arXiv subjects

Bruno Bouchard

Publications and source records attributed to Bruno Bouchard.

47 records · Page 3Linked to original sources

arXiv2014

BSDEs with weak terminal condition

We introduce a new class of Backward Stochastic Differential Equations in which the $T$-terminal value $Y_{T}$ of the solution $(Y,Z)$ is not fixed as a random variable, but only satisfies a weak constraint of the form $E[Ψ(Y_{T})]\ge m$, for some (possibly random) non-decreasing map $Ψ$ and some threshold $m$. We name them \textit{BSDEs with weak terminal condition} and obtain a representation of the minimal time $t$-values $Y_{t}$ such that $(Y,Z)$ is a supersolution of the BSDE with weak terminal condition. It provides a non-Markovian BSDE formulation of the PDE characterization obtained for Markovian stochastic target problems under controlled loss in Bouchard, Elie and Touzi \cite{BoElTo09}. We then study the main properties of this minimal value. In particular, we analyze its continuity and convexity with respect to the $m$-parameter appearing in the weak terminal condition, and show how it can be related to a dual optimal control problem in Meyer form. These last properties generalize to a non Markovian framework previous results on quantile hedging and hedging under loss constraints obtained in Föllmer and Leukert \cite{FoLe99,FoLe00}, and in Bouchard, Elie and Touzi \cite{BoElTo09}.

math.PR↗

arXiv2013

Portfolio management under risk contraints - Lectures given at MITACS-PIMS-UBC Summer School in Risk Management and Risk Sharing

The aim of these lectures at MITACS-PIMS-UBC Summer School in Risk Man- agement and Risk Sharing is to discuss risk controlled approaches for the pricing and hedging of financial risks. We will start with the classical dual approach for financial markets, which al- lows to rewrite super-hedging problems in terms of optimal control problems in standard form. Based on this, we shall then consider hedging and pricing prob- lems under utility or risk minimization criteria. This approach will turn out to be powerful whenever linear (or essentially linear) problems are considered, but not adapted to more general settings with non-linear dynamics (e.g. large investor models, high frequency trading with market impact features, mixed finance/insurance issues). In the second part of this lecture, we will develop on a new approach for risk control problems based on a stochastic target formulation. We will see how flexible this approach is and how it allows to characterize very easily super- hedging prices in term of suitable Hamilton-Jacobi-Bellman type partial differ- ential equations (PDEs). We will then see how quantile hedging and expected loss pricing problems can be embeded into this framework, for a very large class of financial models. We shall finally consider a simple example of optimal book liquidation in which the control is a continuous non-decreasing process, as an illustration of possible practical developments in optimal trading under risk constraint.These lectures are organized in small chapters, each of them being focused on a particular aspect.

math.PR↗

arXiv2013

No-arbitrage of second kind in countable markets with proportional transaction costs

Motivated by applications to bond markets, we propose a multivariate framework for discrete time financial markets with proportional transaction costs and a countable infinite number of tradable assets. We show that the no-arbitrage of second kind property (NA2 in short), recently introduced by Rasonyi for finite-dimensional markets, allows us to provide a closure property for the set of attainable claims in a very natural way, under a suitable efficient friction condition. We also extend to this context the equivalence between NA2 and the existence of many (strictly) consistent price systems.

q-fin.CP↗

arXiv2013

Robust no-free lunch with vanishing risk, a continuum of assets and proportional transaction costs

We propose a continuous time model for financial markets with proportional transactions costs and a continuum of risky assets. This is motivated by bond markets in which the continuum of assets corresponds to the continuum of possible maturities. Our framework is well adapted to the study of no-arbitrage properties and related hedging problems. In particular, we extend the Fundamental Theorem of Asset Pricing of Guasoni, Rásonyi and Lépinette (2012) which concentrates on the one dimensional case. Namely, we prove that the Robust No Free Lunch with Vanishing Risk assumption is equivalent to the existence of a Strictly Consistent Price System. Interestingly, the presence of transaction costs allows a natural definition of trading strategies and avoids all the technical and un-natural restrictions due to stochastic integration that appear in bond models without friction. We restrict to the case where exchange rates are continuous in time and leave the general càdlàg case for further studies.

