In this project, the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets and the sensitivity analysis of utility-based prices for the case of utility functions defined on the whole real line is considered. The research focuses on necessary and sufficient conditions under which the mathematical statements hold. In particular, we are looking for the largest class of contingent claims that can be analyzed in this framework. The risk-tolerance wealth process turns out to be the appropriate numeraire for pricing. In addition, a class of multi-dimensional stochastic games is studied.
In complete markets, described by models like the Black and Scholes model, prices are uniquely defined by "no-arbitrage" considerations. In more realistic models, which are incomplete, prices of contingent claims can only be defined taking into account the preferences and wealth of specific investors. The dependence of prices on the number of contingent claims for investors who can borrow any amount of cash from the money market (infinite credit line) is studied. The pricing techniques can be applied to a large number of incomplete models, including options on non-traded assets and energy derivatives.
