资产定价中的一些数学问题的研究
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摘要
这篇学位论文研究了从资产定价中提出的一些数学问题,包括四个部分.
在第一部分中,我们考虑期权定价.在股票价格过程服从一般的几何Riemannian Brown-ian运动的假设下,我们得到了欧式期权价格所满足的偏微分方程；并讨论了该方程的解的存在唯一性.实数R上的恰当的Riemannian度量可以产生高峰厚尾的收益率分布,也可以解释价格偏移和波动率微笑.
在第二部分中,我们讨论信念异质性泡沫(heterogeneous beliefs bubbles).我们在资产红利满足CIR模型的假设下,通过粘性解的方法得到了极小均衡价格的显示表达式.进而,得到了极小信念异质性泡沫的显示解.
在第三部分中,我们严格分析了一个项目的进入和退出决策问题.和经典的成本是常数的假设不同,我们假设成本是产品价格的线性函数.在这个假设下,我们给出了项目是永远不值得投资的条件；此外,如果产品价格高于某个值,项目可能会被终止.这些结论在经典的假设下是不存在的.如果项目是值得投资的,我们给出了进入和退出决策的精确解,并且得到了项目价值的显示表达式.
在第四部分中,我们研究了动态资产配置和静态资产配置的等价性问题.在风险资产的价格过程是由一个Poisson过程驱动的假设下,通过限制效用函数和投资策略,我们利用变分方法得到了动态资产配置和静态资产配置等价的一个充要条件.然后,我们提供了该等价性的一个简单的充分条件.
This thesis focuses on some mathematical problems from asset pricing. It consists of four parts.
In the first part, we consider option pricing. We provide a partial differential equation for European options on a stock whose price process follows a general geometric Riemannian Brow-nian motion. The existence and the uniqueness of solutions to the partial differential equation are investigated. A proper Riemannian metric on R can make the distribution of the stock return rates induced by our model have the character of leptokurtosis and fat-tail; besides, it can also explain price bias and volatility smile.
In the second part, we discuss heterogeneous beliefs bubbles. Under the assumption that the dividends of an asset satisfy a CIR model, we obtain a closed formula for the minimal equilibrium price by the method of viscosity solutions. Then the minimal heterogeneous beliefs bubbles are determined explicitly.
In the third part, we rigorously analyze entry and exit decisions of a project. Instead of supposing that the costs are constant in classical research, we assume that they are linear with respect to the price of the commodity produced by the project. Under this assumption, we obtain a condition which guarantees that investing in the project is worthless; besides, the project may be terminated when the commodity price is greater than a certain value. In contrast, there are no such results provided that the costs are constant. Moreover, we offer an explicit solution of entry and exit decisions if the project is worthy to be invested in.
In the fourth part, we consider the equivalence of dynamic and static asset allocations in the case where the price of the risk asset (a stock, for example) is driven by a Poisson process. By restricting utility functions and trading strategies and using the variation method, we obtain a necessary and sufficient condition for the equivalence of dynamic and static asset allocations. Then we provide a simple sufficient condition for the equivalence.
在第一部分中,我们考虑期权定价.在股票价格过程服从一般的几何Riemannian Brown-ian运动的假设下,我们得到了欧式期权价格所满足的偏微分方程；并讨论了该方程的解的存在唯一性.实数R上的恰当的Riemannian度量可以产生高峰厚尾的收益率分布,也可以解释价格偏移和波动率微笑.
在第二部分中,我们讨论信念异质性泡沫(heterogeneous beliefs bubbles).我们在资产红利满足CIR模型的假设下,通过粘性解的方法得到了极小均衡价格的显示表达式.进而,得到了极小信念异质性泡沫的显示解.
在第三部分中,我们严格分析了一个项目的进入和退出决策问题.和经典的成本是常数的假设不同,我们假设成本是产品价格的线性函数.在这个假设下,我们给出了项目是永远不值得投资的条件；此外,如果产品价格高于某个值,项目可能会被终止.这些结论在经典的假设下是不存在的.如果项目是值得投资的,我们给出了进入和退出决策的精确解,并且得到了项目价值的显示表达式.
在第四部分中,我们研究了动态资产配置和静态资产配置的等价性问题.在风险资产的价格过程是由一个Poisson过程驱动的假设下,通过限制效用函数和投资策略,我们利用变分方法得到了动态资产配置和静态资产配置等价的一个充要条件.然后,我们提供了该等价性的一个简单的充分条件.
This thesis focuses on some mathematical problems from asset pricing. It consists of four parts.
