Applications Of Martingale And Stochastic Control Theory In Portfolio Selection And Option Pricing

Portfolio selection and option pricing are the main research topics in mod-ern mathematical finance. The classical portfolio selection theories are generally established in the framework of Markowitz’s mean-variance or von Neumann-Morgenstern’s expected utility where the rational optimal investment behaviors for risk averse investors are investigated. Then, the classical option pricing the-ory is mainly focused on obtaining reasonable option price by applying the no-arbitrage principle.This dissertation is devoted to the further development of them from different aspects. On the one hand, the optimal investment for "irrational" investors is investigated for non-expected utility via the martingale approaches. On the other hand, by employing the stochastic dynamic programming techniques, we study the pricing of options written on non-traded assets and dynamic trading strategies for stocks and options.Firstly, we consider the optimal portfolio selection models for loss averse investors. In section2.1, we study a general dynamic portfolio selection model for loss-averse investors with wealth constraints in a complete market. By applying the martingale methods, we derive the optimal terminal wealth of the investors with or without the benchmark floor constraints. Simultaneously, the existence and uniqueness of the corresponding Lagrange multiplier are proved. We analyze the properties of these solutions, and compare them to that of the general risk averse investors. In section2.2and2.3, the similar problems are analyzed subtly in the incomplete market and jump-diffusion market, respectively.Secondly, we investigate the optimal portfolio selection model with VaR con-straint in the framework of RDEU. Via the quantile formulation approach, the optimal terminal wealth is obtained. An example is presented to obtain the opti- mal wealth processes and the investment strategies.Lastly, we construct a model for pricing of options written on non-traded assets and trading strategies. Suppose the stock and option can be continuously traded without friction. Under the exponential utility maximization criterion, a relation between the optimal positions for the stock and option is derived by using the stochastic dynamic programming techniques. The dynamic option pricing-equations are also established. In particular, the properties of the associated solutions are discussed and their explicit representations are demonstrated using the Feynman-Kac formula. We further compares the equilibrium price to the existing price notions, such as the marginal price and indifference price.