| In this paper we address problems of partial-hedging and utility maximization, in a general continuous-time multi-currency market with proportional transaction costs. After setting up the market model and describing its basic properties in Chapter 2, we study in Chapter 3 two partial-hedging problems for a cash-settled contingent claim: the problem of minimizing expected shortfall, and the problem of maximizing probability of perfect hedge. With the help of tools from non-smooth convex analysis, we establish the existence of optimal trading strategies and describe them in terms of appropriate dual optimization problems.; In Chapter 4, we consider a utility maximization problem under the same market model. Unlike the existing literature in mathematical finance, our utility function is the so-called “direct utility” depending on terminal consumption, rather than the “indirect utility” that depends on terminal wealth. We analyze the consumer's choice at the terminal time and compare our direct utility approach with the traditional indirect utility approach. We show that in a market with transaction costs, direct utility is much easier to handle than indirect utility, and that the standard tools from convex analysis and the smooth calculus of variations still suffice to establish the existence result. |
