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Derivatives and Risk Management in Theory and Practice
23-24 March 2009
Dr Eric Benhamou, Pricing Partners
Slides
Eric Benhamou is the CEO of Pricing Partners
(www.pricingpartners.com), a start-up specializing
in software and service for independent valuation of financial derivatives products. Current coverage
includes interest rates, credit, equity, inflation, foreign exchange, commodities, insurance derivatives
and hybrids. Eric Benhamou is also known for its endeavor to gather financial institutions, start-ups
and public research centers on collaborative innovation in financial mathematics. Previously, he headed
the fixed income quantitative research at Ixis CIB, joining from Goldman Sachs. He is a regular speaker
at professional conferences and has published various articles on subjects like advanced Monte Carlo simulation,
inflation derivatives and other option pricing results. A former alumnus of the Ecole Polytechnique, the ENSAE,
he holds a Ph.D. in financial mathematics from the London School of Economics.
Abstract
PSmart Expansion and Fast Calibration for Jump Diffusion
Using Malliavin calculus techniques, we derive an analytical formula for the price of European options,
for any model including local volatility and Poisson jump process.
We show that the accuracy of the formula depends on the smoothness of the payoff function.
Our approach relies on an asymptotic expansion related to small diffusion and small jump frequency/size.
Our formula has excellent accuracy (the error on implied Black-Scholes volatilities
for call option is smaller than 2 bp for various strikes and maturities).
Additionally, model calibration becomes very fast.
This is joint work with E. Gobet and M. Miri.
Philipp Beyer, Universty of Konstanz and Postbank
Philipp Beyer is studying towards his diploma degree in economics at the University of Konstanz
under supervision of Prof. Jens Jackwerth. The topic of the thesis is the pricing of exotic options using Lévy processes.
He is currently completing an internship at the quantitative analysis team at Deutsche Postbank AG.
Abstract
Model Risk in Pricing and Hedging Exotic Equity Derivatives
We consider the following models for equity derivatives:
- Heston Stochastic Volatility Model
- Merton Jump Diffusion
- Variance Gamma (VG)
- VG with a Gamma Ornstein-Uhlenbeck clock (VG-OU)
- VG wiht a CIR clock (VG-CIR)
- Normal Inverse Gaussian (NIG)
- Normal Inverse Gaussian with a Gamma Ornstein-Uhlenbeck clock (NIG-OU)
- Normal Inverse Gaussian with a CIR clock (NIG-CIR)
First, we consider the calibration to quoted option prices within these models.
We use the calibrated models to compute the prices of some exotic options including
cliquet options and compare the prices generated by each model.
We proceed by computing the forward characteristic functions for the models
and study the forward implied volatility surface for each model.
The different shapes of the forward volatility surfaces resulting
from applying different models are studied.
Furthermore, we show how the intial model parameters affect the shape
of the forward volatility surface within a single model.
Our conclusion is that if pricing and risk manage exotic options is
considered one has to keep track of the forward volatility surface
and make sure not only which model to use but adjust the parameters
within a chosen model reasonably.
This is joint work with Jörg Kienitz (Deutsche Postbank AG).
Dr Peter Carr, Bloomberg L.P.
Dr. Peter Carr is the Head of Quantitative Financial Research at Bloomberg LP,
where his group is responsible for all facets
of the business operation relating to modeling and analytics.
He is also the Director of the Masters in Math Finance program at NYU's Courant Institute.
Prior to his current positions,
he headed equity derivative research groups for six years at
Banc of America Securities and at Morgan Stanley.
His prior academic positions include 4 years as an adjunct professor
at Columbia University and 8 years as a finance professor at Cornell University.
Since receiving his PhD. in Finance from UCLA in 1989,
he has published extensively in both academic and industry-oriented journals.
He is currently the treasurer of the Bachelier Finance Society
and a practitioner director for the Financial Management Association.
Peter is also an associate editor for 6 academic journals
related to mathematical finance and derivatives.
He has given numerous talks at both practitioner and academic conferences.
