鞅方法和随机控制理论在投资组合和期权定价中的应用
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摘要
投资组合选择和期权定价是现代数理金融理论的两大研究主题,经典投资组合选择问题的研究通常建立在Markowitz均值-方差或von Neumann-Morgenstern期望效用框架下,探讨风险厌恶投资者的理性最优投资行为；而经典期权定价问题的研究则是利用无套利定价原理,探讨合理的期权价格.
本文将利用鞅方法和随机控制理论从不同的角度对这两个问题展开讨论,一方面在非von Neumann-Morgenstern期望效用理论框架下,研究“非理性”投资者的最优投资行为；另一方面将投资问题与期权定价问题建立在一个模型框架下.同时分析投资者的最优股票、期权投资以及期权定价问题.
首先,系统研究了损失厌恶投资者的最优投资组合选择模型,其中,在连续时间完全市场框架下,分别考虑财富非负和带有基准下限约束的情形,讨论了一般价值函数下的损失厌恶投资者的最优投资问题,分析了损失厌恶投资者的投资组合保险策略,讨论了解的存在唯一性,并和经典期望效用最大化意义下的结果做了对比分析；另外,对于不完全市场以及跳-扩散市场情形下的最优投资问题,也做了细致的分析.
其次,在秩相依期望效用理论框架下,研究了带有财富VaR约束的最优投资组合选择模型,利用分位数函数技术对模型进行求解,获得了最优期末财富,并分析了一个例子,得到了最优财富过程及最优资产配置策略.
最后,研究了标的资产为不可交易资产的期权定价以及最优股票、期权投资问题.在风险偏好是指数形式的假定下,建立了股票和期权最优配置之间的动态关系,获得了动态期权定价公式.特别地,利用Feynman-Kac公式,给出了封闭形式的价格表示；同时,分析了解的性质,讨论了动态均衡价格与边际价格和效用无差异价格之间的关系.
Portfolio selection and option pricing are the main research topics in mod-ern mathematical finance. The classical portfolio selection theories are generally established in the framework of Markowitz's mean-variance or von Neumann-Morgenstern's expected utility where the rational optimal investment behaviors for risk averse investors are investigated. Then, the classical option pricing the-ory is mainly focused on obtaining reasonable option price by applying the no-arbitrage principle.
This dissertation is devoted to the further development of them from different aspects. On the one hand, the optimal investment for "irrational" investors is investigated for non-expected utility via the martingale approaches. On the other hand, by employing the stochastic dynamic programming techniques, we study the pricing of options written on non-traded assets and dynamic trading strategies for stocks and options.
Firstly, we consider the optimal portfolio selection models for loss averse investors. In section2.1, we study a general dynamic portfolio selection model for loss-averse investors with wealth constraints in a complete market. By applying the martingale methods, we derive the optimal terminal wealth of the investors with or without the benchmark floor constraints. Simultaneously, the existence and uniqueness of the corresponding Lagrange multiplier are proved. We analyze the properties of these solutions, and compare them to that of the general risk averse investors. In section2.2and2.3, the similar problems are analyzed subtly in the incomplete market and jump-diffusion market, respectively.
Secondly, we investigate the optimal portfolio selection model with VaR con-straint in the framework of RDEU. Via the quantile formulation approach, the optimal terminal wealth is obtained. An example is presented to obtain the opti- mal wealth processes and the investment strategies.
Lastly, we construct a model for pricing of options written on non-traded assets and trading strategies. Suppose the stock and option can be continuously traded without friction. Under the exponential utility maximization criterion, a relation between the optimal positions for the stock and option is derived by using the stochastic dynamic programming techniques. The dynamic option pricing-equations are also established. In particular, the properties of the associated solutions are discussed and their explicit representations are demonstrated using the Feynman-Kac formula. We further compares the equilibrium price to the existing price notions, such as the marginal price and indifference price.
本文将利用鞅方法和随机控制理论从不同的角度对这两个问题展开讨论,一方面在非von Neumann-Morgenstern期望效用理论框架下,研究“非理性”投资者的最优投资行为；另一方面将投资问题与期权定价问题建立在一个模型框架下.同时分析投资者的最优股票、期权投资以及期权定价问题.
首先,系统研究了损失厌恶投资者的最优投资组合选择模型,其中,在连续时间完全市场框架下,分别考虑财富非负和带有基准下限约束的情形,讨论了一般价值函数下的损失厌恶投资者的最优投资问题,分析了损失厌恶投资者的投资组合保险策略,讨论了解的存在唯一性,并和经典期望效用最大化意义下的结果做了对比分析；另外,对于不完全市场以及跳-扩散市场情形下的最优投资问题,也做了细致的分析.
其次,在秩相依期望效用理论框架下,研究了带有财富VaR约束的最优投资组合选择模型,利用分位数函数技术对模型进行求解,获得了最优期末财富,并分析了一个例子,得到了最优财富过程及最优资产配置策略.
最后,研究了标的资产为不可交易资产的期权定价以及最优股票、期权投资问题.在风险偏好是指数形式的假定下,建立了股票和期权最优配置之间的动态关系,获得了动态期权定价公式.特别地,利用Feynman-Kac公式,给出了封闭形式的价格表示；同时,分析了解的性质,讨论了动态均衡价格与边际价格和效用无差异价格之间的关系.
