A Conference in Honour of Walter Schachermayer — Vienna University, July 12-16, 2010


Acciaio, Beatrice (University of Vienna, AT)
"Risk assessment for uncertain cash flows: Model ambiguity, discounting ambiguity, and the role of bubbles"
[slides available]
We study the risk assessment of uncertain cash flows in terms of dynamic convex risk measures for processes. These risk measures take into account not only the amounts but also the timing of a cash flow. We discuss their robust representation in terms of suitably penalized probability measures on the optional sigma-field. This yields an explicit analysis both of model and discounting ambiguity. We focus on supermartingale criteria for different notions of time consistency. In particular we show how bubbles may appear in the dynamic penalization, and how they cause a breakdown of asymptotic safety of the risk assessment procedure. This is joint work with Hans Föllmer and Irina Penner.

Bank, Peter (TU Berlin, DE)
"Convex Duality and Intertemporal Consumption Choice"
[slides available]
We show how to develop a theory of convex duality for optimal investment in incomplete markets when utility is obtained from a consumption stream, rather than from terminal wealth as in the celebrated paper by Kramkov and Schachermayer (1999). Joint work with Helena Kauppila.

Barndorff-Nielsen, Ole E. (University of Aarhus, DK)
"Ambit Processes: Aspects of Theory and Applications"
[slides available]
Ambit processes is a class of stochastic processes designed for the description and analysis of tempo-spatial dynamical developments. The focus in the talk will be on stationary regimes, and recent results in the theory of ambit fields and processes and applications to turbulence and to energy markets will be presented.

Beiglböck, Mathias (University of Vienna, AT)
"A direct proof of the Bichteler-Dellacherie Theorem and connections to arbitrage"
We give an elementary proof of the celebrated Bichteler-Dellacherie Theorem which states that the class of stochastic processes allowing for a useful integration theory consists precisesly of those processes which can be written in the form S=M+A, where M is a local martingale and A is a finite variation process. In other words, S is a good integrator if and only if it is a semi-martingale. We obtain this decompostion rather directly from an elementary discrete time Doob-Meyer decomposition. As a by-product we obtain a characterization of semi-martingales in terms of "no free lunch", thus extending a result from (DS94). Joint work with Walter Schachermayer and Bezirgen Veliyev.

Björk, Tomas (Stockholm School of Economics, SE)
"Time inconsistent stochastic control"
[slides available]
We present a theory for stochastic control problems which, in various ways, are time inconsistent in the sense that they do not admit a Bellman optimality principle. We attach these problems by viewing them within a game theoretic framework, and we look for subgame perfect Nash equilibrium points.
For a general controlled Markov process and a fairly general objective functional we derive an extension of the standard Hamilton-Jacobi-Bellman equation, in the form of a system of non-linear equations, for the determination for the equilibrium strategy as well as the equilibrium value function. All known examples of time inconsistency in the literature are easily seen to be special cases of the present theory. We prove that for every time inconsistent problem, there exists an associated time consistent problem such that the optimal control and the optimal value function for the consistent problem coincides with the equilibrium control and value function respectively for the time inconsistent problem. We also study some concrete examples.

Cancelled: Bühlmann, Hans (ETH Zurich, CH)
"On the Prudence of the Actuary and the Courage of the Entrepreneur (Gambler)"
The modeling of the insurance company's business as a stochastic process is standard. The strategic question to be answered is then: "How do we decide, whether the business is good or bad?" In the standard modeling context this means to define a preference ordering on the (properly defined) set of stochastic processes.
Traditional actuarial practice uses the probability of ruin to define the preference ordering, the more modern approach (although already proposed by De Finetti in 1957) uses the value of the firm (expected discounted sum of dividends) for this purpose. Both approaches have originated a huge literature of mathematical and economic papers.
One finds however few answers in these papers to the question, how the implications of these approaches reflect in the "daily operations" of the insurance company.
I shall treat three fundamental aspects: Capitalization, Premium Calculation, Risk Acceptance and Transfer.
Both De Finetti and Borch, who were the most prominent advocates for using the value of the firm as preference criterion have suggested that in this approach one needs an additional control to guarantee stability of the business operation.

Campi, Luciano (University Paris-Dauphine, FR)
"Efficient trading strategies in financial markets with proportional transaction costs"
[slides available]
In this talk, we characterize efficient contingent claims - that is, optimal portfolios for at least one rational agent - in a very general financial market model of foreign currencies with proportional transaction costs and multidimensional utility functions. In our setting, transaction costs may be random, time-dependent and have jumps. Thanks to the dual formulation of expected multivariate utility maximization problem established in Campi and Owen (2010), we provide a complete characterization of efficient portfolios, generalizing earlier results of Dybvig (1988) and Jouini and Kallal (2001). We show in particular that an efficient portfolio is cyclic anticomonotonic with respect to some dual variable, that can be viewed as generalized (strictly) consistent price process. After defining a utility price in this multidimensional setting, we propose a measure of strategy inefficiency as well as a method for computing it effectively, which is based on the notion of cyclic anticomonotonicity. This is based on a joint work with Elyes Jouini and Vincent Portes.

