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研究生:戴天時
研究生(外文):Dai, Tian-Shyr
論文名稱:使用格子點模型評價亞式選擇權
論文名稱(外文):Pricing Asian Options on Lattices
指導教授:呂育道呂育道引用關係
指導教授(外文):Lyuu, Yuh-Dauh
學位類別:博士
校院名稱:國立臺灣大學
系所名稱:資訊工程學研究所
學門:工程學門
學類:電資工程學類
論文種類:學術論文
論文出版年:2003
畢業學年度:92
語文別:英文
論文頁數:83
中文關鍵詞:格子樹模型亞式選擇權
外文關鍵詞:LatticeAsian optionMultiresolution latticeRange bound algorithmIntegral lattice
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路徑相依選擇權(Path-dependent options)的履約報酬會受到標的物的歷史價格的影響,這種選擇權在金融市場上扮演了重要的角色。不幸的是,我們無法有效率且精確地評價部分路徑相依選擇權,亞式選擇權就是一個最明顯的例子。亞式選擇權的履約報酬受標的物的歷史(算術)平均價格的影響,如何精確地評價亞式選擇權成為財務工程學界及實務界上長久未決的重要問題。直到今日,還是沒有人提出簡易的解析公式解來評價亞式選擇權。學術期刊上有許多文獻探討如何用近似的方法來評價;然而,多數已知的辦法都已被證實是不精確地或是沒有效率地。
亞式選擇權可用格子樹(如二元格子樹)模型來評價,其評價的結果會隨著格子樹模型的期數增加,而逼近到正確的選擇權價格。不幸的是,如果在格子樹模型上評價而不採用任何的近似方法,則其評價所耗費的時間會隨著格子樹期數的增加而成指數的增長。雖然有不少的學術文獻提出有效率地近似演算法,然而他們所提供的方法都沒辦法確保評價結果的精確度。為了方便起見,下文中提到“精確格子樹演算法”時,其意義為在格子樹上作評價時不使用任何的近似法。
我的博士論文使用格子樹模型來評價亞式選擇權。我提出兩種不同的作法來達成提供精確答案和以及有效率地評價的要求。在第一種作法中,我提出一個三元格子樹模型。這個格子樹模型是針對亞式選擇權的條款而設計,所以在這個格子樹模型上執行的精確格子樹演算法的計算效率可以大幅提高。經過嚴謹的分析,我證明其計算時間低於指數增長的函數(意即為sub-exponential time algorithm)。在第二種做法中,我們提供一項執行效率更好的近似演算法,這個評價演算法可提供選擇權價格(使用精確格子樹演算法得到的結果)的上界和下界。當評價歐式的亞式選擇權(European-style Asian options)時,我們證明該近似演算法的評價結果會隨格子樹期數增加而逼近到正確的價格。而在實際的數值實驗中,也驗證了該演算法可精確地評價美式的亞式選擇權。
Path-dependent options are options whose payoff depends nontrivially on the price history of a asset.
They play an important role in financial markets.
Unfortunately, some of them are known to be difficult to price in terms of speed and/or accuracy.
The Asian option is one of the most prominent examples.
The Asian option is an option whose payoff depends on the arithmetic average price of the asset.
How to price such a derivative efficiently and accurately has been a long-standing research and practical problem.
Up to now, there is still no simple exact closed form for pricing Asian options.
Numerous approximation methods are suggested in the academic literature.
However, most of the existing methods are either inefficient or inaccurate or both.
Asian options can be priced on the lattice.
A lattice divides the time interval between the option initial date and the maturity date into $n$
equal time steps.
The pricing results converge to the true option value as $n\rightarrow \infty$.
Unfortunately, only exponential-time algorithms are currently available
if such options are to be priced on a lattice without approximations.
Although efficient approximation methods are available,
most of them lack convergence guarantees or error controls.
A pricing algorithm is said to be {\em exact} if no approximations are used in backward inductions.
This dissertation addresses the Asian option pricing problem with the lattice approach.
