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題名 單一資產與複資產的美式選擇權之評價
The Valuation of American Options on Single Asset and Multiple Assets
作者 劉宣谷
Liu, Hsuan Ku
貢獻者 劉明郎
Liu, Ming Long
劉宣谷
Liu, Hsuan Ku
關鍵詞 自由邊界問題
美式選擇權
美式交換型選擇權
混合型非線性整數規劃問題
非線性規劃問題
Free boundary problem
American option
American exchange option
nonlinear mixed integer programming
nonlinear programming
日期 2007
上傳時間 17-Sep-2009 13:49:47 (UTC+8)
摘要 過去的三十年間由於評價美式選擇權所產生的自由邊界問題已經有相當的研究成果。本論文將證明自由邊界問題的解為遞增函數。更進一步提出自由邊界凹性的嚴謹証明。利用我們的結論可以得知美式選擇權的最佳履約邊界對時間而言為嚴格遞減的凹函數。這個結果對可用來求導最佳履約邊界的漸近解。

對於美式交換選擇權，我們將其自由邊界問題轉換成單變數的積分方程，同時提供一個永續型美式交換選擇權的評價公式。對於有限時間的美式交換選擇權的最佳履約邊界，我們將提供一個接近到期日的漸近解並發展一個數值方法求其數值解。數值計算的結果顯示漸近解在接近到期日時與數值解非常接近。

對於評價美式選擇權，我們提出使用混合整數非線性規劃(MINLP)的模型，這個模型的最佳解同時提供賣方的完全避險策略、買方的最佳交易策略與美式選擇權的公平價格。因為求算MINLP模型的解需耗用大量的計算時間，我們證明此模型和其非線性規劃的寬鬆問題有相同的最佳解，所以只需求算寬鬆問題即可。觀察數值結果亦顯示非線性規劃的寬鬆問題可以大幅的降低計算的時間。此外，當市場的價格低於公平價格時，我們提出一個最小化賣方期望損失的數學規劃模型，此模型的解提供賣方最小化其期望損失的避險策略。
In the past three decades, a great deal of effort has been made on solving the free boundary problem (FBP) arising from American option valuation problems. In this dissertation, we show that the solutions, the price and the free boundary, of this FBP are increasing functions. Furthermore, we provide a rigorous verification that the free boundary of this problem is concave. Our results imply that the optimal exercise boundary of an American call is a
strictly decreasing concave function of time. These results will provide a useful information to obtain an asymptotic formula for the optimal exercise boundary.


For pricing of American exchange options (AEO), we convert the associated FBP into a single variable integral equation (IE) and provide a formula for valuating the perpetual AEO.
For the finite horizon AEO, we propose an asymptotic solution as time is near to expiration and develop a numerical method for its optimal exercise boundary.
Compared with the computational results, the values of our asymptotic solution are close to the computational results as time is near to expiration.


For valuating American options, we develop a mixed integer nonlinear programming (MINLP) model. The solution of the MINLP model provides a hedging portfolio for writers, the optimal trading strategy for buyers, and the fair price for American options at the same time. We show that it can be solved by its nonlinear programming (NLP) relaxation. The numerical results reveal that the use of NLP relaxation reduces the computation time rapidly. Moreover, when the market price is less than the fair price, we propose
a minimum expected loss model. The solution of this model provides a hedging strategy that minimizes the expected loss for the writer.
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[2] G. Barles and H. M. Soner, Option Pricing with Transaction Costs and a Nonlinear Black-Scholes Equation, Finance and Stochstics 2 (1998) 369-397.
[3] G. Barone-Adesi and R. Whaley, Efficient Analytic Approximation of American Option Values, Journal of Finance 42 (1987) 301-320.
[4] M. Benaroch and R. J. Kauffman, Justifying Electronic Banking Network Expansion Using Real Options Analysis, MIS Quarterly 24 (2000) 197-225.
[5] P. T. Berg, Deductibles and the Inverse Gaussian Distribution, Astin Bulletin 24 (1994) 319-323.
[6] F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81 (1973) 637–654.
