Research Of Some Problems On Real Option And Partial Information

In this dissertation, we consider several problems origining from real options andpartial information under incomplete markets. Based on statistical filtering, stochasticcontrol, equilibrium pricing and consumption utility indifference pricing approaches,we set up formal models of mathematical finance to quantitatively explore the pricing,timing, and financing of real option, and give the sensitivity analysis of asset pricingand risk hedging to the factors of partial information and idiosyncratic risk.The studies of this dissertation include the following aspects:(1) We extend the real options theory. Supposing the mean appreciation rateof cash flows generated by an irreversible investment project is an unobservablenormal random variable, we study the problems of real option where the goal of aninvestor is to maximize expected discounted utility of infinite lifetime consumption.Respectively for a self-insurance model and a hedge model, we get the differentKalman filtering estimations of the mean appreciation rate. Based on ConstantAbsolute Risk Aversion (CARA) utility, we derive the certainty equivalent value ofcash flows after investment, and then obtain a semi-closed-form solution of afree-boundary Partial Differential Equation (PDE) system, which are satisfied by theimplied value and the optimal investment threshold of the option to invest. We showthat the solutions are independent of the utility time-discount rate. We providenumerical results by finite difference method, compare the results with those under afully observable case and propose a method to measure the value of information. Thetheory analysis and numerical calculations show that partial information leads to laterinvestment and a significant loss of the implied value of the real option. The lossreaches the maximum value at the investment threshold and it increases with theinvestor’s risk aversion, estimation risk and project’s volatility, any larger value ofwhich leads to a smller option value and a bigger investment threshold. In addition,the results show that the consumption utility indifference pricing approach takes intoaccount non-systematic risk premium, as well as the systematic risk premium. But, therisk attitude of the investor has impacts on the strategies of investment, consumptionand asset pricing only if the investor is exposed to non-systematic risk.(2) We study the impact of random conversion of macroeconomic conditions onthe problems of irreversible investment. Suppose the stochastic cash flows after investment not only depend on the enterprise’s income risk, but also depend on thecurrent macroeconomic conditions (boom or recession). The macroeconomicconditions obey a continuous-time Markov chain with two states, and influence thecash flows multiply. Through stochastic control theory and modified smooth fittechnology, we obtain the closed form solutions for the implied value and the optimalinvestment threshold of the real option according to the macroeconomic state. Inaddition, the numerical results present the effects of the random conversion ofmacroeconomic conditions and the discount rate, and the parameters such as meanappreciation rate and volatility on the implied option value and the optimal stopping.(3) We study the pricing and timing of real option based on the criteria ofmaximizing the probability of survival. The firm with uninsured risk of cash flowsafter investment makes decisions on the exercising of the option so as to maximize thesurvival probability. We derive closed-form solutions for the firm’s survivalprobability, investment threshold and the corresponding implied value of real optionby solving the HJB equation based on the theory of stochastic control. The numericalresults present the effects of wealth and volatility on the survival probability and realinvestment. It shows that the firm’s survival probability increases with wealth and theimplied option value increases firstly and then decreases with the increasing of thewealth. The firm significantly enhances the survival probability due to the investmentopportunity. A strong discover of our study different to the classic option pricingtheory is that the higher the uncertainty of the cash flow, the lower the option value.(4) We focus on the role of contingent convertible bond (CCB) as a debtfinancing instrument in an investment-based framework. We model the firm’sinvestment behavior as a real option which is exercised under firm-value andequity-value maximizing policies respectively. We attempt to clarify how CCB affectsthe firm’s investment policy, agency cost of debt and optimal capital structure throughcomparing with straight bond financing. Numerical analysis demonstrates that underan endogenous conversion rule, CCB financing leads to overinvestment, a higherleverage and a bigger agency cost. At the same time, CCB financing decreasesbankruptcy risk, and increases the firm’s value. In particular, for an exogenousconversion, the CCB financing considerably decreases the agency cost as well. Thereexists a unique exogenous conversion ratio such that the agency cost reaches theminimum value zero.(5) Introducing the problem of partial information into the pricing model of thehedge fund. Suppose the expected return rate of a hedge fund is not observable but obey a two-point distribution. The fund manager can dynamically update his beliefabout the true value of the expected return based on the realization of the net assetvalue of the hedge fund. We study the impact of the uncertainty of the expected returnrate on the valuation of the fund manager’s various fees and the investor’s claim. Theresults show that partial information has significant impact on the management fees,performance fees and the claims. Specifically, a non-updating fund manager who justtakes the expected return as a constant always underestimates the values, and moreoften than not, the amount underestimated is significant. The closer the net asset valuegets to the high-water mark or the larger the uncertainty of the expected return is, thebigger the amount underestimated will become.