This thesis aims at investigating hedging and portfolio optimisation problems under weak stochastic target constraints. Our first contribution consists in the representation of the hedging price of some contingent claims under both probabilistic and expected shortfall ("weak") constraints holding on a set of dates. We consider a Markovian and complete market framework and favour a dual approach. This work is an extension to Föllmer and Leukert (1999,2000). We then extend the previous results to the case where the wealth process diffusion is semi-linear in the control/strategy variable. The previous convex duality machinery does not apply anymore and we rely on PDE arguments. Bouchard, Elie and Touzi (2009) already proved the PDE characterisation of such price functions but a comparison result, necessary to build a convergent numerical scheme, is still missing in the literature. We will prove that such a result actually holds. The main difficulty arises from the discontinuity of the operators involved in the PDE characterisation of the price function. An application to the quantile hedging of Bermudan options is provided. Our third contribution relies on the PDE characterisation of the problem of portfolio optimisation under a European quantile hedging constraint. We extend the results of Bouchard, Elie and Imbert (2010) to the case where the constraint holds in a weaker sense. The study is based on a reformulation of the initial constraint into an obstacle and almost-sure stochastic target one. This reduction is done by the introduction of an additional controlled state variable coming from the diffusion of the probability of reaching the target (see Bouchard, Elie and Touzi (2009)) and by means of the Geometric Dynamic Programming principle of Soner and Touzi (2002). However this additional controlled state variable raises non-trivial boundary conditions that have to be characterised. We also have to handle the discontinuity of the operators involved in the characterisation.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:689135 |
| Date | January 2016 |
| Creators | Bouveret, Géraldine |
| Contributors | Chassagneux, Jean-Francois ; Brigo, Damiano |
| Publisher | Imperial College London |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/10044/1/33726 |
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