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Various dependence prescriptions
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Dybvig (1988) introduced the interesting problem of how to construct in the cheapest possible way a terminal wealth with desired distribution. This idea has induced a series of papers concerning generality, consequences and applications. As the optimized claims typically follow the trend in the market, they are not useful for investors who wish to...
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Citations
... 1 This idea of "quantile formulation" finds its pedigree in a more general "rearrangement formulation" for this kind of optimization problems (Bernard et al., 2014a(Bernard et al., , 2015b(Bernard et al., , 2014bBurgert and Rüschendorf, 2006;Rüschendorf, 1983). ...
In this paper, we consider the optimal investment problem with both probability distor-tion/weighting and general non-concave utility functions with possibly finite number of inflection points, and propose a general approach for solving this problem. Existing literature has shown the equivalent relationships (strong duality) between the concavified problem and the original one by either assuming the presence of probability weighting or the non-concavity of utility functions, but not both. In this paper, we combine both features and propose a step-wise relaxation Lagrange method to handle the optimization problems under general non-concave utility functions and probability distortion functions. The necessary and sufficient conditions on eliminating the duality gap for the Lagrange method have been provided under this circumstance. We have applied this solution method to solve several representative examples in mathematical behavioral finance: the CPT model which has inverse S-shaped probability distortion and a S-shaped utility function (i.e. one inflection point), Value-at-Risk based risk management (VAR-RM) model with probability distortion, Yarri's dual model and the goal reaching model. We obtain the closed-form optimal trading strategy for a special example of the CPT model, where a "distorted" Merton line has been shown exactly. The slope of the "distorted" Merton line is given by an inflation factor multiplied by the standard Merton ratio, and an interesting finding is that the inflation factor is solely dependent on the probability distortion rather than the utility function.
... 1 This idea of "quantile formulation" finds its pedigree in a more general "rearrangement formulation" for this kind of optimization problems (Bernard et al., 2014a(Bernard et al., , 2015b(Bernard et al., , 2014bBurgert and Rüschendorf, 2006;Rüschendorf, 1983). ...
In this paper, we consider the optimal investment problem with both probability distor- tion/weighting and general non-concave utility functions with possibly finite number of inflection points. Our model contains the model under cumulative prospect theory (CPT) as a special case, which has inverse S-shaped probability weighting and S-shaped utility function (i.e. one inflection point). Existing literature have shown the equivalent relationships (strong duality) between the concavified problem and the original one by either assuming the presence of probability weighting or the non-concavity of utility functions, but not both. In this paper, we combine both features and propose a step-wise relaxation method to handle general non-concave utility functions and probability distortion functions. The nec- essary and sufficient conditions on eliminating the duality gap for the Lagrange method based on the step-wise relaxation have been provided under this circumstance. We have applied this solution method to solve in closed-form several representative examples in mathematical behavioral finance including the CPT model, Value-at-Risk based risk management (VAR-RM) model with probability distortions, Yarri’s dual model and the goal reaching model. We obtain a closed-form optimal trading strategy for a special example of the CPT model, where a “distorted” Merton line has been shown exactly. The slope of the “distorted” Merton line is given by an inflation factor multiplied by the standard Merton ratio, and an interesting finding is that the inflation factor is solely dependent on the probability distortion rather than the non-concavity of the utility function.
... In particular, a generalized version of quantile formulation incorporating dependence structure was proposed in Bernard et al. (2015b). See also Bernard et al. (2014b) for the applications to solving expected utility maximization problems, Recently, Rüschendorf and Vanduffel (2020) reformulated the optimal portfolio selection problem as an optimization problem on real functions under monotonicity conditions and obtained a unified method to solve several problems of interest such as the Merton problem, the rank-dependent utility theory (RDUT) problem , a general form of the CPT problem as well as Yaari's optimal investment problem. Boudt et al. (2020) derived explicit solutions to the optimal payoff problem for a general Yaari investor in a variety of relevant cases. ...
