Jakša Cvitanić

Utility maximization in incomplete markets with random endowment

Abstract. This paper solves the following problem of mathematical finance: to find a solution to the problem of maximizing utility from terminal wealth of an agent with a random endowment process, in the general, semimartingale model for incomplete markets, and to characterize it via the associated dual problem. We show that this is possible if the dual problem and its domain are carefully defined. More precisely, we show that the optimal terminal wealth is equal to the inverse of marginal utility evaluated at the solution to the dual problem, which is in the form of the regular part of an element of

Option pricing, interest rates and risk management

({\bf L}^\infty)^*

A closed-form solution to the problem of super-replication under transaction costs

(the dual space of

Reflected forward-backward SDEs and obstacle problems with boundary conditions

{\bf L}^\infty

Optimal risk-sharing with effort and project choice

).

Monte Carlo computation of optimal portfolios in complete markets

Introduction Part I. Option Pricing: Theory and Practice: 1. Arbitrage theory Yu. M. Kabanov 2. Market models with frictions: arbitrage and pricing issues E. Jouini and C. Napp 3. American options: symmetry properties J. Detemple 4. Purely discontinuous asset price processes D. Madan 5. Latent variable models for stochastic discount factors R. Garcia and E. Renault 6. Monte Carlo methods for security pricing P. Boyle, M. Broadie and P. Glasserman Part II. Interest Rate Modeling: 7. A geometric view of interest rate theory T. Bjork 8. Towards a central interest rate model A. Brace, T. Dun and G. Barton 9. Infinite dimensional diffusions, Kolmogorov equations and interest rate models B. Goldys and M. Musiela 10. Libor market model with semimartingales F. Jamshidian 11. Modeling of forward Libor and swap rates M. Rutkowski Part III. Risk Management and Hedging: 12. Credit risk modeling, intensity based approach T. Bielecki and M. Rutkowski 13. Towards a theory of volatility trading P. Carr and D. Madan 14. Shortfall risk in long-term hedging with short-term futures contracts P. Glasserman 15. Numerical comparison and local risk-minimisation and mean-variance hedging D. Heath, E. Platen and M. Schweizer 16. A guided tour through quadratic hedging approaches M. Schweizer Part IV. Utility Maximization: 17. Theory of portfolio optimization in markets with frictions J. Cvitanic 18. Bayesian adaptive portfolio optimization I. Karatzas and X. Zhao.

On optimal terminal wealth under transaction costs

Abstract. We study the problem of finding the minimal price needed to dominate European-type contingent claims under proportional transaction costs in a continuous-time diffusion model. The result we prove has already been known in special cases – the minimal super-replicating strategy is the least expensive buy-and-hold strategy. Our contribution consists in showing that this result remains valid for general path-independent claims, and in providing a shorter and more intuitive, financial mathematics-type proof. It is based on a previously known representation of the minimal price as a supremum of the prices in corresponding shadow markets, and on a PDE (viscosity) characterization of that representation.

GENERALIZED NEYMAN-PEARSON LEMMA VIA CONVEX DUALITY ⁄

In this paper we study a class of forward-backward stochastic differential equations with reflecting boundary conditions (FBSDER for short). More precisely, we consider the case in which the forward component of the FBSDER is restricted to a fixed, convex region, and the backward component will stay, at each fixed time, in a convex region that may depend on time and is possibly random. The solvability of such FBSDER is studied in a fairly general way. We also prove that if the coefficients are all deterministic and the backward equation is one-dimensional, then the adapted solution of such FBSDER will give the viscosity solution of a quasilinear variational inequality (obstacle problem) with a Neumann boundary condition. As an application, we study how the solvability of FBSDERs is related to the solvability of an American game option.

Optimal Allocation to Hedge Funds : An Empirical Analysis

We consider first-best risk-sharing problems in which “the agent” can control both the drift (effort choice) and the volatility of the underlying process (project selection). In a model of delegated portfolio management, it is optimal to compensate the manager with an option-type payoff, where the functional form of the option is obtained as a solution to an ordinary differential equation. In the general case, the optimal contract is a fixed point of a functional that connects the agents and the principals maximization problems. We apply martingale/duality methods familiar from optimal consumption-investment problems.

Optimal compensation with adverse selection and dynamic actions

We introduce a method that relies exclusively on Monte Carlo simulation in order to compute numerically optimal portfolio values for utility maximization problems. Our method is quite general and only requires complete markets and knowledge of the dynamics of the security processes. It can be applied regardless of the number of factors and of whether the agent derives utility from intertemporal consumption, terminal wealth or both. We also perform some comparative statics analysis. Our comparative statics show that risk aversion has by far the greatest influence on the value of the optimal portfolio.