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Derivatives and Risk Management in Theory and Practice
26-27 March 2007
Alexander Antonov, Numerix
Slides
Alexander Antonov got his PhD degree from the Landau Institute
for Theoretical Physics in 1997 and joined NumeriX LLC in 1998 where
he currently works as a Vice President Quantitative Research.
His activity is concentrated on modeling and numerical methods
for interest rates, cross currency, and credit. He is a regular
speaker for NumeriX at international conferences.
Abstract
Skew and smile calibration using Markovian projection
We briefly remind a subject of Markovian projection from the Gyongy theorem to Piterbarg’s research.
Then we address a universal approach to the Markovian projection to two main processes
in the Mathematical finance: a displaced diffusion and a Heston process.
The first process is able to capture option skew effects and the second one
can reproduce both skew and smile.
We develop a systematic view on the Markovian projection of these two processes,
and work out a set of computationally efficient formulas valid for a large class
of non-Markovian underlying processes.
We illustrate the theory with multiple examples including calculation of FX options
in cross-currency models,
swaption pricing in LIBOR Market Models and a spread of two correlated Heston models.
Theory and applications of the Markovian projection to a displaced diffusion can be found in
Antonov and Misirpashaev, "Markovian Projection onto a Displaced Diffusion: Generic
Formulas with Applications" (October 9, 2006).
Available at SSRN:
http://ssrn.com/abstract=937860
Jörg Behrens, Ernst & Young Switzerland
Slides
Jörg is partner of Ernst & Young Switzerland
and head of Central Europe Financial Risk Management.
His special focus is on risk quantification,
in particular in the areas of Solvency II and Economic Capital Management.
Jörg also leads EY internal thought leadership projects
to develop products and solutions in these areas.
Prior to joining Ernst & Young,
Jörg has lead the Quantitative Risk Team of Andersen in Zurich,
a position he assumed after 7 years with UBS in investment banking
and risk management based in London and Zurich.
He holds a Ph.D. in particle physics and until 2005
has been a member of the Swiss Standard Setting Board,
an expert panel lead by the Swiss Insurance regulator
to advise on the Swiss Solvency Test,
the Swiss implementation of Solvency II.
Abstract
Economic Capital Models under Solvency II
Solvency II places various challenges to the insurance industry.
Despite of similarities with Basel II and experience with internal insurance models,
various modeling and implementation challenges remain.
We select some technical aspects of Solvency II
and discuss options for implementation within an internal solvency model.
Denis Belomestny, Weierstrass Institute Berlin
Slides
Denis Belomestny got his PhD degree in Mathematics and Statistics from
Moscow State University in 2002. Since 2003 he is a researcher at the
Weierstrass Institute Berlin in the project Applied Mathematical Finance.
His activity in this project is mainly
concerned with the development of new Monte Carlo algorithms for
pricing exotic interest rate derivatives as well as with
the development of new statistical methods for robust
and efficient calibration of financial models.
Abstract
True upper bounds for Bermudan products via non-nested Monte Carlo
We present a generic non-nested Monte Carlo procedure for computing
true upper bounds for Bermudan products,
given an approximation of the Snell envelope.
The pleonastic true stresses that, by construction,
the estimator is biased above the Snell envelope.
The key idea is a regression estimator for the Doob martingale part
of the approximative Snell envelope, which preserves the martingale property.
The so constructed martingale may be employed for computing dual upper bounds
without nested simulation.
In general, this martingale can also be used as a control variate
for simulation of conditional expectations.
In this context, we develop a variance reduced version
of the nested primal-dual estimator (Anderson & Broadie (2004))
and nested consumption based (Belomestny & Milstein (2006)) methods.
Numerical experiments indicate the efficiency of
the non-nested Monte Carlo algorithm and the variance reduced nested one.
This is joint work with Christian Bender and John G.M. Schoenmakers.
Oliver Caps, Dresdner Bank
Slides
Dr. Oliver Caps is a senior quantitative analyst
in the model validation team at Dresdner Bank
and develops valuation models for exotic interest rate and hybrid products.
He holds a Ph.D. in mathematics and an MBA.
Currently, his main interests are multi-factor interest rate models
and smile modelling with stochastic volatility.
Abstract
On the Valuation of Power-Reverse Duals and Equity-Rates Hybrids
For the valuation of cross-asset products
like the popular power-reverse dual variants or equity-rates hybrids
it is important to take into account pricing effects of asset smiles,
stochastic rates or interest rate smiles appropriately.
