Scientific Program

July, 9-13

Schedule

	Mon	Tue	Wed	Thu	Fri
10:00-10:50	Kardaras	Hynd	Elie	Cruzeiro	Muhle-Karbe
11:00-11:50	Larsen	dos Reis	Mendes	Ruf	Kort
12:00-14:30	Lunch	Lunch	Lunch	Lunch	Lunch
14:30-15:20	Decamps	Tehranchi	Free	Pulido	Zambrini
15:30-16:20	Robertson	Rasonyi	Free	Cipriano	Eberlein

Speakers and Abstracts

• Cipriano - On the 2D-dimensional Stochastic Navier-Stokes Equations with Navier friction condition

Abstract: We consider the 2D- Navier-Stokes equations in a bounded domain, with Navier friction –type boundary conditions and perturbed by a stochastic driving force. Motivated by the studies of the turbulent flows we prove existence of solution for the stochastic Navier-Stokes equations and study the inviscid limit.

• Cruzeiro - A stochastic approach to the deterministic Navier-Stokes equations

Abstract: We show how solutions of the Navier-Stokes equations can be derived via a stochastic variational principle. The underlying stochastic Lagrangian flows thus satisfy a generalized stochastic "geodesic" equation, which can also be described as a forward-backward s.d.e.

• Decamps - A Bayesian adaptive singular control problem arising from corporate finance.

Abstract: Stochastic optimization problems that combine features of both bounded variation control and stopping are relatively scarce in the applied probability literature. Recently, mixed stochastic control problems have emerged from corporate finance in continuous time. Bounded variation control problems arise in the corporate finance literature when we consider the optimal liquidity management (dividend and issuance policies) of a firm subject to costly external financing. On the other hand, discretionary stopping in stochastic control arises naturally in solvency risks problems where the shareholders have to determine endogenously the optimal time to liquidate a project. In this paper, we analyse the interaction between dividend policy (the bounded variation control) and optimal abandonment (discretionary stopping) of a liquidity constrained firm whose profitability is partially known. This lead us to study a mixed singular/optimal stopping problem under partial information on the drift of a diffusion that we solve explicitely in some special cases.

• Eberlein - Market models for credit risky portfolios

Abstract: In this paper we specify market models for credit portfolios in a top-down setting driven by time-inhomogeneous Lévy processes. We provide conditions for the absence of arbitrage, explicit examples and an affine setup which includes contagion. As a major application we derive pricing formulas for single tranche CDOs (STCDOs) and options on STCDOs. A calibration to iTraxx data using an extended Kalman filter is performed based on a longtime observation period. This is joint work with Zorana Grbac and Thorsten Schmidt.

• Elie - Exact replication under portfolio constraint: a viability approach for BSDE

Abstract: In this talk, we consider the problem of super-replicating a given contingent claim, whenever the incompleteness of the market is due to the presence of closed convex constraints on the portfolio strategies, written in terms of number of shares. In the dimension 1 Black Scholes model, Broadie, Shreve and Soner observed that the price under constraint of a given claim is simply the unconstrained price of a more expensive claim, defined as the facelift transform of the one of interest. For a given model and convex constraint set in dimension d, we exhibit a necessary and sufficient condition under which the latter is true for a large class of European options. Our argumentation relies on the use of viability arguments for BSDEs together with localization procedures. Several financial examples will be considered in this talk. This is a joint work with Jean-Francois Chassagneux, Imperial College and Idris Kharroubi, University Paris-Dauphine.

• Hynd - Option pricing in the large risk aversion, small transaction cost limit

Abstract: We study the asymptotic analysis of solutions of a sequence of PDE with gradient constraint arising in an option pricing model with transaction costs. The limit of these solutions satisfies a nonlinear Black-Scholes type equation. An interesting feature of this work is that the nonlinearity in the Black-Scholes type equation comes about as a solution of a nonstandard eigenvalue PDE problem.

• Kardaras - A guide through market viability for frictionless markets

Abstract: In this talk, we elaborate on the notions of no-free-lunch that have proved essential in the theory of financial mathematics---most notably, arbitrage of the first kind. Focus will be given in most recent developments. The precise connections with the semimartingale property of asset-price processes, as well as existence of deflators, numeraires and pricing probabilities via use of Foellmer's exit measure are explained. Furthermore, the consequences that these notions have in the valuation of illiquid assets in the market will be briefly explored.

