|
|
Derivatives and Risk Management in Theory and Practice
27-28 March 2006
Prof Joachim K Anlauf, University of Bonn
Slides
Joachim K. Anlauf is professor at the Institut für Informatik II,
Rheinische Friedrich-Wilhems-Universität in Bonn.
Abstract
Pricing of Derivatives by Fast, Hardware Based Monte Carlo Simulation
In this talk we will show how typical problems of financial engineering can be
solved completely in hardware (as opposed to software solutions running on
traditional computers). As an example we have chosen the pricing of derivatives
by Monte Carlo simulations. It turns out that so called FPGA’s (field programmable
gate arrays) can be configured to run the whole algorithm in parallel, where the
calculation is accelerated by two orders of magnitude measured against a
state-of-the-art personal computer. These results are achievable using a single
FPGA-chip mounted on a PCI-card that is plugged into a standard PC.
The talk is organized as follows:
After introducing the architecture of FPGA’s and the way of configuring (programming)
them, the example application will be presented. Some remarks about the implementation
of the Monte Carlo simulation will show that we are able to exploit the parallelism of
many algorithms with the help of FPGA’s. Depending on the scenario a huge speedup can
be obtained. Advances in programming languages and compilation tools make it reasonable
to apply these techniques to many problems of financial engineering, including the
traditionally very time consuming risk management calculations.
Dr Andreas Binder, MathConsult Linz
Slides
Andreas Binder obtained his Ph.D. in applied mathematics (with a thesis on continuous
casting of steel) in 1991 from the University of Linz, Austria, one of the world´s
leading centres of inverse problems research. Since 1996, he has been managing director
of MathConsult, a company developing software solutions for engineering applications and
for the finance industry.
Abstract
Can you feel the heat? Inverse problems in finance and thermal processes
Calibration - or parameter identification - in computational finance is an inverse
problem, which is typically ill-posed in the sense of Hadamard, which means that
arbitrarily small perturbations or noise in the data may lead to arbitrarily large
changes in model paramaters if this type of problems is not handled carefully.
We describe some model problems from engineering applications and from finance and
show the common difficulties. We present the basic (and some advanced) concepts
of regularization techniques like Tikhonov regularization or Landweber iteration.
Some examples show the key features of regularization and its limitations.
Dr Oliver Brockhaus, Commerzbank
Slides
Dr Oliver Brockhaus is head of equity financial engineering at Commerzbank Corporates &
Markets. Before Commerzbank, he was a quant at Deutsche Bank, JPMorgan, Hypovereinsbank
and Calyon. His interests include smile and correlation modelling for equities and credit.
Abstract
Implied Sampling: Properties and Pitfalls
Sampling according to the equity distribution implied by Vanilla options has
become a market standard. The main advantage is that this method allows to
efficiently simulate path dependent payoffs. We discuss the approach, its
limitations and generalisations. In particular, we study the model prices
for various products in comparison with alternative (computationally less
efficient) methods.
Prof Peter Carr, Bloomberg
Slides
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
recently won awards from Wilmott Magazine for Cutting Edge Research and from Risk
Magazine for Quant of the Year.
Abstract
Vanilla No Touch Duality
Vanilla No Touch Duality
Dr Matthias Fengler, Sal. Oppenheim
Slides
Dr. Matthias Fengler is a quantitative analyst at Sal. Oppenheim, Frankfurt. He obtained
his PhD in Quantitative Finance from the Humboldt-Universität zu Berlin and is author
of the book Semiparametric
Modeling of Implied Volatility recently published in the Lecture Notes in Finance,
Springer-Verlag.