In this talk we discuss requirements for a flexible hybrid model
and describe a stochastic volatility (Heston) model
for asset processes (like FX or equity) with stochastic rates
(Markovian HJM dynamics like the Hull-White dynamics)
satisfying these needs and allowing for an efficient calibration.
We describe how several numerical problems
can be managed when implementing the model,
show calibration results,
and give an impression of the price impact for hybrid products.
Sergio Dutra, Commerzbank
Slides
Sergio Dutra has a PhD in physics from Imperial College, London. He worked
as a research scientist in the field of quantum optics for
about 7 years writing 26 papers in peer-reviewed journals, one chapter in a
book, and a textbook. His first quantitative finance job was
at ABN AMRO in the model validation group where he stayed a year. After
that he moved to a front-office position at CommerzBank where
he has been working since about two years.
Abstract
Realistic Interest-rate Smile in a Cross-Currency Markov Functional Model
This general multicurrency extension preserves the simple driftless
dynamics of the hidden Markovian state variables that is one of the main
reasons behind the practical usefulness of the original single currency
Markov functional model. This model describes each interest rate market in a
completely non-parametric way leading to realistic modelling of
interest-rate smile, whilst still allowing practical calibration. It models
the FX rate in a semiparametric way allowing a simple and transparent
interpretation for the correlations between the driving Brownian factors in
terms of correlations between actual financial quantities observable in the
market. As an example, the model is applied to multi-callable quanto swaps.
Ernst Eberlein, University of Freiburg
Slides
Ernst Eberlein is Professor of Stochastics and Mathematical Finance at the University
of Freiburg. He is a cofounder of the Freiburg Center for Data Analysis and
Modeling (FDM), an elected member of the International Statistical Institute and at
present Executive Secretary of the Bachelier Finance Society. His current research
and his consulting activities focus on statistical analysis and realistic modeling of
financial markets, risk management, as well as pricing of derivative products.
Abstract
A Cross-Currency Lévy Market Model
The Lévy Libor or market model which was introduced in Eberlein and Özkan (2005) is extended
to a multi-currency setting. As an application we derive closed form pricing formulas for crosscurrency
derivatives. Foreign caps and floors and cross-currency swaps as well as quanto caplets are
studied in detail. Numerically efficient pricing algorithms based on bilateral Laplace transforms are
derived. A calibration example is given for a two-currency setting (EUR, USD). This is joint work
with Nataliya Koval.
Gabriele Guehring, d-fine
Slides
Gabriele Gühring is manager at d-fine GmbH
where she focuses on implementing trading and treasury systems
as well as on internal models for measuring market risk.
She holds a MSc in mathematical finance from Oxford University
and a PhD in mathematics from the University of Tübingen.
Before joining d-fine GmbH she was working at
Landesbank Baden-Württemberg (LBBW) in the department of Bond Research and Economics.
Abstract
Stochastic processes for implied volatilities
Pricing models specifying a stochastic process for the implied volatility are investigated.
No-arbitrage arguments lead to an expression specifying the drift of this stochastic implied volatility process.
The drift restriction formula connects implied volatility,
spot volatility and the volatility of the implied volatility process.
It is possible to show that this formula is also satisfied
for every explicitly specified stochastic spot volatility process.
The proof of this result gives an explicit
formula for the volatility of the implied volatility in terms of the volatility of the spot volatility.
Consequences in case the implied volatility or the spot volatility
are uncorrelated with the asset process are derived.
We show that a spot volatility process which is not correlated
with the process for the asset price, does not imply
that the process for the implied volatility
is also not correlated with the asset price.
Nevertheless, we obtain a symmetric smile function in both cases.
Susanne Griebsch, Frankfurt School of Finance & Management
Susanne Griebsch is currently a PhD-student at the
Frankfurt School of Finance & Management, conducting research at the
Centre for Practical Quantitative Finance and at Lucht Probst Associates (LPA), Frankfurt.
She graduated in Mathematics at the J.W. Goethe University, Frankfurt.
Her ongoing research focuses on stochastic volatility modeling
in foreign exchange markets as well as on pricing of Exotic options.
Abstract
Closed-Form Exotic Option Pricing in the Heston Model
In our study we focus on closed-form option pricing under stochastic volatility models,
particularly with regard to the Heston model.