He is also credited with numerous contributions to quantitative finance including:
co-inventing the variance gamma model,
inventing static and semi-static hedging of exotic options,
and popularizing variance swaps and corridor variance swaps.
Peter has won awards from Wilmott Magazine for
Cutting Edge Research and from Risk Magazine for Quant of the Year.
Abstract
Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions:Disentangling the
Multi-dimensional Variations in S\&P 500 Index Options
The level of an equity index and the volatility of an equity index interact through several distinct
channels. First, holding business risk fixed, an increase in the level of
financial leverage raises the level of the equity volatility. Second,
regardless of the level of financial leverage, a positive shock to business
risk increases the cost of capital and reduces the valuation of future cash
flows, generating an instantaneous negative correlation between asset returns
and asset volatility. Finally, the market experiences both small continuous
movements and large market disruptions. The large and negative market
disruptions often generate self-exciting behaviors. The occurrence of one
disruption induces more disruptions to follow, thus raising market volatility.
We propose an equity index dynamics that capture all three channels of
interactions through the separate modeling of the asset return dynamics and the
financial leverage variation. We analyze how the different sources of
variations impact the index options behaviors differently across a wide range
of strikes, maturities, and calendar days.
This is joint work with Liuren Wu.
Dr Kai Detlefsen, Commerzbank Financial Engineering
1996-2003: Diploma in Maths at Humboldt University Berlin (Prof. Foellmer)
2003-2004: Master in Statistics at Humboldt University Berlin (Prof. Haerdle)
2004-2007: Phd in Economics at Humboldt University Berlin (Prof. Haerdle)
2007-now : Financial Engineer for Commerzbank's Corporates & Markets Equity trading divison.
Research interest: Dividend modelling, stochastic volatiltiy models, risk measures.
Abstract
Approximations of the Forward Smile
We explore the prices of forward starting options via a Taylor approximation.
The resulting representation explains the main risks in ratchets, ie identifies important greeks.
Using approximations for implied volatility and skew, we look at qualitative features of
forward implied volatilities and skews in the Heston model.
Prof Raquel M. Gaspar, ISEG, Technical University Lisbon
Slides
Raquel M. Gaspar holds a PhD degree in Finance from the Stockholm School of Economics,
where she was supervised by Professor Tomas Bjork and specialized herself in mathematical finance,
concretely, in interest rate and credit risk markets and models.
She also holds a Post-graduation degree in Risk Management and Derivatives from IDEFE,
NovaForum and IMC, a Master in Applied Mathematics to Economics and Management
from ISEG and has done her undergraduate studies in Economics at Universidade Nova de Lisboa.
Her research has been presented in conferences worldwide and published
both in academic journals and industry oriented books.
She is a 10 years experienced lecturer, at various levels - undergraduate, master,
PhD and Executive Education - both in Portugal and abroad.
Currently, she is Assistant Professor at ISEG, Technical University of Lisbon
where she belongs to the scientific commissions of both Finance and Mathematical Finance masters.
Besides her academic career she collaborates with the industry, mainly as consultant, since 1998.
Abstract
Convexity Adjustments - A Unified Framework
This article aims to clarify the notion of convexity in fixed income markets. The main challenge
is to provide a unified framework for all the different “convexity adjustments” that exist out
there. We explain the basic and appealing idea behind the use of convexity adjustments and
focus on the situations we believe are of particular importance to practitioners: yield convex-
ity adjustments, forward versus futures convexity adjustments, timing and quanto convexity
adjustments.
We claim that the appropriate way to look into any of these adjustments is as a side effect of a
measure change, as proposed by Plesser (2003). When using the appropriate setup, there may
be no immediate urge to do Taylor approximations or fall into too unrealistic assumptions. By
using one unified framework, we hope to clarify some issues and help the reader realize that
some of the assumptions that are sometimes imposed may be unnecessary.
