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Derivatives and Risk Management in Theory and Practice
April 3-5 2002
Arun Bagchi, University of Twente
Abstract
Parameter Estimation in Continuous-Time Financial Data: Application
to Exponential-Affine Term Structures
The exponential-affine term structure model is a class of models
in which the yields to maturity are affine functions of some
state vector x(t). Since the interest rate factors
x(t) are not directly observed, unknown parameters
in these models need to be estimated on the basis of observing
the bond prices of different maturities. Although the state space
model is set-up in continuous time, all existing parameter
estimation techniques discretize the observation equation in time
in order to use known statistical/filtering methods. We resolve
this incongruity in the present paper by working throughout with
the original continuous-time formulation. We explain the maximum
likelihood parameter estimation methodology in this framework and
discuss modifications needed when the observation noise
covariance is unknown. Finally we illustrate the methods by means
of extensive simulation studies. Application to real treasury data
will also be discussed.
Hans-Peter Deutsch, Andersen
Slides
Dr Hans-Peter Deutsch is a Partner at Andersen and head of Andersen's Financial and
Commodity Risk Consulting (FCRC) in Germany, which he founded in 1997 and developed
from scratch to the over hundred people strong consulting practice it is today. He is also Faculty
Member and Member of the Advisory Board of the Mathematical Finance Programme at the
University of Oxford (see
http://www.conted.ox.ac.uk/mathsfinance/)
England, and Director of
the German Chapter of GARP, the Global Association of Risk Professionals (see
http://www.garp.com/).
He has worked with clients in several IT-based trading and risk
management projects, including software selection and development, pricing and risk
management models for derivatives, and is author of several books (see for instance
http://www.palgrave.com/catalogue/catalogue.asp?Title_Id=0333977068)
and many publications
in this area and a regular speaker at conferences. Before joining Andersen, Hans-Peter headed
trading system development at a major German Bank and served as a consultant with Andersen
Consulting (now Accenture). He holds a Ph.D. in theoretical physics and is also author of about
20 international scientific publications in this field, mainly on Monte Carlo simulations of
stochastic processes.
Abstract
Second Order Approximations for Fast Value-at-Risk Computations
As soon as a portfolio contains significant optionality standard (delta normal)
variance-covariance Value-at-Risk calculations lead to very inaccurate results
while full fledge Monte Carlo Simulations with full re-pricing often involve
unacceptably long computing time. A natural compromise is to extend the delta
normal variance-covariance methods to include 2nd order terms of the portfolio
value's taylor expansion. This talk highlights the algebraic, analytical and
statistical tasks involved when doing such a delta-gamma approximation and
presents several different alternative ways for calculating a Value at Risk in
such a framework.
Jürgen Hakala, Commerzbank
Slides
Jürgen Hakala is Head of Quantitative Reserach at Commerzbank
Treasury and Financial Products since 4 years.
His research areas are models and products for foreign exchange derivatives and
hybrid interest rate and foreign exchange models. Computational Finance is
a key element for all his activities.
He received a masters degree in theoretical physics from the University of Karlsruhe
and a Ph.D. in mathematics from the University of Bonn at the institute for Neural Networks.
Abstract
Stochastic volatility models: A Finite Difference Approach
Many exotic options are very sensitiv to changes in implied
volatility. In such cases it is essential to
have a model of the underlying which reflects the change of volatility with time quite well.
One approach is Heston's stochastic volatility model which assumes that the volatility of
an asset is a stochastic process itself. However for exotic options no closed form solutuions
are known. Instead a PDE derived from the stochastic model is solved.
We present some ideas and first results on how to improve the speed
to calculate quite accurate prices using the Finite Difference Method.
This is a joint presentation of Jürgen Hakala and Tino Kluge.
Wolfgang Härdle, Humboldt University of Berlin
Slides
Abstract
The Dynamics of Implied Volatilities: A Common Principle Component Approach
It is common practice to identify the number and sources of shocks that move implied
volatilities across space and time by applying Principal Components Analysis PCA) to
pooled covariance matrices of changes in implied volatilities. This approach, however, is
likely to result in a loss of information, since the surface structure of implied
volatilities in the maturities and moneyness dimension is neglected. In this paper we
propose to estimate the implied volatility surface at each point in time
nonparametrically and to analyze the implied volatility surface slice by slice with a
common principal components analysis (CPCA). As opposed to traditional PCA, the basic
assumption of CPCA is that the space spanned by the eigenvectors is identical across
groups, whereas variances associated with the components are allowed to vary. This allows
us to study a p variate random vector of k groups, say the ''volatility smile'' at
p different grid points of moneyness for k maturities, simultaneously. Our evidence
suggests that surface dynamics can indeed be traced back to a common eigenstructure
between covariance matrices of the surface ''slices'', which allow for the usual shift,
slope, and twist interpretation of shocks to implied volatilities. This insight is a
suitable starting point for VaR Monte Carlo Simulations of delta-gamma neutral, vega
sensitive option portfolios.
