金融投资决策模型研究
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摘要
本文运用随机最优控制理论、随机分析中的It(?)公式及非线性滤波技术，研究投资者极大化终止时刻期望效用的最优投资策略问题。论文的前面部分，分别在完备信息和不完备信息情形下，运用It(?)公式与非线性滤波技术，导出Hamilton-Jacobi-Bellman(HJB)方程，得到了最优投资决策的一阶条件，并根据It(?)公式，提出了求解该一阶条件中的偏微分方程的成功率较高的试探求解法。在投资者具有幂函数效用情形下，运用这种新的试探求解法，本文导出了完备信息下的最优投资策略。特别地，在投资者具有对数效用函数情形下，分别就完备信息与部分信息情形，直接运用It(?)公式，得到了最优投资决策的简单计算公式，通过比较完备信息与不完备信息情形下投资者的最优目标函数的差异，得到了(内部交易者获得的)信息价值的简洁公式，从而给出了信息价值的精确度量，并且该度量值易于投资者操作使用。最后，讨论贷款利率高于存款利率的投资策略问题，分别就部分信息对数效用和完备信息幂函数效用，代替复杂的随机动态规划方法，运用It(?)公式和非线性规划技术，同样得到了最优投资策略的简单解。
This paper utilizes stochastic optimal control theory, Ito formula in stochastic analysis and nonlinear filter technique to maximize the expected utility from the terminal wealth. Firstly, the paper provides a simple review to some problems on investment and consumption. Secondly, the paper derives HJB equation by means of It: formula and nonlinear filter technique, and obtains some necessary conditions of the optimal strategy. Furthermore, the paper puts forward a successful trial solution to solve the partial differential equation for that necessary conditions. By utilizing this trial solution, the paper obtains the optimal strategy for the power utility under full information. Especially, the paper provides the optimal solution for logarithmic utility under both lull and partial information, by utilizing Ito formula only. By comparing optimal target function, the paper shows a concise formula to the valuation of information. In the end, assuming the borrowing rate is bigger than saving rate , the paper provides explicit solutions to both logarithmic utility with partial information and power utility with full information by Ito's formula other than the complicated dynamic programming. All of the strategies are suitable for operating online.
This paper utilizes stochastic optimal control theory, Ito formula in stochastic analysis and nonlinear filter technique to maximize the expected utility from the terminal wealth. Firstly, the paper provides a simple review to some problems on investment and consumption. Secondly, the paper derives HJB equation by means of It: formula and nonlinear filter technique, and obtains some necessary conditions of the optimal strategy. Furthermore, the paper puts forward a successful trial solution to solve the partial differential equation for that necessary conditions. By utilizing this trial solution, the paper obtains the optimal strategy for the power utility under full information. Especially, the paper provides the optimal solution for logarithmic utility under both lull and partial information, by utilizing Ito formula only. By comparing optimal target function, the paper shows a concise formula to the valuation of information. In the end, assuming the borrowing rate is bigger than saving rate , the paper provides explicit solutions to both logarithmic utility with partial information and power utility with full information by Ito's formula other than the complicated dynamic programming. All of the strategies are suitable for operating online.
引文
1 Cox J, Huang C. Optimal Consumption and Portfolio Policies when Asset Prices Follow a Diffusion Process. J. of Econ. Theory, 1989, 49: 33-83
2 Cox J, Huang C. A Variational Problem Arising in Financial Economics. J. of Math. Econ., 1991, 20: 465-487
3 Delbaen F, Schachermayer W. A General Version of the Fundamental Theorem of Asset Pricing. Mathematische Annalen, 1994, 300: 463-520
4 Fllmer H, Schweizer M. Hedging of Contingent Claims under Incomplete Information. Davis M H A, Elliott R J eds., Applied Stochastic Analysis, London: Gordon and Breach, 1990
5 Gouriéroux C, Lanrent J P, Pham H. Mean-variance Hedging and Numéraire.Mathematical Finance, 1998, 8: 179-200
6 Harrison J M, Pliska S R. Martingales and Stochastic Integrals in the Theory of Continuous Trading. Stochastic Process Appl., 1981, 11: 215-260
7 Harrison J M, Pliska S R. A Stochastic Calculus Model of Continuous Trading: Complete Markets. Stochastic Process Appl., 1983, 15: 313-316
8 Karatzas I, Lehoczky J, Sethi S, Shreve S. Explicit Solution of a General Consumption-Investment Problem. Math. Oper. Res., 1986, 11: 261-294
