  A speculative agent with Prospect Theory preference chooses the optimal time
to purchase and then to sell an indivisible risky asset to maximize the
expected utility of the round-trip profit net of transaction costs. The
optimization problem is formulated as a sequential optimal stopping problem and
we provide a complete characterization of the solution. Depending on the
preference and market parameters, the optimal strategy can be "buy and hold",
"buy low sell high", "buy high sell higher" or "no trading". Behavioral
preference and market friction interact in a subtle way which yields surprising
implications on the agent's trading patterns. For example, increasing the
market entry fee does not necessarily curb speculative trading, but instead it
may induce a higher reference point under which the agent becomes more
risk-seeking and in turn is more likely to trade.