In this paper, we consider the optimal investment problem with both probability distor- tion/weighting and general non-concave utility functions with possibly finite number of inflection points. Our model contains the model under cumulative prospect theory (CPT) as a special case, which has inverse S-shaped probability weighting and S-shaped utility function (i.e. one inflection point). Existing literature have shown the equivalent relationships (strong duality) between the concavified problem and the original one by either assuming the presence of probability weighting or the non-concavity of utility functions, but not both. In this paper, we combine both features and propose a step-wise relaxation method to handle general non-concave utility functions and probability distortion functions. The nec- essary and sufficient conditions on eliminating the duality gap for the Lagrange method based on the step-wise relaxation have been provided under this circumstance. We have applied this solution method to solve in closed-form several representative examples in mathematical behavioral finance including the CPT model, Value-at-Risk based risk management (VAR-RM) model with probability distortions, Yarri’s dual model and the goal reaching model. We obtain a closed-form optimal trading strategy for a special example of the CPT model, where a “distorted” Merton line has been shown exactly. The slope of the “distorted” Merton line is given by an inflation factor multiplied by the standard Merton ratio, and an interesting finding is that the inflation factor is solely dependent on the probability distortion rather than the non-concavity of the utility function.
... When the optimal solution exists, we show that is unique and the payoff of the optimal portfolio is a digital option: it leads to certain winning payoff in some market scenarios and to zero otherwise. We then extend our model to the setting in Bernard, Boyle and Vanduffel (2014); Bernard, Rüschendorf and Vanduffel (2014), and Bernard et al. (2015), where the agent focuses on payoffs that have a particular dependence structure, modeled by a copula, with a given benchmark payoff. We derive similar results in this extended model. ...
... Consider the setting in Bernard, Boyle and Vanduffel (2014); Bernard, Rüschendorf and Vanduffel (2014), and Bernard et al. (2015): There is a benchmark payoff A and the agent wants to choose payoff X that has a given dependence structure with A, and this dependence structure is specified by a given copula function C; i.e., X and A follow the joint distribution: ...
... We follow the approach in Bernard, Rüschendorf and Vanduffel (2014) to convert (3.2) into a problem of finding the optimal quantile. More precisely, similar to the quantile approach, we first find the payoff X that minimizes the cost E[ρX] given the marginal distribution F X of X and the copula constraint (3.1). ...
We consider portfolio optimization under a preference model in a single-period, complete market. This preference model includes Yaari's dual theory of choice and quantile maximization as special cases. We characterize when the optimal solution exists and derive the optimal solution in closed form when it exists. The payoff of the optimal portfolio is a digital option: it yields an in-the-money payoff when the market is good and zero payoff otherwise. When the initial wealth increases, the set of good market scenarios remains unchanged while the payoff in these scenarios increases. Finally, we extend our portfolio optimization problem by imposing a dependence structure with a given benchmark payoff and derive similar results.
... The basic reason for doing so is that under law-invariant preferences optimal payoffs will have their lowest outcomes when the economy is in a downturn (Dybvig (1988a)), and arguably this feature does not fit with the aspirations of many investors. We model state-dependence using dependence constraints, i.e., we prescribe the copula between the payoff and some benchmark asset (Bernard et al. (2014b), Bernard et al. (2015b)). We provide a reduction result (a quantile formulation) that allows dealing with optimal portfolio selection problems for investors with concave preferences and who have an additional state-dependent constraint. ...
... To account for this (undesirable) feature, we may want to provide the investor with the opportunity to maintain a desired dependence with a benchmark asset (state-dependent constraint). This idea was developed in Bernard et al. (2014a), Bernard et al. (2014b) and Bernard et al. (2015b). Specifically, Bernard et al. (2015b) determine the optimal payoff for an expected utility maximizer under such dependence constraint. ...
... If X T is a solution to problem (U C) then denote by U = τ ϕ T |A T the conditional distributional transform and X T = F −1 X T |A T (1−U )). Then by arguments as in the proof of Theorem 3.1 in Bernard et al. (2014b), we obtain that ( X T , A T ) ∼ (X T , A T )), i.e., X T is admissible. Furthermore, since U , X T are independent we obtain that X T , ϕ T are anti-monotonic, conditionally on A T = a. ...
In the framework of continuous-time market models with specified pricing density, optimal payoffs under increasing law-invariant preferences are known to be anti-monotonic with the pricing density. Consequently, optimal portfolio selection problems can be reformulated as optimization problems on real functions under monotonicity conditions. We solve two basic types of these optimization problems, which makes it possible to obtain in a fairly unified way the optimal payoff for several portfolio selection problems of interest. In particular, we completely solve the optimal portfolio selection problem for an investor with preferences as in cumulative prospect theory or as in Yaari’s dual theory. Extending previous work, we also characterize optimal payoffs when the payoff is required to have a fixed copula with some benchmark (state-dependent constraint). Specifically, we show that if one can determine the optimal payoff under a concave law-invariant objective, then one can also determine the optimal payoff when adding the state-dependent constraint. In the final part of the paper, we consider an extension to (incomplete) market models in which the pricing density is not completely specified. When a sufficient number of payoffs have a known market price, we show that optimal payoffs are anti-monotonic to some pricing density that we explicitly derive from these market prices. As examples, we deal with some exponential Lévy market models and some market models involving Itô processes.