9 Karatzas I, Lehoczky J, Shreve S. Optimal Consumption and Portfolio Decisions for a "Small Investor" on a Finite Horizon. SIAM J. Control Optimiz., 1987, 25: 1557-1586
10 Kramkov D. Optional Decomposition of Supermartingales and Hedging Contingent Claims in Incomplete Markets. Probability Theory and Related Fields, 1996,105: 459-479
11 Lakner P. Utility Maximization with Partial Information. Stochastic Process Appl., 1995, 56: 247-273
12 Lakner P. Optimal Trading Strategy for an Investor: The Case of Partial Information. Stochastic Process Appl., 1998, 76: 77-97
13 Liptser R S, Shiryayev A N. Statistics of Random Process. New York: SpringerVerlag, 1977
14 Merton R C. Portfolio Selection under Uncertainty: The continuous-time case.Rev. Econ. Statist., 1969, 51: 247-257
15 Merton R C. Optimal Consumption and Portfolio Rules in a Continuous Time Model. J. Econ. Theory, 1971, 3: 373-413
16 Merton R C. Continuous-Time Finance. Cambridge, 1990
17 Pham H, Rheinlnder T, Schweizer M. Mean-Variance Hedging for Continuous Processes: New Proofs and Examples. Finance and Stochastics, 1998, 2: 173-198
18 Pliska S. A Stochastic Calculus Model of Continuous Trading: Optimal Portfolios. Math. Oper. Res., 1986, 11: 371-382
19 Protter P. Stochastic Integration and Differential Equations: A New Approach, Applications of Mathematics. Berlin: Springer, 1990
20 Rogers,L.C.G.,Shi,Z.,The value of an Asian option,Journal of Applied Probability, 32(1995),1077-1088
21 日本银行研修社，最新国际金融工具，中国金融出版社，北京，1993．
22 Samuelson,P. A.,Rational theory of warrant pricing, Ind. Management Rev. 6(1965), 13-39
23 Samelson,P. A., Lifetime portfolio selection by dynamic stochastic programming, Review of Economics and statistics, 51(1969),239-246
24 Schweizer, M.,Risk-mininmlity and orthgonality of martingales, Stochastics Stochastic Rep., 30(1990), 123-131
25 Schweizer, M., Option hedging for semimartingales, Stochastic Process. Appl.,37(1991),339-363
26 Schweizer, M., Mean-variance hedging for general claims, Ann. Appl. Probab. 2(1992), 171-179
27 Sharpe,W.F.,Capital asset price: A theory of market equilibrium under conditions of risk, Journal of Finance, 19(1964), 425-442
28 Sharpe,W.F.,Portfolio Theory and Capital Markets,New York,McGraw Hill, 1970
29 Shirakawa, H. & Konno,H., Pricing of options under proportional transaction costs, Working paper, 1995
30 Shreve, S. E. & Soner, H. M., Optimal investment and consumption with transaction costs, The Ann. Os Appl. Probab., 4(1994), 609-692
31 Vila,J.-L. & Zariphopoulou,T., Optimal consumption and portfolio choice with borrowing contraints, J. Econ. Theory, 77(1997), 402-431
32 汪丁丁，回顾“金融革命”，经济研究，12(1997)，69-77
33 Wilmott, P., Dewynne,J. & Howison,S., Option Pricing: mathematical models and computation. Oxford Financial Press, UK,1993
34 严加安，彭实戈，方诗赞，吴黎明，随机分析选讲，科学出版社，北京，1997
35 Yang Z J, Ma C Q. Optimal Trading Strategy with Partial Information: The Simplified and Generalized Models. International Journal of Theoretical &: Applied Finance, 2001, 4: 759-772
36 徐东，杨招军，黄立宏，金融与随机分析评述，系统工程，2003年第3期
37 杨昭军，李致中，邹捷中．部分信息下的最优投资消费策略显式解．应用概率统计，2001，16：390-398
38 Yang, Z. J. & Zou, J. Z., The Hedging Strtegy of an Asion Option, Symposium on Markov Process & controlled Markov Chain(1999)
39 Yong, J., Finding adapted solutions of forward-backward stochastic differential equations—method of continuation, Prob. Th. & Rel. Fields, 107(1997),537-572
40 Yong,J.,Linear forward-backward stochastic differential equations, Appl. Math.Optim.,39(1999),93-119
41 雍炯敏，数学金融学中的若干问题，数学的实践与认识，2(1999)，97-108
42 Zariphopoulou,T.,Consumption-investment models with constraints,SIAM J. Control Optimiz., 32(1994), 59-84
43 周立，金融衍生工具发展与监管，中国发展出版社，北京，1997
2 Cox J, Huang C. A Variational Problem Arising in Financial Economics. J. of Math. Econ., 1991, 20: 465-487
3 Delbaen F, Schachermayer W. A General Version of the Fundamental Theorem of Asset Pricing. Mathematische Annalen, 1994, 300: 463-520
4 Fllmer H, Schweizer M. Hedging of Contingent Claims under Incomplete Information. Davis M H A, Elliott R J eds., Applied Stochastic Analysis, London: Gordon and Breach, 1990