Two different methods are suggested to meet the efficiency and accuracy requirements.
First, a new trinomial lattice for pricing Asian options is suggested.
This lattice is designed so the computational time can be dramatically reduced.
The resulting exact pricing algorithm
is proven to be the first exact lattice algorithm to break the exponential-time barrier.
Second, a polynomial time approximation algorithm is developed.
This algorithm computes the upper and the lower bounds of the option value
of the exact pricing algorithm.
When the number of times steps ($n$) of the lattice becomes larger,
this approximation algorithm is proven to converge to the true option value for
pricing European-style Asian options.
Extensive experiments also reveal that the algorithm works well for American-style Asian options.
1 Introduction 1
1.1 Se ingheGround............................1
1.2 Opions ..................................1
1.3 AsianOpions...............................2
1.4 TheLa iceApproach ..........................4
1.5 Pricing Asian Options with the Lattice Approach and Its Problems .4
1.6 The Contributions of this Dissertation .................4
1.6.1 The Subexponen ial-Time Lattice Algorithm ..........5
1.6.2 TheRange-BoundAlgorihms..................7
1.7 Srucuresof hisDisseraion......................7
2 Preliminaries 9
2.1 Basic Assumptions ............................9
2.2 OpionBasics...............................10
2.2.1 De finiionsofOpions ......................10
2.2.2 WhoNeedsOpions .......................12
2.2.3 HedgingandHedgers.......................13
2.2.4 Pricing an Option with Arbitrage-Free Base Pricing Theory .14
2.3 AsianOpions...............................17
2.3.1 De finiions.............................17
2.3.2 Pricing Asian Option by Applying Risk Neutral Variation ..17
2.3.3 AdvanagesofAsianOpions ..................18
2.4 ReviewofLieraure ...........................18
2.4.1 ApproximaionAnalyicalFormulae...............19
2.4.2 MoneCarloSimulaion .....................20
2.5 La iceandRelaedPDEApproach...................21
2.5.1 TheSrucureofCRRBinomialLa ice ............22
2.5.2 HowoConsrucaLa ice ...................23
2.5.3 Pricing an Ordinary Option with he Lattice Approach ....23
2.5.4 ThePDEApproach........................25
2.6 Pricing the Asian option with Lattice Approach and Its Problems ..25
2.6.1 Approximation Algorithms and Its Problems ..........27
3 The Integral Trinomial Lattice 30
3.1 ASimpleInuiion ............................30
3.2 TheMuliresoluionLa iceModel ...................31
3.3 AnOverviewof heNewlyProposedLa ice..............34
3.3.1 TheSrucureofaTrinomialla ice...............36
3.4 La iceConsrucion ...........................37
3.5 Proof....................................39
3.5.1 Validiyof heLa ice ......................40
3.5.2 Running-Time Analysis .....................44
4 The Range Bound Algorithms 45
4.1 TheRangeBoundParadigm.......................45
4.2 An Overview of he Proposed Range Bound Algorithms ........46
4.3 Preliminaries for European-Style Asian Options ............48
4.4 The Design and Error Analysis of he Basic Range-Bound Algorithm 49
4.4.1 Description of the Algorithms ..................49
4.4.2 An Optimal Choice of the Number of Buckets .........52
4.4.3 ErrorAnalysis...........................53
4.5 Tigh er Range Bounds ..........................54
4.5.1 A Tighter Upper-Bound Algorithm ...............54
4.5.2 A Tighter Lower-Bound Algorithm ...............55
4.6 Numerical Results for European-Style Asian Options .........57
4.7 Algorithms for American-Style Asian Options .............61
4.7.1 UsefulTerminology........................62
4.7.2 Full-RangeAlgorihms......................63
4.7.3 The First Upper-Bound Algorithm ...............64
4.7.4 TheSecondUpper-BoundAlgorihm..............65
4.7.5 ALower-BoundAlgorihm....................68
4.8 TheRange-BoundProofs.........................69
4.9 Numerical Results for American-Style Asian Options .........72
5Conclusions 77
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