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1 國立交通大學 財務金融研究所 碩士論文 Modified Richken and Trevor ree 於 GARCH Opion 評價之應用 Value he GARCH Opion applying he Modified Richken and Trevor ree 研究生 : 黃靖謙 指導教授 : 王克陸教授 中華民國九十七年七月

2 Modified Richken and Trevor Tree 於 GARC H Opion 評價之應用 Value he GARCH Opion applying Modified Richken and Trevor ree 研究生 : 黃靖謙 指導教授 : 王克陸博士 Suden:Ching-Chien Huang Advisor:Dr. Keh-Luh Wang 國立交通大學 財務金融研究所碩士班 碩士論文 A Thesis Submied o Graduae Insiue of Finance College of Managemen Naional Chiao Tung Universiy in parial Fulfillmen of he Requiremens for he Degree of Maser of Science in Finance June 008 Hsinchu, Taiwan, Republic of China 中華民國九十七年七月

3 Modified Richken and Trevor Tree 於 GARCH Opion 評價之應用 學生 : 黃靖謙 指導教授 : 王克陸教授 國立交通大學財務金融研究所碩士班 摘要一般而言, 用來評價選擇權的方式大部分為 Black-Scholes Model 與數值分析方法, 其中數值分析方法又分為多種不同的模型 例如 : 蒙地卡羅法 二項式法等等 雖然 Black-Scholes Model 在早期被各界廣泛採用, 但它的缺點是有太多的假設, 隨著今日日新月異的多種選擇權的發明,Black-Scholes Model 在實證分析時出現了一些不合理的問題 ; 我們可以知道 Black-Scholes Model 面對這些新奇選擇權的評價時並不適用 Duan (1995) 發表了 GARCH 選擇權定價模型, 論文中指出根本資產之價格動態過程, 在服從 GARCH 模型的行程下, 引入經濟學上 均衡概念的主張, 經過適當的風險測度轉換之後, 可以導出歐式選擇 i

4 權的價格 但是在此條件狀態下的選擇權訂價理論, 其數值分析方法, 仍不夠完備, 以致於實務上未能完全地擷取而加以運用 其問題的主要癥結在於 GARCH 模型, 其本質上必然會產生路徑相依 (pah dependence) 的問題, 導致運算與處理上的困難程度增加 而所謂的路徑相依, 是指在選擇權存續期間, 其價格會受到標的資產價格本身波動性的影響 反之, 路徑獨立 (pah independence) 是指選擇權價格只受到標的資產其到期日時之價格影響 GARCH 模型的路徑相依的性質, 會使得欲用樹狀圖來刻劃價格的波動過程中, 各時點的可能狀態個數會因時間的往前推移, 而呈現指數的遞增情形, 而使得樹狀圖陣列非常的龐大, 使得 GARCH 選擇權定價模型在實務上的應用並不理想 而 Richken 和 Trevor (1999) 針對在非連續時間的 GARCH 模型, 對歐式選擇權和美式選擇權的訂價, 建構一個所謂的樹狀演算法 且說明此一樹狀演算法可以進一步擴展到標的資產服從一般化 GARCH 模型之下, 建立出有效的運算方法, 此一具體運算方法, 不僅適用於 GARCH 模型之下選擇權的訂價, 而且, 也可以用來處理很多雙變數的擴散模型 RT 演算法的優點在於可以捕捉各個時點的條件變異數, 可以解決 GARCH 模型路徑相依的問題, 使得評價能更有效率 於 1999,S. Figlewski 與 B. Gao 提出了適應性網狀模型 (Adapive Mesh Model, AMM), 同時解決了分配誤差 (disribuion errors) 與非線 ii

