Abstract:
Strategic asset/liability management (ALM) for sovereigns requires
analytical tools not usually provided by typical ALM approaches to financial
institutions. Recognizing this deficiency, the World Bank undertook a
research project that I directed for the development of tools for sovereign
ALM. In this talk I will discuss the results of and continuing developments
from that project. One of the developments is an open and flexible
technology using dynamic stochastic programming that allows the inclusion of
special objectives and policy, legal, and regulatory constraints. Other
tools provide insight and intuition into an otherwise very complex process
and spur new ways of thinking about risk. Lastly, I will discuss how this
technology is applicable to pension funds and insurance companies.
Abstract:
Determination of a price for an asset that is not marketed is a fundamental
problem of modern finance. It arises in the context of valuing firms that
are not yet public, determining the value of projects, valuing the risk of
insurance contracts, valuing commodity transactions, valuing loans subject
to credit risk, and pricing various complex securities. In additional, it is
desirable to determine the best hedge associated with a position in a
non-marketed asset and measure the remaining risk associated with the hedged
position. Researchers and professionals have several pricing concepts and
tools to assist in the determination of value in these situations. Two
fundamental pricing methods are the Capital Asset Pricing Model and the
Black-Scholes equation. Both have advantages and disadvantages but each has
limited scope. A general result that extends these results is desired.
There are several criteria that serve to guide
the pricing of general assets. One is that the new asset, together with its
price, should not induce an arbitrage in the market. Another is that a price
should be consistent with the price of asset's payoff projected onto the
marketed space. Another criterion is that the price should be a universal
zero-level price, which means that all individuals should agree that the
price is optimal. These criteria are inter-related and in some important
cases in a discrete-time framework there are prices that satisfy all three
criteria.
In a continuous-time framework these criteria
are all satisfied by a special price process derived from an extended
version of the Black-Scholes equation. This equation allows one to price
assets that are not marketed or assets that are derivatives of processes
that are observed but not marketed. The derivation of the equation is
simplified by use of a new operational calculus that quickly produces the
standard Black-Scholes equation and leads to the extended version. Examples
and applications of this general equation will be discussed.
(RiskLab talk)
Tuesday, September 10, 2002, 12.15 - 13.30
(ETHZ,
ML E12, entrance Clausiusstr. 2, upstairs 1.5 floors)
Abstract:
Many investors have a long-term perspective, but most standard portfolio
theories are based on single-period models that do not incorporate dynamic
phenomena. A dynamic view significantly changes the way one invests. For
example, while volatility is considered undesirable in the short run, it can
be beneficial in the long run, producing outstanding returns. The basic
principle of long-term investing was discovered by Kelly in 1956. It implies
that investors should maximize the expected logarithm of the single-period
return to maximize growth. Significant progress has been made since Kelly's
original work. The theory has been applied to stock portfolio design. Simple
methods for evaluating of portfolios in terms of their growth potential have
also been discovered. A significant advance is the extension of the theory
to multi-dimensions, based on the logarithm of a matrix as first defined by
Bellman. This theory allows one to use the full benefit of the theory in
more complex investment situations. An even more powerful extension applies
to nonlinear dynamics, such as when commissions are important or in the
context of complex trading schemes. These theories will be discussed and
their application to investment illustrated. The underlying theories have
application in other fields and thus are of general interest to operations
researchers and applied mathematicians.
(RiskLab talk)
Wednesday, August 21, 2002, 15.15 h
(ETHZ,
HG G26.5)
Abstract:
Put some weight on the Wiener measure depending on
Brownian exponential functionals up to time T.
Then the knowledge of the
asymptotics of these functionals as T tends to infinity may be used
to prove that the weighted measures converge weakly, and to describe the
limit laws. (Joint work with Y. Harija, RIMS, Kyoto)
(Seminar on Financial and Insurance Mathematics)
Thursday, August 15, 2002, 17.15 h
(ETHZ,
HG G26.5)
Abstract:
Benoit Mandelbrot suggested that Weierstrass' nowhere differentiable
function can be modified and randomized so as to approximate
fractional Brownian motion, which is a Gaussian self-similar process
whose paths are almost surely non-differentiable. The randomization
involves introducing independent and identically distributed random
variables with finite variance in the definition of the Weierstrass
function. We will show how one then obtains fractional Brownian motion
in the limit.