Carmona, René (Princeton University, USA)

Cerny, Ales (Cass Business School, London, UK)
"Admissible Strategies in Semimartingale Portfolio Optimization"
The choice of admissible trading strategies in mathematical modelling of financial markets is a delicate issue, going back to Harrison and Kreps (1979). In the context of optimal portfolio selection with expected utility preferences this question has been a focus of considerable attention over the last twenty years. We propose a novel notion of admissibility that has many pleasant features - admissibility is characterized purely under the objective measure; the wealth of any admissible strategy is a supermartingale under all pricing measures; local boundedness of the price process is not required; neither strict monotonicity, strict concavity nor differentiability of the utility function are necessary; the definition encompasses both the classical mean-variance preferences and the monotone expected utility. For utility functions finite on the whole real line, our class represents a minimal set containing simple strategies which also contains the optimizer, under conditions that are milder than the celebrated reasonable asymptotic elasticity condition on the utility function.

Cooper, James B. (Johannes Kepler Universität Linz, AT)
"What is a quantum field? or How to do analysis in the space of observables"
The family of observables (unbounded self-adjoint or normal operators on a Hilbert space) plays the role of the real (resp. complex) numbers in quantum theory. It is therefore surprising that there is no systematic treatment of analysis in this space. It is the purpose of our talk to present such a theory. We shall concentrate on two central problems: what is an analytic function between sets of observables and what is a distribution with values in the space of observables? We describe the basic mathematical conepts and theories which are required, discuss some earlier definitions from the literature and present examples and results on the topic of the title.

Cuchiero, Christa (ETH Zurich, CH)
"Affine processes on non-canonical state spaces"
Based on stochastic invariance, we consider necessary conditions on possible state spaces and admissible parameter sets for affine processes. As a particular example for non-canonical state spaces, we present affine processes on symmetric cones.

Czichowsky, Christoph (ETH Zurich, CH)
"Time-Consistent Mean-Variance Portfolio Selection in Discrete and Continuous Time"
[slides available]
It is well known that mean-variance portfolio selection is a time inconsistent control problem in the sense that the dynamic programming principle fails. We present a time-consistent formulation of this problem which is based on local mean-variance efficiency. We start in discrete time, where the formulation is straightforward, and then find the natural extension to a general continuous-time semimartingale setting. This generalises recent results by Basak and Chabakauri (2009) and Björk and Murgoci (2008) where the treatment relies on an underlying Markovian framework. As a new feature we justify the continuous-time formulation by showing that it coincides with the continuous-time limit of the discrete-time formulation. The proof of this convergence exploits a global characterisation of the locally optimal strategy in terms of the Föllmer-Schweizer decomposition of the mean-variance tradeoff process.

Davis, Mark (Imperial College London, UK)
"Arbitrage Bounds for Weighted Variance Swap Prices"
[slides available]
This paper builds on earlier work by Davis and Hobson (Mathematical Finance, 2007) giving model-free - except for a 'frictionless markets' assumption - necessary and sufficient conditions for absence of arbitrage given a set of current-time put and call options on some underlying asset. Here we suppose that the prices of a set of put options, all maturing at the same time, are given and satisfy these conditions. We now add a path-dependent option, specifically a weighted variance swap, to the set of traded assets and ask what are the conditions on its time-0 price under which consistency with absence of arbitrage is maintained. We work under the extra modelling assumption that the underlying asset price process has continuous paths. In general, we find that there is always a non-trivial lower bound to the range of arbitrage-free prices, but only in certain cases is there a finite upper bound. In the case of, say, the vanilla variance swap, a finite upper bound exists when there are additional traded European options which constrain the left wing of the volatility surface in appropriate ways. This is joint work with Vimal Raval and Jan Obloj.

Delbaen, Freddy (ETH Zurich, CH)
"Mod-φ convergence and applications"
(Joint work with Kowalski and Nikeghbali)
We introduce some precisions on the weak convergence of probability laws and will give some applications in number theory and random matrix theory.

Dolinsky, Yan (ETH Zurich, CH)
"Error Estimates for Multinomial Approximations of American Options in a Class of Jump Diffusion Models"
[slides available]
We derive error estimates for multinomial approximations of American options in a class of multidimensional jump-diffusion models. We assume that the payoffs are Markovian and satisfy Lipschitz type conditions. Error estimates for such type of approximations were not obtained before. Our main tool is the strong approximations theorems for i.i.d. random vectors which were obtained in [2]. For the multidimensional Black-Scholes model our results can be extended also to a general path dependent payoffs which satisfy Lipschitz type conditions. For the case of multinomial approximations of American options for the Black-Scholes model our estimates are a significant improvement of those which were obtained in [1] (for game options in a more general setup).
[1] Yu.Kifer, Optimal stopping and strong approximation theorems, Stochastics 79 (2007), 253-273.
[2] A.I Sakhanenko, A New Way to Obtain Estimates in the Invariance Principle, High Dimensional Probability II, (2000) 221-243.

El Karoui, Nicole (École Polytechnique, Palaiseau, FR)
"An Exact Representation of Non Linear Utility Stochastic PDEs"
[slides available]
(Joint work with M'rad Mohamed.) We propose a new approach to forward dynamic utilities, recently introduced by M. Musiela and T. Zariphopoulou, to model possible changes over the time of individual preferences of an agent. In particular, there is no-prespecified trading horizon. These utilities satisfy a property of consistency with a given incomplete financial market which gives them similar properties to the function values of classical portfolio optimization. First, we derive a non linear stochastic PDEs satisfied by Itô type consistent stochastic utilities and their dual convex functions, from Itô-Ventcel formula. Then, under some regularity assumptions and the hypothesis that the optimal wealth (resp. optimal state price density) is a increasing function of the initial capital, we characterise all consistent utilities with a given increasing optimal wealth process, from the composition of the dual optimal process and the inverse of the optimal wealth. This allows us to reduce the resolution of fully nonlinear second order utility SPDE to the existence of monotone solutions of two stochastic differential equations. We also express the volatility of consistent utilities as an operator of the first and the second order derivatives of the utility in terms of the optimal primal and dual policies. We also provide some examples based on power utilities with random coefficient.

