Stochastic Portfolio Optimization Theory in Mathematical Finance

To our best knowledge, there is at the moment no compatible book in the Springer Finance Series which gives to students, academics and practitioners working on technical questions in the analysis of the financial markets a standard machinery of Stochastic Portfolio Management. Beside the rigorous presentation of the theoretical models, we pay attention to rich collection of examples which might be of particular interest for practical implementations and applications of those models, and give extensive discussion of drawbacks of some applications in practical analysis of the financial markets in the real world. We need to emphasis that this manuscript has to be seen as a supplementary textbook to the standard textbooks in the Springer Finance Text- book Series as Karatzas, I. and Shreve, S. E. (1991) Brownian Motion and Stochastic Calculus Springer-Verlag, NY and Karatzas, I. and Shreve, S. E. (1998) Methods of Mathematical Finance Springer-Verlag, NY.In this manuscript we begin with some basic introductory definitions of the wealth process, give notion of the hedging problem. In Chapter 1 we begin with introduction to the duality in formulation of the set of super-hedging strategies and derive the martingale representation of the problem. Here, we cover the cases when the markets are complete, which corresponds to the standard class-room example. In the next section we present an extension of this model to the incomplete markets where we state the set of conditions for portfolio constraints. Here, we give the PDE formulation of the problem, give the Feynman-Kac representation in the smooth and in the viscosity sense. Particularly, in the second part of this chapter we pay main attention to the introduction to students a standard machinery of the viscosity solution, i.e., give the notion of the problem, discuss main properties, recall the existence and uniqueness results. The next section of this chapter deals with pricing equation under portfolio constraints in the incomplete case. Here, we introduce to the students the main result for the derivation of a PDE corresponding to the problem using the dynamic programming principle. Here, we compare different approaches of solving problems, and point out some drawback in their practical implementations. Then, we give the rigorous discussion of the Hamilton-Jacobi-Bellman pricing problem. We begin with statement of the dynamic programming principle from the chapter before to find the discontinuous viscosity solution of the problem and to state a suitable boundary conditions; we give the notion of super- and sub-solutions to the problem, we prove the comparison principle for the viscosity solution of the problem. At the end of this section, we present a rich collection of the examples including the case of non-hedgeable stochastic volatility models. We close this chap- ter with discussion of the approximate hedging technics, which include the quantile hedging strategy, hedging under expected loss constraints. The next Chapter 2 is devoted to presentation of the stochastic hedging approach. Here, we begin with presentation of the machinery to define the model in which the trading strategies may have an impact on the wealth process. Here, we introduce to students geometric dynamic programming principle and work with them together through this chapter on solving the American type options problem. Here, we give rigorously introduction to machinery of solving this set of problem, we begin with formulation of the corresponding PDE, state viscosity super- and sub-solution property, work on the proper statement of the boundary condition. Some generalizations of those results and rich collection of examples including the Black-Scholes model, close this Chapter.Chapter 3 is devoted to a rich collection of Exercises and Practical Problems with the hint of proofs. In Appendix we review some standard machinery for solving the viscosity solutions.