Abstract
Better than its Reputation: An Empirical Hedging Analysis of the Local
Volatility Model for Barrier Options
We discuss the pricing and hedging of barrier options
within the framework of the local volatility model. While there is an ample
literature on pricing issues, we focus on the dynamic hedging under
alternative stickiness assumptions on the implied volatilility dynamics
and different hedging strategies. Alternative stickiness assumptions on the
implied volatilility dynamics lead
to different computational procedures for the delta:
The delta can be computed assuming that the local volatility surface is
fixed (sticky-local-volatility or model-consistent delta), or assuming that
the implied volatility surface is fixed (sticky-strike delta), or assuming
that the implied volatility surface floats with the underlying spot value
(sticky-moneyness delta). Using data of the EUREX for options on the DAX, we
compare the three delta concepts in an empirical hedging analysis for
barrier options with a maturity of one and two years. We find that delta
hedging alone does not lead to satisfactory results with the sticky-strike
assumption performing best. However, when we use plain vanilla options as
additional hedging instruments, the hedging performance can be improved
considerably. We analyze two different dynamic hedging strategies involving
plain vanilla options and demonstrate that the resulting hedging errors are
distributed around zero with a small variance. Several non-parametric tests
on the empirical time series of hedging errors confirm that the
sensitivities computed under the sticky-strike assumption yield the best
hedging results, while model-consistent hedges have the largest variance.
This is joint work with Bernd Engelmann (Quanteam) and Peter Schwendner (Sal. Oppenheim).
Dr Christian Fries, DZ Bank
Slides
Christian Fries is head of model development, rates & hybrids at DZ
Bank’s risk control and a lecturer at University of Frankfurt. He
obtained a Ph.D. in mathematics (PDEs) from RWTH Aachen. His current
research interests are hybrid interest rate models and Monte Carlo
methods.
Abstract
Proxy simulation schemes using likelihood ratio weighted Monte Carlo for generic
robust Monte-Carlo sensitivities and high accuracy drift approximation with
applications to the LIBOR Market Model
We consider a generic framework for generating likelihood ratio weighted Monte Carlo simulation
paths, where we use one simulation scheme K° (proxy scheme) to generate realizations
and then reinterpret them as realizations of another scheme K* (target scheme) by adjusting
measure (via likelihood ratio) to match the distribution of K* such that
E[f(K*) | Ft] = E[f(K°) · w | Ft]. (1)
This is done numerically in every time step, on every path.
This makes the approach independent of the product (the function f in (1)) and even of the
model, it only depends on the numerical scheme.
The approach is essentially a numerical version of the likelihood ratio method and
Malliavin’s Calculus reconsidered on the level of the discrete numerical simulation
scheme. Since the numerical scheme represents a time discrete stochastic process sampled
on a discrete probability space the essence of the method may be motivated without a deeper
mathematical understanding of the time continuous theory (e.g. Malliavin’s Calculus).
The framework is completely generic and may be used for high accuracy drift approximations
and the robust calculation of partial derivatives of expectations w.r.t. model parameters
(i.e. sensitivities, aka. Greeks) by applying finite differences by reevaluating the expectation
with a model with shifted parameters. We present numerical results using a Monte-Carlo simulation
of the LIBOR Market Model for benchmarking.
This is joint work with Jörg Kampen of Heidelberg University.
Alexander Giese, Hypovereinsbank
Slides
Alexander Giese is Co-Head of Quantitative Research for Equity, Indices and Portfolio
Strategies in the Global Derivatives Team of HVB Corporates & Markets.
Abstract
On the Pricing of Auto-Callable Equity Structures in the Presence of Stochastic
Volatility and Stochastic Interest Rates
Auto-callable equity structures have become very popular in the last few
years. The characteristic feature of these structured products is that
depending on the path of the equity underlying the product is automatically
called and the notional is redeemed early on pre-prescribed dates known as
the auto-call dates. Clearly, auto-callable equity structures carry exposure
to the implied volatility skew, the volatility of the interest rates and to
the correlation between equity and interest rates. In order to take these
risk factors into account when pricing auto-callable equity structures, we
develop option pricing models that admit stochastic volatility, stochastic
interest rates and correlation between equity and interest rates. Using
these hybrid models we analyze the impact of the various risk factors on the
price of auto-callable structures.