For fixed income markets, convexity has emerged as an intriguing and challenging notion. Tak-
ing this effect into account correctly could provide financial institutions with a competitive
advantage. The idea underlying the notion of a convexity adjustment is quite intuitive and
can be easily explained in the following terms. Many fixed income products are non-standard
with respect to aspects such as the timing, the currency or the rate of payment. This leads to
complex pricing formulas, many of which are hard to obtain in closed-form. Examples of such
products include in-arreas or in-advance products, quanto products, CMS products, or equity
swaps, among others. Despite their non-standard features, these products are quite similar
to plain vanilla ones whose price can either be directly obtained from the market or at least
computed in closed-form. Their complexity can be understood as introducing some sort of bias
into the pricing of plain vanilla instruments. That is, we may decide to use the price of plain
products and adjust it somehow to account for the complexity of non-standard products. This
adjustment is what is known as convexity adjustment.
Dr Andreas Grau, Thetaris GmbH
Slides
Andreas J. Grau is Thetaris’ Chief Executive Officer.
He is an expert in Monte Carlo methods for complete
as well as incomplete markets and is responsible for
the numerical methods implemented in the Theta Suite.
Andreas Grau holds degrees in engineering and
computer science as well as a Dr. rer. nat. in financial mathematics.
Abstract
Computer Aided Finance - Another Journey in the Quest for the Holy Grail of Financial Engineering
With a unified theory for pricing any derivative still eluding the financial engineering
community, a new approach to product modelling is presented that may help the
industry deal with the new trends towards ever-increasing volume and complexity of
the products traded. We propose a product description language that is both simple
and general. It is suitable for computer processing, enabling tools to automatically derive
pricing algorithms. Important features of this language are shown and examples
for a wide range of derivatives presented. This is joint work with
Stefan Dirnstorfer.
Dr Jörg Kienitz, Postbank
Dr. Jörg Kienitz is head of quantitative analysis,
a unit within the treasury department of Deutsche Postbank AG.
After finishing his Ph.D. in stochastics and probability theory he worked for Reuters
and in IT since joining Deutsche Postbank AG in 2004. His team is responsible for pricing
and analysing structured investment products, product development, derivatives pricing and asset allocation.
Jörg frequently lectures at conferences and in academia at university level including the universities of Bonn,
Duisburg and Oxford. He is the co-author of the book Monte Carlo frameworks (Building customisable high performance C++ applications)
published by Wiley in 2009. Jörg also works as a consultant giving training courses on quantitative methods
for finance in Frankfurt, Paris and London.
Abstract
CMS - First, Second and Third Generation Products
The demand for structured interest-rate products has
led to a wide range of products based on CMS rates.
Such products involve caps and floors on the coupon.
Therefore, we need formulae to price such options,
risk manage the positions and construct hedges.
We start by describing first generation products.
The coupons of these basic CMS products are linked
to an n-year swap rate. We review the pricing and
hedging and show how the full swaption-smile is incorporated.
We proceed by looking at second generation products.
Expressing views on a steepening or a flattening of the curve
involve coupons linked to an n-year and m-year swap rate.
We consider again caps and floors and use an analytic solution
taking into account the smile of the single rates as well
as the fact that the market prices each strike with different volatility.
This is commonly known as the correlation smile.
Finally, we extend the solution to the case of third generation products.
Such products are used to trade the curvature
and involve coupons linked to an l-year, m-year and n-year swap rate.
Such products may be used for expressing the shape
of the curve from the central banks driven front
end to the pension funds driven long end.
Again analytic solutions for caps and floors are derived.
Finally, we compare the prices for caps and floors obtained
from our analytic solutions to those obtained by using
some well known (calibrated) term structure models.
This is joint work with
Manuel Wittke (University of Bonn).
Dr Tilman Huhne, d-fine GmbH
Slides
Tilman Huhne is a senior manager with d-fine GmbH.
He is responsible for d-fine's energy and commodity risk consulting unit
and engages in a broad range of respective topics. Currently his main areas
of interest are modelling and pricing of energy and commodity derivatives,
risk quantification and aggregation at portfolio level, development of trading
and hedging strategies along the supply chain, commodity hedge accounting and integrated risk reporting.
Mr. Huhne holds a PhD in Physical Chemistry from the LMU Munich, Germany and an MSc
in Mathematical Finance from the University of Oxford, UK.