[7] M. Broadie and J. Detemple, The Valuation of American Options on Multiple Assets, Mathematical Finance 7 (1997) 241-286.
[8] A. Brooke, D. Kendrick, and A. Meeraus, GAMS–A User’s Guide, Scientific Press, Redwood City, CA, 1988.
[9] P. R. Carr, R. Jarrow, and R. Myneni, Alternative Characterizations of American Put Options, Mathematical Finance 2 (1992) 87–106.
[10] C. F. Chang, Recovering Risk-Neutral Probability Distribution from Market Option Prices, Master Degree Thesis, National Chengchi University, Taiwan, 2006.
[11] X. Chen and J. Chadam, A Mathematical Analysis for the Optimal Exercise Boundary of American Put Option, SIAM Journal of Mathematical Analysis 5 (2007) 1613-1641.
[12] X. Chen, J. Chadam, L. Jiang, and W. Zheng, Convexity of the Exercise Boundary of the American Put on a Zero Dividen Asset, Mathematical Finance, 18(2008) 185-197.
[13] I. J. Chen, Option Pricing by Game Theory, Master Degree Thesis, National Chengchi University, Taiwan, 2007.
[14] J. Cox, S. Ross, and M. Rubinstein, Option Pricing: A Simplified Approach, Journal of Financial Economics 7 (1979) 263-384.
[15] M. A. H. Dempster and J. P. Hutton, Pricing American Stock Option by Linear Programming, Mathematical Finance 9 (1999) 229-254.
[16] M. A. H. Dempster, and D. G. Richards, Pricing American Stock Option Fitting the Smile, Mathematical Finance 10 (2000) 157-177.
[17] E. Ekstr¨om, Convexity of the Optimal Stopping Boundary for the American Put Option, Journal of Mathematical Analysis and Applications 299 (2004) 147-156.
[18] J. D. Evans, R. Kuske, and J. B. Keller, American Options with Dividends Near Expiry, Mathematical Finance 12 (2002) 219-237.
[19] D. Flamours and D. Giamouridis, Estimating Implied PDFs from American Option on Futures: A New Semiparametric Approach, The Journal of Future Market 22 (2002) 1-30.
[20] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall Inc (1964).
[21] A. Friedman, Analyticity of the Free Boundary for the Stefan Problem, Archive for Rational Mechanics and Analysis 61 (1976) 97-125.
[22] A. Friedman and R. Jensen, Convexity of the Free Foundary in the Stefan Problem and in the Dam Problem, Archive for Rational Mechanics and Analysis 67 (1977) 1-24.
[23] R. Gesker and H. E. Johnson, The American Put Option Valued Analytically, Journal of Finance 39 (1984) 1511-1524.
[24] H. U. Gerber and E. S. W. Shiu, Martingale Approach to Pricing Perpetual American Option, Astin Bulletin 24 (1994) 195-220.
[25] H. U. Gerber and E. S. W. Shiu, Martingale Approach to Pricing Perpetual American Options on Two Stocks, Mathematical Finance 6 (1996) 303-322.
[26] J. Goodman and D. N. Ostrov, On the Early Exercise Boundary of the American Put Option, SIAM Journal of Applied Mathematics 62 (2002) 1832-1835.
[27] S. R. Grenadier and A. M. Weiss, Investment in Technological Innovations: An Option Pricing Approach, Journal of Financial Economics 44 (1997) 397-41.
[28] Z. Guo and S. H. Yen, Determining Institutional Investor’s Dynamic Asset Allocation, Journal of Financial Studies 14 (2006) 77-93.
[29] S. D. Hodges and A. Neuberger, Hedging Option Portfolios in the Presence of Transaction Costs, Review of Futures Markets 8 (1989) 222–239.
[30] T. Hoggard, E. Whalley and P. Wolmott, Hedeging Option Portfolio in the Presence of Transaction Costs, Advances in Futures and Options Research 7 (1994) 21–35.
[31] J. Z. Huang, M. G. Subrahmanam, and G. G. Yu, Pricing and Hedging American Options: A Recursive Integration Method, The Review of Financial Studies 9 (1996) 277-300.