Embrechts, Paul (ETH Zurich, CH)
"From financial mathematics to quantitative risk management"
[slides available]
Financial mathematics has had a remarkable success in becoming a mature, academic field of research, and this with considerable applications to finance and economics. The credit (subprime) crisis has however shown that more attention has to be given to a more holistic approach to (quantitative) risk management, this both from a practical as well as methodological point of view. In this talk I will give examples of questions we need to address going forward; these examples will very much be based on personal experience gathered over the years leading up to the crisis, as well as from more recent discussions with regulators and the banking industry. This talk is in part based on a RiskLab paper written jointly with Catherine Donnelly "The devil is in the tails: actuarial mathematics and the subprime mortgage crisis", ASTIN Bulletin 40(1), pp. 1-33, 2010.

Émery, Michel (University of Strasbourg, FR)
"Effectivity in proofs that some filtrations are generated by Brownian motions"
One of the pillars of stochastic calculus is the notion of a filtration. A classification of stochastic filtrations modulo isomorphisms is far beyond horizon, but one can sometimes prove that a given filtration is generated by some Brownian motion; such a generating BM is then more or less explicitly constructed or described. Where does the difference in effectivity come from, and what makes some filtrations more difficult to be shown to be Brownianly generated?

Fajardo, José (Getulio Vargas Foundation, Rio de Janeiro, BR)
"Skewness Measures and Implied Volatility Skew"
We study different notions of skewness through a parameter that we find relevant in order to quantify and explain the different types of skewness encountered in market models. Then, we study the implied volatility skew. We assume that stock is driven by a time-dependent Lévy process.

Feehan, Paul (Rudgers University, New Brunswick, USA)
"American-style options, stochastic volatility, and degenerate parabolic variational inequalities"
[slides available]
Elliptic and parabolic partial differential equations arising in option pricing problems involving the Cox-Ingersoll-Ross or Heston stochastic processes are well-known to be degenerate parabolic. We provide a report on our work on the existence, uniqueness, and regularity questions for variational inequalities involving degenerate parabolic differential operators and applications to American-style option pricing problems for the Heston model. This is joint work with Panagiota Daskalopoulos at Columbia University.

Filipović, Damir (Swiss Finance Institute, École Polytechnique Fédéral de Lausanne, CH)
NEW (different talk): "Equivalent Measure Changes for Jump-Diffusions"
[slides available]
A classical problem in mathematical finance is the question whether the stochastic exponential of a given martingale is a true martingale or not. In this talk I present some explicit sufficient conditions in the standard situation of a jump diffusion factor model, which often can be directly verified. This is based on a joint paper with Patrick Cheridito and Marc Yor.
Originally planned: "Quadratic Variance Swap Models: Theory and Evidence"
We introduce a quadratic term structure model for the variance swap rates. The latent multivariate state variable is shown to follow a quadratic process characterized by linear drift and quadratic diffusion functions. The univariate case turns out to be a parsimonious and flexible class of models. We provide a complete classification and canonical representation, and discuss model identification. We fit the model to the cross section of variance swap rates and returns of the S&P 500 Index, and perform a specification analysis. This is joint work with Elise Gourier and Loriano Mancini.

Cancelled: Föllmer, Hans (Humboldt Universität zu Berlin, DE)
"Asymptotic Arbitrage, Large Deviations, and Model Ambiguity"
We first summarize joint work with Walter Schachermayer on asymptotic arbitrage and large deviations and then discuss some further developments, in particular results of Thomas Knispel on optimal long run investment under model ambiguity and some first steps towards a theory of large deviations for risk measures.

Friz, Peter (TU and WIAS Berlin, DE)
"Ordinary, partial and backward stochastic differential equations driven by rough signals"
[slides available]
We shall discuss the applicabilty of rough paths methods in the novel context of partial and backward stochastic differential equations. In the case of non-linear noise terms, there are connections to the theory of "quadratic" PDEs and BSDEs. Joint work with M. Caruana, H. Oberhauser and J. Diehl.

Gerhold, Stefan (Vienna University of Technology, AT)
"Refined volatility expansion in the Heston model"
[slides available]
It is known that Heston's stochastic volatility model exhibits moment explosion, and that the critical moment can be obtained by solving (numerically) a simple equation. This yields a leading order expansion for the implied volatility at large strikes (Roger Lee's moment formula). Motivated by recent tail-wing refinements of this moment formula, we first derive a novel tail expansion for the Heston density, and then show the validity of a refined volatility expansion. Our methods and results may prove useful beyond the Heston model: the entire analysis is based on affine principles; at no point do we need knowledge of the (explicit, but cumbersome) closed form expression of the Fourier transform of log-spot. This is joint work with P. Friz, A. Gulisashvili, and S. Sturm.

Ghoussoub, Nassif (University of British Columbia and Banff International Research Station, Vancouver, CA)
"Self-dual Variational Calculus"
[slides available]
How to solve PDEs by completing squares?
Motivated in part by the basic equations of quantum field theory (e.g. Yang-Mills, Ginzburg-Landau, etc...), we identify a large class of partial differential equations that can be solved via a newly devised "selfdual variational calculus". In both stationary and dynamic cases, solutions of such equations are not derived from standard critical point theory applied to Euler-Lagrange functionals, but from the minimization of appropriately chosen self-dual functionals.
The class contains many of the basic families of linear and nonlinear, stationary and evolutionary partial differential equations: Transport equations, Nonlinear Laplace equations, Cauchy-Riemann systems, Navier-Stokes equations, but also - infinite dimensional - gradient flows of convex potentials (e.g., heat equations), nonlinear Schrödinger equations, Hamiltonian systems, and many other parabolic-elliptic equations.
Besides existence and uniqueness issues, "Self-dual Variational Calculus" provides a natural and unifying approach for dealing with nonlinear inverse and control problems, as well as with the homogenization of monotone vector fields.