Abstract
Commodity Derivatives - Modelling and Pricing in Practice
Taking a practicioner's viewpoint we survey modelling and pricing approaches
for commodity derivatives currently employed by banks and industrial companies.
After giving a brief overview of the underlying commodity markets,
we focus on spot and forward curve price dynamics and identify key risk drivers for the asset groups energy,
agricultural commodities, and industrial metals. Based on this we review popular modelling approaches that
take into account empirical commodity price features such as seasonality, volatility dynamics, convenience yield dynamics and spikes.
We comment on validation approaches and selection criteria for derivative pricing models.
Dr Jan Maruhn, UniCredit Markets & Investment Banking
Jan Maruhn is working as a quantitative researcher in the
Financial Engineering Equities and Hybrids team
(Structured Products Development) of UniCredit Markets and Investment Banking.
He obtained his PhD in mathematics from the University of Trier, Germany.
His research interests include the application of nonlinear
and stochastic optimization techniques as well as
numerical methods in general to problems arising in mathematical finance.
Abstract
Nonparametric Local Volatility Models and their Calibration
In contrast to parametric local volatility models,
which by definition possess a certain degree of smoothness,
nonparametric model variants suffer of a high degree of ill-posedness
which may result in rugged local volatility surfaces.
To obtain a smooth output surface,
one either has to sufficiently presmooth the input data
or to directly control variations of the surface during the calibration process.
In this talk we illustrate new variants for both methods.
On the one hand we present a presmoothing approach
which is phrased in the implied volatility
rather than the usually considered call price space.
On the other hand we show that adjoint techniques,
which have recently been introduced by Giles and Glasserman in the finance community,
are very efficient methods to speed up calibrations
with direct control of the ruggedness of the local volatility surface.
The presented results are joint work with Christian Boehm, Andre Loerx and Ekkehard Sachs.
Dr Tamas Mayer, Ernst & Young Zürich
Dr. Tamas Mayer is working as a Senior Consultant Financial Services Risk Management
at Ernst & Young Zürich. He studied physics at the University of Zürich, where he
received his PhD in 2005. He also holds an advanced degree in mathematical finance (MAS Finance UZH/ETH Zürich).
His areas of interest include the valuation of exotic
derivatives, market risk models as well as insurance modeling.
Abstract
Risk Sharing in Insurance Groups
In this talk we investigate solvency models of insurance groups, consisting of a parent company and
its subsidiaries. We develop a model for the Swiss Solvency Test (SST), which takes regulatory
requirements and other relevant effects
into account. These include the credit risk related to the parent company, the limited liability of the
parent company with respect to its subsidiaries, a possible ordination between the subsidiaries and
a realistic premium principle for risk transfer. We study the capital and risk
transfer structure of small insurance groups, in particular we investigate the impact of the above
mentioned effects on the group solvency requirement.
This is a joint work with Dr. Andreas Kull (AXA Winterthur),
Dr. Philipp Keller (Ernst & Young Zürich) and Helga Portmann (Bundesamt für Gesundheit, BAG Schweiz)
Prof Anthony Neuberger, University of Warwick
Anthony Neuberger is a Professor at the University of Warwick where he heads up the Finance Group.
His research interests include option pricing theory, corporate hedging and risk management,
investment and pensions policy. Prior to coming to Warwick, he was for some years at the London Business School,
and prior to that he was a civil servant working
in the UK Department of Energy and Cabinet Office.
Abstract
The Covariance between Returns and Implied Volatility
The skew in implied volatility has variously been interpreted as evidence of asymmetric risk preferences,
transaction costs, the presence of jumps and the correlation between returns and the volatility of returns.
We explore the relation between the skew and contracts on the covariance between returns and implied volatility.
Dr Bereshad Nonas, Commerzbank Financial Engineering
Bereshad Nonas is working as Financial Engineer for Commerzbank's Corporates & Markets
Fixed Income trading divison.
The recent focus of his research has been on single and multifactor Markov Functional
models and their usage for exotic derivatives.
He joined the team from the bank's model validation group where he was looking after
Foreign Exchange and Fixed Income products.