5 Gouriéroux C, Lanrent J P, Pham H. Mean-variance Hedging and Numéraire.Mathematical Finance, 1998, 8: 179-200
6 Harrison J M, Pliska S R. Martingales and Stochastic Integrals in the Theory of Continuous Trading. Stochastic Process Appl., 1981, 11: 215-260
7 Harrison J M, Pliska S R. A Stochastic Calculus Model of Continuous Trading: Complete Markets. Stochastic Process Appl., 1983, 15: 313-316
8 Karatzas I, Lehoczky J, Sethi S, Shreve S. Explicit Solution of a General Consumption-Investment Problem. Math. Oper. Res., 1986, 11: 261-294
9 Karatzas I, Lehoczky J, Shreve S. Optimal Consumption and Portfolio Decisions for a "Small Investor" on a Finite Horizon. SIAM J. Control Optimiz., 1987, 25: 1557-1586
10 Kramkov D. Optional Decomposition of Supermartingales and Hedging Contingent Claims in Incomplete Markets. Probability Theory and Related Fields, 1996,105: 459-479
11 Lakner P. Utility Maximization with Partial Information. Stochastic Process Appl., 1995, 56: 247-273
12 Lakner P. Optimal Trading Strategy for an Investor: The Case of Partial Information. Stochastic Process Appl., 1998, 76: 77-97
13 Liptser R S, Shiryayev A N. Statistics of Random Process. New York: SpringerVerlag, 1977
14 Merton R C. Portfolio Selection under Uncertainty: The continuous-time case.Rev. Econ. Statist., 1969, 51: 247-257
15 Merton R C. Optimal Consumption and Portfolio Rules in a Continuous Time Model. J. Econ. Theory, 1971, 3: 373-413
16 Merton R C. Continuous-Time Finance. Cambridge, 1990
17 Pham H, Rheinlnder T, Schweizer M. Mean-Variance Hedging for Continuous Processes: New Proofs and Examples. Finance and Stochastics, 1998, 2: 173-198
18 Pliska S. A Stochastic Calculus Model of Continuous Trading: Optimal Portfolios. Math. Oper. Res., 1986, 11: 371-382
19 Protter P. Stochastic Integration and Differential Equations: A New Approach, Applications of Mathematics. Berlin: Springer, 1990
20 Rogers,L.C.G.,Shi,Z.,The value of an Asian option,Journal of Applied Probability, 32(1995),1077-1088
21 日本银行研修社，最新国际金融工具，中国金融出版社，北京，1993．
22 Samuelson,P. A.,Rational theory of warrant pricing, Ind. Management Rev. 6(1965), 13-39
23 Samelson,P. A., Lifetime portfolio selection by dynamic stochastic programming, Review of Economics and statistics, 51(1969),239-246
24 Schweizer, M.,Risk-mininmlity and orthgonality of martingales, Stochastics Stochastic Rep., 30(1990), 123-131
25 Schweizer, M., Option hedging for semimartingales, Stochastic Process. Appl.,37(1991),339-363
26 Schweizer, M., Mean-variance hedging for general claims, Ann. Appl. Probab. 2(1992), 171-179
27 Sharpe,W.F.,Capital asset price: A theory of market equilibrium under conditions of risk, Journal of Finance, 19(1964), 425-442
28 Sharpe,W.F.,Portfolio Theory and Capital Markets,New York,McGraw Hill, 1970
29 Shirakawa, H. & Konno,H., Pricing of options under proportional transaction costs, Working paper, 1995
30 Shreve, S. E. & Soner, H. M., Optimal investment and consumption with transaction costs, The Ann. Os Appl. Probab., 4(1994), 609-692
31 Vila,J.-L. & Zariphopoulou,T., Optimal consumption and portfolio choice with borrowing contraints, J. Econ. Theory, 77(1997), 402-431
32 汪丁丁，回顾“金融革命”，经济研究，12(1997)，69-77
33 Wilmott, P., Dewynne,J. & Howison,S., Option Pricing: mathematical models and computation. Oxford Financial Press, UK,1993
34 严加安，彭实戈，方诗赞，吴黎明，随机分析选讲，科学出版社，北京，1997
35 Yang Z J, Ma C Q. Optimal Trading Strategy with Partial Information: The Simplified and Generalized Models. International Journal of Theoretical &: Applied Finance, 2001, 4: 759-772
36 徐东，杨招军，黄立宏，金融与随机分析评述，系统工程，2003年第3期
37 杨昭军，李致中，邹捷中．部分信息下的最优投资消费策略显式解．应用概率统计，2001，16：390-398
38 Yang, Z. J. & Zou, J. Z., The Hedging Strtegy of an Asion Option, Symposium on Markov Process & controlled Markov Chain(1999)
39 Yong, J., Finding adapted solutions of forward-backward stochastic differential equations—method of continuation, Prob. Th. & Rel. Fields, 107(1997),537-572
40 Yong,J.,Linear forward-backward stochastic differential equations, Appl. Math.Optim.,39(1999),93-119
41 雍炯敏，数学金融学中的若干问题，数学的实践与认识，2(1999)，97-108
42 Zariphopoulou,T.,Consumption-investment models with constraints,SIAM J. Control Optimiz., 32(1994), 59-84
43 周立，金融衍生工具发展与监管，中国发展出版社，北京，1997