q-fin.PR↗

arXiv2012

Weak Dynamic Programming for Generalized State Constraints

We provide a dynamic programming principle for stochastic optimal control problems with expectation constraints. A weak formulation, using test functions and a probabilistic relaxation of the constraint, avoids restrictions related to a measurable selection but still implies the Hamilton-Jacobi-Bellman equation in the viscosity sense. We treat open state constraints as a special case of expectation constraints and prove a comparison theorem to obtain the equation for closed state constraints.

math.OC↗

arXiv2008

Strong Approximations of BSDEs in a domain

We study the strong approximation of a Backward SDE with finite stopping time horizon, namely the first exit time of a forward SDE from a cylindrical domain. We use the Euler scheme approach of Bouchard and Touzi, Zhang 04}. When the domain is piecewise smooth and under a non-characteristic boundary condition, we show that the associated strong error is at most of order $h^{\frac14-\eps}$ where $h$ denotes the time step and $\eps$ is any positive parameter. This rate corresponds to the strong exit time approximation. It is improved to $h^{\frac12-\eps}$ when the exit time can be exactly simulated or for a weaker form of the approximation error. Importantly, these results are obtained without uniform ellipticity condition.

math.PR↗

arXiv2006

Optimal consumption in discrete-time financial models with industrial investment opportunities and nonlinear returns

We consider a general discrete-time financial market with proportional transaction costs as in [Kabanov, Stricker and Rásonyi Finance and Stochastics 7 (2003) 403--411] and [Schachermayer Math. Finance 14 (2004) 19--48]. In addition to the usual investment in financial assets, we assume that the agents can invest part of their wealth in industrial projects that yield a nonlinear random return. We study the problem of maximizing the utility of consumption on a finite time period. The main difficulty comes from the nonlinearity of the nonfinancial assets' return. Our main result is to show that existence holds in the utility maximization problem. As an intermediary step, we prove the closedness of the set $A_T$ of attainable claims under a robust no-arbitrage property similar to the one introduced in [Schachermayer Math. Finance 14 (2004) 19--48] and further discussed in [Kabanov, Stricker and Rásonyi Finance and Stochastics 7 (2003) 403--411]. This allows us to provide a dual formulation for $A_T$.

math.PR↗

arXiv2006

Maturity randomization for stochastic control problems

We study a maturity randomization technique for approximating optimal control problems. The algorithm is based on a sequence of control problems with random terminal horizon which converges to the original one. This is a generalization of the so-called Canadization procedure suggested by Carr [Review of Financial Studies II (1998) 597--626] for the fast computation of American put option prices. In addition to the original application of this technique to optimal stopping problems, we provide an application to another problem in finance, namely the super-replication problem under stochastic volatility, and we show that the approximating value functions can be computed explicitly.

math.PR↗

arXiv2005

On the Hedging of American Options in Discrete Time Markets with Proportional Transaction Costs

In this note, we consider a general discrete time financial market with proportional transaction costs as in Kabanov and Stricker (2001), Kabanov et al. (2002), Kabanov et al. (2003) and Schachermayer (2004). We provide a dual formulation for the set of initial endowments which allow to super-hedge some American claim. We show that this extends the result of Chalasani and Jha (2001) which was obtained in a model with constant transaction costs and risky assets which evolve on a finite dimensional tree. We also provide fairly general conditions under which the expected formulation in terms of stopping times does not work.

math.PR↗

arXiv2005

Explicit characterization of the super-replication strategy in financial markets with partial transaction costs

We consider a multivariate financial market with transaction costs and study the problem of finding the minimal initial capital needed to hedge, without risk, European-type contingent claims. The model is similar to the one considered in Bouchard and Touzi (2000), except that some of the assets can be exchanged freely, i.e. without paying transaction costs. In this context, we generalize the result of the above paper and prove that the value of this stochastic control problem is given by the cost of the cheapest hedging strategy in which the number of non-freely exchangeable assets is kept constant over time.

math.PR↗

arXiv2005

No-arbitrage in discrete-time markets with proportional transaction costs and general information structure

We discuss the no-arbitrage conditions in a general framework for discrete-time models of financial markets with proportional transaction costs and general information structure. We extend the results of Kabanov and al. (2002), Kabanov and al. (2003) and Schachermayer (2004) to the case where bid-ask spreads are not known with certainty. In the "no-friction" case, we retrieve the result of Kabanov and Stricker (2003).

math.PR↗