【作者】 宋丹丹；

【导师】 杨招军；

【作者基本信息】 湖南大学 ， 金融学， 2013， 博士

【摘要】 本学位论文针对非完备市场中实物期权与部分信息相关的若干问题，运用统计滤波技术、随机控制理论、均衡定价和消费效用无差别定价方法，建立严格的数理金融模型，研究实物期权定价、择时、融资结构的数量规律，给出资产定价和风险对冲对部分信息、特质风险等因素的敏感性分析。本文研究包括如下几个方面：（1）扩展经典的实物期权理论。假设不可逆投资项目随机收益流的平均增长率是不可观测的正态随机变量，研究最大化无限生命期内的期望消费折扣总效用的实物期权问题。分别针对自保（self-insurance）模型和对冲模型，得到了关于平均增长率的滤波估计，然后在常绝对风险厌恶（CARA）效用函数下，推导出了相应的投资后收益流的确定性等价价值，以及投资期权的隐含价值和最优投资触发水平所满足的具有自由边界的半闭式偏微分方程系统，结果独立于效用的时间折扣因子。运用有限差分方法，获得了数值解，与完备信息下的结果进行对比，给出了信息价值的测算方法。理论分析和数值结果表明部分信息致使延迟投资，并导致投资期权隐含价值的显著损失，该损失在投资触发水平处达到最大，且随着投资者风险厌恶指数、估计风险、项目收益波动率的增加而提高。结果还表明消费效用无差别定价准则不仅考虑了系统风险溢价、也考虑了非系统风险溢价。但是，投资者的风险态度只有在存在非系统风险时，才会对投资消费决策和资产定价产生影响。（2）研究了宏观经济条件的随机转换对不可逆投资决策问题的影响。假设不可逆投资项目执行后的收益流不仅依赖于企业自身的收益风险，也依赖于企业所处的外部宏观经济状态（繁荣或萧条）。宏观经济状态服从连续时间的两状态马尔科夫链，并以成倍的形式影响项目投资后收益流的价值。通过最优控制与修正的光滑拟合技术，得到对应于每个宏观经济状态下的投资期权隐含价值以及最优投资时刻的闭式解。数值分析给出了宏观经济条件和贴现率的随机转变以及平均增长率、波动率等参数与最优投资和隐含价值的关系。（3）基于最大化生存概率准则，研究了实物期权的定价与择时问题。假设投资收益流面临着不可对冲的随机风险，企业的目标是如何通过最优执行该期权来最大化其生存概率。运用随机控制理论，通过求解HJB方程，获得了企业的生存概率，实物期权的投资阈值以及相应的期权隐含价值的闭式解，数值分析给出了生存概率和实物投资随财富总量以及项目波动率的变化规律。结果显示：企业的生存概率随财富的增加而增大，期权价值随财富的增加先增大后减小，同时企业拥有项目投资机会显著增强了企业的生存概率。与经典期权定价理论不同的是，收益流的波动率越大，实物期权的隐含价值越小。（4）在实物投资框架中，研究了或有可转换债券（CCB）作为一种新型的债务融资工具对公司所起到的作用。公司的项目投资行为作为一种实物期权分别以公司价值最大化和股权价值最大化为目标。通过与普通债券融资相比较，分析了CCB对公司最优投资策略、债务代理成本以及公司最优资本结构的影响机制。数值分析表明在内生的转换法则下，一方面，CCB融资会导致公司过度投资，高杠杆以及较大的代理成本。另一方面，CCB融资有效地降低了破产风险，增加了公司的总价值。特别地，对于外生的转换边界，CCB融资显著的降低了代理成本，并存在唯一的外生转换比率使得代理成本降为最小值零。（5）将部分信息的问题引入到对冲基金的定价模型中，假设对冲基金的期望收益率不可观测但是已知服从两点分布。基金管理者可以根据可观测的对冲基金的净资产价值动态更新他对期望收益率真值的估计。分析了期望收益率的不确定性对基金管理者的各种薪酬水平和投资者权益价值的影响。结果表明：部分信息对管理费、激励费和投资者权益价值产生了显著的经济学影响。特别地，一个没有进行信息更新，仅仅认为期望回报率是个常数的基金管理者总是低估这些价值，而且往往被低估的程度非常严重。净资产价值越接近高收益线（high-water mark），或者期望回报率的不确定性越大，基金价值被低估的程度越明显。更多还原

【Abstract】 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.更多还原