Incomplete markets, transaction costs and liquidity effects

A angent's optimization problem of the expected terminal wealth utility in a trinomial tree economy is solved. At each transaction date, the agent can trade in a riskless asset, a primitive asset subject to constant proportional transaction costs, and a contingent claim characterized by some parameters k whose bid and ask price is defined by allowing for different equivalent martingale measures. In addition to the classical portofolio choice problem, the characteristic of the contingent claim k is determined endgenously in the optimization problem. Under suitable conditions, it is proved that the optimal demand of the agent in the primitive risky asset is zero independantly of the terminal wealth utility function : the agent prefers not to trade in the asset subject to transaction costs, which prevents the market from being complete, rather than trading in both assets. Next the optimal choice of the contingent claim is characterized and te results are applied to European call and put options with fixed maturity and varying exercise price k.