Abstracts

• Dr. Carole Bernard, University of Waterloo

Path-dependent Inefficient Strategies and How to Make Them Efficient


We make the following assumptions. (1) Agents’ preferences depend only on the probability distribution of terminal wealth. (2) Agents prefer more to less. (3) The market is perfect and frictionless. (4) The market is arbitrage-free and could be incomplete. Under these assumptions, we show that in general path-dependent strategies are inefficient and not optimal. In addition, we characterize the ones that are cost-efficient. We obtain an explicit formula for the efficiency cost of a strategy as well as for the payoff of the cost-efficient derivative that should be preferred by all investors. Finally, we show that in the Black and Scholes framework, the necessary and sufficient conditions for a strategy to be cost-efficient is that its terminal payoff is an increasing function of the stock price. We illustrate the sudy by exhibiting the specific form of a derivative that dominates the lookback option, the geometric Asian option or the barrier option.

This is joint work with Prof. Phelim Boyle.



• Christoph Becker, Frankfurt School of Finance & Management

State-Dependent Dependencies: A Continuous-Time Dynamics for Correlations


We propose a new asset price model in continuous time where correlations and volatilities are functions of the current state of the market. The state of the market is based on a window of past asset realisations, the length of this window being a measure for the memory of the market. The approach is motivated by empirical findings from regression analyses in discrete time. A maximum likelihood approach is developed to estimate the parameters of the model from discrete asset realisations. We find strong empirical evidence that correlations increase in bear markets and for the existence of financial contagion in international markets. We analyse the severity of financial contagion dependent on market conditions. We explore consequences of market-state dependent volatilities and correlation in financial risk management and option pricing theory. We investigate the variance as a measure of portfolio risk and compare the variance from a model with constant correlation with the variance of a model with state dependent correlation. We propose a measure for losses in diversification due to a potential correlation breakdown.

This is joint work with Prof. Wolfgang Schmidt.



• Dr. Andreas Binder, MathConsult

Using Different Error Functionals in the Calibration of Stochastic Volatility Models


Stochastic volatility models and models including jump processes like the Heston and the Bates model gain more and more interest in the community. For practical purposes, it is essential that a fast and stable calibration routine is available. This calibration is quite frequently intrinsically instable due to the inverse problem nature of the taks. In this talk, we study the use of different error functionals (L1,L2 norm) and minimization algorithms (local and global) for solving the inverse problem. We also report the influence of the different parameter sets obtained on the price of exotic options. In order to speed up the calculations especially when using global optimization techniques we ported the code to run on GPUs.

This is a joint work with M. Aichinger and J. Fürst.



• Alexander Giese, Unicredit Markets and Investment Banking

Structured Equity Derivatives with Issuer Risk


During the recent financial crisis the credit spreads of banks skyrocketed from a few basis points at the beginning of 2007 to several hundreds of basis points end of 2008. As a result, the issuer risk has become a very important pricing factor in the valuation of equity linked structured notes issued by banks. One standard approach of incorporating issuer risk into the pricing of equity products assumes independence between the equity underlyings and the credit risk of the issuer and simply multiplies the equity dependent cash flows with the survival probability of the issuer. Since equity underlyings and credit spreads are highly negatively correlated, significant mispricing can be the result of applying such an approach. During the talk, we introduce several hybrid equity credit models which allow for equity credit correlation. Using these hybrid models we analyse the impact of the equity credit correlation on the fair values of representative equity linked structures with issuer risk.



• Dr. Jürgen Hakala, EFG Financial Products

Auto-Differentiation in Finance: A Casestudy


Auto-Differentation is a programming technique that uses function-composition and the mechanical application of the chain rule to obtain derivative expressions by the evaluation of a multivariate function. We show that this technique is a useful tool for selected applications in finance: model calibration – replacing the finite difference Jacobian by AD. Monte Carlo Simulation – augmenting the pathwise, and/or likelihood ratio method



• Dr. Christian Kahl, Commerzbank

Modelling Credit-Hybrid Products


We present an extended multi-factor stochastic hazard rate model, where pricing of contingent claims is done via a partial-(integro) differential equation, by introducing a default copula. This lattice copula is then compared to correlating the default event times, which is the common approach within a Monte Carlo approach. Analytical results for the short time step limit of the partial-(integro) differential equation implementation are derived and linked to the lower tail dependency of the respective copula.



• Sebastien Kayrouz, Murex

Logical SpaceTM


Time interpolation in the varied forms of strike or moneyness space are not logical, interpolation in delta space raises questions and encounters computational problems. We aim to present a new “Logical SpaceTM” for volatility modelling, applicable to all asset classes and adding transparency to skewness and leptokurtosis.

This is a joint presentation with Dr.Gerd Zeibig.



