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 隨著衍生性商品在非流動性市場的交易量增加，當評價其價值時，必須將市場的流動性納入考量。Krakovsky(1999)將市場的流動性納入Black-Scholes(1973)的分析中，並推導出考慮流動性之Black-Scholes選擇權評價模型，此為一非線性的偏微分方程式，對於簡單的歐式選擇權也難以求出其封閉解，故本文嘗試使用擾動法將此非線性的偏微分方程式重新整理，並利用有限差分法之理論來分析所求得之數值解。
 Pricing market liquidity into derivatives has become essential as the use of equity and fixed-income derivatives in illiquid markets has increased. However, the Liquidity-adjusted Black-Scholes equation is non-linear and cannot be solved in a closed form even for simple European-style options due to the complexity of the gamma term. In the paper, we try to apply perturbation method in the Liquidity-adjusted Black-Scholes equation, and then use finite difference methods to solve the partial differential equation numerically.
 1. INTRODUCTION ........ 4 2. LIQUIDITY - ADJUSTED BLACK - SCHOLES EQUATION ........ 5 3. METHODOLOGY 8 3.1 Perturbation Method ........ 8 3.2 The initial and Boundary Conditions ........ 11 3.3 Logarithmic Transform ........ 13 3.4 The Weighted Average Method ........ 14 3.5 Algorithm ........ 17 4. NUMERICAL RESULT 21 4.1 Call Option price, Delta, and Gamma ........ 21 4.2 Put-Call Parity ........ 24
 @Article{Krakovsky,author = {A. Krakovsky},title = {Pricing Liquidity into Derivatives},journal = {Risk},year = 1999,month = {December},volume = 12(12),pages = {65--67}}@InCollection{ZvanVetzalForsyth1998book,author = {R. Zvan and K. Vetzal and P. Forsyth},title = {Swing Low, Swing High},booktitle = {Hedging with Trees},publisher = {Risk books},year = 1998,pages = {75--79}}@Book{Lomov,author = {S.A. Lomov},title = {Introduction to the General Theory of Singular Perturbations},publisher = {American Mathematical Society},year = 1992}@Book{Nayfeh,author = {A.H. Nayfeh},title = {Introduction to perturbation Techniques},publisher = {Wiley Classics Library},year = 1981}@Book{Morton,author = {K.W. Morton and D.F. Mayers},title = {Numerical Solution of Partial Differential Equations},publisher = {Cambridge University},year = 1994}@Book{Burden,author = {R.L. Burden and J.D. Faires},title = {Numerical Analysis},publisher = {Brooks/Cole},year = 1997,edition = {sixth}}@Book{TavellaRandall2000,author = {D. Tavella and C. Randall},title = {Pricing Financial Instruments: The Finite Difference Method},publisher = {John Wiley \& Sons.},year = 2000}@Book{Hull,author = {J.C. Hull},title = {Options, Futures, and Other Derivatives},publisher = {Prentice-Hall},year = 2000,edition = {fourth}}@Article{Black,author = {F. Black and M. Scholes},title = {The pricing of options and corporate liabilities},journal = {Journal of Political Economy},year = 1973,volume = 81,pages = {637--659}}@Article{Kangro,author = {R. Kangro and R. Nicolaides},title = {Far field boundary conditions for Black-Scholes equations},journal = {SIAM journal on numerical analysis},year = 2000,volume = 38(4),pages = {1357--1368}}@Article{HullWhite1990,author = {J.C. Hull and A. White},title = {Valuing Derivative Securities Using the Explicit FiniteDifference Method},journal = {Journal of Financial and Quantitative Analysis},year = 1990,month = {March},pages = {87--100}}@Article{ABWhite1979,author = {A.B. White},title = {On selection of equidistributing meshes for two-point boundary-value problems},journal = {SIAM J. Numer. Anal.},year = 1979,volume = 16,pages = {472--502}}@Article{DewynneWilmott1993,author = {J.N. Dewynne and P. Wilmott},title = {Partial to the Exotic},journal = {Risk},year = 1993,volume = 6,month = {March},pages = {38--46}}[13]蘇敏娟, "考慮流動性之選擇權評價模型：顯性有限差分法之探討",國立中正大學應用數學研究所, 民國89年.[14]林淑卿, "Black-Scholes選擇權評價模型之數值探討",國立中正大學應用數學研究所, 民國89年.[15]王錦秋,"Moving Mesh and Grid Adaption : Survey and Application",國立中正大學應用數學研究所, 民國88年.[16]王儷容．沈中華譯, "金融期貨與選擇權市場",五南圖書出版公司, 民國87年.
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