Dr Simon Johnson, Commerzbank
Slides
Dr Simon Johnson is co-head of credit and interest rate Financial Engineering at
Commerzbank Corporates & Markets. Before Commerzbank, he was a senior quant at NumeriX
and a quant at Reech Capital plc. He started his career as a technology consultant at
The Technology Partnership plc. His interests include term structure models of interest
rates and smile modelling.
Abstract
Numerical Methods for the Markov Functional Model
Some numerical methods for efficient implementation of the 1- and 2-factor
Markov Functional models of interest rate derivatives are proposed. These
methods allow a sufficiently rapid implementation of the standard calibration
method, that joint calibration to caplets and swaptions becomes possible within
reasonable CPU time. Prices for Bermudan swaptions generated within the Markov
Functional model are found to be very close to market consensus prices. Bermudans
are therefore a good example of a product ideally suited to this model.
Prof Christoph Kühn, Frankfurt MathFinance Institute
Slides
Christoph Kühn is Juniorprofessor at the Frankfurt MathFinance Institute.
He holds a diploma in mathematical economics from the University of
Marburg and a
PhD in mathematics from Munich University of Technology.
His main research interests are pricing and hedging of derivatives in
incomplete markets
and the microstruture of financial markets.
Abstract
Optimal Portfolios in Markets with a Large Investor
A large investor is somebody whose trades move market prices significantly.
Put differently, he is faced with an illiquid financial market.
The first part of this talk is about a microeconomic motivation
of illiquid market models with both a permanent and a non-permanent
price impact caused by a transaction of a large investor.
Then, we discuss the large investor's utility maximization problem.
Jan Maruhn, University of Trier
Slides
Jan Maruhn works as a research associate at the University of Trier in
the numerical analysis group lead by Prof. Dr. Sachs. His research
interests include the application of robust optimization as well as
nonlinear and stochastic programming techniques to optimization problems
arising in finance. Currently, he is particularly interested in the
development of numerical algorithms for the computation of static hedge
portfolios for barrier options.
Abstract
Eliminating Model Parameter Uncertainty from Static Hedge Portfolios:
The Case of Barrier Options
The static hedging approaches for barrier options developed in the
literature so far can perform very poorly if applied to a real world
setting. One of the main reasons for this bad performance is that the
approaches do not take model parameter uncertainty into account. During the
talk we present a new approach to derive static super-replication strategies
in general financial market models. Furthermore, by using appropriate
optimization methods, the strategies can be robustified with respect to
changes in the model parameters. We will illustrate the concept and
numerical results for the Black Scholes model as well as general stochastic
volatility models. As it turns out, the resulting hedging strategies have
attractive properties and are only marginally more expensive than the
barrier option itself. This is joint work with E. Sachs.
Prof Gunter Meissner, Hawaii Pacific University
Slides
Gunter Meissner is Founder and President of Derivatives Software. After a lectureship
in Mathematics and Statistics at the Economic Academy Kiel he joined Deutsche Bank in
1990, trading interest rate futures, swaps and options. Gunter Meissner became Head of
Product Development in 1994, responsible for originating algorithms for new derivative
products. In 1995/1996 he was Head of Options at Deutsche Bank Tokyo. Presently, he is
also Professor for Finance at the Hawaii Pacific University
(www.hpu.edu).
Abstract
Valuing credit default swaps with counterparty risk – A combined copula-LMM approach
The paper derives a model with a closed form solution for valuing credit
default swaps including reference asset – counterparty default correlation.
The default correlation between the reference asset and the counterparty is
incorporated in two quadruple trees. One tree represents the default swap
payoff of the default swap seller; the other tree represents the default swap
premium payments of the default swap buyer. Swap valuation techniques are
then applied to derive the fair default swap price.
The model incorporates two correlation approaches used in today’s credit
practice, the Gaussian copula approach and the discrete correlation approach.
The Gaussian copula results in a higher credit default swap premium than the
discrete approach, since it produces lower joint default probabilities.
