Convex Duality in Mean-Variance Hedging Under Convex Trading Constraints

Czichowsky, Christoph; Schweizer, Martin

doi:10.1239/aap/1354716590

Cited by 12 publications

(11 citation statements)

References 29 publications

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“…Remark 2.17. Let us briefly compare Theorem 2.16 to the main result of Theorem 3.12 in [7]. The latter imposes the extra assumption that E(N) satisfies the reverse Hölder inequality R 2 (P ), and proves that the space G T (Θ(C)) is then closed in L 2 (P ).…”

Section: Formulation Of the Problem And Preliminariesmentioning

confidence: 88%

“…of terminal constrained gains is convex and closed in L 2 (P ). Such constrained meanvariance hedging problems in a general semimartingale framework have been studied in [7]. As explained there, one can formulate constraints on trading strategies and then adapt closedness results from the unconstrained case to obtain closedness under constraints as well.…”

Section: Formulation Of the Problem And Preliminariesmentioning

confidence: 99%

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Cone-Constrained Continuous-Time Markowitz Problems

Czichowsky

Schweizer

2012

SSRN Journal

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The Markowitz problem consists of finding in a financial market a self-financing trading strategy whose final wealth has maximal mean and minimal variance. We study this in continuous time in a general semimartingale model and under cone constraints: Trading strategies must take values in a (possibly random and timedependent) closed cone. We first prove existence of a solution for convex constraints by showing that the space of constrained terminal gains, which is a space of stochastic integrals, is closed in L 2 . Then we use stochastic control methods to describe the local structure of the optimal strategy, as follows. The value process of a naturally associated constrained linear-quadratic optimal control problem is decomposed into a sum with two opportunity processes L ± appearing as coefficients. The martingale optimality principle translates into a drift condition for the semimartingale characteristics of L ± or equivalently into a coupled system of backward stochastic differential equations for L ± . We show how this can be used to both characterise and construct optimal strategies. Our results explain and generalise all the results available in the literature so far. Moreover, we even obtain new sharp results in the unconstrained case.

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Section: Formulation Of the Problem And Preliminariesmentioning

confidence: 88%

Section: Formulation Of the Problem And Preliminariesmentioning

confidence: 99%

Cone-Constrained Continuous-Time Markowitz Problems

Czichowsky

Schweizer

2012

SSRN Journal

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“…Constrained dynamic mean‐variance portfolio selection problems with various constraints have been attracting increasing attention in the last decade, e.g., Li, Zhou, and Lim (), Zhu, Li, and Wang (), Bielecki et al. (), Sun and Wang (), Labbé and Heunis (), and Czichowsky and Schweizer (). Recently, Czichowsky and Schweizer () further considered cone‐constrained continuous‐time mean‐variance portfolio selection with semimartingale price processes.…”

Section: Optimal Mean‐variance Policy In a Discrete‐time Cone‐constramentioning

confidence: 99%

Mean‐variance Policy for Discrete‐time Cone‐constrained Markets: Time Consistency in Efficiency and the Minimum‐variance Signed Supermartingale Measure

Cui

2015

Mathematical Finance

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The discrete‐time mean‐variance portfolio selection formulation, which is a representative of general dynamic mean‐risk portfolio selection problems, typically does not satisfy time consistency in efficiency (TCIE), i.e., a truncated precommitted efficient policy may become inefficient for the corresponding truncated problem. In this paper, we analytically investigate the effect of portfolio constraints on the TCIE of convex cone‐constrained markets. More specifically, we derive semi‐analytical expressions for the precommitted efficient mean‐variance policy and the minimum‐variance signed supermartingale measure (VSSM) and examine their relationship. Our analysis shows that the precommitted discrete‐time efficient mean‐variance policy satisfies TCIE if and only if the conditional expectation of the density of the VSSM (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 finding indicates that the TCIE property depends only on the basic market setting, including portfolio constraints. This motivates us to establish a general procedure for constructing TCIE dynamic portfolio selection problems by introducing suitable portfolio constraints.

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“…We paraphrase Czichowsky and Schweizer () who describe a semimartingale financial market with general convex constraints on trading strategies. Under their construction, we define

S = {(S_{t})}_{0 \leq t \leq T}

to be an

ℜ^{n}

‐valued semimartingale model of the discounted prices of n ‐risky assets.…”

Section: Multiple Objective Volatility Hedging Of MV Portfoliosmentioning

confidence: 99%

“…The space

Θ_{S} (M)

of permissible integrands imposes a square‐integrability condition on the stochastic integral process

\int ϑ d S

, and the argument M in brackets indicates the existence of trading constraints in the sense that

ϑ_{t} (ω)

must lie in a convex‐closed subset

M (ω, t)

ℜ^{n}

. Czichowsky and Schweizer () contribute the fact that this setup is the most general formulation for MV hedging under constraints (for evidence of a congruent NHF trading strategy, see Kajiji and Forman, ).…”

Section: Multiple Objective Volatility Hedging Of MV Portfoliosmentioning

confidence: 99%

On multiobjective combinatorial optimization and dynamic interim hedging of efficient portfolios

Dash

Kajiji

2014

Int Trans Operational Res

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A dynamic portfolio policy is one that periodically rebalances an optimally diversified portfolio to account for time-varying correlations. In order to sustain target-level Sharpe performance ratios between rebalancing points, the efficient portfolio must be hedged with an optimal number of contingent claim contracts. This research presents a mixed-integer nonlinear goal program (MINLGP) that is directed to solve the hierarchical multiple goal portfolio optimization model when the decision maker is faced with a binary hedging decision between portfolio rebalance periods. The MINLGP applied to this problem is formed by extending the separable programming foundation of a lexicographic nonlinear goal program (NLGP) to include branchand-bound constraints. We establish the economic efficiency of applying this normative approach to dynamic portfolio rebalancing by comparing the risk-adjusted performance measures of a hedged optimal portfolio to those of a naively diversified portfolio. We find that a hedged equally weighted small portfolio and a hedged efficiently diversified small portfolio perform similarly when comparing risk-adjusted return metrics. However, when percentile risk measures are used to measure performance, the hedged optimally diversified portfolio clearly produces less expected catastrophic loss than does its nonhedged and naively diversified counterpart.

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Convex Duality in Mean-Variance Hedging Under Convex Trading Constraints

Cited by 12 publications

References 29 publications

Cone-Constrained Continuous-Time Markowitz Problems

Cone-Constrained Continuous-Time Markowitz Problems

Mean‐variance Policy for Discrete‐time Cone‐constrained Markets: Time Consistency in Efficiency and the Minimum‐variance Signed Supermartingale Measure

On multiobjective combinatorial optimization and dynamic interim hedging of efficient portfolios

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