Three essays on valuation and investment in incomplete markets

Incomplete markets provide many challenges for both investment decisions and valuationproblems. While both problems have received extensive attention in complete markets,there remain many open areas in the theory of incomplete markets. We present the resultsin three parts. In the first essay we consider the Merton investment problem of optimal portfoliochoice when the traded instruments are the set of zero-coupon bonds. Working withina Markovian Heath-Jarrow-Morton framework of the interest rate term structure driven byan infinite dimensional Wiener process, we give sufficient conditions for the existence anduniqueness of an optimal investment strategy. When there is uniqueness, we provide a characterizationof the optimal portfolio. Furthermore, we show that a specific Gauss-Markov random field model can be treated within this framework, and explicitly calculate the optimalportfolio. We show that the optimal portfolio in this case can be identified with the discontinuitiesof 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. Weappeal to the indifference pricing framework instead of the classic Black-Scholes method dueto the natural incompleteness in such a market model. Because we price time-sensitive interestrate claims, the units in which we price are very important. This will require us to takecare in formulating the investor’s utility function in terms of the units in which we express thewealth function. This leads to new results, namely a general change-of-numeraire theorem inincomplete markets via indifference pricing. Lastly, in the third essay, we propose a methodto price credit derivatives, namely collateralized debt obligations (CDOs) using indifference.We develop a numerical algorithm for pricing such CDOs. The high illiquidity of the CDOmarket coupled with the allowance of default in the underlying traded assets creates a veryincomplete market. We explain the market-observed prices of such credit derivatives via therisk aversion of investors. In addition to a general algorithm, several approximation schemesare proposed.