Optimal strategies in incomplete financial markets
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This thesis analyzes the optimal strategies of rational agents in incomplete financial markets. The incompleteness may arise from the stochastic volatility of stock prices, in which case we study the optimal pricing and hedging strategies of an option trader. We introduce a new concept that we call the relative indifference price, which is the price at which a trader is indifferent to trade in an additional option, given that he is currently holding and dynamically hedging a portfolio of options. We find that the appropriate volatility risk premium depends on the trader's risk aversion coeffcient and his portfolio position before selling or buying the additional option. More generally, the incompleteness of the market may arise from both the drift and volatility of the stock being driven by a correlated factor. In this setting, we study the optimal consumption and investment policies of CARA, conservative CRRA and aggressive CRRA agents. In particular, we provide interpretations of the non-myopic investment in terms of martingale measures and the risk monitoring strategy of a path-dependent option.