Journal: FINRISK Working Paper Series

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FINRISK

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20102
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Publications 1 - 2 of 2
  • Convex Duality in Mean-Variance Hedging Under Convex Trading Constraints
    Item type: Working Paper
    Czichowsky, Christoph; Schweizer, Martin (2010)
    FINRISK Working Paper Series
    We study mean-variance hedging under portfolio constraints in a general semi-martingale model. The constraints are formulated via predictable correspondences, meaning that the trading strategy is restricted to lie in a closed convex set which may depend on the state and time in a predictable way. To obtain the existence of a solution, we first establish the closedness in L2 of the space of all gains from trade (i.e., the terminal values of stochastic integrals with respect to the price process of the underlying assets). This is a first main contribution which enables us to tackle the problem in a systematic and unified way. In addition, using the closedness allows us to explain and generalise in a systematic way the convex duality results obtained previously by other authors via ad hoc methods in specific frameworks.
  • Time-Consistent Mean- Variance Portfolio Selection in Discrete and Continuous Time
    Item type: Working Paper
    Czichowsky,, Christoph (2010)
    FINRISK Working Paper Series
    It is well known that mean-variance portfolio selection is a time-inconsistent optimal control problem in the sense that it does not satisfy Bellmans optimality principle and therefore the usual dynamic programming approach fails. We develop a time-consistent formulation of this problem, which is based on a local notion of optimality called local mean-variance efficiency, in a general semimartingale setting. We start in discrete time, where the formulation is straightforward, and then find the natural extension to continuous time. This generalises recent results by Basak and Chabakauri (2010), Björk and Murgoci (2008) and Björk, Murgoci and Zhou (2010), where the treatment as well as the notion of optimality rely on an underlying Markovian framework. We justify the continuous-time formulation by showing that it coincides with the continuous-time limit of the discrete-time formulation. The proof of this convergence is based on a global description of the locally optimal strategy in terms of the structure condition and the Föllmer–Schweizer decomposition of the mean-variance tradeoff process. As a byproduct, this also gives new convergence results for the Föllmer–Schweizer decomposition, i.e. for locally risk minimising strategies.
Publications 1 - 2 of 2