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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:
Mutual funds are often restricted to allocate certain percentages of fund assets to certain securities that have different degrees of illiquidity. However, the existing literature on how mutual funds should trade has largely ignored the coexistence of position limits and differential illiquidity and thus the optimal trading strategy for a typical mutual fund is largely unknown. In this paper, we use a novel approach to study the optimal trading strategy of mutual funds who face both position limits and differential illiquidity. We show the existence, uniqueness, smoothness, and characterization of the optimal trading strategy. We provide extensive analytical comparative statics for the optimal trading strategy and an efficient numerical method for solving a class of similar problems. We find that the presence of position limits can significantly increase liquidity premium and surprisingly, liquidity premium can be higher when the limits are less stringent. We find that myopically choosing the "optimal'' strategy is costly. We also examine the optimal choice of position limits and empirical implications on performance evaluation.

Illiquidity, Portfolio Constraints, Liquidity Premium, Transaction Costs

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