8,190 research outputs found
Applications of Stochastic Control to Portfolio Selection Problems
Portfolio selection is an important problem both in academia and in practice. Due to its significance, it has received great attention and facilitated a large amount of research. This thesis is devoted to structuring optimal portfolios using different criteria.
Participating contracts are popular insurance policies, in which the payoff to a policyholder is linked to the performance of a portfolio managed by the insurer. In Chapter 2, we consider the portfolio selection problem of an insurer that offers participating contracts and has an S-shaped utility function. Applying the martingale approach, closed-form solutions are obtained. The resulting optimal strategies are compared with two portfolio insurance hedging strategies, e.g. Constant Proportion Portfolio Insurance strategy and Option Based Portfolio Insurance strategy. We also study numerical solutions of the portfolio selection problem with constraints on the portfolio weights.
In Chapter 3, we consider the portfolio selection problem of maximizing a performance measure in a continuous-time diffusion model. The performance measure is the ratio of the overperformance to the underperformance of a portfolio relative to a benchmark. Following a strategy from fractional programming, we analyze the problem by solving a family of related problems, where the objective functions are the numerator of the original problem minus the denominator multiplied by a penalty parameter. These auxiliary problems can be solved using the martingale method for stochastic control. The existence of a solution is discussed in a general setting and explicit solutions are derived when both the reward and the penalty functions are power functions.
In Chapter 4, we consider the mean-risk portfolio selection problem of optimizing the expectile risk measure in a continuous-time diffusion model. Due to the lack of an explicit form for expectiles and the close relationship with the Omega measure, we propose an alternative optimization problem with the Omega measure as an objective and show the equivalence between the two problems. After showing the solution for the mean-expectile problem is not attainable but the value function is finite, we modify the problem with an upper bound constraint imposed on the terminal wealth and obtain the solution via the Lagrangian duality method and pointwise optimization technique. The global expectile minimizing portfolio and efficient frontier are also considered in our analysis.
In Chapter 5, we consider the utility-based portfolio selection problem in a continuous-time setting. We assume the market price of risk depends on a stochastic factor that satisfies an affine-form, square-root, Markovian model. This financial market framework includes the classical geometric Brownian motion, the constant elasticity of variance (CEV) model and the Heston's model as special cases. Adopting the Backward Stochastic Differential Equation (BSDE) approach, we obtain the closed-form solutions for power, logarithm, or exponential utility functions, respectively.
Concluding remarks and several potential topics for further research are presented in Chapter 6
Mean-Variance Policy for Discrete-time Cone Constrained Markets: The Consistency in Efficiency and Minimum-Variance Signed Supermartingale Measure
The discrete-time mean-variance portfolio selection formulation, a
representative of general dynamic mean-risk portfolio selection problems, does
not satisfy time consistency in efficiency (TCIE) in general, i.e., a truncated
pre-committed efficient policy may become inefficient when considering the
corresponding truncated problem, thus stimulating investors' irrational
investment behavior. We investigate analytically effects of portfolio
constraints on time consistency of efficiency for convex cone constrained
markets. More specifically, we derive the semi-analytical expressions for the
pre-committed efficient mean-variance policy and the minimum-variance signed
supermartingale measure (VSSM) and reveal their close relationship. Our
analysis shows that the pre-committed discrete-time efficient mean-variance
policy satisfies TCIE if and only if the conditional expectation of VSSM's
density (with respect to the original probability measure) is nonnegative, or
once the conditional expectation becomes negative, it remains at the same
negative value until the terminal time. Our findings indicate that the property
of time consistency in efficiency only depends on the basic market setting,
including portfolio constraints, and this fact motivates us to establish a
general solution framework in constructing TCIE dynamic portfolio selection
problem formulations by introducing suitable portfolio constraints
Continuous time mean-variance portfolio selection with nonlinear wealth equations and random coefficients
This paper concerns the continuous time mean-variance portfolio selection
problem with a special nonlinear wealth equation. This nonlinear wealth
equation has nonsmooth random coefficients and the dual method developed in [7]
does not work. To apply the completion of squares technique, we introduce two
Riccati equations to cope with the positive and negative part of the wealth
process separately. We obtain the efficient portfolio strategy and efficient
frontier for this problem. Finally, we find the appropriate sub-derivative
claimed in [7] using convex duality method.Comment: arXiv admin note: text overlap with arXiv:1606.0548
Continuous-time Mean-Variance Portfolio Selection with Stochastic Parameters
This paper studies a continuous-time market {under stochastic environment}
where an agent, having specified an investment horizon and a target terminal
mean return, seeks to minimize the variance of the return with multiple stocks
and a bond. In the considered model firstly proposed by [3], the mean returns
of individual assets are explicitly affected by underlying Gaussian economic
factors. Using past and present information of the asset prices, a
partial-information stochastic optimal control problem with random coefficients
is formulated. Here, the partial information is due to the fact that the
economic factors can not be directly observed. Via dynamic programming theory,
the optimal portfolio strategy can be constructed by solving a deterministic
forward Riccati-type ordinary differential equation and two linear
deterministic backward ordinary differential equations
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
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A review of portfolio planning: Models and systems
In this chapter, we first provide an overview of a number of portfolio planning models
which have been proposed and investigated over the last forty years. We revisit the
mean-variance (M-V) model of Markowitz and the construction of the risk-return
efficient frontier. A piecewise linear approximation of the problem through a
reformulation involving diagonalisation of the quadratic form into a variable
separable function is also considered. A few other models, such as, the Mean
Absolute Deviation (MAD), the Weighted Goal Programming (WGP) and the
Minimax (MM) model which use alternative metrics for risk are also introduced,
compared and contrasted. Recently asymmetric measures of risk have gained in
importance; we consider a generic representation and a number of alternative
symmetric and asymmetric measures of risk which find use in the evaluation of
portfolios. There are a number of modelling and computational considerations which
have been introduced into practical portfolio planning problems. These include: (a)
buy-in thresholds for assets, (b) restriction on the number of assets (cardinality
constraints), (c) transaction roundlot restrictions. Practical portfolio models may also
include (d) dedication of cashflow streams, and, (e) immunization which involves
duration matching and convexity constraints. The modelling issues in respect of these
features are discussed. Many of these features lead to discrete restrictions involving
zero-one and general integer variables which make the resulting model a quadratic
mixed-integer programming model (QMIP). The QMIP is a NP-hard problem; the
algorithms and solution methods for this class of problems are also discussed. The
issues of preparing the analytic data (financial datamarts) for this family of portfolio
planning problems are examined. We finally present computational results which
provide some indication of the state-of-the-art in the solution of portfolio optimisation
problems
Efficient Frontier for Robust Higher-order Moment Portfolio Selection
This article proposes a non-parametric portfolio selection criterion for the static asset allocation problem in a robust higher-moment framework. Adopting the Shortage Function approach, we generalize the multi-objective optimization technique in a four-dimensional space using L-moments, and focus on various illustrations of a four-dimensional set of the first four L-moment primal efficient portfolios. our empirical findings, using a large European stock database, mainly rediscover the earlier works by Jean (1973) and Ingersoll (1975), regarding the shape of the extended higher-order moment efficient frontier, and confirm the seminal prediction by Levy and Markowitz (1979) about the accuracy of the mean-variance criterion.Efficient frontier, portfolio selection, robust higher L-moments, shortage function, goal attainment application.
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