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Abstract:
In trading stocks investors naturally aspire to "buy low and sell high (BLSH)". This paper formalizes the notion of BLSH by formulating stock buying/selling in terms of four optimal stopping problems involving the global maximum and minimum of the stock prices over a given investment horizon. Assuming that the stock price process follows a geometric Brownian motion, all the four problems are solved and buying/selling strategies completely characterized via a free-boundary PDE approach.

Black -- Scholes market, optimal stopping, stock goodness index, value function, free-boundary PDE (variational inequality)

Abstract:
This paper formulates a single-period portfolio choice model under Kahneman and Tversky’s cumulative prospect theory, featuring a reference point in wealth, S-shaped utility (value) functions with loss aversion, and distortions in probability. An analytical treatment of the model is carried out. A new measure of loss aversion for large payoffs, called the loss aversion degree (LAD), is introduced. The issue of model well-posedness is brought about, of which the LAD is shown to be a critical determinant. The sensitivity of the prospective value function with respect to the stock allocation is then investigated, which demonstrates as a by-product that this function is neither concave nor convex. Optimal solutions are finally derived explicitly for the cases when the reference point is the risk-free return and when the utility function is piece-wise linear. These results are in turn employed to demonstrate that optimal risky exposures monotonically decrease as the LAD increases.

Portfolio choice, single period, cumulative prospect theory, reference point, loss aversion, S-shaped utility function, probability distortion, well-posedness

Abstract:
Partly motivated by a deeper understanding of the role human greed has played in the current financial crisis, this paper quantifies the notion of greed, and explores its connection with leverage and potential losses, in the context of a continuous-time behavioral portfolio choice model under (cumulative) prospect theory. We argue that the reference point is the critical exogenous parameter in defining greed. An asymptotic analysis on optimal trading behaviors when the pricing kernel is lognormal and the $S$-shaped utility is a two-piece CRRA shows that both the level of leverage and the magnitude of potential losses will grow unbounded if the greed grows uncontrolled. However, the probability of ending with gains does not diminish to zero even as the greed approaches infinity. This explains why a sufficiently greedy behavioral agent, despite the risk of catastrophic losses, is still willing to gamble on potential gains because they have a positive probability of occurrence whereas the corresponding rewards are huge. As a result an effective way to contain human greed, from a regulatory point of view, is to impose a priori bounds on leverage and/or potential losses.

Prospect theory, greed, leverage, gains and losses, reference point, portfolio choice

Abstract:
We fill a gap in the proof of a (rather critical) lemma, Lemma B.1, in Jin and Zhou (Mathematical Finance, Vol. 18 (2008), pp. 385–426). We also correct a couple of other minor errors in the same paper.

portfolio selection, continuous time, cumulative prospect theory, behavioral criterion, S-shaped function, probability distortion