... w.r.t ≤ sm in dependence on the constraints C i = C Xi,Z . This also yields ordering criteria for the solution EF −1 X1|Z (U )F −1 X2|Z (1 − U ) of the minimization problem min EX 1 X 2 | X i ∼ F i , Z ∼ G , C Xi,Z = C i (0.14) 4 which has an application in the theory of cost-ecient claims with xed payo structure, see Bernard et al. (2014b). ...
... As an interesting modication of this problem, Bernard et al. (2014b) andBernard et al. (2015) consider a payo X * T with xed dependence structure w.r.t. a random benchmark A T , described by a copula C ∈ C 2 , i.e. ...
... The following result describes cost-ecient claims with given payo-structure. Theorem 6.3 (Bernard et al. (2014b)) Let X T be a claim with C X T ,A T ∼ C . Then (i) Y * T := F −1 X T (1 − τ ξ T ) is a cost-ecient claim, i.e. it solves optimization problem (6.11). ...
For the class of partially specified risk factor models, general ordering results in dependence on the partial specifications are established. In particular, the Schur-order on copula derivatives and the sign-change order are introduced to derive lower orthant ordering conditions on the ✳-product of copulas and supermodular ordering conditions on the upper resp. lower product of bivariate copulas which describe the dependence structure of general specified resp. conditionally comonotonic resp. conditionally countermonotonic random vectors. In the case of internal factor models, it is shown that the standard bivariate dependence orders yield a strongly simplified ordering result for the risk bounds.
The characterization of the supermodular order for multivariate normal distributions is extended to the class of elliptically contoured distributions. As a consequence, for elliptical C-vine models, the upper bound in supermodular order is improved compared to the conditionally comonotonic case if the specifying partial correlations are bounded.
Applications to real market data show the considerable improvement of the standard risk bounds and, in the theory of cost-efficient claims, the dependence of the prices of constrained cost-efficient claims on the specifications.
... Recently an interesting modification of the cost efficiency method has been introduced in Bernard et al. (2014b) and Bernard et al. (2015) allowing to include an essential insurance aspect by specifying the states in which payments are obtained. This is done in mathematical terms by specifying the joint dependence of the payoff X T with a benchmark A T which is observable in the market as e.g. an index or a specific option. ...
... As an interesting modification of this problem (Bernard et al. 2014b) and Bernard et al. (2015) consider a payoff X * T with fixed dependence structure with respect to a random benchmark A T , described by a copula C , i.e. (3.27) In this case X * T is called cost-efficient claim with fixed payoff structure. ...
... The following result describes cost-efficient claims with given payoff-structure (see Bernard et al. (2014b)). ...
Motivated by the problem of sharp risk bounds in partially specified risk factor models and by the method of cost-efficient payoffs with given payoff structure we introduce and describe some stochastic odering problems for conditionally comonotonic resp. antimonotonic random variables. The aim is to describe the influence of the specified dependence of the components of the random vector X with a benchmark Z on the risk bounds in a risk portfolio resp. on the gain of cost efficiency of the optimal payoffs. We obtain in particular explicit results in dependence on distributional parameters for elliptical models in the case of risk bounds and for the multivariate Samuelson model in the case of cost efficient payoffs.
... The basic reason for doing so is that under law-invariant preferences optimal payoffs will have their lowest outcomes when the economy is in a downturn (Dybvig (1988a)), and arguably this feature does not fit with the aspirations of many investors. We model state-dependence using dependence constraints, i.e., we prescribe the copula between the payoff and some benchmark asset (Bernard et al. (2014b), Bernard et al. (2015b)). We provide a reduction result (a quantile formulation) that allows dealing with optimal portfolio selection problems for investors with concave preferences and who have an additional state-dependent constraint. ...