Norbert Hofmann, Goethe-University
Slides
Norbert Hofmann is scientific assistant to Prof.Dr.P.E.Kloeden in the
Section: Numerics, dynamics and optimization at Goethe-University
in Frankfurt. He formerly worked at the Weierstrass-Institute of Applied
Analysis and Stochastics in Berlin, at the University of Erlangen-Nuernberg
and as a visiting research fellow at the Australian National University in
Canberra (Australia). His research area is stochastic numerics. Particularly
he is interested in numerical methods for stochastic differential equations.
Computational Finance turned out to be an important application of his work.
He has written papers on the application of weak approximation of stochastic
differential equations in option pricing and on the approximation of large
portfolios.
Abstract
Approximating the square root process
We consider numerical methods for the simulation of paths of the
square root process. We introduce a special implicit method that
reflects the positivity of the exact dynamics. Moreover we show
that the new method is suited to overcome stability problems.
By means of simulation studies we compare the new method with the
Euler scheme. It turns out that modifications of the Euler scheme fail.
This is joint work with Eckhard Platen (University of Technology
Sydney, Australia).
Tino Kluge, Chemnitz University of Technology
Tino Kluge is a student of Mathematics and Technology at Chemnitz University
of Technology. He recently worked in the Quantitative Research department
at Commerzbank as an intern where he pursued a project about
stochastic volatility models and finite difference methods.
Abstract
Stochastic volatility models: A Finite Difference Approach
Many exotic options are very sensitiv to changes in implied
volatility. In such cases it is essential to
have a model of the underlying which reflects the change of volatility with time quite well.
One approach is Heston's stochastic volatility model which assumes that the volatility of
an asset is a stochastic process itself. However for exotic options no closed form solutuions
are known. Instead a PDE derived from the stochastic model is solved.
We present some ideas and first results on how to improve the speed
to calculate quite accurate prices using the Finite Difference Method.
This is a joint presentation of Jürgen Hakala and Tino Kluge.
Lutz Molgedey, Andersen
Slides
Abstract
Libor market model with stochastic time homogenous mean reverting volatility
We present a variant of the Libor market model with stochastic volatility. As
for the usual deterministic volatility Libor market model we require the
parametrization of the model to be as time homogenous as possible. Here, this is
achieved by using time homogenous mean reversion levels and speeds for the
stochastic volatilities of the respective forward rates. Correct (perfect)
pricing of the (at-the-money) caplets corresponds then to non-stationary initialvalues
of the forward rate volatilities. However, demanding a time homogenous
model restricts possible caplet smile surfaces. Those restrictions (and
advantages) will be discussed in the talk.
Jörn Rank, Andersen
Slides
Jörn Rank is a senior consultant at Andersen's Financial and Commodity Risk
Consulting Group. During his time at Andersen, he has worked with several German
banks. The main part of his work was the implementation of trading and risk
management systems. Before he joined Andersen in 1998, he worked for a few
months at Commerzbank in Frankfurt. Jörn holds a Ph.D. in theoretical physics
from the University of Bielefeld and a diploma in Mathematical Finance from the
University of Oxford. He is author of several international scientific
publications on high energy physics.
Abstract
Improving VaR Calculatuions by Using Copulas and Non-Gaussian Margins
Apart from historical simulation, most Value-at-Risk (VaR) methods assume a
multivariate normal distribution of the risk factors. In this work we present
the application of copulas for the calculation of the VaR. This enables us to
use arbitrary distribution functions for the risk factros. The risk factors
themselves are linked together by a copula function that describes the
dependence structure between them. We discuss the modification of the
Monte-Carlo (MC) method of the VaR calculation under this generalization. Usinga
financial portfolio based on historical FX rates over a period of ten years,
we compare the backtesting results obtained from the "traditional" MC method
with the one from the "copula" MC method, using various copulas and various
distribution functions for the margins.
L. C. G. Rogers, University of Bath
Chris Rogers is Professor of Probability at the University of Bath. He is
the author of more than 100 publications, including the famous two-volume
work, Diffusions, Markov Processes, and Martingales with David Williams. His
Finance papers include the potential approach to term structure of interest
rates, complete models of stochastic volatility, portfolio turnpike
theorems, improved binomial pricing, infrequent portfolio review
high-frequency data modelling. Chris is co-editor of Finance and Stochastics
and an associate editor of several journals, including Mathematical Finance.