In the first part, we consider option pricing. We provide a partial differential equation for European options on a stock whose price process follows a general geometric Riemannian Brow-nian motion. The existence and the uniqueness of solutions to the partial differential equation are investigated. A proper Riemannian metric on R can make the distribution of the stock return rates induced by our model have the character of leptokurtosis and fat-tail; besides, it can also explain price bias and volatility smile.
In the second part, we discuss heterogeneous beliefs bubbles. Under the assumption that the dividends of an asset satisfy a CIR model, we obtain a closed formula for the minimal equilibrium price by the method of viscosity solutions. Then the minimal heterogeneous beliefs bubbles are determined explicitly.
In the third part, we rigorously analyze entry and exit decisions of a project. Instead of supposing that the costs are constant in classical research, we assume that they are linear with respect to the price of the commodity produced by the project. Under this assumption, we obtain a condition which guarantees that investing in the project is worthless; besides, the project may be terminated when the commodity price is greater than a certain value. In contrast, there are no such results provided that the costs are constant. Moreover, we offer an explicit solution of entry and exit decisions if the project is worthy to be invested in.
In the fourth part, we consider the equivalence of dynamic and static asset allocations in the case where the price of the risk asset (a stock, for example) is driven by a Poisson process. By restricting utility functions and trading strategies and using the variation method, we obtain a necessary and sufficient condition for the equivalence of dynamic and static asset allocations. Then we provide a simple sufficient condition for the equivalence.
引文
[1]E. Allen, Modeling with Ito Stochastic Differential Equations, Springer, P.O. Box 17,3300 AA Dordrecht, The Netherlands,2007.
[2]David Applebaum, Levy processes and stochastic calculus (second edition), Cambridge Uni-versity Press, The Edinburgh Building, Cambridge CB2 8RU, UK8,2009.
[3]George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists (6th ed.), Elsevier Academic Press,30 Corporate Drive, Suite 400, Burlington, MA 01803, USA; 525 B Street, Suite 1900, San Diego, California 92101-4495, USA; 84 Theobald's Road, London WC1X 8RR, UK,2005.
[4]L. Bachelier, Theorie de la speculation, Annales Scientifiques de l'E. N. S.17 (1900),21-28.
[5]Avner Bar-Ilan and William C. Strange, Investment lags, The American Economic Review 86 (1996), no.3,610-622.
[6]Fischer Black and Myron Scholes, The Valuation of Options and Corporate Liabilities, Jour-nal of Political Economy 81 (1973),637-654.
[7]John Black, Nigar Hashimzade, and Gareth D. Myles, Oxford:A Dictionary of Economics, Oxford University Press, Great Clarendon Street, Oxford OX2 6DP,2009.
[8]Svetlana Boyarchenko and Sergei Levendorskii, Irreversible decisions under uncertainty: Optimal stopping made easy, Spriger, Verlag Berlin Heidelberg,2007.
[9]Kjell Arne Brekke and Bernt (?)ksendal, Optimal switching in an economic activity under uncertainty, SIAM Journal on Control and Optimization 32 (1994), no.4,1021-1036.
[10]Terence Chan, Pricing Contingent Claims on Stocks Driven by Levy Processes, The Annals of Applied Probability 9(1999),504-528.
[11]Xi Chen and Robert V. Kohn, Asset price bubbles from heterogeneous beliefs about mean reversion rates, Finance and Stochastics 15 (2011), no.2,221-241.
[12]Yoon K. Choi and Seok Weon Lee, Investment and abandonment decisions with uncertain price and cost, Journal of Business Finance & Accounting 27 (2000), no.1-2,195-213.
[13]Marius Costeniuc, Michaela Schnetzer, and Luca Taschini, Entry and exit decision problem with implementation delay, Journal of Applied Probability 45 (2008), no.4,1039-1059.
[14]John C. Cox and Chi-fu Huang, Optimal consumption and portfolio policies when asset prices follow a diffusion process, Journal of Economic Theory 49 (1989), no.1,33-83.
[15]John C. Cox, Jonathan E. Ingersoll Jr., and Stephen A. Ross, A Theory of the Term Structure of Interest Rates, Econometrica 53 (1985), no.2,385-407.
[16]Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc.27 (1992), no.1,1-67.
[17]Annie Cuyt, Vigdis Brevik Petersen, Brigitte Verdonk, Haakon Waadeland, and William B. Jones, Handbook of Continued Fractions for Special Functions, Springer,2008.
[18]Avinash Dixit, Entry and Exit Decisions under Uncertainty, Journal of Political Economy 97 (1989), no.3,620-638.