• Prof. Steve Kou, Columbia University

Clustering Defaults and Pricing of Collateralized Debt Obligations


The past several years have been an eventful period for the U.S. financial markets, mainly due to the crisis in subprime credit markets and the difficulty in modeling collateralized debt obligations (CDOs). In this paper we shall propose a model for CDOs that can incorporate clustering defaults. The model is based on Polya processes and the cumulative intensity of counting processes. Empirical evidences suggest that the model can calibration the current CDO data very well.



• Prof. Dilip Madan, University of Maryland

Capital Requirements,Acceptable Risks and the Value of the Taxpayer Put


Limited liability for the firm in the presence of unbounded liabilities delivers a free put option to the firm that is rarely valued and accounted for. We christen this put option the taxpayer put. In addition the optimality of free markets is called into question by the introduction of adverse risk incentives exaggerated by compensation aligned to stock market values. In such a context we introduce the concept of socially acceptable risks, operationalized by a positive expectation after distortion of the distribution function for risky cash flows. This results in a definition of capital requirements making the risks undertaken acceptable to the wider community. Enforcing such capital requirements can mitigate the perverse risk incentives introduced by limited liability provided that the set of acceptable risks is suitably conservatively de.ned. Additionally the value of the free taxpayer put may be substantially reduced. We illustrate all computations for the six major US banks at the end of 2008.



• Dr Fabio Mercurio, Bloomberg

Libor Market Models with Stochastic Basis


We start by describing the major changes that occurred in the quotes of market rates after the 2007 subprime mortgage crisis. We then show how to price interest rate swaps under the new market practice of using different curves for generating future LIBOR rates and for discounting cash flows. Straightforward modifications of the market formulas for caps and swaptions will also be derived.
Finally, we will introduce a new LIBOR market model, which will be based on modeling the joint evolution of FRA rates and forward rates belonging to the discount curve. We will start by analyzing the basic lognormal case and then add stochastic volatility.



• Dr. Attilio Meucci, Bloomberg

Managing diversification


We propose a unified, fully general methodology to define, analyze and act on diversification in any environment, including long-short trades in highly correlated markets. First, we build the diversification distribution, i.e. the distribution of the uncorrelated bets in the portfolio that are consistent with the portfolio constraints. Next, we summarize the wealth of information provided by the diversification distribution into one single diversification index, the effective number of bets, based on the entropy of the diversification distribution. Then, we introduce the mean-diversification efficient frontier, a diversification approach to portfolio optimization. Finally, we describe how to perform mean-diversification optimization in practice in the presence of transaction and market impact costs, by only trading a few optimally chosen securities.



• Dr. Rolf Poulsen, University of Copenhagen

Empirical Performance of Models for Barrier Option Valuation


In this paper the empirical performance of alternative models for barrier option valuation is studied. Five commonly used models are compared: the Black-Scholes model, the constant elasticity of variance model, the Heston stochastic volatility model, the Merton jump-diffusion model, and the infinite activity Variance Gamma model. We employ time-series data from the USD/EUR exchange rate market, and use plain vanilla option prices as well as a unique data-set of observed market values of barrier options. The different models are calibrated to plain vanilla option prices, and cross-sectional and prediction errors for plain vanilla and barrier option values are investigated. For plain vanilla options, the Heston and Merton models have similar and superior performance both in cross-section and for prediction horizons of up one week. For barrier options, the performances of continuous-path models (Black-Scholes, constant elasticity of variance, and Heston) is a mixed picture, while both models with jumps (Merton and Variance Gamma) perform markedly worse.



• Dimitri Reiswich, Frankfurt School of Finance & Management

Potential PCA Interpretation Problems for Volatility Smile Dynamics


The typical factor loadings found in PCA analysis for financial markets are commonly interpreted as a level, skew, twist and curvature effect. Lord and Pelsser question whether these effects are an artefact resulting from a special structure of the covariance or correlation matrix. They show that there are some special matrix classes, which automatically lead to a prescribed change of sign pattern of the eigenvectors. In particular, PCA analysis on a covariance or correlation matrix which belongs to the class of oscillatory matrices will always show n-1 changes of sign in the n-th eigenvector. This is also the case in most PCA results and raises the question whether the observed effects have a valid economic interpretation. We extend this line of research by considering an alternative matrix structure which is consistent with foreign exchange option markets. For this matrix structure, PCA effects which are interpreted as shift, skew and curvature can be generated from unstructured random processes. Furthermore, we find that even if a structured system exists, PCA may not be able to distinguish between these three effects. The contribution of the factors explaining the variance in the original system is incorrect.