He is a frequent speaker at industry conferences and courses, and consults
for a number of financial clients.
Abstract
Monte Carlo valuation of American options
This paper introduces a `dual' way to price
American options, based on simulating the path of the option
payoff, and of a judiciously-chosen Lagrangian martingale.
Taking the pathwise maximum of the payoff less the martingale
provides an upper bound for the price of the option, and
this bound is sharp for the optimal choice of Lagrangian
martingale. As a first exploration of this method, three
examples are investigated numerically; the accuracy
achieved with
even very simple-minded choices of Lagrangian martingale
is surprising. The method also leads naturally to candidate
hedging policies for the option, and estimates of the risk
involved in using them.
Uwe Schmock, ETH and University of Zürich
Uwe Schmock is currently director of the program Master of Science
in Finance, which is offered jointly by the Swiss Federal Institute
of Technology (ETH) and the University of Zürich.
Uwe studied mathematics and physics at the Technical University of
Berlin, Germany, and at the California Institute of Technology.
He holds a diploma and a Ph.D. in mathematics from the TU Berlin.
He formerly worked for five years as a postdoc at the University of
Zürich, for four years as Credit Suisse Research Fellow at ETH
Zürich, and for more than two years as Research Director of the finance
competence center RiskLab within the Department of Mathematics at the
ETH Zürich. The Swiss RiskLab is financially supported by
Credit Suisse Group, Swiss Re, UBS AG, and ETH Zürich. Uwe's
research interests include applications of large and moderate deviations
theory, securitisation, applications of extreme value theory, model
risk, risk capital allocation, and mathematical finance in general.
Abstract
Term structure models for credit risks
This talk gives an overview on the available theoretical
methods for pricing defaultable bonds. We review popular models for
the term structure of risk-free interest rates, introduce hazard
rates and loss fractions and thereby motivate default-adjusted
interest rates. Several modelling assumptions for recovery at default
are mentioned. We proceed by discussing the term structure of credit
spreads and their dependence on risk-free interest rates. We conclude
with remarks on the modelling of dependent defaults.
Ingo Schneider, BHF-Bank
Slides
Ingo Schneider is Fixed Income Derivatives Trader at BHF-Bank, (ING-Group) Frankfurt.
He trades interest rate derivatives, develops and
implements OTC interest rate derivative products and works on term structure models.
He received his diploma and doctorate degree in Physics from Goethe-University in Frankfurt.
Abstract
Using Finite Differences for Pricing Options
This is a pratical session which emphasizes the details
implementing a numerical method. We will show a step by step
implementation of various finite difference schemes in order to
solve a Partial Differential Equation. Based on a a paper about
"Pricing Arithmetic Average Asian Options (Vecer 2001) we apply
the finite difference methodologie and compare the results with
Monte Carlo.
The interested participant is asked to bring his laptop with EXCEL and
VBA installed.
Peter Schwendner, Sal. Oppenheim jr. & Cie
Peter Schwendner holds a Ph.D. in theoretical physics
from the University of Goettingen. Since 1998, he works at the
Equity Trading & Products Department of Sal. Oppenheim,
where he develops and implements equity derivative products.
Some joint work with Bernd Engelmann and others
can be downloaded from www.oppenheim.de/quant.
Abstract
Quantitative Aspects of Equity Derivatives Trading
Academics usually analyse the pricing of very complicated derivatives,
regardless if they are actively traded or not.
But also plain vanilla option books can be a rich playground for quantitative concepts.
In this talk, I present some quant aspects of the risk management of a large
number of simple equity derivatives.
I will refer to some joint work with colleagues and friends.
Contents:
- What is special about equity derivatives in comparison to other asset
classes?
- Retail derivatives business model
- Hedging of equity derivatives books using EUREX options
- Fast pricing of a large number of options for market-making using price caching
- Implementation of Monte Carlo simulation for value-at-risk estimation
Tino Senge, Commerzbank
Slides
Tino Senge is a Quantitative Research Specialist at Commerzbank Treasury
and Financial Products in Frankfurt where he is working on
models for pricing foreign exchange derivatives. Tino has studied
Mathematics in Jena (Germany) and Cork (Ireland). Before joining
Commerzbank he had worked with Commerz Financial Products and DG Bank.
His current research interests include jump-diffusion model
models for modeling the volatility smile in foreign exchange markets and
its application to the pricing of exotic derivatives.
