We present a fairly general theory of time inconistent control. The theory is then applied to solve a mean variance portfolio optimization problem where the risk aversion is a function of current wealth.

We review the technically demanding requirements of automated options market making and propose a possible solution. Given the mid prices of European options on a rectangular grid of strikes and maturities, we construct an exactly consistent Markov martingale for the underlying asset price which evolves in continuous time with a continuous state space. In contrast to other approaches based on entropy or mixing, the approach just requires univariate search. The solution combines ideas from the Local Vol model with ideas from the Variance Gamma model.

In option pricing theory, partial (integro-)differential equations are usually associated with Markovian models, but since the seminal work of Dupire (1994) we know that Markovian projection methods can be used to derive PDEs for call option prices in many non-Markovian models where the price follows a Brownian martingale under the risk-neutral measure.

We give conditions under which the flow of marginal distributions of a discontinuous semimartingale X can be matched by a Markov process whose infinitesimal generator is expressed in terms of the local characteristics of X. Using this Markovian projection result, we derive a forward partial integro-differential equation for option prices in a large class of (non-Markovian) semimartingale models with jumps.

Our work extends the results of Gyongy (1986) and Dupire (1994) to the case of discontinuous semimartingales.

Joint work with Amel Ben-Tata (Universite de Paris VI).

I will review some recent results on existence, uniqueness and approximation of backward SDEs of mean-field type. I will also show that they apprear in a natural way as adjoint processes in a stochastic maximum principle for optimal control of ordinary SDEs of mean-field type. An application to portfolio selection will be highlighted.

The talk is based on joint work with R. Buckdahn, J. Li, S. Peng and D. Andersson

We consider the effects of the inclusion of finite liquidity into the Black-Scholes-Merton financial model, which generally result in highly nonlinear partial differential equations (PDEs). We investigate in detail a model studied by Schöbucher and Wilmott (2000). First, we consider a first-order feedback model, which leads to the exceptional case of a linear PDE. Numerical results, and more particularly an asymptotic-analysis approach close to option expiry, reveal subtle differences from the Black-Scholes-Merton model. Second, we go on to consider a full-feedback model in which price impact is fully incorporated into the model. Here, standard numerical techniques lead to spurious results in even the simplest cases. An asymptotic approach, valid close to expiry, is (again) mounted, and this reveals two distinct classes of behaviour. The first may be attributed to the infinite gamma associated with standard option payoff conditions, for which it is necessary to admit solutions with discontinuous first derivatives; perhaps even more disturbingly, negative option values are a frequent occurrence. The second failure (applicable to smoothed payoff functions) is caused by a singularity in the coefficient of the diffusion term in the option-pricing equation. Our conclusion is that several classes of model in the literature involving finite liquidity irretrievably break down (i.e. there is insufficient ‘financial modelling’ in the pricing equation). We also provide results from solving the analogous stochastic differential equations, which also indicate a number of spurious behaviours. Our analysis should provide guidance in developing models in the future that avoid such pitfalls.

We present an extension of Ito calculus to functionals and establish that under certain conditions, path-dependent option prices satisfy a PDE. We apply this result to compute the sensitivity of path-dependent option prices to perturbations of the local volatility surface. We show that the coefficients of the portfolio of European options with the same volatility risk profile can be obtained as the source term of a PDE.

The famous Dupire formula yields a time-dependent local volatility given call option prices for all strikes and maturities. We investigate whether one can find a time-homogeneous model that produces given call prices for all strikes but for one fixed maturity. This is an on-going project with Johan Tysk and David Hobson.

We examine the British payoff mechanism (introduced in Peskir and Samee, 2008) in the context of path dependent options. In particular, we focus on the 'British Asian' and the 'British Russian' option. Such options provide their holder with an endogenous protection against unfavourable stock price movements. The price of such options can be characterised as the unique solution to a parabolic free-boundary problem, whose properties and solution we investigate. Finally, we provide a preliminary financial analysis of both options and conclude that in many circumstances these options can be considered an attractive alternative to existing path dependent options. This is joint work with Goran Peskir and Farman Samee.

In this talk we discuss Bernstein estimates for weakly fully non-linear elliptic systems. We are particularly interested in systems that arise in the stochastic optimal control problems of hybrid systems. For these a generalization of Bernstein estimates for first and second derivatives of classical solutions will be presented.

In this talk we will present the optimal multi-modes switching problem in its general setting. In using probabilistic tools we show the existence of an optimal strategy. In the Markovian case, this problem is related to a system of PDEs with obstacles which depend on the solution. We show that this system has a unique solution in viscosity sense. Finally we address the switching problem under knightian uncertainty.