Before that he worked in the risk methodology group. He holds a PhD in Theoretical
Physics form the Technical Unversity (RWTH) of Aachen, Germany.
Abstract
The SABR Model in Approximation
We look at the properties of the stochastic volatility CEV model first introduced by P. Hagan et al. in 2002
that by now is being widely used in the financial community as a way of describing implied volatility surfaces.
The success of the model is based on using dynamics that are close to market behaviour as well as on an analytic
approximation that allows fast calculation of European option values. Unfortunately this approximation is known
to fail under certain input parameter combinations.
We put the original approximation in context with more recent literature and present a new workaround that builds
on the current solutions but provides more stable results for a wider range of parameters.
Prof Wim Schoutens, K.U.Leuven
Slides
Wim Schoutens has a degree in Computer Science and a PhD in Science, Mathematics. He is a
research professor in the Department of Mathematics at the Catholic University of Leuven,
Belgium.
He is a regular independent consultant and trainer to the banking industry on equity modeling,
structured products, credit derivatives, and other financial engineering problems.
His research interests cover all areas of financial Mathematics, in particular Lévy jump models. Wim
is author of the Wiley book “Lévy Processes in Finance: Pricing Financial Derivatives” and editor
(together with A.E. Kyprianou and Paul Wilmott) of the Wiley-book “Exotic Option Pricing and
Advanced Lévy Models”.
He recently has published in leading journals i.a. on advanced equity models, model risks, hedging
of variance swaps, jump driven credit models, multivariate financial engineering, pricing and
hedging of credit derivatives (CDSs, CDOs, CMS, CPPIs, CPDOs, ABSs, …)
He currently teaches several courses related to financial engineering in different Master programs.
He is a regular lecturer for the financial industry of in-house courses and public courses.
Abstract
Implied Levy Volatility
We introduce implied Levy volatility and study its use and behavior.
The concept of implied volatility in the Black-Scholes model
is one of the key points to its success and its wide spread use.
The implied volatility is very intuitive to use.
However, the Black-Scholes model is not really founded by empirical historical data;
stock returns tend to be more skewed and have fatter tails
than the normal distribution can provide.
Hence blind trust in a single implied volatility number
and all the numbers derived from that,
like deltas and other hedge parameters could be dangerous.
Here we try to develop a similar concept but now under a Levy framework
and therefore based on empirical more founded distributions.
More precisely, we introduce Levy implied time and space volatility
and make a study about the shape of implied Levy volatilities.
Further, we analyze its performance in delta hedging strategies
for a battery of Levy settings.
Typically in the Levy setting there are some additional degrees of freedom,
i.e. parameters that can be set freely.
We look for the historical optimal settings on the
basis of a delta hedge study of short term ATM vanilla on the SP500.
We show that under such parameter settings the model perform systematically better.
We illustrate this by looking to the daily hedge error
distribution and by noting that for the Levy models under investigation
its empirical mean is closer to zero and its the empirical variance
is smaller than under the Black-Scholes setting.
Prof Steven E. Shreve, Carnegie Mellon University
Steven E. Shreve is Orion Hoch 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 Mixture Model for Exotic Options
The Black-Scholes option pricing formula is based on the assumption
that the underlying asset price has a log-normal distribution
under a so-called risk-neutral (or martingale) probability measure.
However, this assumption leads to option prices that
do not agree with those observed in the market.
A solution to this dilemma adopted in practice is to assume
that the distribution of the underlying asset price
is a mixture of log-normal distributions.
In this talk, we discuss how to construct a dynamic model
that is consistent with this practice,
and then extend the construction of the model to price exotic options.
The essence of the extended construction is to create a Markov process
so that the joint distribution of the process
and its maximum-to-date agrees with the joint distribution
of a given Ito process and its maximum-to-date at each fixed time.
This is joint work with Gerard Brunick.
Prof George Skiadopoulos, University of Piraeus
Slides
George Skiadopoulos is Assistant Professor in the Department of Banking and Financial Management of the University of Piraeus.
He is also an Associate Research Fellow at the Financial Options Research Centre (FORC) of the University of Warwick.
