76 research outputs found
General indifference pricing with small transaction costs
We study the utility indifference price of a European option in the context
of small transaction costs. Considering the general setup allowing consumption
and a general utility function at final time T, we obtain an asymptotic
expansion of the utility indifference price as a function of the asymptotic
expansions of the utility maximization problems with and without the European
contingent claim. We use the tools developed in [54] and [48] based on
homogenization and viscosity solutions to characterize these expansions.
Finally we study more precisely the example of exponential utilities, in
particular recovering under weaker assumptions the results of [6].Comment: 43 page
A mathematical treatment of bank monitoring incentives
In this paper, we take up the analysis of a principal/agent model with moral
hazard introduced in [17], with optimal contracting between competitive
investors and an impatient bank monitoring a pool of long-term loans subject to
Markovian contagion. We provide here a comprehensive mathematical formulation
of the model and show using martingale arguments in the spirit of Sannikov [18]
how the maximization problem with implicit constraints faced by investors can
be reduced to a classical stochastic control problem. The approach has the
advantage of avoiding the more general techniques based on forward-backward
stochastic differential equations described in [6] and leads to a simple
recursive system of Hamilton-Jacobi-Bellman equations. We provide a solution to
our problem by a verification argument and give an explicit description of both
the value function and the optimal contract. Finally, we study the limit case
where the bank is no longer impatient
Contracting theory with competitive interacting agents
In a framework close to the one developed by Holmstr\"om and Milgrom [44], we
study the optimal contracting scheme between a Principal and several Agents.
Each hired Agent is in charge of one project, and can make efforts towards
managing his own project, as well as impact (positively or negatively) the
projects of the other Agents. Considering economic Agents in competition with
relative performance concerns, we derive the optimal contracts in both first
best and moral hazard settings. The enhanced resolution methodology relies
heavily on the connection between Nash equilibria and multidimensional
quadratic BSDEs. The optimal contracts are linear and each agent is paid a
fixed proportion of the terminal value of all the projects of the firm.
Besides, each Agent receives his reservation utility, and those with high
competitive appetence are assigned less volatile projects, and shall even
receive help from the other Agents. From the principal point of view, it is in
the firm interest in our model to strongly diversify the competitive appetence
of the Agents.Comment: 36 page
Second Order Backward Stochastic Differential Equations with Quadratic Growth
We extend the wellposedness results for second order backward stochastic
differential equations introduced by Soner, Touzi and Zhang \cite{stz} to the
case of a bounded terminal condition and a generator with quadratic growth in
the variable. More precisely, we obtain uniqueness through a representation
of the solution inspired by stochastic control theory, and we obtain two
existence results using two different methods. In particular, we obtain the
existence of the simplest purely quadratic 2BSDEs through the classical
exponential change, which allows us to introduce a quasi-sure version of the
entropic risk measure. As an application, we also study robust risk-sensitive
control problems. Finally, we prove a Feynman-Kac formula and a probabilistic
representation for fully nonlinear PDEs in this setting.Comment: 31 page
Existence and uniqueness results for BSDEs with jumps: the whole nine yards
This paper is devoted to obtaining a wellposedness result for
multidimensional BSDEs with possibly unbounded random time horizon and driven
by a general martingale in a filtration only assumed to satisfy the usual
hypotheses, i.e. the filtration may be stochastically discontinuous. We show
that for stochastic Lipschitz generators and unbounded, possibly infinite, time
horizon, these equations admit a unique solution in appropriately weighted
spaces. Our result allows in particular to obtain a wellposedness result for
BSDEs driven by discrete--time approximations of general martingales.Comment: 48 pages, final version, forthcoming in the Electronic Journal of
Probabilit
Moral Hazard in Dynamic Risk Management
We consider a contracting problem in which a principal hires an agent to
manage a risky project. When the agent chooses volatility components of the
output process and the principal observes the output continuously, the
principal can compute the quadratic variation of the output, but not the
individual components. This leads to moral hazard with respect to the risk
choices of the agent. We identify a family of admissible contracts for which
the optimal agent's action is explicitly characterized, and, using the recent
theory of singular changes of measures for It\^o processes, we study how
restrictive this family is. In particular, in the special case of the standard
Homlstr\"om-Milgrom model with fixed volatility, the family includes all
possible contracts. We solve the principal-agent problem in the case of CARA
preferences, and show that the optimal contract is linear in these factors: the
contractible sources of risk, including the output, the quadratic variation of
the output and the cross-variations between the output and the contractible
risk sources. Thus, like sample Sharpe ratios used in practice, path-dependent
contracts naturally arise when there is moral hazard with respect to risk
management. In a numerical example, we show that the loss of efficiency can be
significant if the principal does not use the quadratic variation component of
the optimal contract.Comment: 36 pages, 3 figure
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