Goldammer, Verena (Vienna University of Technology, AT)
"Modelling of Dependent Credit Rating Transitions"
[slides available]
For modeling dependent credit rating transitions we internalize the effect of shocks on the continuous-time rating by using a marked point process. Such processes model random event times, and a random mark specifies the possible simultaneous credit rating transitions at each event time. Within this framework we study a special model in more detail, that is a Markov jump process and all obligors with the same rating are only allowed to change to the same rating class at the same time. In this framework clustering of defaults is possible and we can generate different shapes of loss distributions which is shown by a numerical simulation of a credit portfolio. Furthermore we calibrate the model to historical rating transitions. For a specialization of our general model, we are able to give an analytic expression for the maximum likelihood estimator and show that the estimator is asymptotically efficient.

Guasoni, Paolo (University of Boston, USA)
"Relaxed Utility Maximization"
[slides available]
For a relaxed investor - one whose relative risk aversion vanishes as wealth becomes large - the utility maximization problem may not have a solution in the classical sense of an optimal payoff represented by a random variable. This nonexistence puzzle was discovered by Kramkov and Schachermayer, who introduced the reasonable asymptotic elasticity condition to exclude such situations. Utility maximization becomes well-posed again representing payoffs as measures on the sample space, including those allocations singular with respect to the physical probability. The expected utility of such allocations is understood as the maximal utility of its approximations with classical payoffs - the relaxed expected utility. We decompose relaxed expected utility into its classical and singular parts, representing the singular part in integral form, and proving the existence of optimal solutions for the utility maximization problem, without conditions on the asymptotic elasticity. This is joint work with Sara Biagini (University of Pisa).

Jouini, Elyčs (CEREMADE et Institut de Finance Dauphine, FR)
"Financial Markets Equilibrium with Heterogeneous Agents"
(with Jaksa Cvitanic, Semyon Malamud and Clotilde Napp) This paper presents an equilibrium model in a pure exchange economy when investors have three possible sources of heterogeneity. Investors may differ in their beliefs, in their level of risk aversion and in their time preference rate. We study the impact of investors heterogeneity on the properties of the equilibrium. In particular, we analyze the consumption shares, the market price of risk, the risk free rate, the bond prices at different maturities, the stock price and volatility as well as the stock's cumulative returns, and optimal portfolio strategies. We relate the heterogeneous economy with the family of associated homogeneous economies with only one class of investors. We consider cross sectional as well as asymptotic properties.

Kallsen, Jan (Christian-Albrechts-Universität zu Kiel, DE)
"On shadow prices in portfolio optimization"
This talk deals with the dual approach to portfolio optimisation with transaction costs. According to a general principle, there exists a shadow price process (or consistent price system) such that optimal trading without transaction costs relative to this fictitious asset amounts to trading the original price process with transaction costs. In this talk we consider several aspects of existence and computation of shadow price processes.

Karatzas, Ioannis (Columbia University, New York, USA)
"Probabilistic Aspects of Arbitrage"
[slides available]
Consider the logarithm log(1/U(T,z)) of the highest return on investment that can be achieved relative to a market with Markovian weights, over a given time-horizon [0,T] and with given initial market weight configuration Z(0) = z. We characterize this quantity (i) as the smallest amount of relative entropy with respect to the Föllmer exit measure, under which the market-weight process Z(.) is a diffusion with values in the unit simplex and the same covariance structure but zero drift; and (ii) as the smallest "total energy" expended by the respective drift, over a class of probability measures which are absolutely continuous with respect to the exit measure and under which Z(.) stays in the interior of the unit simplex at all times, a.s. The smallest relative entropy, or total energy, corresponds to the conditioning of the exit measure on the event that Z(.) stays in the interior of the unit simplex throughout the time interval [0,T]; whereas, under this "minimal energy" measure, the portfolio generated by the function U(.,.) has the numeraire and relative log-optimality properties. This same portfolio also realizes the highest possible relative return on investment with respect to the market. (Joint work with D. Fernholz.)

Kardaras, Constantinos (Boston University, USA)
"Forward-convex convergence in probability of sequences of nonegative random variables"
[link to paper/arXiv as talk on blackboard]
For a sequence in nonnegative random variables, we provide simple necessary and sufficient conditions to ensure that each sequence of its forward convex combinations converges in probability to the same limit. These conditions correspond to a measure-free version of the notion of uniform integrability and are related to the numeraire problem of mathematical finance.

Keller-Ressel, Martin (ETH Zurich, CH)
"Affine Processes are Regular"
[slides available]
Affine Processes are a class of Markov processes with important applications in finance and other fields. These processes have been described and fully characterized by Duffie, Filipovic and Schachermayer (2003) under the condition of 'regularity'. Here, a Markov process is called regular, if its characteristic function is differentiable in time with (space-)continuous derivatives. In this talk I present a joint work with Walter Schachermayer and Josef Teichmann that shows that for affine processes on the canonical state space this condition is automatically fulfilled, i.e. that every affine process is regular. The regularity problem has interesting connections to Hilbert's fifth problem on the differentiability of continuous transformation groups, from which we borrow some tools that are unusual in the field of stochastics. I also report some preliminary results on the regularity problem for affine processes with general (non-canonical) state space.