He holds a Ph.D. in Financial Derivatives from the University of Warwick, and an M.Sc.
in Econometrics and Mathematical Economics from the London School of Economics.
During the period 1995-99, he worked as a Research Fellow in FORC undertaking projects supported
by the Centre's Corporate Members such as Deutsche Morgan Grenfell, Foreign and Colonial,
HSBC, Kleinwort Benson Securities, Price Waterhouse & Coopers, Robert Fleming, SBC Warwburg,
Tokyo Mitsubishi International, and the Central Bank of Austria.
From 1999 until 2000 he worked in the Research and Development Department
of the Athens Derivatives Exchange (ADEX). He has also acted
as a consultant to hedge funds and leading Greek financial institutions.
His research interests include asset allocation, alternative investments,
option pricing and hedging under the presence of implied volatility smiles,
and risk management focusing on Value-at-Risk.
He has published in academic journals and books, such as the Energy Economics,
European Financial Management Journal, International Journal of Theoretical and Applied Finance,
Journal of Alternative Investments, Journal of Banking and Finance,
Journal of Futures Markets, Journal of Risk Finance, RISK,
Review of Derivatives Research, and is a speaker in international conferences.
He has also taught a number of executive training courses in Greece and abroad.
He is a member of the editorial board of the Journal of Business Finance and Accounting (JBFA)
and serves in the Academic Advisory Council of the Professional Risk Managers International Association (PRMIA).
Abstract
Asset Allocation with Option-Implied Distributions: A Forward-Looking Approach
We address the empirical implementation of the static asset allocation
problem by employing forward-looking information from market option prices. To
this end, constant maturity one-month S&P 500 implied distributions are
extracted and subsequently transformed to the corresponding risk-adjusted ones. The
optimal portfolio strategy is obtained for the cases where direct maximisation of
the expected utility and its truncated Taylor series expansion is performed separately.
We find that the use of the risk-adjusted implied distributions makes the investor
significantly better o¤ compared with the case where she uses the historical
distribution of returns to calculate her optimal strategy. The results hold under a
number of evaluation metrics and utility functions and carry through even when
transaction costs are taken into account. This is joint work with
Alexandros Kostakisy and Nikolaos Panigirtzoglou.
Check the working papers on
http://web.xrh.unipi.gr/faculty/gskiadopoulos/
Dr Radu Tunaru, Cass Business School
With a background in Mathematics and Statistics
(Diploma in Mathematics, 5 years full-time, PhD in Probability and Statistics, PhD in Statistical Modeling)
Radu Sebastian Tunaru has been specialising in Financial Engineering and Financial Mathematics since 1999.
He worked as a Lecturer in Operations Research and Probability (1994-1996),
Research Fellow in Finance and Econometrics (1999) and
Senior Lecturer in Financial Mathematics (2000-2005).
His experience in the finance industry includes working as a quant for
Bank of Montreal in Structured Credit Investments
dealing with the cash-flow risk management models for two SIVs and the launch of one CDPC,
and for Merrill Lynch in Structured Finance, EMEA RMBS.
Currently a Senior Lecturer in Financial Mathematics,
he is teaching Financial Engineering and Advanced Mathematical Finance courses
and doing research on statistical credit arbitrage, Mathematical Finance,
numerical methods for pricing derivatives, pricing freight derivatives and pricing property derivatives.
He is the recipient of Multinational Finance Journal Best Paper Award, vol 5, 2001,
for the paper “Emerging Markets: Investing with Political Risk”,
Eastern Finance Association prize for the Outstanding Paper in International Finance for the paper
“Modelling Political Risk with a Doubly Stochastic Poisson Process”, Charleston USA 2001;
and SMEED prize awarded for the best young researcher in the field, 31st UTSG Annual Conference, York 1999.
He has published over 30 articles and book chapters.
Abstract
Deterministic Approximation Algorithms for European Options Pricing
European options pricing can be recast as an expectation calculation.
I show how to generate closed form approximation algorithms
for pricing any European contingent claim.
Our results can be applied for any models where the distribution
of the underlying is known at maturity.