Abstract
Jump-Diffusion Models in Foreign Exchange Markets
Jump-diffusion models to price FX options will be considered. Different
distributions for the jump size will be stated and analysed. Features and
limitations of the obtained models will be presented. Besides theoretical
results and notes on implementation, the question will be discussed whether
this models can be used to price options taking the volatility smile in
todays FX markets into account.
Steven E. Shreve, Carnegie Mellon University
Slides
Steven E. Shreve is a Professor of Mathematics at Carnegie Mellon University. Steven is the author with
I. Karatzas, of two books related to finance: ``Brownian Motion and Stochastic Calculus'' and ``Methods
of Mathematical Finance,'' co-editor of the proceedings ``Mathematical Finance, Vol. 65, Institute for
Mathematics and its Applications,'' and advisory editor of the journal ``Finance and Stochastics.''
Steve began research on the capital asset pricing model in 1980, and has worked in various aspects of
mathematical finance since then, including the effect of transaction costs on option pricing, the
effect of unknown volatility on option prices, pricing and hedging of exotic options, and models of
credit risk. In 1991 he founded the Ph.D. program in Mathematical Finance at Carnegie Mellon, and in
1994 was one of the founders of the Master's program in Computational Finance.
Abstract
A Unified Model for Credit Derivatives
This is joint work with Alain Belanger of Scotia Capital Markets and
Dennis Wong of Bank of America. A framework is provided for pricing
derivatives on defaultable bonds and other credit-risky contingent
claims. The framework includes structural models (those in which the
time of default is determined by the value of the issuing firm), general
reduced-form models (those in which default is exogenous), and
reduced-form models in which default can occur only at
specific times, such as coupon payment dates. Within the general
framework, multiple recovery conventions for contingent claims
are considered: recovery of a fraction of par, recovery of a fraction of
a no-default version of the same claim, and recovery of a fraction
of the pre-default value of the claim. These recovery conventions are
matched to appropriate default protection contracts. A
stochastic-integral
representation for credit-risky contingent claims is provided, and the
integrand for the credit exposure part of this representation
is identified. In the case of intensity-based reduced-form
models, credit spread and credit-risky term structure are studied.
Gerhard Stahl, Federal Banking Supervisory Office
Abstract
Modelling Event Risk
The talk will review the on-going joint work with E. Platen, Sydney, on
modelling specific market risk for equities. The talk will cover the
following topics: benchmarked prices as basic inputs of risk models,
modelling event risk based on t-distributions and regulatory implications.
Finally, an empirical study for the relevant equity marktes provides
insights on the validity of the proposed models.
Hermann Stahl, Commerzbank
Slides
Hermann Stahl is a lawyer and admitted to the bar in Frankfurt am Main and
New York. He heads an area within Commerbank's central legal department
which deals with derivatives, trading and exchanges. Her received his
education in law and related subjects at Universität Bayreuth, Bayreuth,
Germany, Washington & Lee University, Lexington, Virginia, USA, and the
University of Illinois at Urbana-Champaign, Champaign, Illinois, USA.
Abstract
Documentation of OTC Derivatives and other Financial Instruments
The markets for OTC derivatives and securities repurchase and lending have
created their own documentation standards. Transactions are documented with
trade confirmations which refer to a master agreement. The master agreement
provides for the legal and credit terms and integrates all transactions into
the master agreement by forming one single agreement. Organizations on
national and international level have produced master agreements for various
types of business and published sets of definitions that simplify the task of
documenting individual trades.
A positive side effect of master agreements is that the credit exposure that
both parties have under the various transactions may be netted for capital
adequacy purposes.
Felix Streichert, University Tübingen
Slides
Felix Streichert is currently Research Assistant at the Department of
Computer Architecture within the Wilhelm-Schickard-Institute of the
Eberhard-Karls-University T?bingen. He holds a diploma in Technical
Cybernetics from the University of Stuttgart. His research interests are
Evolutionary Algorithms in general, Hybrid Evolutionary Algorithms and
financial applications of Evolutionary Algorithms.
Abstract
Evolutionary Alogrithms and Finanical Applications
Evolutionary Algorithms (EA) constist of serveral heuristics which are able
to solve optimization tasks by imitating some aspects of natural evolution.
They may use different levels of abstraction, but they are always working on
whole populations of possible solutions for a given task. EAs are an
approved set of heurisitcs which are flexible to use and postulate only
neglectible requirements on the optimization task.
As a practical application technical trading rules found by the use of EA
will be presented.