We will describe what happens if one
introduces in the above randomization strongly dependent random variables
instead of independent ones or if one uses infinite variance random variables
instead of finite variance ones. This is joint work with Vladas Pipiras.
References:
V. Pipiras and M. S. Taqqu (2000).
Convergence of the Weierstrass-Mandelbrot process to fractional
Brownian motion. Fractals, 8:369-384.
V. Pipiras and M. S. Taqqu (2000).
Convergence of weighted sums of random variables with long-range
dependence. Stochastic Processes and their Applications, 90(1):157-174.
V. Pipiras and M. S. Taqqu (2000).
The Weierstass-Mandelbrot process provides a series approximation
to the harmonizable fractional stable motion.
In C. Bandt, S. Graf, and M. Zaehle, editors.
Fractal Geometry and Stochastics II, pages 161-179.
Birkhäuser.
(Seminar on Financial and Insurance Mathematics)
Friday, August 9, 2002, 15.15 h
(ETHZ,
HG G26.3)
Abstract:
We model in a game theoretic context managerial intervention (impulse-type
controls with random outcome) directed towards value enhancement in the
presence of uncertainty and spillover effects. Two firms face real
investment opportunities, and before making the irreversible investment
decisions, they have options to enhance value by doing more R&D and/or
acquiring more information. Due to spillovers, firms act strategically by
optimizing their behavior, conditional on the actions of their counterpart.
They face two decisions that are solved for interdependently in a two-stage
game. The first-stage decision is: what is the optimal level of coordination
between them? The second-stage decision is: what is the optimal effort for a
given level of the spillover effects and the cost of information
acquisition? For the solution we adopt an option pricing framework that
allows analytic tractability. This single period two-stage game is extended
to a multiperiod stochastic game setting where firms face (path-dependency
inducing) switching costs that make strategy revisions harder.
(RiskLab talk)
Friday, June 28, 2002, 15.15 h
(ETHZ,
HG G26.3)
Abstract:
We first introduce the new theory of portfolio selection suggested by
Thorsten Hens and Klaus Schenk-Hoppé (2001) in a recent working paper. This
theory is based on evolutionary reasoning in simple repeated market
situations. The ultimate success of a strategy is measured by the wealth
share the strategy is eventually able to conquere in an evolutionary process
of market selection. For the case of identical and constant saving rates,
Evstigneev, Hens and Schenk-Hoppé (2001) identify a simple portfolio
strategy as being the unique strategy that p-almost surely will gain the
total market wealth.
The goal of our
project is to extend the evolutionary model to the case where investors face
to exogeneous saving rate, or liquity shocks and to analyze whether one can
identify also in this case a unique portolio strategy which gains the total
market wealth in the long-run. In the second part of the talk we present a
reward-risk approach for portfolio selection in a two-period model. The
analysis of the two-period case should viewed as a first step in the study
of the portfolio selection mechanism.
(Informal RiskLab talk)
Tuesday, June 25, 2002, 12.15 h
(ETHZ,
Hermann-Weyl-Zimmer, HG G43)
Abstract:
The value proposition of a reinsurer to his customer is his diversified
portfolio, which allows him to take the direct insurer's risk at a lower
cost of capital. Reinsurers also need to be protected against huge
fluctuations of the sum of the claims in a given time interval (say
1 year) as recent events have shown in particular. Dependencies in
extreme situations heavily impact the amount of capital required and
consequently the cost of a risk in an individual reinsurance contract.
Therefore we need to develop a measure for the diversification effect in the
extreme situations.
The goal of my talk is to present a capital
allocation concept by taking dependencies and diversification in the tails
into account. I will briefly recall some theory behind that concept which
has recently been discussed in various academic papers. In the main part of
the talk I describe how this concept can be used in practice for pricing
reinsurance contracts.
(This talk is also suitable for students in financial and insurance
mathematics)
(Lunchtime seminar on Financial and Insurance Mathematics)
Monday, June 24, 2002, 12.15 h
(ETHZ, HG G26.5)
Abstract:
The project
Impact of
Asymmetric Information on Banks Capital Allocation and Hedging
Decisions analyzes the effect of asymmetric information on banks'
hedging and capital decisions. In the current paper a bank
financed with deposits and equity is analyzed in a one period
model. The bank's motivation for risk management is that deposits
can lead to bank runs. In the advent of such a run, information
asymmetry leads to a loss in the bank assets' value when the bank
needs to sell them. The hedging strategy that maximizes the value
of equity is derived. We identify conditions under which well
known results such as complete hedging, maximal speculation or
irrelevance of the hedging decisions can be obtained.
