Non-concave and behavioural optimal portfolio choice problems
Meireles Rodrigues, Andrea Sofia
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Our aim is to examine the problem of optimal asset allocation for investors exhibiting a behaviour in the face of uncertainty which is not consistent with the usual axioms of Expected Utility Theory. This thesis is divided into two main parts. In the first one, comprising Chapter II, we consider an arbitrage-free discrete-time financial model and an investor whose risk preferences are represented by a possibly nonconcave utility function (defined on the non-negative half-line only). Under straightforward conditions, we establish the existence of an optimal portfolio. As for Chapter III, it consists of the study of the optimal investment problem within a continuous-time and (essentially) complete market framework, where asset prices are modelled by semi-martingales. We deal with an investor who behaves in accordance with Kahneman and Tversky's Cumulative Prospect Theory, and we begin by analysing the well-posedness of the optimisation problem. In the case where the investor's utility function is not bounded above, we derive necessary conditions for well-posedness, which are related only to the behaviour of the distortion functions near the origin and to that of the utility function as wealth becomes arbitrarily large (both positive and negative). Next, we focus on an investor whose utility is bounded above. The problem's wellposedness is trivial, and a necessary condition for the existence of an optimal trading strategy is obtained. This condition requires that the investor's probability distortion function on losses does not tend to zero faster than a given rate, which is determined by the utility function. Provided that certain additional assumptions are satisfied, we show that this condition is indeed the borderline for attainability, in the sense that, for slower convergence of the distortion function, there does exist an optimal portfolio. Finally, we turn to the case of an investor with a piecewise power-like utility function and with power-like distortion functions. Easily verifiable necessary conditions for wellposedness are found to be sufficient as well, and the existence of an optimal strategy is demonstrated.