5 性誤差 (non-lineariy errors), 並且提升了評價模型運算的效率 由於 AMM 在評價上表現出不錯的彈性以及效率, 後來, 有不少研究將 AMM 應用於權證的評價上 本篇論文將 AMM 中的概念應用在 RT 模型上, 我們稱之為 AMM-RT 模型 由於非線性誤差大部分出現在執行價格附近, 因此 AMM 執行價格增加網格節點的密度來提升估計的精確度和減少非線性誤差 我們將這種想法應用於 RT 模型的到期日前一天, 在到期日的前一日與到期日之間, 我們仍然使用 RT 模型的演算法, 將這段期間切割的較細 這樣的方式可以達到跟 AMM 一樣的效用, 同時也可以如同之前的 RT 模型一樣具有捕捉條件變異數的能力 波動性 (volailiy) 對於任何一種金融商品而言, 都有相當顯著的關係存在, 因此我們選擇 RT 模型搭配 GARCH 模型來預測選擇權價格, 然而, 我們又希望增加其精確度與減少其誤差, 故到期日前一天增加切割期數以期能達到我們想要的效果 本論文將嘗試分別以傳統的 BS 模型 ( 在不同的 volailiy 下 ) 與 RT 模型以及 AMM-RT 模型再搭配 GARCH (1,1) 模型去模擬並比較股票選擇權價格 iii

6 Value he GARCH Opion applying he Modified Richken and Trevor ree Suden:Ching-Chien Huang Advisors:Dr. Keh-Luh Wang Insiue of Finance, Naional Chiao Tung Universiy ABSTRACT Evaluaing sock opion price wih radiional predicive echniques have proven o be difficul. GARCH opion pricing model proposed by Duan has been proven o be more suiable for he ask. BS model have so many assumpions ha i canno be suiable in some exoic opion. GARCH opion pricing model solve he problem which may occur while using he BS model. This hesis focuses on he sock opion price esimaing based on GARCH (1, 1) model, which have been surveyed by earlier researcher as well as he comparison beween each model is discussed. Derived from he firs GARCH opion price model proposed by Duan (1995), he Richken-Trevor Model offers more accurae pricing han CRR model and radiional rinomial ree model. AMM proposed by S. Figlewski and B. Gao adds he mesh poin densiy parially o modify he inefficiency and calculaing error of he CRR and rinomial laice model, which iv

7 addresses he problems of disribuion errors and non-lineariy errors as well as upgrade he efficiency of he pricing model. We apply he idea of AMM in he dae T (i.e. he day before he mauriy day). Raher han he fine mesh srucure like AMM, we develop anoher fine mesh by he same approach of RT model. We jus increase he number of ime sep by changing parameer m (Here m is he segmenal level of he las rading day; m=, 3, 5 will be discussed) in he las dae T. We call his jusified model Modified RT Model (AMM-RT) in his hesis. The same as AMM, he AMM-RT model solve he nonlineariy error around he srike price while evaluaing exoic price like, barrier opion. By his modified RT model, we also solve he nonlineariy error as well as increase he accuracy. In his hesis, we demonsrae a comparison of accuracy beween BS model (wih differen volailiy), RT model and AMM-RT model. Wih heir abiliy o discover paerns in nonlinear and chaoic financial sysems, he GARCH opion pricing model wih AMM-RT algorihm no only offer he abiliy o predic marke direcions more accuraely han curren echniques bur also reduce he complexiy of compuing of he original RT model. Numerical analysis via above mehods are discussed and compared wih performance. Finally, fuure direcions for applying he AMM-RT model o he financial markes are also dissered. v