Knispel, Thomas (Leibniz Universität Hannover, DE)
"Optimal long term investment under model ambiguity"
[slides available]
We study criteria for optimal portfolio management that allow for model ambiguity and take a long term view. The analysis is carried out for an incomplete market model, where the dynamics of asset prices is affected by an external stochastic factor process of diffusion type. To cope with model ambiguity, we admit an entire class Q of possible prior models for the dynamics of stock prices and take a worst-case approach to portfolio optimization. The class Q corresponds to certain perturbations of the drift terms of both the assets and the factor process. In this situation we focus on three problems that are closely related:
- the asymptotic maximization of robust expected HARA utility as time tends to infinity,
- a robust outperformance criterion, where the investor aims at maximizing, in the long run, the worst-case probability that the portfolio's growth rate exceeds a given target,
- and the asymptotic minimization of robust downside risk, defined as the worst-case probability that the portfolio's growth rate falls below a given target.
The asymptotic analysis provides useful insight for investors with long but finite time horizon. Moreover, the asymptotic criteria allow for stationary optimal policies and will thus be more tractable than their finite horizon counterparts. Our method involves the duality approach to robust utility maximization for a varying time horizon, dynamic programming methods, and large deviations techniques.

Kramkov, Dmitry (Carnegie Mellon University, Pittsburgh, USA)
"On financial models with price impact"
[slides available]
We begin with a review of different approaches in mathematical finance and financial economics to the construction of financial models, where the dependence of market prices on investor's strategy, called a price impact or a demand pressure, is not negligible. We then present some results on a particular type of such a model obtained jointly with Peter Bank.

Kupper, Michael (Humboldt Universität zu Berlin, DE)
"Risk Preferences and their Robust Representation"
Due to the plurality of interpretations of risk, we concentrate on context invariant features related to this notion: diversification and monotonicity. We define and study general properties of three key concepts, risk order, risk measure and risk acceptance family and their one-to-one relations. Our main result is a uniquely characterized dual robust representation of lower semi continuous risk orders. We then illustrate this approach in different settings. In the setup of random variables, where risk perception can be interpreted as a model risk, we give a robust representation for numerous risk measures: various certainty equivalents, or a general version of Aumann and Serrano's economic index. In the setup of lotteries where risk perception can be seen as a distributional risk, we show that the Value at Risk is a risk measure on this level (not for random variables) and give a robust representation. Finally risk perception on state dependent lotteries à la Anscombe and Aumann, where results clarifying the interplay between model risk and distributional risk are given, in particular for the Cerreia-Vioglio, Maccheroni, Marinacci, and Montrucchio uncertainty preferences or Maccheroni, Marinacci, and Rustichini variational preferences. It is based on joint work with Samuel Drapeau.

Kusuoka, Shigeo (University of Tokyo, JP)
"Numerical Computation of Expectation of Diffusion Processes"
We will discuss about numerical computation of expectation of diffusion processes with (and without) Dirichlet boundary condition. We first show some regularity results on diffusion semigroup and then we try to show how one can apply these results to numerical computation. We plan to use Ninomiya-Victour method mainly as an example.

Kwapień, Stanisław (University of Warsaw, PL)
"On Hoeffding Decomposition in Lp"
[slides available]
We give a new proof of a result by J. Bourgain which says that if (Ω,F,P) is a product of probability spaces then Vd - the orthonormal in L2(Ω,F,P) projection on the space spanned by those X ∈ L2(Ω,F,P) which depend on most of d-variables is a bounded operator in Lp(Ω,F,P) for 1.

Larsen, Kasper (Carnegie Mellon University, Pittsburgh, USA)
"Horizon dependency for the utility maximization problem"
[link to paper/arXiv as talk on blackboard]
This paper studies the utility maximization problem with changing time horizons in the incomplete Brownian setting. We show that the dual and primal value functions as well as the optimal terminal wealth are left continuous with respective to the time horizon T > 0. We exemplify that the expected utility stemming from applying the T-horizon optimizer on a time interval [0, S] with S < T may not converge as S ↑ T to the T-horizon value. In other words, exiting an optimal strategy before maturity can have severe costs for the investor. Finally, we provide necessary and sufficient conditions preventing the existence of this phenomenon.

Larsson, Martin (Cornell University, New York, USA)
"Price Bubbles in Variance and Volatility Swaps" (poster presentation)
By the Fundamental Theorem of Asset Pricing, absence of arbitrage (in a suitable sense) is equivalent to the existence of an equivalent local martingale measure. Sometimes the price process of the underlying becomes a strict local martingale, which can be interpreted as the presence of a price bubble. Working under No Free Lunch With Vanishing Risk together with the so-called No Dominance condition, we analyze bubbles in variance and volatility swaps in a framework where dynamic trading in the stock and static positions in European puts and calls are allowed. We also highlight some pitfalls that can arise if one takes the commonplace approach of approximating the variance swap payoff by the final value of the quadratic variation of the log-price. This is joint work with Robert Jarrow, Younes Kchia and Philip Protter.

Lu, Dan (Leipzig University, DE)
"Pricing and hedging of hybrid credit risk products: a filtering approach" (poster presentation)
This paper proposes an information-based approach for structural credit risk models. We assume the asset value process V of a firm follows a geometric brownian motion. However, the process V is not directly observable for investors in secondary markets. Their information set consists of the default history, dividend payments and noisy observation of the asset value process V. In this setting the pricing of credit derivatives leads to a challenging nonlinear filtering problem. We derive the dynamics of the conditional density of the asset value process. Contrary to the full information case, there exists a default intensity process. That intensity is calculated in terms of the conditional density. Moreover, the innovation approach is used to derive the dynamics of market prices.