In Heston's model closed-form formulas exist only for a few options.
Most of these closed-form solutions are constructed from characteristic functions
for the calculation of the product's expected payoff values.
In our work we follow this approach and derive multivariate
characteristic functions dependent
on at least two spot values for different points in time.
The derived characteristic functions
are then applied to closed-form option pricing of
Fader options and discretely monitored Barrier options.
The derived formulas are evaluated with different numerical methods
and compared to Monte Carlo values with regard to accuracy and computational times.
Reinhard Hirsch, d-fine
Slides
Reinhard Hirsch is Head of the Corporate Risk Consulting Business Unit,
which includes the Commodity and Energy Risk Consulting Business at d-fine GmbH.
Prior to this, he was responsible for commodity price risk management,
commodity contract management and real option based portfolio management
in corporate planning, strategic marketing and purchasing at Bayer AG.
Before moving to the chemical industry he was responsible for risk methodology
at Deutsche Bank's Global Risk Management Group.
He has written several scientific publications on Monte Carlo simulations
and gives lectures on commodities and mathematical finance
at international universities and business schools.
Reinhard holds a PhD in theoretical physics.
Abstract
Modelling and Forecasting of Prices and Forward Curves for Energy and Commodities
Energy and Commodities Derivatives are an old but also very actual topic
with a strong growing market and an exiting development during the last years.
All kinds of Investors from pension funds to private persons
are today exploring at least the exchange traded commodities.
Enormous profits combined with huge losses of hedge funds
in these markets punctuate the need of detailed understanding and modelling
the pice processes and risk measurement for commodities.
We give an brief introduction to the characteristics
of energy and commodity markets and show an analysis
of the special elements of risk measurement and modelling
of price processes and the forward curve dynamics of crude oil,
refined products and base metals.
The forecasting power of these models is analysed for crude oil
and compared to multivariate models for the forecasting of
price levels and the shape of the forward curve,
i.e. backwardation / contango strength of the forward curve.
Martin Keller-Ressel, Vienna University of Technology
Slides
Martin Keller-Ressel is a PhD-student of Josef Teichmann at the research
unit 'Financial and Actuarial Mathematics' (FAM) at TU Wien. His research
interests are the calibration of option pricing models with jumps,
equilibrium models for option pricing and interest rate modelling in the
framework of affine processes.
Abstract
Yield Curve Shapes and the Asymptotic Short Rate Distribution in Affine One-Factor Models
We consider a model for interest rates where the short rate is
given by a time-homogenous, one-dimensional affine process in the
sense of Duffie, Filipovic and Schachermayer. We show that in such a model
yield curves can only be normal, inverse or humped (i.e. endowed
with a single local maximum). Each case can be characterized by
simple conditions on the present short rate. We give
conditions under which the short rate process will converge to a
limit distribution and describe the limit distribution in terms of
its cumulant generating function. We apply our results to the
Vasicek model, the CIR model, a CIR model with added jumps and a
model of Ornstein-Uhlenbeck type.
Jörg Kienitz, Postbank
Slides
Dr. Jörg Kienitz works for Deutsche Postbank AG since October 2004. He
joined the Postbanks Treasury Department as a Quantitative Analyst
and is now responsible for the team which develops pricing and
simulation tools.
Jörg studied mathematics with focus on probability theory and
stochastic analysis at Bristol and Bielefeld where he receive his PhD
in January 2001. Before joining Deutsche Postbank he worked for Reuters
and Postbank Systems. Jörg frequently lectures at universities
on financial mathematics and works as a lecturer for IIR/IFF Germany.
A book on Monte Carlo methods using C++ together with Daniel Duffy of
Datasim is under preparation.
Abstract
Monte Carlo Simulation Software and Application to CPPI
Monte Carlo Simulation has become a key technology in the financial
sector. It can be applied in a variety of settings. To cope a wide range
of applications an efficient, robust and generic Monte Carlo Engine is
necessary.
We consider a CPPI approach to a basket of IR linked funds as an
example. CPPI is an abbrevation for "Constant Proportion Portfolio
Insurance". It is a portfolio management technique aimed on the one hand
side at maximizing returns for the investor and on the other hand side
protecting the principal. It has been applied in the equity and hedge
fund business since the early 80's. It now has been expanded to other
asset classes like commodities, interest rates or credit. Our example
will focus on the simulation of the basket and on selecting the
"optimal" starting configuration.