8 Acknowledgemen Tha I exis is a perpeual surprise which is life - Rabindranah Tagore 我的碩士求學過程 現在回想起來感覺像是一場冒險旅程 旅程中充滿各種的關卡 每個關卡都需要一把鑰匙, 而通過之後是絕無僅有的寶物與更多的抉擇 冒險中的鑰匙是我的老師 朋友 以及家人 甚至於一些從未見過的朋友 寶物是我從錯誤中學習的人生經驗 而抉擇是我充滿未知的人生 在此, 我想感謝王克陸老師 在過去近兩年中, 王老師扮演著亦師亦友的角色 王老師幫助我的不僅是學業, 更多的是在我碰到難關時給我極大的幫助 從老師身上看到的是學者與家庭的典範 王老師不僅在研究上提供我很好的舞台, 也讓我在國內的學習環境多了國際視野與各種難得的經驗 同時謝謝所長鍾惠民老師, 鐘老師每次遇到我時總會不斷的給我鼓勵, 提供我源源不決的信心跟肯定, 我很想感謝鐘老師與我一起經歷求學的酸甜苦辣 另外, 我非常感謝博士班學長劉炳麟 炳麟學長除了給予我研究的方向之外, 也花了極大的心力與我一起討論研究方法以及協助我處理程式方面的問題, 很感謝炳麟學長的傾囊相授 我也從學長身上我學到積極的研究精神以及謙虛的處事態度 我也必須要感謝電子所的吳東佑學弟, 謝謝他能夠在兩年期間課業上助我一臂之力 也要感謝財金所的秋男, 感謝他在資料蒐集上的幫助, 謝謝他在初稿完成前最後幾天, 毫無怨言的幫我整理繁雜的資料 最後我想謝謝我的父母 女友泰琳和從電研所博一認識到現在的鄭淳護博士多年的支持與幫助, 學長很像是我旅途中一起度難關的伙伴 也謝謝財金所的同學文誠 以文 詩政 建佑 妤芳 佳彣等 並謝謝財金所所有夥伴們 同時在此感謝所有口試委員們的寶貴意見 vi

9 T o my parens, broher and friends. vii

10 Conen Absrac and Acknowledgemen... i~vii Chaper 1 Inroducion Overview Sock Opion Pricing... 4 Chaper Lieraure Review Sock Opion Pricing GARCH Opion Pricing Model Richken and Trevor Tree Adapive Mesh Model Chaper 3 Mehodology Emperical Procedure Chaper 4 Numerical Illusraion Dae analysis Numerical Analysis Chaper 5 Conclusion Bibolography Personal Informaion... 59

11 CHAPTER 1 Inroducion 1.1 Overview The prevailing noion in sociey is ha wealh brings comfor and luxury, so i is no surprising ha here has been so much work done on ways o predic he markes. Various echnical, fundamenal, and saisical indicaors have been proposed and used wih varying resuls. However, no one echnique or combinaion of echniques has been successful enough o consisenly "bea he marke". Wih he developmen of GARCH opion pricing model, researchers and invesors can wish ha he marke myseries can be unraveled. This hesis is an invesigaion of GARCH opion pricing model combining differen laice model wih an emphasis on sock price volailiy predicion. Because i is ofen imporan o obain price fas, he efficien numerical algorihms play a vial role in derivaives pricing when prices changes quickly in sock marke. In financial economerics, General Auoregressive Condiionally Heeroskedasic (GARCH) processes are wildly used o model he reurns a regular inervals on socks, currency and oher asses. 1

12 Specifically, he GARCH process ypically represens he incremens, ln S 1 ln S, of he logarihms of he asse price a dae 1,, 3. These models capure many of so-called sylized feaures of such daa, e.g. ail heaviness, volailiy clusering and dependence wihou correlaion. Many financial ime series daa suffer from he sochasic change in volailiy over ime. For mos financial commodiies, reurn innovaion will influence fuure volailiies. This issue has become an imporan and imperaive empirical fac. Mandelbro (1963) showed ha large absolue reurns are more likely o follow he large absolue reurn innovaion, which is called volailiy clusering. The volailiy will be influenced by he exrinsic environmen changes. If he news is bad, he volailiy will be larger. Black (1976) called his phenomenon leverage effec. This implies ha here is a negaive correlaion beween asse reurn innovaion and volailiy innovaion. (Bollerslev, Chou, and Kroner, 199) Using ineracion effec beween reurns and volailiy is very imporan in he opion price model. In 1973, Black and Scholes use he hisory volailiy o calculae he opion value. On he assumpion of seing he volailiy as consan, hey ignore he issue abou he volailiy iself changes wih he ime. Alhough BS Model is wildly used, many empirical analysis showed ha BS model will bring he issues of pricing error, for example: underesimae he value of ou-of-money-opion and volailiy smile. Duan (1995) was he firs o propose a GARCH opion pricing model. He indicaes ha opion can be priced when he dynamics of he price of he underlying asse comply wih he GARCH process. Unforunaely, pleny of he pah dependence of he pricing models prefer o use Mnoe Carlo simulaion over