[32] S. D. Jacka, Optimal Stopping and the American Put, Mathematical Finance 1 (1992) 1-14.
[33] J. C. Jackwerth and M. Rubinstein, Recovering Probability Distributions from Option Prices, Journal of Finance 51 (1996) 1611–1631.
[34] R. Jarrow and A. Rudd, Option Pricing, Jones-Irwin, Homewood, IL (1983).
[35] N. Ju and R. Zhong, An Approximate Formula for Pricing American Options, Journal of Derivatives 7 (1999) 31–40.
[36] I. J. Kim, An Analytic Valuation of American Option, The Review of Financial Studies 3 (1990) 547–572.
[37] C. Knessl, Asymptotic Analysis of the American Call Option with Dividends, European Journal of Applied Mathematics 13 (2002) 587–616.
[38] A. J. King, Duality and Martingales: A Stochastic Programming Perspective on Contingent Claims, Mathematical Programming Ser. B 91 (2002) 543–562.
[39] I. I. Koloder, Free Boundary Problem for the Heat Equation with Applications to Problems of Change of Phase, Communications in Pure and Applied Mathematics 9 (1956) 1–31.
[40] D. B. Kotlow, A Free Boundary Problem Connected with Optimal Stopping Problem for Difussion Process, Transactions of the American Mathematical Society 184 (1973) 457–478.
[41] R. S. Kuske and J. B. Keller, Optimal Exercise Boundary for an American Put, Applied Mathematical Finance 5 (1998) 107–116.
[42] Y. K. Kwok, Mathematical Models of Financial Derivatives, Springer (1998).
[43] H. E. Leland, Option Pricing and Replication with Transaction Costs, Journal of Finance 40 (1985) 1283–1301.
[44] Y. J. Liao, Recovering Risk-Neutral Probability via Biobjective Programming Model, Master Degree Thesis, National Chengchi University, Taiwan, 2006.
[45] K. F. Liu, Building a Consistent Pricing Model from Observed Option Prices via Linear Programming, Master Degree Thesis, National Chengchi University, Taiwan, 2005.
[46] M. L. Liu and H. K. Liu, An Asymptotic Solution for an Integral Equation Arising form a Finite Maturity American Exchange Option, in review process, 2007.
[47] M. L. Liu and H. K. Liu, Solving a Two Variables Free Boundary Problem Arising in a Perpetual American Exchange Option Pricing Model, accepted by Taiwanese Journal of Mathematics, 2007.
[48] M. L. Liu, and H. K. Liu, Optimal Trading Strategy of Option Portolfios, in review process, 2007.
[49] M. L. Liu, G. I. Yang, and H. K. Liu, Linear Programming Model for Option Trading Strategy, INFORMS Taiwan Chapters Annual Meeting, Taipei, Taiwan, 2005.
[50] M. L. Liu and H. K. Liu, Value an Ameircna Option by Duality Theory, in review process, 2007.
[51] M. L. Liu and H. K. Liu, Pricing Contingent Claim Using Duality of the Stochastic Programming, International Sympsium on Management Engineering, Kitakyushu, Janpan, 2007.
[52] J. Lee and D. A. Paxson, Valuation of R&D Real American Sequential Exchange Options, R&D Management 31 (2001) 191–200.
[53] F. A. Longstaff and E. S. Schwartz, Valuing American Options by Simulation: A Simple Least-Squares Approach, The Review of Financial Studies 14 (2001) 113-147.
[54] D. G. Luenberger, Investment Science Oxford University Press, New York, 1998.
[55] W. Margrabe, The Pricing of an Option to Exchange One Asset for Another, Journal of Finance 33 (1978) 177-186.
[56] W. MacMillan, Analytic Approximation for the American Put Option, Advance in Futures and Options Research 1 (1986) 119–141.
[57] R. McDonald and D. Siegel, The Value of Waiting to Invest, Quarterly Journal of Economics 101 (1986) 707-727.