The formulas are proved with results from probability theory,
mainly based on weak convergence of probability measures and central limit theorems.
The algorithms proposed here are based on the binomial distribution,
negative binomial distribution and the Poison generalized binomial distribution for
single-asset options and on multinomial distribution for the multi-asset options.
Moreover, the formulas derived in this fashion are deterministic and can handle complex payoffs.
Since no simulation is involved some of the common pitfalls
associated with Monte Carlo techniques are avoided.
Furthermore, computational costs can be saved
when calculating derivatives prices contingent on the same underlying.
The algorithms are related to grid sets that are derived from weak convergence results.
We prove that those sets are dense in the space of outcomes
of the random quantity defining the model
under which the no-arbitrage pricing is realised.
For example, under a Black-Scholes model,
the grid is dense in the set of real numbers
representing the possible states of the Wiener process.
We apply our algorithms to spread options and compare
our method with other methods in the literature.
The numerical results indicate a very good precision.
In addition, we show how the same formulas can be applied
to calculate other integrals in finance related to a stochastic frontier model.
Carlos Veiga, Frankfurt School of Finance & Management
Slides
Carlos has started the PhD program on March 2007 having as advisor Prof. Uwe Wystup.
He is Portuguese, born in 1976, and moved to Frankfurt to undertake this program.
Before joining the Frankfurt School of Finance & Management,
he worked eight years at Millennium bcp's investment bank on the equity derivatives trading desk.
The desk's main business lines are the issuing,
market-making and hedging of certificates, warrants and structured products.
His academic career was developed at Universidade Nova de Lisboa, Portugal
and includes an Economics Degree (4 years) from the Faculty of Economics
and a Master in Statistics and Optimization (2 years) from the Faculty of Science and Technology.
Carlos is working on systematically rating structured products.
Abstract
Closed Formula for Options with Discrete Dividends and its Derivatives
We present a closed pricing formula for European options under
the Black-Scholes model and formulas for its partial derivatives. The
formulas are developed making use of Taylor series expansions and by
expressing the spatial derivatives as expectations under special measures,
as in Carr, together with an unusual change of measure technique
that relies on the replacement of the initial condition.
The closed formulas are attained for the case where no dividend payment policy is
considered. Despite its little practical relevance, a digital dividend
policy case is also considered which yields approximation formulas. The
results are readily extensible to time dependent volatility models but
no so for local-vol type models. For completeness, we reproduce the
numerical results in Vellekoop and Nieuwenhuis using the formulas here
obtained. The closed formulas presented here allow a fast calculation
of prices or implied volatilities when compared with other valuation
procedures that rely on numerical methods.
Dr Thomas Weber, Weber und Partner
Since several years Thomas Weber is working in close cooperation with SciComp,
a software technology provider for pricing derivatives.
Before that he was working in the risk methodology group at
Deutsche Bank and as an independent consultant in the risk controlling area
for major banks and cooperations in Germany.
Thomas earned his PhD from the University of Mannheim (Prof. Bühler).
There he researched on interest rate derivatives with a special focus on HJM models.
Abstract
Speeding Up - Parallelisation Of Derivative Pricing Models
Latest developments with low cost GPUs makes parallel methods
available for the masses and accelerates the execution of a
derivative pricing model by factors of 20-100x. But to benefit
from the new hardware opportunity new numerical techniques
have to be developed and/or adapted.
These techniques will be discussed and results will be
shown for Monte Carlo simulation, PDE solutions, and calibration.
Manuel Wittke, University of Bonn
Manuel Wittke is currently working on his PhD at the University of Bonn under supervision of Prof. Klaus Sandmann.
Before he started as a research assistant, he studied economics at the universities of Konstanz
and Bonn with a strong focus on financial economics.
His main research interests lie on contingent claims on multi-asset processes under stochastic interest rates and volatilities,
on the influence of the cross-asset correlation structure and numerical valuation methods.
He starts working as a quantitative analyst at Deutsche Postbank AG in April 2009.
Abstract
CMS - First, Second and Third Generation Products
The demand for structured interest-rate products has
led to a wide range of products based on CMS rates.
