355 research outputs found

### A Note on the Quantile Formulation

Many investment models in discrete or continuous-time settings boil down to
maximizing an objective of the quantile function of the decision variable. This
quantile optimization problem is known as the quantile formulation of the
original investment problem. Under certain monotonicity assumptions, several
schemes to solve such quantile optimization problems have been proposed in the
literature. In this paper, we propose a change-of-variable and relaxation
method to solve the quantile optimization problems without using the calculus
of variations or making any monotonicity assumptions. The method is
demonstrated through a portfolio choice problem under rank-dependent utility
theory (RDUT). We show that this problem is equivalent to a classical Merton's
portfolio choice problem under expected utility theory with the same utility
function but a different pricing kernel explicitly determined by the given
pricing kernel and probability weighting function. With this result, the
feasibility, well-posedness, attainability and uniqueness issues for the
portfolio choice problem under RDUT are solved. It is also shown that solving
functional optimization problems may reduce to solving probabilistic
optimization problems. The method is applicable to general models with
law-invariant preference measures including portfolio choice models under
cumulative prospect theory (CPT) or RDUT, Yaari's dual model, Lopes' SP/A
model, and optimal stopping models under CPT or RDUT.Comment: to appear in Mathematical Financ

### Investment under Duality Risk Measure

One index satisfies the duality axiom if one agent, who is uniformly more
risk-averse than another, accepts a gamble, the latter accepts any less risky
gamble under the index. Aumann and Serrano (2008) show that only one index
defined for so-called gambles satisfies the duality and positive homogeneity
axioms. We call it a duality index. This paper extends the definition of
duality index to all outcomes including all gambles, and considers a portfolio
selection problem in a complete market, in which the agent's target is to
minimize the index of the utility of the relative investment outcome. By
linking this problem to a series of Merton's optimum consumption-like problems,
the optimal solution is explicitly derived. It is shown that if the prior
benchmark level is too high (which can be verified), then the investment risk
will be beyond any agent's risk tolerance. If the benchmark level is
reasonable, then the optimal solution will be the same as that of one of the
Merton's series problems, but with a particular value of absolute risk
aversion, which is given by an explicit algebraic equation as a part of the
optimal solution. According to our result, it is riskier to achieve the same
surplus profit in a stable market than in a less-stable market, which is
consistent with the common financial intuition.Comment: 17 pages, 1 figur

### A Note on the Monge-Kantorovich Problem in the Plane

The Monge-Kantorovich mass-transportation problem has been shown to be
fundamental for various basic problems in analysis and geometry in recent
years. Shen and Zheng (2010) proposed a probability method to transform the
celebrated Monge-Kantorovich problem in a bounded region of the Euclidean plane
into a Dirichlet boundary problem associated to a nonlinear elliptic equation.
Their results are original and sound, however, their arguments leading to the
main results are skipped and difficult to follow. In the present paper, we
adopt a different approach and give a short and easy-followed detailed proof
for their main results

### Optimal stopping under probability distortion

We formulate an optimal stopping problem for a geometric Brownian motion
where the probability scale is distorted by a general nonlinear function. The
problem is inherently time inconsistent due to the Choquet integration
involved. We develop a new approach, based on a reformulation of the problem
where one optimally chooses the probability distribution or quantile function
of the stopped state. An optimal stopping time can then be recovered from the
obtained distribution/quantile function, either in a straightforward way for
several important cases or in general via the Skorokhod embedding. This
approach enables us to solve the problem in a fairly general manner with
different shapes of the payoff and probability distortion functions. We also
discuss economical interpretations of the results. In particular, we justify
several liquidation strategies widely adopted in stock trading, including those
of "buy and hold", "cut loss or take profit", "cut loss and let profit run" and
"sell on a percentage of historical high".Comment: Published in at http://dx.doi.org/10.1214/11-AAP838 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Moral-hazard-free insurance: mean-variance premium principle and rank-dependent utility theory

This paper investigates a Pareto optimal insurance problem, where the insured
maximizes her rank-dependent utility preference and the insurer is risk neutral
and employs the mean-variance premium principle. To eliminate potential moral
hazard issues, we only consider the so-called moral-hazard-free insurance
contracts that obey the incentive compatibility constraint. The insurance
problem is first formulated as a non-concave maximization problem involving
Choquet expectation, then turned into a concave quantile optimization problem
and finally solved by the calculus of variations method. The optimal contract
is expressed by a second-order ordinary integro-differential equation with
nonlocal operator. An effective numerical method is proposed to compute the
optimal contract assuming the probability weighting function has a density.
Also, we provide an example which is analytically solved

### Continuous-Time Markowitz's Model with Transaction Costs

A continuous-time Markowitz's mean-variance portfolio selection problem is
studied in a market with one stock, one bond, and proportional transaction
costs. This is a singular stochastic control problem,inherently in a finite
time horizon. With a series of transformations, the problem is turned into a
so-called double obstacle problem, a well studied problem in physics and
partial differential equation literature, featuring two time-varying free
boundaries. The two boundaries, which define the buy, sell, and no-trade
regions, are proved to be smooth in time. This in turn characterizes the
optimal strategy, via a Skorokhod problem, as one that tries to keep a certain
adjusted bond-stock position within the no-trade region. Several features of
the optimal strategy are revealed that are remarkably different from its
no-transaction-cost counterpart. It is shown that there exists a critical
length in time, which is dependent on the stock excess return as well as the
transaction fees but independent of the investment target and the stock
volatility, so that an expected terminal return may not be achievable if the
planning horizon is shorter than that critical length (while in the absence of
transaction costs any expected return can be reached in an arbitrary period of
time). It is further demonstrated that anyone following the optimal strategy
should not buy the stock beyond the point when the time to maturity is shorter
than the aforementioned critical length. Moreover, the investor would be less
likely to buy the stock and more likely to sell the stock when the maturity
date is getting closer. These features, while consistent with the widely
accepted investment wisdom, suggest that the planning horizon is an integral
part of the investment opportunities.Comment: 30 pages, 1 figur

### Multidimensional indefinite stochastic Riccati equations and zero-sum linear-quadratic stochastic differential games with non-markovian regime switching

This paper is concerned with two-player zero-sum linear-quadratic stochastic
differential games in a regime switching model. The controlled inhomogeneous
system coefficients depend on the underlying noises, so it is a non-Markovian
regime switching model. Based on a new kind of multidimensional indefinite
stochastic Riccati equation (SRE) and a multidimensional linear backward
stochastic differential equation (BSDE) with unbounded coefficients, we can
provide optimal feedback control-strategy pairs for the two players in a
closed-loop form. The main contribution of this paper, which is important in
its own right from the BSDE theory point of view, is to prove the existence and
uniqueness of the new kind of multidimensional indefinite SRE. Interestingly,
the components of the solution can take positive, zero and negative values
simultaneously. We also obtain the corresponding optimal feedback
control-strategy pairs for homogeneous systems under closed convex cone control
constraints. Finally, these results are applied to portfolio selection problems
with different short-selling prohibition constraints in a regime switching
market with random coefficients

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