| dc.description.abstractEN | This thesis consists of three chapters on continuous-time finance, all centred on the problem of optimal investment. A common theme throughout is that the classical benchmark of a frictionless and complete market is often too restrictive for economically relevant applications. The first chapter studies equilibrium returns when investors act strategically and face trading frictions, whereas the latter two chapters analyse optimal investment and utility-based valuation in incomplete semimartingale markets with non-replicable streams. In particular, Chapter II studies an Itô financial market in which the returns of risky assets are determined endogenously through a market-clearing condition amongst heterogeneous risk-averse investors with quadratic preferences and random endowments. Investors act strategically by taking into account the effect of their orders on the assets’ drift. A frictionless market and one with (exogenous) quadratic transaction costs are analysed and compared. In the former, we derive the unique Nash equilibrium at which investors’ demand processes reveal hedging needs that differ from their true ones, resulting in a deviation of the Nash equilibrium from its competitive counterpart. In the presence of price impact and transaction costs, we characterise the Nash equilibrium via the unique solution of a system of FBSDEs and obtain a closed-form expression for the corresponding equilibrium returns. We show that, under common risk aversion and in the absence of noise traders, transaction costs and price impact do not affect equilibrium returns. When noise traders are present, comparing each market structure with its own frictionless benchmark shows that the transaction cost
component of the Nash equilibrium coincides with the competitive one for two investors, but is dampened when more than two investors participate. Comparing instead the full frictional competitive and Nash equilibria, the return gap is shaped jointly by the level of noise trader demand and the trading rate pressure induced by transaction costs. Chapter III considers an incomplete financial market with general continuous semimartingale dynamics in which a log-utility investor receives, in addition to an initial capital, units of a non-replicable endowment process. Using duality methods, we derive a fourth-order expansion of the primal value function with respect to the number ϵ of endowment units held by the investor. This expansion, in turn, provides the basis for a second-order approximation of the corresponding optimal wealth process. The key quantities governing these asymptotics are expressed through Kunita–Watanabe projections, in close analogy with the structure that appears in lower-order results of the same kind. The treatment covers finite and infinite horizons within a unified framework. Chapter IV turns to the valuation problem associated with a small non-replicable liability streaminthesamegeneralsemimartingalesetting—underlog-utilitypreferences. Thereinthe notion of utility-based certainty equivalent is explored, namely the cash amount that offsets the investor’s utility loss from bearing such a position. Since explicit valuation formulas are typically unavailable in incomplete markets, the analysis develops a fourth-order asymptotic expansion of this quantity with respect to the size of said stream. The resulting formula makes precise how market incompleteness affects prices beyond the leading terms and shows that higher-order effects can become economically relevant even for moderate positions. The chapter also treats the infinite-horizon case. Beyond its independent theoretical interest, this offers a convenient approximation mechanism for investors with long maturities. In that regime, the infinite-horizon formulation yields a simpler and more tractable benchmark for utility-based pricing, while still capturing the main economic effects of long-dated non-replicable cash flows. | el |