[58] W. R. Melick and C. P. Thomas, Recovering an Asset’s Implied PDF from Option Prices: AnApplication to Crude Oil during the Gulf Crisis, Journal of Financial and Quantitative Ahalysis 32 (1997) 91-115.
[59] R. Merton, The Theory of Rational Option Pricing, The Bell Journal of Economics and Management science 4 (1973) 141–183.
[60] V. Messia and V. Bosetti, Uncertianty and Option Value in Land Allocation Problems Annals of Operations Research 124 (2003) 165-181.
[61] C. S. Pedersen, Separating Risk and Return in the CAPM: A General Utility-Based Model, European Journal of Operational Research 123 (2000) 628-639.
[62] T. Pennanen and A. King, Arbitrage Pricing of American Contingent Claims in Incomplete Markets-A Convex Optimization Approach, Stochastic Programming E-Print Series 14 (2004).
[63] H. Rhys, J. Song, and I. Jindrichovska, The Time of Real Option Exercise: Some Recent Developments, The Engineering Economist 47 (2002) 436-450.
[64] L. C. G. Rogers, Equivalent Martingale Measures and No-Arbitrage, Stochastics and Stochasitcs Reports 51 (1994) 41-51.
[65] L. C. G. Rogers, Monte Carlo Valuation of American Options, Mathematical Finance 12 (2002) 271-286.
[66] M. Rubinstein, Implied Binomial Trees, Journal of Finance 49 (1994) 771–818.
[67] S. Villeneuve, Exercise Regions of American Options on Several Assets, Finance and Stochstics 3 (1999) 295-322.
[68] T. Wang and R. Neufville, Building Real Options into Physical Systems with Stochastic Mixed-Integer Programming, 8th Real Options Annual International Conference, Monetreal, Canada, 2004.
[69] J. Wu and S. Sen, A Stochastic Programming Model for Currency Option Hedging, Annals of Opretions Research 100 (2000) 227-249.
[70] I. E. Zhang and T. Li, Pricing and Hedging American Options Analytically: A Singular Perturbation Method, working paper (2006).
描述 博士
國立政治大學
應用數學研究所
90751501
96
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0907515013
資料類型 thesis
dc.contributor.advisor 劉明郎zh_TW
dc.contributor.advisor Liu, Ming Longen_US
dc.contributor.author (Authors) 劉宣谷zh_TW
dc.contributor.author (Authors) Liu, Hsuan Kuen_US
dc.creator (作者) 劉宣谷zh_TW
dc.creator (作者) Liu, Hsuan Kuen_US
dc.date (日期) 2007en_US
dc.date.accessioned 17-Sep-2009 13:49:47 (UTC+8)-
dc.date.available 17-Sep-2009 13:49:47 (UTC+8)-
dc.date.issued (上傳時間) 17-Sep-2009 13:49:47 (UTC+8)-
dc.identifier (Other Identifiers) G0907515013en_US
dc.identifier.uri (URI) https://nccur.lib.nccu.edu.tw/handle/140.119/32602-
dc.description (描述) 博士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學研究所zh_TW
dc.description (描述) 90751501zh_TW
dc.description (描述) 96zh_TW
dc.description.abstract (摘要) 過去的三十年間由於評價美式選擇權所產生的自由邊界問題已經有相當的研究成果。本論文將證明自由邊界問題的解為遞增函數。更進一步提出自由邊界凹性的嚴謹証明。利用我們的結論可以得知美式選擇權的最佳履約邊界對時間而言為嚴格遞減的凹函數。這個結果對可用來求導最佳履約邊界的漸近解。

對於美式交換選擇權，我們將其自由邊界問題轉換成單變數的積分方程，同時提供一個永續型美式交換選擇權的評價公式。對於有限時間的美式交換選擇權的最佳履約邊界，我們將提供一個接近到期日的漸近解並發展一個數值方法求其數值解。數值計算的結果顯示漸近解在接近到期日時與數值解非常接近。

對於評價美式選擇權，我們提出使用混合整數非線性規劃(MINLP)的模型，這個模型的最佳解同時提供賣方的完全避險策略、買方的最佳交易策略與美式選擇權的公平價格。因為求算MINLP模型的解需耗用大量的計算時間，我們證明此模型和其非線性規劃的寬鬆問題有相同的最佳解，所以只需求算寬鬆問題即可。觀察數值結果亦顯示非線性規劃的寬鬆問題可以大幅的降低計算的時間。此外，當市場的價格低於公平價格時，我們提出一個最小化賣方期望損失的數學規劃模型，此模型的解提供賣方最小化其期望損失的避險策略。		zh_TW
dc.description.abstract (摘要) In the past three decades, a great deal of effort has been made on solving the free boundary problem (FBP) arising from American option valuation problems. In this dissertation, we show that the solutions, the price and the free boundary, of this FBP are increasing functions. Furthermore, we provide a rigorous verification that the free boundary of this problem is concave. Our results imply that the optimal exercise boundary of an American call is a
strictly decreasing concave function of time. These results will provide a useful information to obtain an asymptotic formula for the optimal exercise boundary.