... To account for this (undesirable) feature, we may want to provide the investor with the opportunity to maintain a desired dependence with a benchmark asset (state-dependent constraint). This idea was developed in Bernard et al. (2014a), Bernard et al. (2014b) and Bernard et al. (2015b). Specifically, Bernard et al. (2015b) determine the optimal payoff for an expected utility maximizer under such dependence constraint. ...
... If X T is a solution to problem (U C) then denote by U = τ ϕ T |A T the conditional distributional transform and X T = F −1 X T |A T (1−U )). Then by arguments as in the proof of Theorem 3.1 in Bernard et al. (2014b), we obtain that ( X T , A T ) ∼ (X T , A T )), i.e., X T is admissible. Furthermore, since U , X T are independent we obtain that X T , ϕ T are anti-monotonic, conditionally on A T = a. ...
The quantile formulation for optimal portfolio selection problems under increasing law-invariant objectives allows to reduce any such problem to an optimization problem on real functions under monotonicity conditions. We solve two basic types of these optimization problems, which makes it possible to solve in a unified way several portfolio selection problems of interest. In particular, we completely solve the optimal portfolio selection problem for an investor with preferences as in Yaari's dual theory of choice. Extending previous work we also derive a reduction result in general form when the payoff is required to have a fixed copula with some benchmark (state-dependent constraint). Specifically, we show that if one can determine the optimal payoff under a concave law-invariant objective, then one can also determine the optimal payoff when adding the state-dependent constraint. Finally, we identify market conditions which ensure attainability of the optimal payoff by means of a static portfolio of puts and calls.
... The basic reason for doing so is that under law-invariant preferences optimal payoffs will have their lowest outcomes when the economy is in a downturn (Dybvig (1988a)), and arguably this feature does not fit with the aspirations of many investors. We model state-dependence using dependence constraints, i.e., we prescribe the copula between the payoff and some benchmark asset (Bernard et al. (2014b), Bernard et al. (2015b)). We provide a reduction result (a quantile formulation) that allows dealing with optimal portfolio selection problems for investors with concave preferences and who have an additional state-dependent constraint. ...
... To account for this (undesirable) feature, we may want to provide the investor with the opportunity to maintain a desired dependence with a benchmark asset (state-dependent constraint). This idea was developed in Bernard et al. (2014a), Bernard et al. (2014b) and Bernard et al. (2015b). Specifically, Bernard et al. (2015b) determine the optimal payoff for an expected utility maximizer under such dependence constraint. ...
... If X T is a solution to problem (U C) then denote by U = τ ϕ T |A T the conditional distributional transform and X T = F −1 X T |A T (1−U )). Then by arguments as in the proof of Theorem 3.1 in Bernard et al. (2014b), we obtain that ( X T , A T ) ∼ (X T , A T )), i.e., X T is admissible. Furthermore, since U , X T are independent we obtain that X T , ϕ T are anti-monotonic, conditionally on A T = a. ...
We show that any problem of optimal payoff (portfolio) choice under an increasing law-invariant objective function can be reduced to an optimization problem for real functions under monotonicity restrictions. We solve some of these optimization problems and apply the results to solve several portfolio selection problems of interest. In particular, we completely describe the optimal payoff for an investor with preferences as in Yaari's Dual Theory of Choice. We extend the reduction result to the case in which the payoff is required to have a fixed copula with some benchmark (state-dependent constraint). Specifically , we show that if one can determine the optimal payoff under a law-invariant objective, then one can also determine the optimal payoff when adding the state-dependent constraint. We also describe a setting with market specified prices that is rich enough to ensure attainability of the optimal payoff by means of a static portfolio of puts and calls.
... This approach was introduced by Takahashi-Yamamoto [21]. See also Bernard et al. [3] and Bernard et al. [4]. ...
In 1988 Dybvig introduced the payoff distribution pricing model (PDPM) as an alternative to the capital asset pricing model (CAPM). Under this new paradigm agents preferences depend on the probability distribution of the payoff and for the same distribution agents prefer the payoff that requires less investment. In this context he gave the notion of efficient payoff. Both approaches run parallel to the theory of choice of von Neumann and Morgenstern [17], known as the Expected Utility Theory and posterior axiomatic alternatives. In this paper we consider the notion of optimal payoff as that maximizing the terminal position for a chosen preference functional and we investigate the relationship between both concepts, optimal and efficient payoffs, as well as the behavior of the efficient payoffs under different market dynamics. We also show that path-dependent options can be efficient in some simple models.
