DH 1st floor SR

In a market with one safe and one risky asset, an investor with a long

horizon and constant relative risk aversion trades with constant

investment opportunities and proportional transaction costs. We derive

the optimal investment policy, its welfare, and the resulting trading

volume, explicitly as functions of the market and preference parameters,

and of the implied liquidity premium, which is identified as the

solution of a scalar equation. For small transaction costs, all these

quantities admit asymptotic expansions of arbitrary order. The results

exploit the equivalence of the transaction cost market to another

frictionless market, with a shadow risky asset, in which investment

opportunities are stochastic. The shadow price is also derived

explicitly. (Joint work with Paolo Guasoni, Johannes Muhle-Karbe, and

Walter Schachermayer)

We propose a dynamic insurance network model that allows to deal with reinsurance counter-party default risks with a particular aim of capturing cascading effects at the time of defaults. We capture these effects by finding an equilibrium allocation of settlements which can be found as the unique optimal solution of a linear programming problem. This equilibrium allocation recognizes 1) the correlation among the risk factors, which are assumed to be heavy-tailed, 2) the contractual obligations, which are assumed to follow popular contracts in the insurance industry (such as stop-loss and retro-cesion), and 3) the interconnections of the insurance-reinsurance network. We are able to obtain an asymptotic description of the most likely ways in which the default of a specific group of insurers can occur, by means of solving a multidimensional Knapsack integer programming problem. Finally, we propose a class of provably strongly efficient estimators for computing the expected loss of the network conditioning the failure of a specific set of companies. Strong efficiency means that the complexity of computing large deviations probability or conditional expectation remains bounded as the event of interest becomes more and more rare.

We develop the idea of using Monte Carlo sampling of random portfolios to solve portfolio investment problems. We explore the need for more general optimization tools, and consider the means by which constrained random portfolios may be generated. DeVroye’s approach to sampling the interior of a simplex (a collection of non-negative random variables adding to unity) is already available for interior solutions of simple fully-invested long-only systems, and we extend this to treat, lower bound constraints, bounded short positions and to sample non-interior points by the method of Face-Edge-Vertex-biased sampling. A practical scheme for long-only and bounded short problems is developed and tested. Non-convex and disconnected regions can be treated by applying rejection for other constraints. The advantage of Monte Carlo methods is that they may be extended to risk functions that are more complicated functions of the return distribution, without explicit gradients, and that the underlying return distribution may be modeled parametrically or empirically based on general distributions. The optimization of expected utility, Omega, Sortino ratios may be handled in a similar manner to quadratic risk, VaR and CVaR, irrespective of whether a reduction to LP or QP form is available. Robustification is also possible, and a Monte Carlo approach allows the possibility of relaxing the general maxi-min approach to one of varying degrees of conservatism. Grid computing technology is an excellent platform for the development of such computations due to the intrinsically parallel nature of the computation. Good comparisons with established results in Mean-Variance and CVaR optimization are obtained, and we give some applications to Omega and expected Utility optimization. Extensions to deploy Sobol and Niederreiter quasi-random methods for random weights are also proposed. The method proposed is a two-stage process. First we have an initial global search which produces a good feasible solution for any number of assets with any risk function and return distribution. This solution is already close to optimal in lower dimensions based on an investigation of several test problems. Further precision, and solutions in 10-100 dimensions, are obtained by invoking a second stage in which the solution is iterated based on Monte-Carlo simulation based on a series of contracting hypercubes.

In this talk, I will present reflected backward stochastic differential equations (reflected BSDEs) and their connection with the pricing of American options. Then I will present a simple perturbative method for studying them. Under the appropriate assumptions on the coefficient, the terminal condition and the lower obstacle, similar to those used by Kobylankski, this method allows to prove the existence of a solution. I will also provide the usual comparison theorem and a new proof for a refined comparison theorem, specific to RBSDEs.

Decision making in the presence of uncertainty is a mathematically delicate topic. In this talk, we consider coherent sublinear expectations on a measurable space, without assuming the existence of a dominating probability measure. By considering discrete-time `martingale' processes, we show that the classical results of martingale convergence and the up/downcrossing inqualities hold in a `quasi-sure' sense. We also give conditions, for a general filtration, under which an `aggregation' property holds, generalising an approach of Soner, Touzi and Zhang (2011). From this, we extend various results on the representation of conditional sublinear expectations to general filtrations under uncertainty.

In this seminar, we discuss three questions related to the finite difference computation of early exercise options, one of which has a useful answer, one an interesting one, and one is open.

We begin by showing that a simple iteration of the exercise strategy of a finite difference solution is efficient for practical applications and its convergence can be described very precisely. It is somewhat surprising that the method is largely unknown.

We move on to discuss properties of a so-called penalty method. Here we show by means of numerical experiments and matched asymptotic expansions that the approximation of the value function has a very intricate local structure, which is lost in functional analytic error estimates, which are also derived.

Finally, we describe a gap in the analysis of the grid convergence of finite difference approximations compared to empirical evidence.

This is joint work with Jan Witte and Sam Howison.