• Prof. Ekkehard Sachs, University of Trier

Adjoint Techniques in Calibration


The pricing of derivatives in the financial markets becomes an increasingly important area of application for numerical analysis and numerical optimization. Various mathematical models are currently under consideration, which can be described by stochastic differential equations, partial differential equations or even explicit solution formulas. All these models contain a number of parameters that need to be fit such that the model output resembles the market data as closely as possible. This constitutes a nonlinear least squares problem and requires efficient and fast solvers from numerical optimization.
Any fast optimization solver relies on accurate gradient information which, if obtained from finite difference approximation, works well as long as the number of parameters is small. However, for a larger number of parameters like time dependent parameters, the computing time requirement for the gradient calculation can be enormous. In this talk we illustrate how to replace the finite difference or sensitivity approach by an adjoint approach which yields a substantial savings in computing time and is applicable in a SDE or PDE framework. Furthermore, we discuss the use of reduced order models. Here e.g. the PDE is replaced by a system of ordinary differential equations which is then used in calibrating the model. Finally, the overall optimization effort in calibrating a PDE model can be reduced to an effort equivalent to a few evaluations of a PDE.



• Dr. Christof Schmidhuber, Fintegral Asset Management

Alternative Beta in Practice


The asset allocation process of an investor typically involves an optimization process: its objective is to maximize expected return subject to constraints such as risk tolerance or A&L matching. Traditional market factors, including equity indices or interest rates, are usually dominant, but sometimes more or less sophisticated trading strategies also play a role to enhance returns. Recently, so-called “alternative market factors” have attracted much attention. They represent systematic trading strategies, such as momentum- or contrarian strategies. Exposure to these new market factors has been called “alternative beta”. It has been advocated that alternative beta can be used to replicate the performance of some of these sophisticated strategies with much improved liquidity and transparency. In this talk we propose a classification scheme for the most important forms of alternative beta. We show how the corresponding alternative market factors can be combined with traditional market factors in order significantly improve the risk-return profiles of investment portfolios. We also investigate the effectiveness of alternative market factors in replicating non-investible hedge fund indices, based on one year of real trading



• Prof. Uwe Schmock, Technical University of Vienna

Generalization of the Dybvig-Ingersoll-Ross Theorem and Asymptotic Minimality


The long-term limit of zero-coupon rates with respect to the maturity does not always exist. In this case we use the limit superior and prove corresponding versions of the Dybvig-Ingersoll-Ross theorem, which says that long-term spot and forward rates can never fall in an arbitrage-free model. Extensions of popular interest rate models needing this generalization are presented. In addition, we discuss several definitions of arbitrage, prove asymptotic minimality of the limit superior of the spot rates, and illustrate our results by several continuous-time short-rate models.
This is joint work with Verena Goldammer.

[paper] [slides]

• Dr. Roland Seydel, d-fine

The risk of default, credit securitization of a bank and impulse control


Financial instruments such as Asset-Backed Securities (ABS) were at the heart of the unfolding financial crisis of 2007 and 2008. These securities bundle loans that banks want to dispose of, e.g., subprime home loans.
Before the crisis, ABS were thought to increase diversification of banks and thus to make the financial system more resilient; although this turned out to be wrong in general, such instruments still are an important tool for managing the risk of an individual bank.
In our talk, we present a model of a bank in a Markov-switching economy that can reduce its loan exposure by discrete impulses. We start with an introduction to the model and its real-world background. The value function of impulse control is associated with the (viscosity) solution of a PDE called quasi-variational inequality (QVI). This QVI is solved numerically, and practical insights and conclusions from the numerical results are discussed.
This talk is based on joint work with Rüdiger Frey.



• Prof. Michel Vellekoop, University of Amsterdam

Early Exercise Premia for Assets with Dividends


Standard option pricing models usually pay no or little attention to the inclusion of realistic dividend structures in the model for the underlying asset prices. In this talk we show how cash dividends can be included in option pricing schemes in a consistent way, and we study the poperties of American options when dividends are included. We derive a generalized version of a well-known integral equation for the early exercise boundary which allows the inclusion of dividends, and use this to illustrate the differences with the case where no dividends are present.



• Prof. Uwe Wystup, MathFinance

Vedic Mathematics: Teaching an Old Dog New Tricks


We show what we all should have learned in high school but didn't: How the authors of the Indian vedas did mental arithmetics: multiplication - vertically and crosswise, division - by one more than the one before, square roots - the duplex method.

[slides]

• Dr. Gerd Zeibig, Murex

Logical SpaceTM


Time interpolation in the varied forms of strike or moneyness space are not logical, interpolation in delta space raises questions and encounters computational problems. We aim to present a new “Logical SpaceTM” for volatility modelling, applicable to all asset classes and adding transparency to skewness and leptokurtosis.

This is a joint presentation with Sebastien Kayrouz.

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