Portfolio choice under convex transaction costs

We consider a portfolio optimization problem for an investor who faces convex transaction costs on trading a stock. The linear component in these costs defines a no–trading zone, while the nonlinear component does not allow trading the stock at an arbitrary rate. The latter component results from a stock illiquidity for a market maker and causes a stock illiquidity for a price–taking investor. We consider the effects that stock illiquidity has on an investor’s optimal trading and his welfare when the stock path is continuous, when the stock may crash, and when there is a second stock that is perfectly liquid. We find that even though the illiquid stock allocation could be of any proportion in the portfolio, an investor has a highest expected utility when this allocation is the same as that in the liquid market. Thus, no flight to liquidity in the presence of liquid securities. This conclusion remains unchanged for any degree of convexity of a nonlinear term in the transaction costs. Moreover, this degree also does not affect the shape of the no–trading zone. We find that portfolio volatility can often be too high, making a risk–averse investor willing to hold a liquid stock with negativ