呂育道臺灣大學：資訊工程學研究所許為元Hsu, Wei-YuanWei-YuanHsu2007-11-262018-07-052007-11-262018-07-052007http://ntur.lib.ntu.edu.tw//handle/246246/53693路徑相依衍生性金融商品的報酬與其標的資產的歷史價值有強烈的相關性。在定價此類型的衍生性金融商品時，需要將歷史價格資料編碼成各種狀態。亞式選擇權是種強烈的路徑相依衍生性金融商品。雖然已存有高效率的數值方法以及逼近公式可用來定價亞式選擇權，但這些方法卻沒有保證一定會收斂到正確價格上。 格點演算法可以用來定價亞式選擇權。將產品從開始至到期日的時間切成$n$期。最好的精確無誤且收斂的演算法可在2^{O(sqrt{n},)}時間計算出結果（精確無誤演算法只使用了將連續模型給離散化的近似估計，非精確無誤則是使用了其它估計法如內插等）。但為了提升效率，所有非精確無誤的格點演算法無法保存全部的價格狀態，這些方法只能保持著有限個數的價格狀態並使用內插法來求取不存在的值。此外，我們更延伸了我們的結果使之能夠為離散型亞式選擇權定價。這之所以重要的原因是市場上所交易的都是離散型亞式選擇權。這篇論文是第一個提出$O(n^2)$時間並且會收斂、用以計算歐式亞式選擇權價值的格點演算法；這也是目前最具效率的演算法並具保證會收斂到正確價格上。這個演算法使用拉式乘數來決定每個格點上要分配多少個價格狀態的最佳分佈。同時，這演算法也很節省記憶體的使用量。 障礙選擇權是另一種重要的路徑相依衍生性金融商品。產品的歷史價格將會決定一個特定的事件是否被觸發，進而影響最後的報酬。我們將障礙選擇權的特性併入亞式選擇權，並延伸我們的$O(n^2)$時間且收斂的格點演算法來計算這種複雜的選擇權。 我們用大量的數值實驗並與目前已存在的偏微分方程法（PDE），解析方法，以及其它格點演算法做比較來驗證我們所提出的演算法是正確、具有高效率、且有高度競爭力的。這些結果，將以格點定價歐式亞式選擇權以及亞式障礙選擇權的複雜度降至與定價單純期權的複雜度同個層級。Path-dependent derivatives have payoffs that depend strongly on the price history of the underlying asset. In pricing such derivatives, the historical information needs to be encoded as part of the state. Asian options are strongly path-dependent derivatives. Although efficient numerical methods and approximate closed-form formulas are available, most lack convergence guarantees. Asian options can be priced on the lattice. Let the time to maturity be partitioned into $n$ periods. The best exact convergent lattice algorithm runs in 2^{O(sqrt{n},)} time. (An exact algorithm is one that does not employ approximations beyond the discretization of the continuous-time model.) All efficient lattice algorithms that are not exact keep only a polynomial number of states and use interpolation to compensate for the less than full representation of the states. Furthermore, we have extended our results in pricing discretely monitored Asian options. This is particularly important because majority of Asian options traded are discretely monitored. This work presents the first $O(n^2)$-time convergent lattice algorithm for pricing European-style Asian options; it is the most efficient lattice algorithm with convergence guarantees. The algorithm relies on the Lagrange multipliers to choose optimally the number of states for each node of the lattice. The algorithm is also memory efficient. Another important type of path-dependent derivative is the barrier options. The price history decides whether a specified event has been triggered or not, and it influences the final payoff. We have combined the barrier feature into Asian options, and provided an extension of the $O(n^2)$-time convergent lattice algorithm to price this difficult option. Extensive numerical experiments and comparison with existing PDE, analytical, and lattice methods confirm the performance claims and the competitiveness of our algorithm. Theses results place the problem of European-style Asian option and Asian barrier option pricing in the same complexity class as that of the vanilla option on the lattice.1 Introduction 1 1.1 Financial Engineering and Asian Options . . . . . . . . . . . . 1 1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Organization of the Dissertation . . . . . . . . . . . . . . . . . 4 2 Background 5 2.1 Option Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Continuous-Time Stock Price Dynamics . . . . . . . . . . . . 7 2.3 The Binomial Option Pricing Model . . . . . . . . . . . . . . . 7 2.4 Pricing Asian Options with BOPM . . . . . . . . . . . . . . . 9 3 Pricing Asian Options Using Lagragian Multipliers 14 3.1 Terminology and Basic Facts . . . . . . . . . . . . . . . . . . . 20 3.2 Description of the Algorithm . . . . . . . . . . . . . . . . . . . 26 3.3 Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.5 Numerical Evaluation . . . . . . . . . . . . . . . . . . . . . . . 37 4 Discrete Asian Options 52 4.1 Pricing Discrete Asian Options with Multinomial Lattices . . . 58 4.2 Numerical Evaluation . . . . . . . . . . . . . . . . . . . . . . . 68 5 Barrier Options: Introduction and Pricing 77 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.2 Trinomial Tree Algorithms . . . . . . . . . . . . . . . . . . . . 83 5.3 Discrete Single-Barrier Options . . . . . . . . . . . . . . . . . 84 5.4 Discrete Double-Barrier Options . . . . . . . . . . . . . . . . . 87 5.5 Numerical Evaluation . . . . . . . . . . . . . . . . . . . . . . . 88 5.6 General Rule for Pricing Discrete Barrier Options . . . . . . . 91 6 Asian Barrier Options 95 6.1 State Allocations, Running Times, and Convergence . . . . . . 96 6.2 Payoffs for States Ending Up In the Money . . . . . . . . . . . 101 6.3 Numerical Evaluation . . . . . . . . . . . . . . . . . . . . . . . 104 7 Conclusions 111 A Theory of Lagrangian Multipliers 122 B Supplementary Materials for Error Analysis 127 B.1 Derivation of the Nodes' State Sizes . . . . . . . . . . . . . . . 127 B.2 Optimality of Our Solution . . . . . . . . . . . . . . . . . . . . 132 B.3 Analysis of the Error Bound . . . . . . . . . . . . . . . . . . . 135 B.4 Polynomial Interpolation . . . . . . . . . . . . . . . . . . . . . 138 B.5 Proof of Eq. (4.9) . . . . . . . . . . . . . . . . . . . . . . . . . 142 B.6 Proof for Convergence of the Multinomial Lattice . . . . . . . 147 C Extension to Multibarrier Options 151 D Keys to More Efficient Computer Codes 158 D.1 Computing Binomial Distributions . . . . . . . . . . . . . . . 158 D.2 Creating Shifted Arrays with Both Positive and Negative Indices160 D.3 Caching Exponential Functions . . . . . . . . . . . . . . . . . 1612329410 bytesapplication/pdfen-US路徑相依衍生性商品格點演算法二元樹多元樹亞式選擇權離散型亞式選擇權亞式障礙選擇權Path dependent derivativeslatticesbinomial modelmultinomial modelAsian optionsdiscrete Asian optionsAsian barrier optionsthesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/53693/1/ntu-96-D89922012-1.pdf