[19]J. Kate Duckworth and Mihail Zervos, An Investment Model with Entry and Exit Decisions, Journal of Applied Probability 37 (2000),547-559.
[20]Steven N. Durlauf and Lawrence E. Blume, The New Palgrave Dictionary of Economics (2nd ed.) Volume I, Palgrave Macmillan, Houndmill, Basingstoke, Hampshite RG21 6XS and 175 Fifth Avenue, New York, N.Y.10010,2008.
[21]Jean-Pierre Fouque, George Papanicolaou, and K. Ronnie Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, The Edinburgh Building, Cambridge CB2 2RU, UK,2000.
[22]Avner Friedman, Stochastic differential equations and applications (volume I), Academic Press, New York, San Francisco, London,1975.
[23]Donatien Hainaut, Entry and exit decisions with switching regime cash flows, International Journal of Managerial Finance 8 (2012), no.1,58-72.
[24]J. Michael Harrison and David M. Kreps, Speculative Investor behavior in a Stock Market with Heterogeneous Expectations, The Quartely Journal Economics 92 (1978), no.2,323-336.
[25]Martin B Haugh and Andrew W Lo, Asset allocation and derivatives, Quantitative Finance 1 (2001), no.1,45-72.
[26]Steven L. Heston, A Closed-Form Solution for Options with Stochastic Volatility with Appli-cations to Bond and Currency Options, The Review of Financial Studies 6 (1993),327-343.
[27]David G. Hobson and L. C. G. Rogers, Complete Models with Stochastic Volatility, Mathe-maticial Finance 8 (1998),27-48.
[28]Elton P. Hsu, Stochastic Analysis on Manifolds, American Mathematical Society, Providence, Rhode Island,2002.
[29]John Hull and Alan White, The Pricing of Options on Assets with Stochastic Volatility, Journal of Finance 42 (1987),281-300.
[30]P. J. Hunt and J. E. Kennendy, Financial Derivatives in Theory and Practice, Revised Edition, John Wiley & Sons, Ltd, The Atrium, Souther Gate, Chicherster, West Sussex PO19 8SQ, England,2004.
[31]Timothy C. Johnson and Mihail Zervos, The explicit solution to a sequential switching prob-lem with non-smooth data, Stochastics:An International Journal of Probability and Stochas-tics Processes 82 (2010), no.1,69-109.
[32]Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus (second edition), Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, Hong Kong, Barcelona,1991.
[33]Fima C. Klebaner, Introduction to Stochastic Calculus with Application, Imperrial College Press,203 Electrical Engineering Building, Imperial College, London SW7 2BT,1998.
[34]Robert V. Kohn and Oana M. Papazoglu, On the equivalence of the static and dynamic asset allocation problems, Quantitative Finance 6 (2006), no.2,173-183.
[35]Hans Christian Kongsted, Entry and exit decisions under uncertainty:The limiting determin-istic case, Economics Letters 51 (1996),77-82.
[36]Sergei Levendorskii, Perpetual American options and real options under mean-reverting processes, Available at SSRN:http://ssrn.com/abstract=714321 or http://dx.doi.org/10.2139/ssrn.714321, May 04,2005.
[37]Jun Liu and Jun Pan, Dynamic derivative strategies, Journal of Financial Economics 69 (2003), no.3,401-430.
[38]Robert C. Merton, Lifetime portfolio selection under uncertainty:The continuous-time case, The Review of Economics and Statistics 51 (1969), no.3,247-257.
[39]David C. Nachman, Spanning and completeness with options, The Review of Financial Stud-ies 1 (1988), no.3,311-328.
[40]Bernt (?)ksendal and Angnes Sulem, Applied stochastic control of jump diffusions (second edition), Spriger, Verlag Berlin Heidelberg,2007.
[41]Robert S. Pindyck, Investments of uncertain cost, Journal of Financial Economics 34 (1993), no.1,53-76.
[42]Philip E. Protter, Stochastic integration and differential equations (second edition, version 2.1), Spriger, Verlag Berlin Heidelberg,2005.
[43]Sheldon M. Ross, An Elementary Introduction to Mathematical Finance:Options and Oth-er Topics, Second Edition, Cambridge University Press, Cambridge, New York, Melbour, Madrid, Cape Town, Singapore, Sao Paul, Delhi,2003.
[44]P. A. Samuelson, Proof that properly anticipated prices fluctuate randomly, Industrial Man-agement Review 6 (1965),41-50.
[45]Atle Seierstad, Stochastic control in discrete and continuous time, Spriger,233 Spring Street, New York, NY 10013, USA,2009.