The model is represented with three LMM (Libor Market Model) processes. One
LMM process simulates risk-free short-term interest rates. Two more LMM processes
generate the reference asset default probabilities and the counterparty default
probabilities. A Visual Basic open source code version of the model is provided.
Keywords: Default swap pricing, copula, reference asset – counterparty default
correlation, Libor Market Model (LMM)
JEL Classification: G12, G13
Co-Authors: Michael Hamp, Janne Kettunen
Prof Thorsten Schmidt, University of Leizpig
Slides
Thorsten Schmidt is juniorprofessor in financial mathematics at the
University of Leipzig.
He has a strong background in statistics and probability theory and is
currently working on pricing and hedging credit risk, infinite dimensional
models and incomplete information issues.
Abstract
Pricing Corporate Securities under Noisy Asset Information
We consider the pricing of corporate securities when investors do not have
full information.
One approach for this is to consider a random default boundary, such that
even if the firm value was known, the time of default would not be
predictable. On the other side, in reality investors do not have access to
the true firm value. This is taken into account using an approach which
considers the firm value unobservable and uses noisy asset information to
obtain a filter problem. The filter problem is solved approximately and
consequences to the pricing of equity and debt are examined.
This is joint work with joint work with Rüdiger Frey of Leipzig University.
Prof Stephen Taylor, Lancaster University
Slides
Stephen Taylor is Professor of Finance at Lancaster University, England. He is the author
of Asset Price Dynamics, Volatility and Prediction, published by Princeton University Press
in 2005. His research into stochastic volatility models and option prices has been published
in several influential papers, in Mathematical Finance, the Journal of Financial and
Quantitative Analysis, the Journal of Econometrics and other premier journals.
Abstract
A Multi-Horizon Comparison of Density Forecasts for the S&P 500 Using
Index Returns and Option Prices
We compare density forecasts of the S&P 500 index from 1991 to 2004, obtained from
option prices and daily and five-minute index returns, over seven horizons ranging
from one day to twelve weeks. Risk-neutral forecasts are derived by estimating the
Heston stochastic volatility model from option prices, which provides a closed-form
density for all future times. Out-of-sample methods, both parametric and non-parametric,
are applied to transform the risk-neutral densities into real-world densities. These
option-based densities are compared with historical densities defined by ARCH models.
We find the best forecasts are produced by the parametric risk-transformation of the
risk-neutral densities, for horizons of one day, one week and two weeks, when forecast
methods are ranked by the out-of-sample likelihood of observed index levels. For longer
horizons, option-based densities continue to outperform the historical densities. A
mixture of the parametric transformation of the risk-neutral densities and the historical
densities obtained from five-minute returns has a higher likelihood than both components
of the mixture, for the one-day and one-week horizons.
The Kolmogorov-Smirnov and Berkowitz diagnostic tests show that the risk-transformed,
option-based densities nearly always pass these tests, and they do so more often than
the other density forecasting methods.
This is joint work with Mark B. Shackleton and Peng Yu.
Prof Robert G Tompkins, HfB - Business School of Finance and Management
Slides
Dr. Robert G. Tompkins was born in Oklahoma, USA and he received his
A.B. (1980), his A.M. (1980) and his MBA (honors) (1986) from the
University of Chicago. He moved to England in 1986 and subsequently
became a British citizen. He earned a Ph.D. (1998) from the
University of Warwick and his Habilitation (2000) from the University
of Technology, Vienna, where Dr. Tompkins lived from 1998 to 2003.
Abstract
Catch the Drift? - Finding the Change of Measure from The Real World to Risk Neutral
To address many of the anomalies in option prices, GARCH models have been proposed for
the pricing of options. Risk neutral drift adjustments often assume normal processes
and employ continuous time limits. Of particular interest is what happens when we consider
discrete time and non-normal processes for the underlying returns.
This research considers the problem of finding the drift adjustment that assures that all
future prices conform to a local martingale. To determine the appropriate drift
adjustment, simulated prices are determined using a Monte Carlo simulation. The logs
of the average prices are estimated and a regression of these versus time is estimated.
