PORTFOLIO OPTIMIZATION UNDER A QUANTILE HEDGING CONSTRAINT

Géraldine Bouveret ()
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Géraldine Bouveret: Smith School, University of Oxford, South Parks Road, Oxford, OX1 3QY, UK

International Journal of Theoretical and Applied Finance (IJTAF), 2018, vol. 21, issue 07, 1-36

Abstract: We study a problem of portfolio optimization under a European quantile hedging constraint. More precisely, we consider a class of Markovian optimal stochastic control problems in which two controlled processes must meet a probabilistic shortfall constraint at some terminal date. We denote by V the corresponding value function. Following the arguments introduced in the literature on stochastic target problems, we convert this problem into a state constraint one in which the constraint is defined by means of an auxiliary value function v characterizing the reachable set. This set is therefore not given a priori but is naturally integrated in v solving, in a viscosity sense, a nonlinear parabolic partial differential equation (PDE). Relying on the existing literature, we derive, in the interior of the domain, a Hamilton–Jacobi–Bellman characterization of V. However, v involves an additional controlled state variable coming from the diffusion of the probability of reaching the target and belonging to the compact set [0, 1]. This leads to nontrivial boundaries for V that must be discussed. Our main result is thus the characterization of V at those boundaries. We also provide examples for which comparison results exist for the PDE solved by V on the interior of the domain.

Keywords: Quantile hedging constraints; optimal control; viscosity solutions (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (2)

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DOI: 10.1142/S0219024918500486

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