Three essays on valuation and investment in incomplete markets
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Incomplete markets provide many challenges for both investment decisions and valuation problems. While both problems have received extensive attention in complete markets, there remain many open areas in the theory of incomplete markets. We present the results in three parts. In the first essay we consider the Merton investment problem of optimal portfolio choice when the traded instruments are the set of zero-coupon bonds. Working within a Markovian Heath-Jarrow-Morton framework of the interest rate term structure driven by an infinite dimensional Wiener process, we give sufficient conditions for the existence and uniqueness of an optimal investment strategy. When there is uniqueness, we provide a characterization of the optimal portfolio. Furthermore, we show that a specific Gauss-Markov random field model can be treated within this framework, and explicitly calculate the optimal portfolio. We show that the optimal portfolio in this case can be identified with the discontinuities of a certain function of the market parameters. In the second essay we price a claim, using the indifference valuation methodology, in the model presented in the first section. We appeal to the indifference pricing framework instead of the classic Black-Scholes method due to the natural incompleteness in such a market model. Because we price time-sensitive interest rate claims, the units in which we price are very important. This will require us to take care in formulating the investor’s utility function in terms of the units in which we express the wealth function. This leads to new results, namely a general change-of-numeraire theorem in incomplete markets via indifference pricing. Lastly, in the third essay, we propose a method to price credit derivatives, namely collateralized debt obligations (CDOs) using indifference. We develop a numerical algorithm for pricing such CDOs. The high illiquidity of the CDO market coupled with the allowance of default in the underlying traded assets creates a very incomplete market. We explain the market-observed prices of such credit derivatives via the risk aversion of investors. In addition to a general algorithm, several approximation schemes are proposed.