize the modification of the optimal consumption process. Using
an asymptotic expansion, they assess the cost of illiquidity to the investor.
The two contributions on interest-rate modelling both build upon the
market-modelling approach for observed effective interest rates by Miltersen,
Sandmann and Sondermann (1997). Bhar et.al. provide an estimation meth
odology for a short-rate model which explicitly recognizes the fact that the
short term interest-rate is unobservable. Their approach aims to connect the
stochastic models for the continuously compounded short rate with the ob
served effective, discretely compounded rates.
Introduction XI
Schlogl analyses this connection in the other direction and shows that
every market model implies a model for the continuously compounded short
rate that is uniquely determined by the interpolation method used for rates
maturing between tenor dates. He provides an interpolation method which
preserves the Markovian properties of discrete-tenor models but allows for
continuous stochastic dynamics of the short rate.
The final set of contributions has its focus on specific pricing problems
that arise in the pricing of exotic options, in particular the connection between
insurance and financial markets, optimal stopping, and barrier features which
all affect the payoff of the option in a nonlinear way.
The connection between the markets for insurance and financial risks
has been a long-standing area of interest to Dieter Sondermann. Nielsen and
Sandmann analyse in their contribution one example where this connection is
particularly evident: equity-linked life and pension insurance contracts. The
authors give results for the existence of a fair periodic premium and provide
approximate and numerical results for their magnitude.
Optimal stopping is the theme of the contributions by Schweizer; Shepp,
Shiryaev and Sulem; and Peskir and Shiryaev. Schweizer analyses the op
timal stopping problems posed by Bermudan options. As Bermuda options
can only be exercised in a subset of the lifetime of the option, the early ex
ercise strategies are subject to this additional restriction. Schweizer shows
under which conditions the problem can be reduced to a modified American
(unrestricted) optimal stopping problem, and how super-replication strategies
can be derived in this setup.
Shepp, Shiryaev and Sulem consider an option that combines American
early exercise, a knockout barrier and lookback-features: the barrier version
of the Russian option. Here, the early exercise strategies are restricted by
the knockout barrier of the option. Despite the complicated structure of the
option, they are able to provide the optimal exercise strategy and the value
function of this derivative.
The following contribution by Schiirger contains an analysis of the distri
bution, moments and Laplace transforms of the suprema of several stochastic
processes - a problem with immediate applications for the pricing of barrier
and lookback options. Schiirger gives explicit formulae for these quantities for
Bessel processes as well as for strictly stable Levy processes with no positive
jumps. For this he uses an elegant transformation from the maximum of a
stochastic process to its first hitting time.
The final contribution again addresses the question of optimal stopping.
Peskir and Shiryaev analyse the Poisson disorder problem, the problem of
detecting a change in the intensity of a Poisson process. In this context they
show that the smooth-pasting condition is not always valid for the optimal
value function if the state vector can be discontinuous.