In a complete financial market every contingent claim can be perfectly replicated by a controlled portfolio of the traded securities and therefore admits a well-defined arbitrage free price. In an incomplete market, as a rule, contingent claims are not replicable. In this case, arbitrage arguments alone are not sufficient to determine unique prices and, hence, more general equilibrium based (or utility based) approach has to be used. This project will study the following related topics of Mathematical Finance: The existence of second derivatives of the value function in the problem of optimal investment and the differentiability of the optimal investment strategy with respect to initial wealth; sensitivity analysis of utility based prices with respect to the number of non-traded contingent claims; the asymptotic analysis of the problems of optimal investment under "small" transaction costs; the "equilibrium" derivation of the interaction between a "large" economic agent and a financial market and the impact of this interaction on the prices of derivative securities. A particular attention is on the derivation of "tight" (ideally, necessary and sufficient) mathematical conditions for the results to hold true.
An important development in world financial markets over the last 25 years is the rapid broadening and expansion of derivatives markets. This became possible primarily because of the creation of the arbitrage-free pricing theory initiated by Black, Scholes and Merton. Several recent events (for example, the near-collapse of Long Term Capital Management) show, however, that the results of the arbitrage-free pricing theory should be used in practice rather cautiously as this theory does not take into account many important features of financial markets such as incompleteness, liquidity constraints, transaction costs and so on. In this project we plan to develop techniques that allow for the computation of the corrections to the arbitrage-free prices of derivatives due to different types of market imperfections.