The key to the simulation of portfolios in different market settings is
a robust and efficient implementation of the Monte Carlo simulation
engine. We will describe a general framework in C++ which can be applied
to this setting and easily extends to other problems, e.g. derivatives
pricing and other portfolio management settings. The framework consist
of a bunch of loosely coupled C++ classes and the application of design
patterns.
Daniel J. Duffy, Jörg Kienitz, Monte Carlo Methods in Quantitative
Finance Generic and Efficient MC Solver in C++, Wilmott Magazine, 2005
Daniel J. Duffy, Jörg Kienitz, Software Frameworks in Quantitative
Finance, Part I Fundamental Pricinples and Applications to Monte Carlo
Methods, Wilmott Magazine, 2007
Daniel J. Duffy, Jörg Kienitz, Efficient and Robust Monte Carlo
Methods in Financial Engineering, Wiley and Sons, forthcoming
Roger Lord, Rabobank International
Slides
Roger Lord is an interest rate quant in the Financial Engineering team of Rabobank International.
He holds an MSc degree in Applied Mathematics from the Eindhoven University of Technology
and an MA degree in Econometrics from Tilburg University.
At present he is finishing his PhD at Erasmus University Rotterdam,
where he has taught several courses within the area of mathematical finance.
Prior to his current position he has worked in the Model Validation department of Rabobank International,
and at Cardano Risk Management.
Abstract
A Fast and Accurate FFT-based Method for Pricing Early-exercise
Options under Lévy Processes
A fast and accurate method for pricing early exercise and certain exotic options
in computational finance is presented in this paper.
The method is based on a quadrature technique and relies heavily on Fourier transformations.
The main idea is to reformulate the well-known
risk-neutral valuation formula by recognising that it is a convolution.
The resulting convolution is dealt with numerically by using the Fast Fourier Transform (FFT).
This novel pricing method, which we dub the Convolution method, CONV method for short,
is applicable to a wide variety of payoffs and only
requires the knowledge of the characteristic function of the model.
As such the method is applicable within exponentially Lévy models,
including the exponentially affine jump-diffusion models.
For an M-times exercisable Bermudan option, the overall complexity
is O(MN log(N)) with N grid points used to
discretise the price of the underlying asset.
It is shown how to price American options efficiently by
applying Richardson extrapolation to the prices of Bermudan options.
Available at SSRN: http://ssrn.com/abstract=966046
Christian Menn, Sal. Oppenheim
Slides
Christian holds a PhD from the School of Business Engineering at University of Karlsruhe.
He worked as a post-doc and visiting assistant professor
at the School of Operations Research at Cornell University.
In July 2006, Christian joined the quantitative research group
at Sal. Oppenheim where he focusses on the model development for exotic equity derivatives.
Abstract
Spot and Derivative Pricing in the EEX Power Market
Using spot and futures price data from the German EEX Power market, we test
the adequacy of various models for electricity spot prices. The models are com-
pared along two different dimensions: (1) We assess their ability to explain the
major data characteristics and (2) the forecasting accuracy for expected future spot
prices is analyzed. We find that the regime switching models clearly outperform its
competitors in almost all respects. Furthermore, for short and
medium-term periods our results underpin the frequently stated hypothesis that
electricity futures quotes are consistently greater than the expected future spot, a
situation which is denoted as contango.
Joint work with Michael Bierbrauer, Svetlozar T. Rachev and Stefan Trück.
Attilio Meucci, Lehman Brothers
Slides
Attilio Meucci holds a BA summa cum laude in Physics
and a PhD in Mathematics from the University of Milan,
an MA in Economics from Bocconi University in Milan, and is CFA chartholder.
Attilio Meucci is a vice president at Lehman Brothers, Inc., New York,
in the research division. Before joining Lehman,
he was a trader at Relative Value International,
a hedge fund in Greenwich, CT. Previously, he was at Bain & Co.,
where he designed solutions for risk management,
portfolio insurance, tactical and strategic asset allocation.
Attilio Meucci is the author of the bestseller
“Risk and Asset Allocation” and several other publications.
He has taught graduate courses on quantitative portfolio management
and risk management in top schools worldwide
and he is frequently invited as a speaker to conferences,
financial institutions and universities.
Find more information on Dr. Attilio Meucci at
www.symmys.com.
