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研究生:邱奕豪
研究生(外文):Yi-Hao Ciou
論文名稱:K步可預期資訊下的最佳停時策略在美式選擇權上的應用
論文名稱(外文):Optimal Look-ahead Stopping Rule and Its Application to American Option
指導教授:鄭宗琳鄭宗琳引用關係
指導教授(外文):Tsung-Lin Cheng
學位類別:碩士
校院名稱:國立彰化師範大學
系所名稱:數學系所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2011
畢業學年度:99
語文別:英文
論文頁數:31
中文關鍵詞:最佳停止規則k步可預期資訊下的停止規則價格策略
外文關鍵詞:Optimal stopping rulek-step look-ahead stopping rulepricing strategy
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一個停止規則是一個非負的整數值隨機變數,而最佳停止規則是使得給定的隨機過程,會有最大期望值的停止規則。向後歸納法,又稱作"Snell envelope",提供一個方法去發現給定過程停止的最佳時間。 在本篇論文,我們考慮更一般情形的停止規則集合,稱作"k步可預期資訊下的停止規則",並且研究它們的基本性質。更進一步地,相似於Bensoussan和 Karatzas分別在西元1984和1988的這兩篇文章,我們推廣美式選權的價格策略到k步可預期資訊下停止規則的集合。
A stopping rule is a nonnegative integer-valued random variable and an optimal stopping rule is a stopping rule which maximizes expectation of the underlying process. Backward induction (Chow, Robbins and Siegmund, 1971), or dubbed as the Snell envelope, provides itself as a rule to
find the optimal time to stop the underlying stochastic sequence. In this thesis, we consider a more general class of stopping rules called "k-step look ahead stopping rule" and study their fundamental properties. Moreover, similar to Bensoussan (1984) and Karatzas (1988), we extend the pricing strategy of American options to the class comprising k-step look ahead stopping rules.
1 Introduction 1
2 A survey on optimal stopping rule and Snell envelope 4
2.1 The nite case of backward induction . . . . . . 4
2.2 Snell envelope . . . . .. . . . 4
2.3 American option . . . . . . . . 5
2.4 What is an equivalent martingale measure? . . . . 6
3 Optimal k-step look-ahead stopping rules 8
4 An application to American option 19
4.1 Application . . . . . . . . . 19
4.2 Simulation . . .. . . . . . . 20
References 23
[1] Bossaerts (1989), Simulation estimators of optimal early exercise, working paper, Carnegie-Mellon University.
[2] Broadie, M. and Glasserman, P. (1997), Pricing American-style securities using simulation, Journal of Economic Daynamics and Control 21 1323-1997.
[3] Broadie, M., Glasserman, P. and Jain G. (1997), Enhanced Monte-Carlo
estimates for American option prices, Journal of derivatives 5 25-44.
[4] Boshuizen, F.A. (1991), Prophet and minimax problems in optimal stopping theory. Ph.D-thesis, Vrije Universiteit, Amsterdam.
[5] Bensoussan, A. (1984), On the theory of option pricing, Acta Appl. Math. 2 139-158.
[6] Chow, Y.S., Robbins, H. and Siegmund, D. (1971), Great expectations: The theory of optimal stopping. Houghton, Boston.
[7] Chow, Y.S., Teicher, H. (1988), Probability Theory, Springer-Verlag. New York, secound edition.
[8] Cox, J., Ingersoll, J.E. and Ross, S.A. (1985), The intertemporal general equilibrium model of asset prices, Econometrica 53 363-384.
[9] Cox, J., and Ross, S.A. (1976), The valuation of options for alternative stochastic processes, Journal of Financial Economics 3 145-166.
[10] Carr, P., (1998), Randomization and the American Put, Review of Financial Studies 11 597-626.
[11] Clement, E., Lamberton, D. and Protter, P.(2002), An analysis of a least squares regression method for American option pricing, Finance and Stochastic 6 449-471.
[12] Carriere, J. (1996), Valuation of early-exercise price of options Using Simulation and nonparametric regression, Mathematics and Economics 19 19-30.
[13] Dynkin, E.B. (1963), Optimal choice of the moment of a Markov process, Dokl. Akad. NaukSSSR 150 238-240.
[14] Grigelionis, B.I. & Shiryaev, A.N.(1966), On the Stefan problem and optimal stopping rules for Markov processes. Theory Probab. Appl. 11 541-558.
[15] Guo, X. and Liu, J. (2005), Stopping at the maximum of Geometric Brownian Motion when singles are received, J. Appl. Prob. 42 826-838.
[16] Hill, T.P. (1987), Expectation inequalities associated with prophet problems, Stoch. Anal. Appl. 5 299-310.
[17] Irle, A. (1979), Monotone stopping problems and continuous time processes, Z. Wahrsch. Verw. Gebiete 48 49-56.
[18] Karatzas, I. (1988), On the pricing of American options, Appl. Math. Optim 17 37-60.
[19] Longsta, F.A. and Schwartz, E.S. (2001), Valuing American options by simulation: a simple least-squares approach, The Review of Financial Studies 14 113-147.
[20] Myneni R. (1992), The price of the American option, Ann. Appl. Prob. 2 1-23.
[21] Mckean, H.P. (1965), A free-boundary problem for the heat equation arising from the problem of mathematical economics, Industrial Managem, Review 6 32-39.
[22] Neveu, J. (1975), Discrete-parameter martingales. North Holland, Amsterdam.
[23] Pedersen, J.L. (1975), Optimal stopping problems for time-homogeneous diusions: a review, Department of Mathematics, ETHZenturn, CH8092 Zurich, Switzerland.
[24] Roters, M. (1998), Optimal stopping in a dice game, J. Appl. Probab. 35 299-235.
[25] Snell, J.L. (1952), Application of martingale system theorem, Trans. Amer. Math. Soc. 73 293-312.
[26] van Moerbeke, P. (1974), Optimal stopping and free boundary problems, Rocky Mountain. J. Math. 4 539-578.
[27] Wesollowski, J., 1999. Poisson Process via Martingale and Related Characteristics. J. Appl. Prob. 36, 919-926.

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