Muhle-Karbe, Johannes (University of Vienna, AT)
"On an Explicit Shadow Price for the Growth-Optimal Portfolio with Transaction Costs"
[slides available]
We consider the maximization of the long-term growth rate in the Black-Scholes model with proportional transaction costs as in Taksar, Klass and Assaf (1988). Similarly as in Kallsen and Muhle-Karbe (2010), we tackle this problem by determining a shadow price, i.e. a frictionless price process within the bid-ask spread of the original market, which leads to the same solution of the optimization problem. Here, it turns out that this shadow price can be calculated explicitly up to determining the root of a deterministic function. This in turn allows to compute asymptotic expansions of arbitrary order both for the no-trade region of the optimal strategy and for the optimal growth rate. This is joint work with Stefan Gerhold and Walter Schachermayer.

Müller, Paul (Johannes Kepler Universität Linz, AT)
"Conditioned Brownian Motion and Uniform square function estimates"
[slides available]
We obtain uniform estimates for Littlewood Paley integrals using conditioned Brownian Motion and uniform estimates for the quadratic variation of a stochastic Integral. Applications are given to Problems in Harmonic Analysis e.g. Interpolation problems, Hardy Spaces and correction theorems.

Nadtochiy, Sergey (Oxford University, UK)
"Tangent Lévy models"
[slides available]
The classical approach to modeling prices of financial instruments is to identify a (small) family of underlying processes, whose dynamics are then described explicitly, and compute the prices of corresponding financial derivatives by taking expectations or maximizing the utility function. However, as certain types of derivatives became liquid, it appeared reasonable to model their prices directly and use these market models to price or hedge exotic securities. This framework was originally advocated by Heath, Jarrow and Morton for the Treasury bond markets. We discuss the characterization of arbitrage-free dynamic stochastic market models based on the European call options of all strikes and maturities. The present work can be viewed as an extension of the dynamic local volatility approach, proposed earlier by Carmona and Nadtochiy. Since the usage of local volatility as a code-book for the option prices is limited (for example, it can only be used if the paths of the underlying are continuous), we attempt to develop a general approach to constructing the market models for call options. In particular, when the underlying is a pure jump process, we introduce the tangent Lévy density as the appropriate code-book, which becomes a substitute to local volatility. We capture the information contained in the surface of option prices in some Lévy density and then prescribe its dynamics via an Itô stochastic process in function space. The main thrust of our work is to characterize consistency between option prices produced by the dynamic Lévy density and their definition as the conditional expectations of corresponding payoffs. We then prove an existence result, providing a simple way to construct and implement a large class of tangent Lévy models (notice that we haven't been able to obtain such a result in the case of dynamic local volatility).

Nutz, Marcel (ETH Zurich, CH)
"Risk Aversion Asymptotics for Power Utility Maximization"
[slides available]
We consider the economic problem of optimal consumption and investment with power utility. We study the optimal strategy as the relative risk aversion tends to infinity or to one. The convergence of the optimal consumption is obtained for general semimartingale models while the convergence of the optimal trading strategy is obtained for continuous models. The limits are related to exponential and logarithmic utility. To derive these results, we combine approaches from optimal control, convex analysis and backward stochastic differential equations.

Orihuela, José (Universidad de Murcia, ES)
"Interplay between Functional Analysis, Optimality and Risk"
[slides available]
We make a tour around the following birthday present for Walter Schachermayer: Theorem.
Let E be a Banach space and V a convex, lower semicontinuous and proper map from E into the real line + infinity. If the subdifferential of V is an onto map then the level sets {V ≤ c} are weakly compact. When the domain of V has interior point the Banach space E must be reflexive. For separable Banach spaces this result was included in the paper by Jouini, Schachermayer and Touzi for their study of risk measures with the Lebesgue property, [2]. The theorem has been recently proved by Delbaen [1], he uses the original version of James compactness Theorem together with a homogenisation trick, for the space E of all Lebesgue integrable functions on a probability space.
[1] F. Delbaen Differentiability Properties of Utility Functions Optimality and Risk- Modern Trends in Mathematical Finance. Springer 2009, 39-48.
[2] E. Jouini, W. Schachermayer and N. Touizi, Lax invariant risk measures have the Fatou property , Advances in Mathematical Economics, Springer 2006, 9 , 49-71.

Preiss, David (University of Warwick, UK)
"Differentiability problems in Banach spaces"
[slides available]
I intend to describe recent work of Joram Lindenstrauss, Jaroslav Tiser and myself that made significant inroads into one of the most puzzling problems of nonlinear geometric functional analysis: does every finite (or even countably infinite) collection of real-valued Lipschitz functions on a Hilbert space have a common point of Frechet differentiability? For example, while previously this was known for one function only, our results imply it also for two functions. We also explain why the still open case of three or more functions is substantially different from what is known so far.

Prokaj, Vilmos (Eötvös Loránd University Budapest, HU)
"Hiding the drift, and the perturbed Tanaka equation"
[slides available]
We deal with the following question of Marc Yor: Assume that we have Brownian motion (BM) with constant drift denoted by S. Is it possible to define a predictable process in terms of S such that the stochastic integral HS is a Brownian motion in its own filtration? That is, can we hide a constant drift. Approximative and weak solution was given previously. The new ingredient for the strong solution, is a result on the perturbed Tanaka-equation. Roughly speaking, this theorem states that the solution of the perturbed Tanaka-equation is pathwise unique, provided that the perturbation is independent and strong enough.