Abstract
Beyond Black-Litterman: Views on Non-Normal Markets - the Copula-Opinion Pooling Approach
The pathbreaking technique by Black and Litterman allows
portfolio managers to smoothly blend their subjective views
on the market with a prior market distribution.
Nevertheless, the BL approach suffers from two drawbacks.
In the first place, in BL both the market prior and the manager's
views are normally distributed.
For most markets the normal assumption is too strong:
fat tails, skewness and high dependence among extreme events characterize
the joint distribution of market risk factors in many contexts.
Secondly, in BL managers express views on the parameters
that determine the market distribution.
In reality, except in normal markets,
it is arguably more natural to express views directly
on the possible realizations of the market.
We rely on the opinion pooling, rather than Bayesian,
theory to expand on BL in the above directions.
We use opinion pooling criteria to determine
the marginal distribution of each view separately,
whereas the joint co-dependence, i.e. the copula,
among the views is directly inherited from the prior market structure.
Finally, a suitable change of coordinates allows us
to translate the joint distribution of the views
into a joint posterior distribution for the market.
First we introduce the theory behind the copula-opinion pooling approach;
then we detail an algorithm to implement the COP approach under virtually
any distributional assumption on the market and the views;
we conclude with an application to the management of a fixed-income portfolio.
Natalie Packham, Frankfurt School of Finance & Management
Slides
Natalie Packham is currently a PhD student at the Frankfurt School of Finance & Management.
She performs research at the Centre for Practical Quantitative Finance,
in particular in the areas of credit derivatives modeling and related computational aspects.
She graduated in Computer Science from the University of Bonn
and holds a Master's degree in Banking & Finance
from Frankfurt School of Finance & Management.
Prior to her PhD studies she worked for Dresdner Kleinwort Wasserstein
as a software enginner for the Fixed Income trading divison.
Abstract
Dependence in Credit Risk and Credit Derivatives Pricing: a Technique for
Multivariate Stratified Sampling
When valuing a claim with Monte Carlo simulation,
the variance of the estimator is a key figure
for assesing the quality of the simulation.
We present a variance reduction technique
for sampling from the joint probability distribution
of several random variables whose estimator is unbiased and consistent.
Furthermore, the strategy does not require
specific knowledge about the problem at hand,
but can be applied in a very general sense.
A practical application is the valuation of claims
that are sensitive to the dependence structure of the underlying securities,
such as first-to-default baskets and CDO tranches.
Jianwei Zhu, Sal. Oppenheim
Slides
Dr. Jianwei Zhu is a VP at Sal. Oppenheim working on implementing
interest rate /cross-asset pricing libaray.
Before joining Sal. Oppenheim,
he was a senior quant for exotic equity derivatives at WestLB in Düsseldorf.
Dr. Jianwei Zhu begun his carreer in the model validation team at Dresdner Bank.
He holds a MSc in mathematic economics from University of Dortmund,
and a PhD in quantitative finance from University of Tübingen.
Dr. Jianwei Zhu published a book entitled "Modular Pricing of Options" in Springer Verlag,
on the application of Fourier analysis to stochastic volatilities,
stochastic interest rates and random jumps.
Abstract
An Extended Libor Market Model with Nested Stochastic Volatility
In this paper we extend standard Libor Market Model (LMM) with nested stochastic volatilities.
The stochastic volatility of each Libor follows a mean-reverting process as in Schoebel and Zhu (1999)
or in Heston (1993) under the individual forward measure of each Libor.
Other than the existing stochastic volatility models,
every volatility in the extended LMM is correlated with its Libor individually,
and the parameters of stochastic volatility are also different over all Libors,
however, are nested by some deterministic functions.
With a nesting function, the same type of parameter
such as mean level in all volatility processes share
a certain term structure. In this model set-up,
we can still derive the stochastic processes for Libors
and volatilities under an arbitrary forward measure.
In line with the stochastic volatility models for equity options,
we obtain a closed-form solution via Fourier transform for caplets and floorlets.
Finally, we use factor representation to express Libors
and swap rates by some independent factors, namely principle components.
The approximated analytical pricing formula for swaption
can then be derived by using the characteristic functions
that are just a product of the characteristic function of each factor.
The numerical implementation of the nested stochastic volatility model is efficient
and identical to the existing stochastic volatility models.
Available at SSRN: http://ssrn.com/abstract=955352
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