For pricing of American exchange options (AEO), we convert the associated FBP into a single variable integral equation (IE) and provide a formula for valuating the perpetual AEO.
For the finite horizon AEO, we propose an asymptotic solution as time is near to expiration and develop a numerical method for its optimal exercise boundary.
Compared with the computational results, the values of our asymptotic solution are close to the computational results as time is near to expiration.


For valuating American options, we develop a mixed integer nonlinear programming (MINLP) model. The solution of the MINLP model provides a hedging portfolio for writers, the optimal trading strategy for buyers, and the fair price for American options at the same time. We show that it can be solved by its nonlinear programming (NLP) relaxation. The numerical results reveal that the use of NLP relaxation reduces the computation time rapidly. Moreover, when the market price is less than the fair price, we propose
a minimum expected loss model. The solution of this model provides a hedging strategy that minimizes the expected loss for the writer.		en_US
dc.description.tableofcontents Title Page i
Abstract iii
Table of Contents v
1 Introduction 1
1.1 Motivations and Research Objectives 1
1.2 Major Results 3
1.2.1 Single Asset American Option Pricing Problems 3
1.2.2 American Exchange Option Pricing Problems 4
1.2.3 Optimization Approaches for Pricing American Style Options 4
1.3 Organization of the Dissertation 5
2 Literature Review 7
2.1 American Option Pricing Problems 7
2.2 American Exchange Option Pricing Problems 9
2.3 Optimization Approaches for Option Pricing Problems 10
2.4 Applications to Real Options 14
3 Single Asset American Style Option Pricing Problems 16
3.1 Free Boundary Problems Arising from Pricing of American Options 17
3.2 Properties of the Solution 20
3.3 Concavity of the Free Boundary 25
3.4 Application to American Call Option 29
3.5 An Asymptotic Solution for the Early Exercise Boundary 33
4 American Exchange Option Pricing Problems 39
4.1 The Formulation of AEO 40
4.2 Properties of the Free Boundary 43
4.3 The Integral Equation 45
4.4 An Asymptotic Solution of Finite-Lived AEO 50
4.5 The Exact Solution of the Perpetual AEO 58
4.6 Integral Recursive Methods 63
4.7 Numerical Results 65
5 Optimization Approaches for Pricing an Option 68
5.1 Notations 69
5.2 Binomial Pricing Approach 70
5.3 MINLP Valuation Models 72
5.4 Writer’s Problems 79
5.5 Numerical Results 81
6 Conclusions and Future Researches 87
6.1 Conclusions 87
6.1.1 Single Asset American Option Pricing Problems 87
6.1.2 American Exchange Option Pricing Problems 88
6.1.3 Optimization Approaches for Pricing an Option 88
6.2 Future Researches 89
6.2.1 Incomplete Markets 89
6.2.2 American Spread Option Valuation Problems 89
6.2.3 Martingale Probability Measure for the American option 90
Bibliography		zh_TW
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dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0907515013en_US
dc.subject (關鍵詞) 自由邊界問題zh_TW
dc.subject (關鍵詞) 美式選擇權zh_TW
dc.subject (關鍵詞) 美式交換型選擇權zh_TW
dc.subject (關鍵詞) 混合型非線性整數規劃問題zh_TW
