The problem of optimal stopping with finite horizon in discrete time

is considered in view of maximizing the expected gain. The algorithm

presented in this talk is completely nonparametric in the sense that it

uses observed data from the past of the process up to time -n+1 (n being

a natural number), not relying on any specific model assumption. Kernel

regression estimation of conditional expectations and prediction theory

of individual sequences are used as tools.

The main result is that the algorithm is universally consistent: the

achieved expected gain converges to the optimal value for n tending to

infinity, whenever the underlying process is stationary and ergodic.

An application to exercising American options is given.

# Past Nomura Seminar

In this talk, we present a pathwise method to construct confidence

intervals on the value of some discrete time stochastic dynamic

programming equations, which arise, e.g., in nonlinear option pricing

problems such as credit value adjustment and pricing under model

uncertainty. Our method generalizes the primal-dual approach, which is

popular and well-studied for Bermudan option pricing problems. In a

nutshell, the idea is to derive a maximization problem and a

minimization problem such that the value processes of both problems

coincide with the solution of the dynamic program and such that

optimizers can be represented in terms of the solution of the dynamic

program. Applying an approximate solution to the dynamic program, which

can be precomputed by any algorithm, then leads to `close-to-optimal'

controls for these optimization problems and to `tight' lower and upper

bounds for the value of the dynamic program, provided that the algorithm

for constructing the approximate solution was `successful'. We

illustrate the method numerically in the context of credit value

adjustment and pricing under uncertain volatility.

The talk is based on joint work with C. Gärtner, N. Schweizer, and J.

Zhuo.

tba

With few exceptions, optimal stopping assumes that the underlying system is stopped immediately after the decision is made.

In fact, most stoppings take time. This has been variously referred to as "time-to-build", "investment lag" and "gestation period",

which is often non negligible.

In this talk, we consider a class of optimal stopping/switching problems with delivery lags, or equivalently, delayed information,

by using reflected BSDE method. As an example, we study American put option with delayed exercise, and show that it can be decomposed

as a European put option and a premium, the latter of which involves a new optimal stopping problem where the investor decides when to stop

to collect the Greek theta of such a European option. We also give a complete characterization of the optimal exercise boundary by resorting to free boundary analysis.

Joint work with Zhou Yang and Mihail Zervos.

The large majority of risk-sharing transactions involve few agents, each of whom can heavily influence the structure and the prices of securities. This paper proposes a game where agents' strategic sets consist of all possible sharing securities and pricing kernels that are consistent with Arrow-Debreu sharing rules. First, it is shown that agents' best response problems have unique solutions, even when the underlying probability space is infinite. The risk-sharing Nash equilibrium admits a finite-dimensional characterisation and it is proved to exist for general number of agents and be unique in the two-agent game. In equilibrium, agents choose to declare beliefs on future random outcomes different from their actual probability assessments, and the risk-sharing securities are endogenously bounded, implying (amongst other things) loss of efficiency. In addition, an analysis regarding extremely risk tolerant agents indicates that they profit more from the Nash risk-sharing equilibrium as compared to the Arrow-Debreu one.

(Joint work with Michail Anthropelos)

Motivated by the European sovereign debt crisis, we propose a hybrid sovereign default model which combines an accessible part which takes into account the movement of the sovereign solvency and the impact of critical political events, and a totally inaccessible part for the idiosyncratic credit risk. We obtain closed-form formulas for the probability that the default occurs at political critical dates in a Markovian CEV process setting. Moreover, we introduce a generalized density framework for the hybrid default times and deduce the compensator process of default. Finally we apply the hybrid model and the generalized density to the valuation of sovereign bond and explain the significant jumps in the long-term government bond yield during the sovereign crisis.

In this talk, I will present a family of forward performance processes in

discrete time. These processes are predictable with regards to the market

information. Examples from a binomial setting will be given which include

the time-monotone exponential forward process and the completely monotonic

family.

Following a quick introduction to derivatives markets and the classic theory of valuation, we describe the changes triggered by post 2007 events. We re-discuss the valuation theory assumptions and introduce valuation under counterparty credit risk, collateral posting, initial and variation margins, and funding costs. A number of these aspects had been investigated well before 2007. We explain model dependence induced by credit effects, hybrid features, contagion, payout uncertainty, and nonlinear effects due to replacement closeout at default and possibly asymmetric borrowing and lending rates in the margin interest and in the funding strategy for the hedge of the relevant portfolio. Nonlinearity manifests itself in the valuation equations taking the form of semi-linear PDEs or Backward SDEs. We discuss existence and uniqueness of solutions for these equations. We present an invariance theorem showing that the final valuation equations do not depend on unobservable risk free rates, that become purely instrumental variables. Valuation is thus based only on real market rates and processes. We also present a high level analysis of the consequences of nonlinearities, both from the point of view of methodology and from an operational angle, including deal/entity/aggregation dependent valuation probability measures and the role of banks treasuries. Finally, we hint at how one may connect these developments to interest rate theory under multiple discount curves, thus building a consistent valuation framework encompassing most post-2007 effects.

Damiano Brigo, Joint work with Andrea Pallavicini, Daniele Perini, Marco Francischello.

We argue that Credit Default Swap (CDS) premia for safe-haven sovereigns, like Germany and the United States, are driven to a large extent by regulatory requirements under which derivatives dealing banks have an incentive to buy CDS to hedge counterparty credit risk of their counterparties.

We explain the mechanics of the regulatory requirements and develop a model in which derivatives dealers, who have a derivatives exposure with sovereigns, need CDS for capital relief. End users without exposure to the sovereigns sell the CDS and require a positive premium equivalent to the capital requirement. The model's predictions are confirmed using data on several sovereigns.

Joint with OMI

A well-known result of Landsberger and Meilijson says that efficient risk-sharing rules for univariate risks are characterized by a so-called comonotonicity condition. In this talk, I'll first discuss a multivariate extension of this result (joint work with R.-A. Dana and A. Galichon). Then I will discuss the restrictions (in the form of systems of nonlinear PDEs) efficient risk sharing imposes on individual consumption as a function of aggregate consumption. I'll finally give an identification result on how to recover preferences from the knowledge of the risk sharing (joint work with M. Aloqeili and I. Ekeland).