Portfolio management and derivative security pricing in markets with stochastic volatility.

摘要

In this thesis we look at two problems of great theoretical and practical interest in modern finance: the optimal investment and consumption problem of an investor under uncertainty and the pricing and hedging of derivative securities. This paper discusses the continuous time optimal portfolio management problem and derivative security pricing in the presence of unhedgeable risks. We relax the assumptions on the dynamics followed by the price of the risky asset. In our formulation, the volatility of the risky asset follows diffusion correlated to the stock price. The economy under consideration has two traded assets: the risky asset or stock and the bond. There are two sources of uncertainty arising from the Brownian motions in the stock and volatility dynamics. Hence our formulation is in an incomplete market setting. As such, many existing models are special cases of our dynamics.; Our setting is interesting from a mathematical and financial perspective. Mathematically, the problem is the formulation of a stochastic optimization problem in a more general setting. In fact, there are two state diffusions, a controlled and an uncontrolled one, and the problems herein are a rare case of fully nonlinear problems for which closed form solutions may be found. Furthermore, the stochastic volatility framework increases the dimensionality of the problem and introduces certain non-linearities to the partial differential equations of the prices. These non-linearities are the offspring of market incompleteness and are further explored and compared against their linear counterparts in complete markets that are compatible with linear pricing rules.; The problem is of interest in finance as it discusses the investor's optimal portfolio construction in a more general and realistic setting. Furthermore, in an incomplete market setting no-arbitrage arguments alone do not provide unique prices for derivatives. Our methodology gives a viable bid-ask spread for prices of European derivatives in a generalized market setting, while being consistent with the no-arbitrage argument in a complete market.