References
[1] B. Elasri, S.Hamadène (2009): The Finite Horizon Optimal Multi-Modes Switching Problem: the Viscosity Solution Approach. AMO '09
[2] Djehiche, B., Hamadène, S. and Popier A. (2007): A Finite Horizon Optimal Multiple Switching Problem. SICON '10.
[3] Hamadène, S. and Jeanblanc, M (2007): On the Starting and Stopping Problem: Application in reversible investments, Math. of Operation Research, 32 (1), 182-192.
[4] Hamadène, S. and Zhang, J. (2007): The Starting and Stopping Problem under Knightian Uncertainty and Related Systems of Reflected BSDEs. Available at arXiv:0710.0908 (October 2007).

It is well-known how to determine the price of perpetual American options if the underlying stock price is a time-homogeneous diffusion. In the talk we consider the inverse problem, i.e. given prices of perpetual American options for different strikes we show how to construct a time-homogeneous model for the stock price which reproduces the given option prices.

This is joint work with Erik Ekström (Uppsala) and Martin Klimmek (Warwick).

This paper takes a step toward defining a continuous time methodology to analyze a class of time inconsistent capital accumulation games when the discount rate is non-constant and commitment is not possible. The subgame perfect and history independent equilibrium consumption policies are characterized with an equation that must be satisfied by the value function. The equation is reminiscent of the classical Hamilton-Jacobi-Bellman equation of optimal control, but with a non-local term that is relevant for the qualitative behavior of the solution. The methodology is applied to an overlapping generations growth model with a utilitarian social planner. The social planner problem is shown to be isomorphic to a time inconsistent capital capital accumulation problem with a non constant discount rate. Existence of multiple continuously differentiable subgame perfect equilibrium consumption policies leading to multiple steady state is established. Decentralization of the equilibrium is possible but requires a distorting positive or negative tax on capital in the long term.

This is joint work with Ivar Ekeland.

The concept of a convex risk measure in finance can be defined in terms of a g-expectation, which is the solution of a backward stochastic differential equations with drift term (also called "driver") given by a convex function g. Therefore the problem of finding the portfolio that minimizes the risk of the terminal financial standing may be regarded as a problem of optimal control of a system of forward-backward stochastic differential equations. This is again linked to a problem of optimal control of (deterministic) PDEs. In this talk we present a solution method for such control problems, and we apply it to an example of risk minimization in a financial market driven by Lévy processes. The talk is based on recent joint work with Agnès Sulem (INRIA, Paris).

We consider the analysis and implementation of efficient numerical solvers for PIDEs arising from asset pricing problems in market models of additive Feller-Lévy type, containing as special cases Lévy Processes as well as additive processes in the sense of Sato.

Sobolev spaces H^m(x)(I) of variable order 0<m(x)<1 on an interval I ⊂ R arise as domains of Dirichlet forms for certain quadratic, pure jump Feller processes X_t ∈ R with unbounded, state-dependent intensity of small jumps. For spline wavelets with complementary boundary conditions, we establish multilevel norm equivalences in H^m(x)(I) and prove preconditioning and wavelet matrix compression results for the variable order pseudodifferential generators A of X.

Sufficient conditions on A to satisfy a Gårding inequality in H^m(x)(I) and time-analyticity of the semigroup T_t associated with the Feller process X_t are established.

As application, wavelet-based algorithms with discontinuous Galerkin time-stepping of log-linear complexity for the valuation of contingent claims on pure jump Feller-Lévy processes X_t with state-depenent jump intensity by numerical solution of the corresponding Kolmogoroff equations which are parabolic pseudodifferential equations of variable order are obtained.

AMS Subject Classification: 35K15, 45K05, 65N30

Joint work with
O. Reichmann, (SAM ETH) and
R. Schneider (TU Berlin, Germany)

One popular approach to the valuation of contingent claims in incomplete financial markets is via exponential utility indifference. The value process for the corresponding stochastic control problem can often be described by a backward stochastic differential equation (BSDE). This is in many cases very useful for proving theoretical properties; but actually solving these equations to obtain more explicit results is difficult, especially if the underlying asset is multidimensional.

With the goal of obtaining more information, we study BSDE transformations that allow us to derive upper and/or lower bounds, in terms of solutions of other BSDEs, that can be computed more explicitly. These ideas and techniques may also be of independent interest for theoretical BSDE studies. We apologise for not using PDEs. The results are joint work with Christoph Frei and Semyon Malamud.

This is a report on a joint work with Erik Ekström. We study the term structure equation for single-factor models that predict non-negative short rates. In particular, we show that the price of a bond or a bond option is the unique classical solution to a parabolic differential equation with a certain boundary behaviour for vanishing values of the short rate. If the boundary is attainable, then this boundary behaviour serves as a boundary condition and guarantees uniqueness of solutions. On the other hand, if the boundary is non-attainable, then the boundary behaviour is not needed to guarantee uniqueness, but it is nevertheless very useful for instance from a numerical perspective.