The slope coefficient of this regression can be interpreted as the change of measure.
A simple experiment with the standard Geometric Brownian Motion (GBM) model finds a slope
coefficient insignificantly different than the usual theoretical drift adjustment. This
approach is compared to the Empirical Martingale Simulation method proposed by Duan and
Simonato (1995) and the results are indistinguishable. This new approach is also related
to the moment matching method of Barraquand (1994), with the additional feature that a
multiplicative adjustment is made which ensures arbitrage is not violated by the estimated
option prices.
Once this approach has been shown to work in simulation of a known model, we turn to the
Real World. For this we consider the British Pound / US Dollar from 1990 to 2004. The
Mixed Unconditional Disturbances (MUD) approach of Tompkins and D’Ecclesia (2006) is
used to determine the empirical distribution of historical returns. This model first
estimates a GARCH (1,1) model and devolatises standardised returns. These returns are
mixed randomly and reprojected to simulate new price series (re-introducing the GARCH(1,1)
volatility). The average log return is regressed against the time horizon of the
simulation and it is found that the drift adjustment that yields asset prices as
exponential martingales is quadratic. The results are compared to what the theoretical
drift adjustment would be for the GARCH pricing model of Heston and Nandi (2000). It is
found that the simulated price series conforms to the first order drift adjustment of
Heston and Nandi (2000), but because the prices are estimated discretely and the
underlying process does not conform to GBM, the higher order drift adjustments appear
related to the variance of the variance.
Finally, as a check, a standard Bootstrapping approach was completed to compare the
results from the MUD simulation. The results of the Bootstrapping method are roughly
in line with the MUD simulation, but the resampling introduces considerable error in
the estimation of the drift coefficients in the regression. The MUD simulation reduced
the error in estimation by a factor of between 8 and 10 times.
JEL classifications: F21, F31, F41
Keywords: Martingale Measure, Girsanov, GARCH, Option Pricing.
Prof Jan Vecer, Columbia University
Slides
Prof Jan Vecer received his PhD in Mathematical Finance
from Carnegie Mellon University. He held academic jobs at the
University of Michigan and Kyoto University before joining the
faculty of the Department of Statistics at Columbia University in
2001. He works in the areas of option pricing and stochastic
optimal control.
Abstract
Trading Maximum Drawdown and Options on Maximum Drawdown
Maximum Drawdown is becoming increasingly important
in the risk management and in the portfolio optimization. In this
talk, we note that the Maximum Drawdown can be traded as a derivative
asset. Several related contracts, such as Call or Put options on the
Maximum Drawdown, or barrier option on the Maximum Drawdown
(Crash option) are also discussed. These contracts can facilitate
risk management for financial institutions concerned with control
of the drawdown of their portfolio.
Dr Ralf Werner, Allianz
Slides
Dr. Ralf Werner currently holds a position as Senior Risk Analyst within the Risk Methodology
team of Allianz, Group Risk Controlling.
His research is mainly focused on various applications of optimization in finance, with
emphasis on non-linear and robust optimization methods.
Abstract
Calibration of the Svensson Model to Simulated Yield Curves
In contrast to existing investigations on the calibration of the Svensson model to
real world yield curves, the calibration to simulated curves faces rather different
hurdles. As simulated yield curves usually come from a mathematical model (e.g.
Black-Karasinski 2-factor model) the availability of reliable data points to derive
the fit is not crucial. In contrast to usual applications, for Monte Carlo simulations
several thousand different yield curves with a broad shape have to be calibrated.
This demands both a very efficient calibration, i.e. acceptable computation time,
as well as the guarantee of convergence to the global optimum.
We investigate the behaviour of different formulations for the calibration problem and
highlight problems using a simple approach based on standard optimization routines. We show
how these results can be improved using a newly developed deterministic adaptive global
optimization routine based on sparse grids, while keeping computation times within
reasonable limits. We close the talk with a description of potential applications
within risk capital models of insurance companies.
This is joint work with Izabella Ferenczi, TU München.
|