776 research outputs found
Continuous-time mean-variance efficiency: the 80% rule
This paper studies a continuous-time market where an agent, having specified
an investment horizon and a targeted terminal mean return, seeks to minimize
the variance of the return. The optimal portfolio of such a problem is called
mean-variance efficient \`{a} la Markowitz. It is shown that, when the market
coefficients are deterministic functions of time, a mean-variance efficient
portfolio realizes the (discounted) targeted return on or before the terminal
date with a probability greater than 0.8072. This number is universal
irrespective of the market parameters, the targeted return and the length of
the investment horizon.Comment: Published at http://dx.doi.org/10.1214/105051606000000349 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
The Informed and Uniformed Agent's Price of a Contingent Claim
The existence of an adapted solution to a backward stochastic differential equation which is not adapted to the filtration of the underlying Brownian motion is proved. This result is applied to the pricing of contingent claims. It allows to compare the prices of agents who have different information about the evolution of the market. The problem is considered in both the classical and the Föllmer-Schweizer hedging case.
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
Interplay between dividend rate and business constraints for a financial corporation
We study a model of a corporation which has the possibility to choose various
production/business policies with different expected profits and risks. In the
model there are restrictions on the dividend distribution rates as well as
restrictions on the risk the company can undertake. The objective is to
maximize the expected present value of the total dividend distributions. We
outline the corresponding Hamilton-Jacobi-Bellman equation, compute explicitly
the optimal return function and determine the optimal policy. As a consequence
of these results, the way the dividend rate and business constraints affect the
optimal policy is revealed. In particular, we show that under certain
relationships between the constraints and the exogenous parameters of the
random processes that govern the returns, some business activities might be
redundant, that is, under the optimal policy they will never be used in any
scenario.Comment: Published at http://dx.doi.org/10.1214/105051604000000909 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
General Stopping Behaviors of Naive and Non-Committed Sophisticated Agents, with Application to Probability Distortion
We consider the problem of stopping a diffusion process with a payoff
functional that renders the problem time-inconsistent. We study stopping
decisions of naive agents who reoptimize continuously in time, as well as
equilibrium strategies of sophisticated agents who anticipate but lack control
over their future selves' behaviors. When the state process is one dimensional
and the payoff functional satisfies some regularity conditions, we prove that
any equilibrium can be obtained as a fixed point of an operator. This operator
represents strategic reasoning that takes the future selves' behaviors into
account. We then apply the general results to the case when the agents distort
probability and the diffusion process is a geometric Brownian motion. The
problem is inherently time-inconsistent as the level of distortion of a same
event changes over time. We show how the strategic reasoning may turn a naive
agent into a sophisticated one. Moreover, we derive stopping strategies of the
two types of agent for various parameter specifications of the problem,
illustrating rich behaviors beyond the extreme ones such as "never-stopping" or
"never-starting"
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
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