• Kort - Strategic Capacity Investment Under Uncertainty

Abstract: This paper considers investment decisions within an uncertain dynamic and competitive framework. Each investment decision involves to determine the timing and the capacity level. In this way we extend the main bulk of the real options theory where the capacity level is given. We consider a monopoly setting as well as a duopoly setting. Our main results are the following. In the duopoly setting we provide a fully dynamic analysis of entry deterrence/accommodation strategies. Contrary to the seminal industrial organization analyses that are based on static models, we find that entry can only be deterred temporarily. To keep its monopoly position for a longer time, the first investor overinvests in capacity. In very uncertain economic environments the first investor eventually ends up being the largest firm in the market. If uncertainty is moderately present, a reduced value of waiting implies that the preemption mechanism forces the first investor to invest so soon that a large capacity cannot be afforded. Then it will end up with a capacity level being lower than the second investor. Joint work with Kuno J.M. Huisman.

• Larsen - Incomplete Continuous-time Securities Markets with Stochastic Income Volatility

Abstract: In an incomplete continuous-time securities market with uncertainty generated by Brownian motions, we derive closed-form solutions for the equilibrium interest rate and market price of risk processes. The economy has a finite number of heterogeneous exponential utility investors, who receive partially unspanned income and can trade continuously on a finite time-interval in a money market account and a single risky security. Besides establishing the existence of an equilibrium, our main result shows that if the investors' unspanned income has stochastic countercyclical volatility, the resulting equilibrium can display both lower interest rates and higher risk premia compared to the Pareto efficient equilibrium in an otherwise identical complete market. Joint with Peter Ove Christensen.

• Mendes - The fractional volatility model: No-arbitrage, leverage and completeness

Abstract: Based on a criterion of mathematical simplicity and consistency with empirical market data, a stochastic volatility model has been obtained with the volatility process driven by fractional noise. Depending on whether the stochasticity generators of log-price and volatility are independent or are the same, two versions of the model are obtained with different leverage behavior. Here, the no-arbitrage and completeness properties of the models are studied. (work with M. J. Oliveira and A.M. Rodrigues)

• Muhle-Karbe - Optimal investment with small transaction costs.

Abstract: Consider an investor with constant absolute risk aversion trading several risky assets with proportional transaction costs. Using informal perturbation arguments, we derive the general structure of the leading-order optimal strategy and associated welfare. The result also applies in the presence of random endowments, thereby leading to utility-based prices and hedging strategies.

• Rasonyi - Optimal investment: from risk-averse to behavioural agents

Abstract: Classical investment problems assume that economic agents are risk-averse. This corresponds to using concave utility functions to describe agents' preferences. More recently, non-concave utilities were proposed and distortions of the probability measure were also considered. We are dealing with optimal investment for an agent whose behaviour is characterized by a possibly non-concave utility function and by probability distortions. This new setting poses several mathematical challenges and exhibits a number of unexpected phenomena. In discrete-time multiperiod models we discuss the well-posedness of this investment problem and show the existence of optimal strategies under suitable conditions. We also have a look at what happens in continuous-time, in particular, we provide a sufficient and (essentially) necessary condition for the Black-Scholes model in the case of power-like utilities and distortion functions.

• Robertson - Utility-Based Pricing in the Large Position, Nearly Complete Limit

Abstract: In this talk, approximations to utility indifference prices for a contingent claim in the large position size limit are provided. Results are valid for general utility functions and semi-martingale models. It is shown that as the position size approaches infinity, all utility functions with the same rate of decay for large negative wealths yield the same price. Practically, this means an investor should price like an exponential investor. In a sizeable class of diffusion models, the large position limit is seen to arise naturally in conjunction with the limit of a complete model and hence approximations are most appropriate in this setting.

• Ruf - Föllmer's measure, Novikov's condition and options on exploding exchange rates

Abstract: In the first part of this talk, I will present a proof of Novikov's condition by means of the Föllmer measure. In the second part, I will discuss an application of the Föllmer measure to Foreign Exchange options. Strict local martingale models have been suggested to model the underlying exchange rate. In such models, put-call parity does not hold if one assumes minimal superreplicating costs as contingent claim prices. I will illustrate how put-call parity can be restored by changing the definition of a contingent claim price. More precisely, I will discuss a change of numeraire technique when the underlying is only a local martingale. Then, the new (Föllmer) measure is not necessarily equivalent to the old measure. If one now defines the price of a contingent claim as the minimal superreplicating costs under both measures, then put-call parity holds. I will discuss properties of this new pricing operator. This talk is based on joint work with Peter Carr and Travis Fisher.

• Zambrini - Geometric approach to Stochastic Control

We summarize recent developments in a research program ("Stochastic Deformation") whose aim is to inject the ideas of Geometric Mechanics in Stochastic Control Theory