Prokaj, Vilmos (Eötvös Loránd University Budapest, HU)
"Hiding a constant drift" (poster presentation)
The presentation deals with the following question: Let B be a Brownian motion in its natural filtration Ft≥0. Is it possible to define an Ft≥0 predictable process H such that the stochastic integral H⋅S with St=Bt+t gives a Brownian motion in its own filtration? That is, can we hide a constant drift. Approximative and weak solution was given previously. The new ingredient for the strong solution, is a result on the perturbed Tanaka-equation. This theorem states that the solution of the perturbed Tanaka-equation dXt=sign(Xt+Nt)dMt is pathwise unique, provided that M,N are orthogonal continuous local martingales, and N dominates M, that is the additive noise N is strong enough.

Protter, Philip E. (Cornell University, New York, USA)
"Absolutely Continuous Compensators"
[slides available]
Often in applications (for example in Survival Analysis and in Credit Risk) one begins with a totally inaccessible stopping time, and then one assumes the compensator of the indicator of the stopping time has absolutely continuous paths. This gives an interpretation in terms of a "hazard function'' process. Ethier and Kurtz have given sufficient conditions for a given stopping time to have an absolutely continuous compensator, and this condition was extended by Yan Zeng in his thesis to a necessary and sufficient condition. We take a different approach and make a simple hypothesis on the filtration under which all totally inaccessible stopping times have absolutely continuous compensators. We show such a property is stable under changes of measure, and under the expansion of filtrations, although there are some delicate points related to progressive expansions; and we detail its limited stability under filtration shrinkage. The talk is based on research performed with Sokhna M'Baye and Svante Janson.

Rásonyi, Miklós (University of Edinburgh, UK)
"Large deviations and asymptotic arbitrage"
[slides available]
In a 2007 paper, H. Föllmer and W. Schachermayer considered a market model where the price process was a diffusion and showed that, under mild conditions, one may realize arbitrage opportunities when the time horizon tends to infinity. They formulated another condition which should imply a stronger form of asymptotic arbitrage where failure probability tends to 0 geometrically fast. The relationship of this conjecture to large deviation theory has been explained, but it was proved only for the case of Ornstein-Uhlenbeck processes. In a discrete-time Markovian model we proved their conjecture and a number of related results on asymptotic arbitrage. This is joint work with Martin L. D. Mbele Bidima.

Cancelled: Rehman, Nasir (AIOU Islamabad, Pakistan)
"Comparison of American Options with Different Strikes, Maturities and Volatilities" (poster presentation)
We establish a comparison result for the value functions of the American put options with different strikes, maturities and volatilities based on systematic use of the Dynamic Programming Principle together with the monotonicity in volatility property for the value functions of the American options.

Rogers, Chris (University of Cambridge, UK)
"Least-Action Filtering"
[slides available]
This talk studies the filtering of a partially-observed multidimensional diffusion process using the principle of least action, equivalently, maximum-likelihood estimation. We show how the most likely path of the unobserved part of the diffusion can be determined by solving a shooting ODE, and then we go on to study the (approximate) conditional distribution of the diffusion around the most likely path; this turns out to be a zero-mean Gaussian process which solves a linear SDE whose time-dependent coefficients can be identified by solving a first-order ODE with an initial condition. This calculation of the conditional distribution can be used as a way to guide SMC methods to search relevant parts of the state space, which may be valuable in high-dimensional problems, where SMC struggles; in contrast, ODE solution methods continue to work well even in moderately large dimension.

Rudloff, Birgit (Princeton University, USA)
"Risk measures for multivariate random variables in markets with transaction costs"
[slides available]
We consider a conical market model (generated, for example, by proportional transaction costs) and extend the notion of set-valued risk measures (Jouini, Meddeb, Touzi 2004, Hamel, Heyde 2010) to the case of random solvency cones at terminal time. This accounts for random exchange rates and/or random transaction costs. Several new features such as market compatibility will be discussed which do not appear (or are trivial) if the solvency cones are constant. Dual representations are given in terms of vector probability measures. This admits an interpretation very close to the scalar case. As examples, we present the set-valued versions of the worst case risk measure and the average value at risk. Related results will be discussed. For example, it can be shown that in analogy to the frictionless case the superhedging price in a conical market (see e.g. Schachermayer 2004, Pennanen, Penner 2009) is a set-valued coherent risk measure, where the supremum in the dual representation is taken w.r.t. the set of equivalent martingale measures. Moreover, we will show that the case of multiple eligible assets perfectly fits into the set-valued framework: The scalar risk measures introduced in Artzner, Delbaen, Koch-Medina 2009 turn out to be scalarizations of set-valued risk measures.

Ruf, Johannes (Columbia University, New York, USA)
"Optimal trading strategies under arbitrage"
[slides available]
Explicit formulas for optimal trading strategies in terms of minimal required initial capital are derived to replicate a given terminal wealth in a continuous-time Markovian context. To achieve this goal this paper does not assume the existence of an equivalent local martingale measure. Instead a new measure is constructed under which the dynamics of the stock price processes simplify. It is shown that delta hedging does not depend on the "no free lunch with vanishing risk" assumption. However, in the case of arbitrage the problem of finding an optimal strategy is directly linked to the non-uniqueness of the partial differential equation corresponding to the Black-Scholes equation. The recently often discussed phenomenon of "bubbles" is a special case of the setting in this paper. Several examples at the end illustrate the techniques described in this work.