Portfolio selection and option pricing are the main research topics in mod-ern mathematical finance. The classical portfolio selection theories are generally established in the framework of Markowitz's mean-variance or von Neumann-Morgenstern's expected utility where the rational optimal investment behaviors for risk averse investors are investigated. Then, the classical option pricing the-ory is mainly focused on obtaining reasonable option price by applying the no-arbitrage principle.
This dissertation is devoted to the further development of them from different aspects. On the one hand, the optimal investment for "irrational" investors is investigated for non-expected utility via the martingale approaches. On the other hand, by employing the stochastic dynamic programming techniques, we study the pricing of options written on non-traded assets and dynamic trading strategies for stocks and options.
Firstly, we consider the optimal portfolio selection models for loss averse investors. In section2.1, we study a general dynamic portfolio selection model for loss-averse investors with wealth constraints in a complete market. By applying the martingale methods, we derive the optimal terminal wealth of the investors with or without the benchmark floor constraints. Simultaneously, the existence and uniqueness of the corresponding Lagrange multiplier are proved. We analyze the properties of these solutions, and compare them to that of the general risk averse investors. In section2.2and2.3, the similar problems are analyzed subtly in the incomplete market and jump-diffusion market, respectively.
Secondly, we investigate the optimal portfolio selection model with VaR con-straint in the framework of RDEU. Via the quantile formulation approach, the optimal terminal wealth is obtained. An example is presented to obtain the opti- mal wealth processes and the investment strategies.
Lastly, we construct a model for pricing of options written on non-traded assets and trading strategies. Suppose the stock and option can be continuously traded without friction. Under the exponential utility maximization criterion, a relation between the optimal positions for the stock and option is derived by using the stochastic dynamic programming techniques. The dynamic option pricing-equations are also established. In particular, the properties of the associated solutions are discussed and their explicit representations are demonstrated using the Feynman-Kac formula. We further compares the equilibrium price to the existing price notions, such as the marginal price and indifference price.
引文
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[2]Alexander, G.J. and Baptista, A.M. (2008). Active Portfolio Management with Benchmarking: Adding a Value-at-risk Constraint. Journal of Economic Dynamics and Control,32(3),779-820.
[3]Ankirclmer, S., Imkeller, P. and Rcis, G. (2010). Pricing and Hedging of Derivatives Based on Nontradable Underlyings. Mathematical Finance,120(2),289-312.
[4]Barberis, N., Huang, M. and Santos, T. (2001). Prospect theory and asset prices[J]. Quarterly Journal of Economics,116(1),1-53.
[5]Bardhan, I. and Chao, X. (1995). Martingale Analysis for Assets with Discontinuous Returns, Mathematics of Operations Research,20(1),243-256.
[6]Basak, S. (1995). A general equilibrium model of portfolio insurance. Review of Financial StudiesjJ], 8(4),1059-1090.
[7]Basak, S. (2002). A comparative study of portfolio insurance. Journal of Economic Dynamics and Control,26,1217-1241.
[8]Basak, S. and Shapiro, A. (2001). Value-at-Risk-Based Risk Management:Optimal Policies and Asset Prices. The Review of Financial Studies,14(2),371-405.
[9]Basak, S., Shapiro, A. and Tepla, L. (2006). Risk Management with Benchmarking. Management Science,52(4),542-557.
[10]Bellamy, N. (2001). Wealth Optimization in an Incomplete Market Driven by a Jump-diffusion Process[J], Journal of Mathematical Economics,35,259-287.
[11]Berkelaar, A.B., Cumperayot P. and Kouwenberg, R. (2002). The Effect of VaR Based Risk Man-agement on Asset Prices and the Volatility Smile [J], European Financial Management,8(2),139-164.
[12]Berkelaar, A.B., Kouwenberg, R. and Post, T. (2004). Optimal portfolio choice under loss aver-sion[J], Review of Economics and Statistics,86(4),973-987.
[13]Bernard, C., He, X.D., Yan, J.A. and Zhou, X.Y. (2011). Optimal Insurance Design under Rank Dependent Utility, Working Paper.
[14]Bielecki, T.R., Jin, H.Q., Pliska, S. and Zhou, X.Y. (2005). Continuous-time Mean-variance Port-folio Selection with Bankruptcy Prohibition[J]. Mathematical Finance,15(2),213-244.
[15]Boyle, P. and Tian, W. (2007). Portfolio Management with Constraints[J]. Mathematical Finance, 17(3),319-343.
[16]Callegaro, G. and Vargiolu, T. (2009). Optimal Portfolio for HARA Utility Functions in a Pure Jump Multidimensional Incomplete Market [J], International Journal of Risk Assessment and Man-agement,11,180-200.
[17]Campbell, R., Husiman, R. and Koedijk, K. (2001). Optimal Portfolio Selection in a Value-at-Risk Framework. Journal of Banking and Finance,25(9),1789-1804.
[18]Carlier, G. and Dana, R.-A. (2011). Optimal Demand for Contingent Claims when Agents have Law Invariant Utilities[J]. Mathematical Finance,21(2),169-201.
[19]Cei, C. (2009). An HJB Approach to Exponential Utility Maximization for jump processes[J], International Journal of Risk Assessment and Managemet,11,104-121.
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