Optimal investment and price dependence in a semi-static market
The author considers the problem of maximizing expected utility from terminal wealth in a semi-static market framework. A specific feature is that a general utility function defined on the positive half-line is used. It is explained that the problem under consideration does not fall under the general umbrella of utility maximization with convex constraints, because it can not be rephrased asking that the portfolio and wealth process lie in some given convex set; rather, it is demanded that the investor chooses his position at time zero arbitrarily and then keeps his position in derivatives unchanged, freely investing in stocks. The existence and uniqueness of the solution is studied as well as the dependence of the value function, of its maximizer and of other quantities of interest on the initial capital and on the price \(p\) of the derivatives. Stability, differentiability, monotonicity and convexity are established. Furthermore, the author proves the convexity in \(p\) of the largest feasible position, defined as the maximum number of shares of derivatives that the agent with given initial wealth can buy at price \(p\) and still be able to invest in the liquid stock market so as to have a nonnegative final wealth.
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