[46]Hiroshi Shirakawa, Evaluation of Investment Opportunity under Entry and Exit Decisions, Mathematical analysis in economics,Surikaisekikenkyusho Kokyuroku 987 (1997),107-124.
[47]Sigbj(?)rn S(?)dal, Entry and Exit Decisions Based on a Discount Factor Approach, Journal of Economic Dynamics & Control 30 (2006),1963-1986.
[48]Elias M. Stein and Jeremy C. Stein, Stock Price Distributions with Stochastic Volatility:An Analytic Approach, The Review of Financial Studies 4 (1991),727-752.
[49]Andrianos E. Tsekrekos, The effect of mean reversion on entry and exit decisions under un-certainty, Journal of Economic Dynamics & Control 34 (2010),725-742.
[2]David Applebaum, Levy processes and stochastic calculus (second edition), Cambridge Uni-versity Press, The Edinburgh Building, Cambridge CB2 8RU, UK8,2009.
[3]George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists (6th ed.), Elsevier Academic Press,30 Corporate Drive, Suite 400, Burlington, MA 01803, USA; 525 B Street, Suite 1900, San Diego, California 92101-4495, USA; 84 Theobald's Road, London WC1X 8RR, UK,2005.
[4]L. Bachelier, Theorie de la speculation, Annales Scientifiques de l'E. N. S.17 (1900),21-28.
[5]Avner Bar-Ilan and William C. Strange, Investment lags, The American Economic Review 86 (1996), no.3,610-622.
[6]Fischer Black and Myron Scholes, The Valuation of Options and Corporate Liabilities, Jour-nal of Political Economy 81 (1973),637-654.
[7]John Black, Nigar Hashimzade, and Gareth D. Myles, Oxford:A Dictionary of Economics, Oxford University Press, Great Clarendon Street, Oxford OX2 6DP,2009.
[8]Svetlana Boyarchenko and Sergei Levendorskii, Irreversible decisions under uncertainty: Optimal stopping made easy, Spriger, Verlag Berlin Heidelberg,2007.
[9]Kjell Arne Brekke and Bernt (?)ksendal, Optimal switching in an economic activity under uncertainty, SIAM Journal on Control and Optimization 32 (1994), no.4,1021-1036.
[10]Terence Chan, Pricing Contingent Claims on Stocks Driven by Levy Processes, The Annals of Applied Probability 9(1999),504-528.
[11]Xi Chen and Robert V. Kohn, Asset price bubbles from heterogeneous beliefs about mean reversion rates, Finance and Stochastics 15 (2011), no.2,221-241.
[12]Yoon K. Choi and Seok Weon Lee, Investment and abandonment decisions with uncertain price and cost, Journal of Business Finance & Accounting 27 (2000), no.1-2,195-213.
[13]Marius Costeniuc, Michaela Schnetzer, and Luca Taschini, Entry and exit decision problem with implementation delay, Journal of Applied Probability 45 (2008), no.4,1039-1059.
[14]John C. Cox and Chi-fu Huang, Optimal consumption and portfolio policies when asset prices follow a diffusion process, Journal of Economic Theory 49 (1989), no.1,33-83.
[15]John C. Cox, Jonathan E. Ingersoll Jr., and Stephen A. Ross, A Theory of the Term Structure of Interest Rates, Econometrica 53 (1985), no.2,385-407.
[16]Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc.27 (1992), no.1,1-67.
[17]Annie Cuyt, Vigdis Brevik Petersen, Brigitte Verdonk, Haakon Waadeland, and William B. Jones, Handbook of Continued Fractions for Special Functions, Springer,2008.
[18]Avinash Dixit, Entry and Exit Decisions under Uncertainty, Journal of Political Economy 97 (1989), no.3,620-638.
[19]J. Kate Duckworth and Mihail Zervos, An Investment Model with Entry and Exit Decisions, Journal of Applied Probability 37 (2000),547-559.
[20]Steven N. Durlauf and Lawrence E. Blume, The New Palgrave Dictionary of Economics (2nd ed.) Volume I, Palgrave Macmillan, Houndmill, Basingstoke, Hampshite RG21 6XS and 175 Fifth Avenue, New York, N.Y.10010,2008.
[21]Jean-Pierre Fouque, George Papanicolaou, and K. Ronnie Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, The Edinburgh Building, Cambridge CB2 2RU, UK,2000.
[22]Avner Friedman, Stochastic differential equations and applications (volume I), Academic Press, New York, San Francisco, London,1975.