(Informal RiskLab talk)
Thursday, June 20, 2002, 17.15 h
(ETHZ,
HG G26.5)
Abstract:
The work which is presented in this talk aims to prove that by exchanging
the two parts obtained after splitting a three dimensional Bessel bridge
from 0 to a >= 0 at an independent random time, uniformly
distributed on [0,1], one obtains a Brownian bridge from 0 to a.
Conversely, we construct a three dimensional Bessel bridge from 0 to
a by exchanging the two parts obtained after splitting a Brownian
bridge from 0 to a at a particular random time which is described
here. For a = 0, the above results are known as Vervaat and
Imhof-Biane transformations.
(Seminar on Financial and Insurance Mathematics)
Tuesday, June 18, 2002, 12.15 h
(ETHZ,
Hermann-Weyl-Zimmer, HG G43)
Abstract:
Definition of classical valuation, fair valuation, GOP valuation.
Analysis of both technical and financial losses with respect to the chosen
valuation basis.
(Lunchtime seminar on Financial and Insurance Mathematics)
Monday, June 17, 2002, 16.15 h
(ETHZ,
HG G26.3)
Abstract:
I study the liquidity risk, measured through the relative
bid-ask spread, on the Swiss market from a risk managers perspective. First,I
present a liquidity adjusted Value-at-Risk measure under the assumption of
perfect correlation between market and liquidity risk. Second, I take a
closer look at the extremal dependence of market and liquidity risk. We
apply a non-parametrical approach to measure bivariate exceedance
probabilities and the respective dependence function. Our analysis for the
Swiss market indicates moderate tail dependence in that roughly 10% of the
exceedances of the 99% quantile are co-exceedances. As our hypothesis tests
for independence are rejected we conclude that extreme dependence between
negative market returns and liquidity is existing in the empirical data and
may be relevant for firm-wide risk management.
(Informal RiskLab talk)
Monday, June 10, 2002, 16.15 h
(ETHZ,
HG G26.3)
Abstract:
The aim of this talk is to give an analytical pricing formula for path
dependent options on yields in the framework of the affine term
structure model with jumps. More precisely, we compute the Laplace
transform with respect to time of an European call option on the maximum
when the instantaneous interest rate process is an Ornstein-Uhlenbeck
type process with backward driven Lévy process having only negative
jumps. We will also discuss possible extensions of the computational
techniques used for more general models with jumps.
Prof. Dr. Albert N. Shiryaev
(Russian Academy of Science)
On stochastic integral representation for stopping
times and partial maxima of a Brownian motion
Abstract:
Let B = B(t) be a standard Brownian motion.
Functionals of the types
S(T) = max{B(t): t <= T}
and
S(g(T)) = max{B(t): t <= g(T)}
with g(T) = sup{t <= T: g(t) = 0} and
S(T(-a)) = max{B(t): t<= T(-a)}, a>0,
with
T(-a) = inf{t: B(t) = -a}
and many similar ones admit, of course, a stochastic integral representation.
We show how to get them by a simple method which illustrates clearly
how stochastic integrals are appearing here.
Abstract:
Moderne Finanzexperten setzen immer komplexer werdende Modelle ein und jeder
Praktiker muß deshalb bis zu einem gewissen Grad mit Begriffen der
Stochastik vertraut sein. Der Bereich des Risiko-Managements erfordert ein
tieferes Verständnis für Konzepte wie z.B. Volatilität,
Korrelation und Extremwerte.
Wenn Modelle auf die Realität angewendet werden und daraus
Schlüsse über die reale Welt gezogen werden, die innerhalb eines
Finanzunternehmens kommuniziert werden sollen, dann muß man sich in der
Sprache der Statistik verständigen. Die Aufgabe des mathematischen
Statistikers ist es nun, die Schnittstelle von Modell und Wirklichkeit zu
überwachen.