Schmidt, Thorsten (TU Chemnitz, DE)
"Dynamic Modelling of CDO Markets"
[slides available]
The ongoing credit crisis highlights the difficulties in the modeling and risk management of collateralized debt obligations. In this talk we will review a top-down approach to CDO markets and analyze conditions for absence of arbitrage in a general model. Thereafter we study the application of the model where on the one side monotonicity of forward rate curves and on the other side the connection to market models play an important role.

Schütt, Carsten (Christian-Albrechts-Universität zu Kiel, DE)
"On the Santalo point"
We construct convex bodies for which the Santalo point and the centroid are far apart.

Shmileva, Elena (St.Petersburg University of Electrical Engineering, RU)
"Law of the iterated logarithm for pure jump Lévy processes"
[slides available]
In the talk, we present results on small deviations for Lévy processes. We also discuss related results on the functional law of the iterated logarithm for these processes, namely, we describe a.s. cluster sets of renormalized Lévy processes.

Sîrbu, Mihai (University of Texas at Austin, USA)
"Optimal investment with high-watermark performance fee"
[slides available]
We consider the problem of optimal investment and consumption when the investment opportunity is represented by a hedge-fund charging proportional fees on profit. The value of the fund evolves as a geometric Brownian motion and the performance of the investment and consumption strategy is measured using discounted power utility from consumption on infinite horizon. The resulting stochastic control problem is solved using dynamic programming arguments. We show by analytical methods that the associated Hamilton-Jacobi-Bellman equation has a smooth solution, and then obtain the existence and representation of the optimal control in feedback form using verification arguments. The presentation is based on joint work with Karel Janecek.

Soner, Mete (ETH Zurich, CH)
"Liquidity based models and problems"
In recent years there has been a renewed interest in financial markets with problems related to the liquidity (or lack of it). Mathematically, the liquidity is modeled by introducing the impact of trading on the stock price processes. In this talk, I will outline several problems such as, optimal execution, option pricing and utility maximization problems in such models.

Sturm, Stephan (Princeton University, USA)
"Is the minimum value of an option on variance generated by local volatility?"
We discuss the possibility of obtaining model-free bounds on volatility derivatives, given present market data in the form of a calibrated local volatility model. A counter-example to a wide-spread conjecture is given. Joint work with M. Beiglböck (Vienna) and P. Friz (Berlin).

Tompkins, Robert (Frankfurt School of Finance & Management, DE)
"Potential PCA Interpretation Problems for Volatility Smile Dynamics"
[slides available]
Principal Component Analysis (PCA) is a common procedure for the analysis of financial market data, such as implied volatility smiles or interest rate curves. Recently, Pelsser and Lord raised the question whether PCA results may not be "facts but artefacts". 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 are incorrect. Finally, for a special case, we provide an analytic correction that recovers correct factor variances from those incorrectly estimated by PCA.

Touzi, Nizar (École Polytechnique, Palaiseau, FR)
"Model independent bound for option pricing: a stochastic control aproach"
(joint with Alfred Galichon and Pierre Henry-Labordere) We develop a stochastic control approach for the derivation of model independent bounds for derivatives under various calibration constraints. Unlike the previous literature, our formulation seeks the optimal no arbitrage bounds given the knowledge of the distribution at some (or various) point in time. This problem is converted into a classical stochastic control problem by means of convex duality. We obtain a general characterization, and provide explicit optimal bounds in some examples.

Urusov, Mikhail (Ulm University, DE)
On the Martingale Property of Exponential Local Martingales
[slides available]
Originally planned: "Optimal Portfolio Liquidation with Dynamic Risk"
[slides available]
We study the problem of optimal liquidation of a large number of shares in an illiquid market. Market impact is modeled via a block-shaped limit order book with infinite resilience. We optimize the value of a certain dynamic risk measure over all adapted strategies and provide explicit formulas for the optimizer. Compared to the popular Almgren and Chriss execution strategy, which results from the deterministic mean-variance optimization, our strategy has an intrinsic time horizon, and the relative selling speed is decreasing in the position size. The latter properties are desirable for practitioners. This is a joint work with Andrey Selivanov.

Veraart, Luitgard (Karlsruhe Institute of Technology, DE)
"The effect of estimation in high-dimensional portfolios"
[slides available]
We study the effect of estimated model parameters in investment strategies on expected log-utility of terminal wealth. The market consists of a riskless bond and a potentially vast number of risky stocks modeled as geometric Brownian motions. The well-known optimal Merton strategy depends on unknown parameters and thus cannot be used in practice. We consider the expected utility of several estimated strategies when the number of risky assets gets large. We suggest strategies which are less affected by estimation errors and demonstrate their performance in a real data example. (Joint work with A. Gandy, Imperial College London).

Yor, Marc (Université Pierre et Marie Curie Paris VI, FR)
"A new look at and extensions of Bougerol's identity"
[handwritten notes available]
An equivalent form of Bougerol's identity may be expressed in terms of the Bessel clock.

Zariphopoulou, Thaleia (University of Oxford, UK)
"Initial investment choice and optimal future allocations"
I will discuss a new perspective on optimal portfolio choice by investigating how knowledge of an investor's desirable initial investment choice can be used to determine his future optimal portfolio allocations. Optimality of investment decisions is built on forward investment performance criteria. The analysis uses the connection between a nonlinear diffusion equation, satisfied by the local risk tolerance, and the backward heat equation. Complete solutions for the case of monotone performance criteria are provided as well as various examples.