臺灣博碩士論文加值系統

從一般投資者的觀點我們會在離散的時間投資某些資產, 因此, 我們通常從市場上觀察到離散的資訊. 在這篇論文裡對於離散型的財務模型我們給投資者三個主要的結果. 另一方面, 在一般財務機構的觀點下, 我們要去面對關於在擁有消費情況下, 如何去降低投資的風險.
最後我們和 Follmer 和 Sondermann (1986) 這篇論文有類似的結果.

From the perspectives of the investors, we invest in some assets in discrete time and thus, we usually observe the discrete information from the market. In this paper we give the investors three main results for our discrete financial model. On the other hand, from the perspectives of the financial institutions, we face the problem about how to reduce the risk of the investment with consumptions. Then we have similar conclusion with Follmer and Sondermann (1986).

Contents

Chapter 1. Introduction 1

Chapter 2. Optimal Strategy with Partial Information in Discrete Simple Model 3
2.1. Model Setup 3
2.2. Trading Strategy and Backward Inductions 5
2.3. Optimal Strategy with Partial Information Under Risk-Neutral Utility 6
2.4. Optimal Strategy with Partial Information Under Risk-Averse Utility 8

Chapter 3. Optimal Hedging Strategy with Partial Information in Discrete Simple
Model 13
3.1. Decomposition of an Attainable Contingent Claim 13
3.2. Optimal Hedging Strategy with Partial Information Under Backward
Induction 15
3.2.1. Risk-Neutral Utility 16
3.2.2. Risk-Averse Utility 17

Chapter 4. Optimal Strategy with Partial Information and Optional Projection
Method in Discrete Simple Model 21
4.1. Model Setup 21
4.2. Optional Projection Method 22
4.3. The Gaussian Case 24

Chapter 5. Risk-Minimizing Hedging Strategy with Consumption in Discrete
Financial Model 29
5.1. Wealth and Cost 29
5.2. Decomposition of a Contingent Claim 30
5.3. A Sequential Procedure for Risk-Minimizing Hedging Strategy 31

Chapter 6. Future Work 37

Bibliography 39

Bibliography
[1] Abhay G. Bhatt. Linear Filtering with Ornstein Ulhenbeck Process as Noise. Submitted for Publication. 2003.
[2] Bernt Oksendal. Stochastic Differential Equations : An Introduction with Applications , Sixth Edition. Springer. 2005.
[3] Damien Lamberton and Bernard Lapeyre. Introduction Stochastic Calculus Applied to Finance, First Edition. Springer. 1996.
[4] Hai-Bo Cheng. Risk-minimizing Duplicating Strategies under Partial Information. Journal of Fudan University, Natural Science. 43, 344-355, 2004.
[5] Hans Follmer and Dieter Sondermann. Hedging of Non-Redundant Contingent Claims. Contributions to Mathematical Economics. 205-223, 1986.
[6] Ioannis Karatzas and Steven E. Shreve. Brownian Motion and Stochastic Calculus, Second Edition. Springer. 1988.
[7] Ioannis Karatzas and Xing Xiong Xue. A Note on Utility Maximization under Partial Observations. Mathematical Finance 1, 57-70, 1991.
[8] Jaksa Cvitanic and Ioannis Karatzas. Convex Duality in Continuous Portfolio Optimization. The Annals of Applied Probability 2, 767-818, 1992.
[9] Martin Baxter and Andrew Rennie. Financial Calculus An Introduction to Derivative Pricing, First Edition. Cambridge University Press. 1996.
[10] Martin Schweizer. Risk-Minimizing Hedging Strategies under Restricted Information. Mathematical Finance. 4, 327-342, 1994.
[11] Martin Schweizer. Variance-Optimal Hedging in Discrete Time. Mathematics of Operations Research. 20, 1-32, 1995.
[12] Peter Lakner. Utility Maximization with Partial Information. Stochastic Processes and Their Applications 56, 247-273, 1995.
[13] Robert J. Elliott, Lakhdar Aggoun and John B.Moore. Hidden Markov Models Estimation and Control, Vol. 1, 1st edn. Springer-Verlag, 2005.
[14] Steven E. Shreve. Stochastic Calculus for Finance : The Binomial Asset Pricing Model, Vol. 1, 1st edn. Springer, 2005.