dc.subject (關鍵詞) 非線性規劃問題zh_TW
dc.subject (關鍵詞) Free boundary problemen_US
dc.subject (關鍵詞) American optionen_US
dc.subject (關鍵詞) American exchange optionen_US
dc.subject (關鍵詞) nonlinear mixed integer programmingen_US
dc.subject (關鍵詞) nonlinear programmingen_US
dc.title (題名) 單一資產與複資產的美式選擇權之評價zh_TW
dc.title (題名) The Valuation of American Options on Single Asset and Multiple Assetsen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) [1] K. J. Arrow and A. C. Fisher, Environmental Preservation, Uncertainty, and Irreversibility, Quarterly Journal of Economics 88 (1974) 312-319.zh_TW
dc.relation.reference (參考文獻) [2] G. Barles and H. M. Soner, Option Pricing with Transaction Costs and a Nonlinear Black-Scholes Equation, Finance and Stochstics 2 (1998) 369-397.zh_TW
dc.relation.reference (參考文獻) [3] G. Barone-Adesi and R. Whaley, Efficient Analytic Approximation of American Option Values, Journal of Finance 42 (1987) 301-320.zh_TW
dc.relation.reference (參考文獻) [4] M. Benaroch and R. J. Kauffman, Justifying Electronic Banking Network Expansion Using Real Options Analysis, MIS Quarterly 24 (2000) 197-225.zh_TW
dc.relation.reference (參考文獻) [5] P. T. Berg, Deductibles and the Inverse Gaussian Distribution, Astin Bulletin 24 (1994) 319-323.zh_TW
dc.relation.reference (參考文獻) [6] F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81 (1973) 637–654.zh_TW
dc.relation.reference (參考文獻) [7] M. Broadie and J. Detemple, The Valuation of American Options on Multiple Assets, Mathematical Finance 7 (1997) 241-286.zh_TW
dc.relation.reference (參考文獻) [8] A. Brooke, D. Kendrick, and A. Meeraus, GAMS–A User’s Guide, Scientific Press, Redwood City, CA, 1988.zh_TW
dc.relation.reference (參考文獻) [9] P. R. Carr, R. Jarrow, and R. Myneni, Alternative Characterizations of American Put Options, Mathematical Finance 2 (1992) 87–106.zh_TW
dc.relation.reference (參考文獻) [10] C. F. Chang, Recovering Risk-Neutral Probability Distribution from Market Option Prices, Master Degree Thesis, National Chengchi University, Taiwan, 2006.zh_TW
dc.relation.reference (參考文獻) [11] X. Chen and J. Chadam, A Mathematical Analysis for the Optimal Exercise Boundary of American Put Option, SIAM Journal of Mathematical Analysis 5 (2007) 1613-1641.zh_TW
dc.relation.reference (參考文獻) [12] X. Chen, J. Chadam, L. Jiang, and W. Zheng, Convexity of the Exercise Boundary of the American Put on a Zero Dividen Asset, Mathematical Finance, 18(2008) 185-197.zh_TW
dc.relation.reference (參考文獻) [13] I. J. Chen, Option Pricing by Game Theory, Master Degree Thesis, National Chengchi University, Taiwan, 2007.zh_TW
dc.relation.reference (參考文獻) [14] J. Cox, S. Ross, and M. Rubinstein, Option Pricing: A Simplified Approach, Journal of Financial Economics 7 (1979) 263-384.zh_TW
dc.relation.reference (參考文獻) [15] M. A. H. Dempster and J. P. Hutton, Pricing American Stock Option by Linear Programming, Mathematical Finance 9 (1999) 229-254.zh_TW
dc.relation.reference (參考文獻) [16] M. A. H. Dempster, and D. G. Richards, Pricing American Stock Option Fitting the Smile, Mathematical Finance 10 (2000) 157-177.zh_TW
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