Based on the Merton's problem and the concept of indifference pricing methodology, the author develops a pair of uncoupled partial differential equations to find the fair price of a single illiquid financial security. The equations are developed by two different methods, and the results are consistent. Furthermore, a vector indifference pricing framework is conjectured for multiple securities valuation. The pricing method is applied on financial contract that could not be traded during valuation period and the liquidity premium is revealed. Especially, this method could be applied on private equity valuation problem.
Another pricing equation using the concept of consistent pricing is also developed in this paper. Moreover by applying variable transformation technique on the Basic Equity Model, an integral from solution is found on the public equity valuation problem under interest rate risk.