Neben diesen allgemeinen Ideen wird speziell das Konzept der Korrelation und
seine Anwendung im Risiko-Management vorgestellt. Die Behauptung, daß
Korrelationen in Zeiten von Marktvolatilitäten höher sind, hat zu
verschiedenen Interpretationen geführt, die am Schluß analysiert
werden.
(Inaugural Lecture in German)
Monday, May 27, 2002, 15:15 h
(ETHZ,
HG E33.1)
Abstract:
Classical theories of financial markets assume an infinitely liquid market
and that all traders are price takers. This theory is a good approximation
for highly liquid stocks, although even there it does not apply well for
large traders, or for modeling transaction costs. We propose a new model
that takes into account illiquidities, while extending the classical model.
Alternatively, or even simultaneously, one can use the model for transaction
costs. In essence, we relax the standard assumption of a competitive market,
where each trader can either buy or sell unlimited quantities of a stock at
the market price. Our approach hypothesizes a stochastic supply curve for a
security's price as a function of trade size. This leads to some interesting
mathematical issues, as well as natural restrictions on hedging strategies.
(Seminar of Financial and Insurance Mathematics)
Thursday, May 16, 2002, 12:15 h
(ETHZ,
Hermann-Weyl-Zimmer, HG G43)
Abstract:
We examine the dynamics of extreme values of overnight borrowing
rates in an inter-bank money market before a financial crisis during
which overnight borrowing rates rocketed up to (simple annual)
4000 percent. It is shown that the generalized Pareto distribution
fits well to the extreme values of the interest rate distribution.
We also provide predictions of extreme overnight borrowing rates
before the crisis. The examination of tails (extreme values) provides
answers to such issues as what are the extreme movements expected
in financial markets; have we already seen the largest moves;
is there a possibility for even larger movements and, are there
theoretical processes that can model the type of fat tails
in the observed data? The answers to such questions are essential
for proper management of financial exposures and laying ground for
regulations.
(Lunchtime Seminar on Financial and Insurance Mathematics)
Tuesday, May 7, 2002, 14:00 h
(ETHZ,
HG D1.2)
Abstract:
It is well known that "the Black-Scholes formula [...] is based on [...]
unrealistically simple assumptions" (F. Black). Relaxing these brings
one to the much-studied problem of pricing and hedging in incomplete markets.
Quadratic approaches have been among the first and most popular in this
area, and they continue to be used in applications and to bring up new
challenges. The goal of this talk is to provide a perspective on the
underlying ideas, questions and results. We shall start with minimal
assumptions on the audience and progress through theory and applications up
to recent developments and still open problems.
(Seminar of Financial and Insurance Mathematics)
Announcement:Workshop I/2002
of CARISMA on Friday, May 3, 2002,
from 10.00 to 17.00 in
Credit Suisse Forum St. Peter,
St. Peterstr. 19, 8001 Zürich.
Monday, April 29, 2002, 16.15 h
(ETHZ,
HG F33.2)
Abstract:
The fast numerical valuation of assets whose prices are driven
by Brownian motion has become standard practice worldwide. In a
Black-Scholes setting, this reduces to the numerical solution of a
parabolic advection-diffusion equation with various initial/boundary
conditions and possibly constraints. When closed form solutions are not
available, numerical methods must be employed; methods for handling such
numerical problems have been well developed. General Lévy processes have
been advocated in recent years for models in option pricing. They offer
more flexibility than Brownian motion and appear superior e.g. for
modeling short-term asset returns whose distributions are heavy-tailed.
Lévy models lead to parabolic integrodifferential equations with nonlocal
integrodifferential operators which, in general, cannot be evaluated
explicitly. The objective of this talk is to show the computational
methodology for the fast numerical solution of pricing problems driven by
general Lévy processes.
(RiskLab talk)
Wednesday, April 17, 2002, 13.15 h
(ETHZ,
HG D7.2)
Henrik Hult
(KTH, Stockholm)
Multivariate extremes for stochastic processes
Abstract:
We will present the latest development in our study of multivariate extremes
for stochastic processes. We consider a regularly varying
d-dimensional additive process. The results will show how the joint
tail behavior of the process is related to its Lévy measure and from
this relation we can derive the tail behavior of the componentwise supremum
of the process and the supremum of its jumps. The results also explain how
extremal events occur in this framework. Throughout the talk we will stress
the usefulness of the concepts and results in portfolio risk management.