[23]Donatien Hainaut, Entry and exit decisions with switching regime cash flows, International Journal of Managerial Finance 8 (2012), no.1,58-72.
[24]J. Michael Harrison and David M. Kreps, Speculative Investor behavior in a Stock Market with Heterogeneous Expectations, The Quartely Journal Economics 92 (1978), no.2,323-336.
[25]Martin B Haugh and Andrew W Lo, Asset allocation and derivatives, Quantitative Finance 1 (2001), no.1,45-72.
[26]Steven L. Heston, A Closed-Form Solution for Options with Stochastic Volatility with Appli-cations to Bond and Currency Options, The Review of Financial Studies 6 (1993),327-343.
[27]David G. Hobson and L. C. G. Rogers, Complete Models with Stochastic Volatility, Mathe-maticial Finance 8 (1998),27-48.
[28]Elton P. Hsu, Stochastic Analysis on Manifolds, American Mathematical Society, Providence, Rhode Island,2002.
[29]John Hull and Alan White, The Pricing of Options on Assets with Stochastic Volatility, Journal of Finance 42 (1987),281-300.
[30]P. J. Hunt and J. E. Kennendy, Financial Derivatives in Theory and Practice, Revised Edition, John Wiley & Sons, Ltd, The Atrium, Souther Gate, Chicherster, West Sussex PO19 8SQ, England,2004.
[31]Timothy C. Johnson and Mihail Zervos, The explicit solution to a sequential switching prob-lem with non-smooth data, Stochastics:An International Journal of Probability and Stochas-tics Processes 82 (2010), no.1,69-109.
[32]Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus (second edition), Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, Hong Kong, Barcelona,1991.
[33]Fima C. Klebaner, Introduction to Stochastic Calculus with Application, Imperrial College Press,203 Electrical Engineering Building, Imperial College, London SW7 2BT,1998.
[34]Robert V. Kohn and Oana M. Papazoglu, On the equivalence of the static and dynamic asset allocation problems, Quantitative Finance 6 (2006), no.2,173-183.
[35]Hans Christian Kongsted, Entry and exit decisions under uncertainty:The limiting determin-istic case, Economics Letters 51 (1996),77-82.
[36]Sergei Levendorskii, Perpetual American options and real options under mean-reverting processes, Available at SSRN:http://ssrn.com/abstract=714321 or http://dx.doi.org/10.2139/ssrn.714321, May 04,2005.
[37]Jun Liu and Jun Pan, Dynamic derivative strategies, Journal of Financial Economics 69 (2003), no.3,401-430.
[38]Robert C. Merton, Lifetime portfolio selection under uncertainty:The continuous-time case, The Review of Economics and Statistics 51 (1969), no.3,247-257.
[39]David C. Nachman, Spanning and completeness with options, The Review of Financial Stud-ies 1 (1988), no.3,311-328.
[40]Bernt (?)ksendal and Angnes Sulem, Applied stochastic control of jump diffusions (second edition), Spriger, Verlag Berlin Heidelberg,2007.
[41]Robert S. Pindyck, Investments of uncertain cost, Journal of Financial Economics 34 (1993), no.1,53-76.
[42]Philip E. Protter, Stochastic integration and differential equations (second edition, version 2.1), Spriger, Verlag Berlin Heidelberg,2005.
[43]Sheldon M. Ross, An Elementary Introduction to Mathematical Finance:Options and Oth-er Topics, Second Edition, Cambridge University Press, Cambridge, New York, Melbour, Madrid, Cape Town, Singapore, Sao Paul, Delhi,2003.
[44]P. A. Samuelson, Proof that properly anticipated prices fluctuate randomly, Industrial Man-agement Review 6 (1965),41-50.
[45]Atle Seierstad, Stochastic control in discrete and continuous time, Spriger,233 Spring Street, New York, NY 10013, USA,2009.
[46]Hiroshi Shirakawa, Evaluation of Investment Opportunity under Entry and Exit Decisions, Mathematical analysis in economics,Surikaisekikenkyusho Kokyuroku 987 (1997),107-124.
[47]Sigbj(?)rn S(?)dal, Entry and Exit Decisions Based on a Discount Factor Approach, Journal of Economic Dynamics & Control 30 (2006),1963-1986.
[48]Elias M. Stein and Jeremy C. Stein, Stock Price Distributions with Stochastic Volatility:An Analytic Approach, The Review of Financial Studies 4 (1991),727-752.
[49]Andrianos E. Tsekrekos, The effect of mean reversion on entry and exit decisions under un-certainty, Journal of Economic Dynamics & Control 34 (2010),725-742.