167 research outputs found
Weighted entropy and optimal portfolios for risk-averse Kelly investments
Following a series of works on capital growth investment, we analyse
log-optimal portfolios where the return evaluation includes `weights' of
different outcomes. The results are twofold: (A) under certain conditions, the
logarithmic growth rate leads to a supermartingale, and (B) the optimal
(martingale) investment strategy is a proportional betting. We focus on
properties of the optimal portfolios and discuss a number of simple examples
extending the well-known Kelly betting scheme.
An important restriction is that the investment does not exceed the current
capital value and allows the trader to cover the worst possible losses.
The paper deals with a class of discrete-time models. A continuous-time
extension is a topic of an ongoing study
Optional decompositions under constraints
Motivated by a hedging problem in mathematical finance, El Karoui and Quenez [7] and Kramkov [14] have developed optional versions of the Doob-Meyer decomposition which hold simultaneously for all equivalent martingale measures. We investigate the general structure of such optional decompositions, both in additive and in multiplicative form, and under constraints corresponding to di_erent classes of equivalent measures. As an application, we extend results of Karatzas and Cvitanic [3] on hedging problems with constrained portfolios
Power Utility Maximization in Discrete-Time and Continuous-Time Exponential Levy Models
Consider power utility maximization of terminal wealth in a 1-dimensional
continuous-time exponential Levy model with finite time horizon. We discretize
the model by restricting portfolio adjustments to an equidistant discrete time
grid. Under minimal assumptions we prove convergence of the optimal
discrete-time strategies to the continuous-time counterpart. In addition, we
provide and compare qualitative properties of the discrete-time and
continuous-time optimizers.Comment: 18 pages, to appear in Mathematical Methods of Operations Research.
The final publication is available at springerlink.co
A model for a large investor trading at market indifference prices. I: single-period case
We develop a single-period model for a large economic agent who trades with
market makers at their utility indifference prices. A key role is played by a
pair of conjugate saddle functions associated with the description of Pareto
optimal allocations in terms of the utility function of a representative market
maker.Comment: Shorten from 69 to 30 pages due to referees' requests; a part of the
previous version has been moved to "The stochastic field of aggregate
utilities and its saddle conjugate", arXiv:1310.728
Dry Markets and Superreplication Bounds of American Derivatives
This paper studies the impact of dry markets for underlying assets on the pricing of American derivatives, using a discrete time framework. Dry markets are characterized by the possibility of non-existence of trading at certain dates. Such non-existence may be deterministic or probabilistic. Using superreplicating strategies, we derive expectation representations for the range of arbitrage-free values of the dervatives. In the probabilistic case, if we consider an enlarged filtration induced by the price process and the market existence process, ordinary stopping times are required. If not, randomized stopping times are required. Several comparisons of the ranges obtained with the two market restrictions are performed. Finally, we conclude that arbitrage arguments are not enough to define the optimal exercise policy.N/
Optimal Investment-Consumption Problem with Constraint
In this paper, we consider an optimal investment-consumption problem subject to a closed convex constraint. In the problem, a constraint is imposed on both the investment and the consumption strategy, rather than just on the investment. The existence of solution is established by using the Martingale technique and convex duality. In addition to investment, our technique embeds also the consumption into a family of fictitious markets. However, with the addition of consumption, it leads to nonreflexive dual spaces. This difficulty is overcome by employing the so-called technique of \relaxation-projection" to establish the existence of solution to the problem. Furthermore, if the solution to the dual problem is obtained, then the solution to the primal problem can be found by using the characterization of the solution. An illustrative example is given with a dynamic risk constraint to demonstrate the method
On the Existence of Shadow Prices
For utility maximization problems under proportional transaction costs, it
has been observed that the original market with transaction costs can sometimes
be replaced by a frictionless "shadow market" that yields the same optimal
strategy and utility. However, the question of whether or not this indeed holds
in generality has remained elusive so far. In this paper we present a
counterexample which shows that shadow prices may fail to exist. On the other
hand, we prove that short selling constraints are a sufficient condition to
warrant their existence, even in very general multi-currency market models with
possibly discontinuous bid-ask-spreads.Comment: 14 pages, 1 figure, to appear in "Finance and Stochastics
Vigilant Measures of Risk and the Demand for Contingent Claims
I examine a class of utility maximization problems with a not necessarily lawinvariant utility, and with a not necessarily law-invariant risk measure constraint. The objective function is an integral of some function U with respect to some probability measure P, and the constraint set contains some risk measure constraint which is not necessarily P-law-invariant. This introduces some heterogeneity in the perception of uncertainty. The primitive U is a function of some given underlying random variable X and of a contingent claim Y on X. Many problems in economic theory and financial theory can be formulated in this manner, when a heterogeneity in the perception of uncertainty is introduced. Under a consistency requirement on the risk measure that will be called Vigilance, supermodularity of the primitive U is sufficient for the existence of optimal continent claims, and for these optimal claims to be comonotonic with the underlying random variable X. Vigilance is satisfied by a large class of risk measures, including all distortion risk measures. An explicit characterization of an optimal contingent claim is also provided in the case where the risk measure is a convex distortion risk measure
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