Josef Teichmann, Technical University Vienna
Born in Lienz, Eastern Tyrol, Austria. Studies of Mathematics in Graz,
Besancon and Vienna. Thesis on Infinite dimensional Lie groups at the
University of Vienna (supervised by Peter Michor). Assistent Professor at
the Department of Financial and Actuarial Mathematics at the Technical
University of Vienna (Walter Schachermayer). Research in Interest Rate
Models, HJM-Theory and Differential Geometry.
Abstract
On finite dimensional Term structure models
We provide the characterization of all
finite-dimensional Heath--Jarrow--Morton models that admit arbitrary
initial yield curves. It is well known that affine term structure
models with time-dependent coefficients (such as the Hull--White
extension of the Vasicek short rate model) perfectly fit any initial
term structure. We find that such affine models are in fact the only
finite-factor term structure models with this property. We also show
that there is usually an invariant singular set of initial yield curves
where the affine term structure model becomes time-homogeneous. We also
argue that other than functional dependent volatility structures -- such as local
state dependent volatility structures -- cannot lead to finite-dimensional realizations.
Finally, our geometric point of view is illustrated by several examples.
Robert Tompkins, Technical University Vienna
Slides
Robert Tompkins is a University Dozent at the Technical Universität, Vienna. He has recently
accepted an Honorary Professorship at the University of Warwick Business School, where he has taught
courses on Financial Markets during the 2000/2001 academic year.
Dr. Tompkins was formerly the Head of International Quantitative Research at Kleinwort Benson
Investment Management. In addition, he remains the Managing Director of the Minerva Group. Prior
to this, he was the Futures and Options Specialist at Merrill Lynch, Europe and an Interest Rate
Options Dealer and Currency Options Trader at two major Chicago banks. He has three degrees from
the University of Chicago, including an MA in Quantitative Methods and an MBA (honours). In addition,
he completed his Ph.D. in Finance at the University of Warwick in 1998 and his Habilitation in
Finanzwirtschaft at the University of Technology, Vienna in 2000.
Robert has authored three books on Options and edited a book on exotic options "From Black Scholes
to Black Holes". Robert is currently writing a series on Exotic Options, which appears in the Austrian
Journal, Bank Archiv. This series will form the basis of a book that will be published by Cambridge
University Press in 2002. He has published widely in RISK Magazine, and a number of academic journals
including Journal of Futures Markets, Journal of Derivatives, Journal of Risk Finance, Journal of Risk,
Quantitative Finance and the European Journal of Finance. Robert's current research interests include
comparisons of established and emerging markets, volatility estimation and forecasting, implied
volatility smile patterns and the hedging of exotic contingent claims.
Abstract
The relation between implied and realised probability density functions
A number of financial regulators [see Neuhaus (1995), Bahra (1996, 1997), McManus (1999) and
Shiratsuka (2001)] have suggested that risk neutral densities (RND) associated with options
markets could provide useful indicators of future market turbulence. Critical to this assumption
is that such RNDs should provide an unbiased forecast of realised probability density functions.
To date, this assumption has not been fully examined.
In this research, we test the ability of RNDs for options on the S&P 500 and the British
Pound / US Dollar to predict future probability densities. We consider three approaches to
estimate the RNDs, which are consistent with approaches proposed and used by financial regulators.
We also provide a number of new testing procedures to assess the efficiency and unbiasness of the
forecasts. These tests provide more power than the usual Komolgorov/Smirnov tests.
Using non-overlapping quarterly data from the mid 1980s to 2000, we find that we can reject the
hypothesis that the RNDs for both the S&P 500 and British Pounds are unbiased forecasts. Even with
a limited number of observations, the tests are powerful enough to allow rejection. These results
are consistent with Weinberg (2001) and are more robust as this work relied upon the use of
overlapping data.
These results tend to support the conclusions of Shiratsuka (2001), that RNDs should not be used
by financial regulators as financial indicators, and that such use could prove counterproductive;
actually increasing future market turbulence rather than alleviating it.
Jürgen Topper, Andersen
Slides
Jürgen Topper joined Andersen in 1997 after finishing a master's degree in
economics at the University of Hannover (Germany). During his university years,
Jürgen worked on several projects for mechanical engineering in academia and
industry on coupling finite element analysis with tools from operations research.
Jürgen's primary areas of interest are exotic derivatives and structured
products. His consulting practice includes numerous international banks and
European corporates.
Abstract
Applying Generalized Passport Options
Except for special cases, generalized passport options do not have closed-form solutions.
Here we show how to derive approximate solutions using finite element methods. We also show that
finite elements offer advantages in computing the hedge parameters. These techniques are applied to
special cases of the generalized passport option which include Asian options and diverse
passport options with caps and/or barriers.
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