Such products involve caps and floors on the coupon.
Therefore, we need formulae to price such options,
risk manage the positions and construct hedges.
We start by describing first generation products.
The coupons of these basic CMS products are linked
to an n-year swap rate. We review the pricing and
hedging and show how the full swaption-smile is incorporated.
We proceed by looking at second generation products.
Expressing views on a steepening or a flattening of the curve
involve coupons linked to an n-year and m-year swap rate.
We consider again caps and floors and use an analytic solution
taking into account the smile of the single rates as well
as the fact that the market prices each strike with different volatility.
This is commonly known as the correlation smile.
Finally, we extend the solution to the case of third generation products.
Such products are used to trade the curvature
and involve coupons linked to an l-year, m-year and n-year swap rate.
Such products may be used for expressing the shape
of the curve from the central banks driven front
end to the pension funds driven long end.
Again analytic solutions for caps and floors are derived.
Finally, we compare the prices for caps and floors obtained
from our analytic solutions to those obtained by using
some well known (calibrated) term structure models.
This is joint work with Jörg Kienitz (Deutsche Postbank AG).
Dr Benedikt Wilbertz, Université Paris 6
Benedikt Wilbertz has studied Mathematics and Business Administration
at the University of Trier, from which he also received a PhD in mathematics.
Currently, his main research interest is the approximation of
probability distributions by means of quantization.
Starting 2009, he will hold a PostDoc position in the
research group "Probabilités numériques et Finance"
of Prof Gilles Pagès at the University Paris 6.
Abstract
Quantization of Probability Distributions and its Applications to Mathematical Finance
Quantization consists in finding the best approximation
to a random variable in the mean sense using only a
finite number of elements. If we equip those
elements with weights corresponding
to the underlying distribution, we arrive at a
deterministic cubature formula, which is optimal for
the class of Lipschitz-continuous functionals.
Constructing a cubature formula in this way,
we are able to numerically compute expectations
with respect to finite and infinite dimensional
random variables very efficiently. This will be
demonstrated in the case of European and American
option pricing for finite dimensional quantization
and the pricing of exotic options for infinite
dimensional quantization. Finally, we present a
hybrid MC-Quantization approach, which combines
the benefits from deterministic as well as randomized cubature.
This is a joint presentation with Gilles Pagès.
Prof Uwe Wystup, Frankfurt School of Finance & Management
Slides
Uwe Wystup is Professor of
Quantitative Finance at Frankfurt School of Finance and Management,
where he is the academic director for the Masters Program in Quantitative Finance.
Before that he worked for Deutsche Bank, Citibank, UBS and Sal. Oppenheim jr. & Cie and as financial engineer and structurer
in the FX Options trading team of Commerzbank.
He is managing director of MathFinance AG and editor of the MathFinance
Newsletter.
Uwe holds a PhD in Mathematical Finance from Carnegie Mellon
University. He specializes in the quantitative aspects of foreign exchange markets,
international treasury management
and structured products.
He published in many scientific journals and wrote two books on Foreign Exchange Risk
and FX Options and Structured Products.
Abstract
Riester-Rente - a Comparative Study
When saving for retirement the market for Riester-savings plans is currently
booming in Germany. Many suppliers try to get their share in the market.
Besides insurance companies banks have also entered the Riester market.
In this paper we compare the performance of four representative investment concepts used
to guarantee the minimum payoff of a Riester plan, which is the
sum of the investors payments. We take a look at
DWS Riesterrente Premium, AXA TwinStar Rente Invest, Nürnberger
Funds-linked Doppel-Invest and Allianz
Riesterrente with Funds and Guaranty. We simulate the
final capital available over an investment horizon of 35 years.
The simulation model is a displaced double-exponential jump diffusion.
We consider optimistic, pessimistic and mixed market scenarios and two types of
investors. As a result we learn that one of the main contributors to the success
of the investment plan is the contract and management fees of the supplier.
Among the considered investment strategies the CPPI-approach performed by DWS
and the variable annuity approach of AXA
outperform classic insurance plans.
This is joint work with Andreas Weber (MathFinance AG).
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