13 rees which would increase he calculaing difficuly. Thus he analyical soluions o prices of opions are no generally available and hence numerical approaches o prices have o be invoked. Richken and Trevor (1999) propose rinomial laice ree o address hese problems. They provide an efficien numerical procedure (a laice approach) for pricing European and American opions under discree-ime GARCH processes. Furhermore, in order o handling American opion, Duan and Simonao (000) proposed anoher numerical algorihm a Markov chain approach almos a he same ime. Because he Mone Carlo esimae is probabilisic and he American opions can be accuraely priced only wih simulaion schemes ha employ advanced echniques, a numerical approach ha processed he American opion more efficienly han previous Mone Carlo simulaion is he binomial ree. Alhough he binomial approach works well under consan volailiy, here will be a formidable challenge o apply his mehod in sochasic volailiy. Rihken and Trevor (1999) consruc a ailored laice approximaion algorihm for he GARCH model by resricing he sorage of condiional variance o he minimum and maximum values a each node of he discreized underlying asse price under he forward building process. S. Figlewski and B. Gao (1999) propose he Adapive Mesh Model (AMM) which adds he mesh poin densiy parially o modify he inefficiency and calculaing error of he CRR and rinomial laice model. In his hesis, we apply he idea of AMM in he dae T (i.e. he day before he mauriy day). Raher han he fine mesh srucure like AMM, we develop anoher fine mesh by he same approach of RT model. We jus increase he number of ime sep by changing parameer m in he las dae T (Here, we 3

14 call he segmenal level of he las rading day m). We call his jusified model Modified RT Model (AMM-RT) in his hesis. The emphasis of his hesis is o compleely invesigae he sock opion price esimaion under Duan s GARCH model in combinaion wih differen algorihms. BS model wih differen volailiy, RT model and AMM-RT model (modified RT model) will be discussed. Moreover, using his modificaion of he laer RT model also makes i possible o apply Duan s GARCH opion pricing model o a broader domain of exchange raded opion conracs. The hesis organized as follows. In secion we will review he basic GARCH opion pricing Model proposed by Duan (1995), he laice algorihm of Richken-Trevor (1999), and Adapive Mesh Model. Secion 3 describes he empirical procedure of our work using AMM-RT o evaluae he arge commodiy price volailiy. The crux of he work, in Secion 4, deails he numerical illusraions of BS model, RT model and our AMM-RT model in concer wih GARCH opion pricing model. This hesis also concludes wih commens on possible fuure work in he area and some conclusions. 1. Research Moivaion There are several moivaions for rying o predic sock marke prices. The mos basic of hese is financial gain. Any sysem ha can consisenly pick winners and losers in he dynamic marke place would make he owner of he sysem very wealhy. Thus, many individuals including researchers, invesmen professionals, and average invesors are coninually looking for his superior sysem which will yield hem high reurns. There is a second moivaion in he research and financial communiies. I has been proposed 4

15 in he Efficien Marke Hypohesis (EMH) ha markes are efficien in ha opporuniies for profi are discovered so quickly ha hey cease o be opporuniies. The EMH effecively saes ha no sysem can coninually bea he marke because if his sysem becomes public, everyone will use i, hus negaing is poenial gain. Doing sock opion price predicions have never been easy even for professional invesors. Sock marke expers are coninuously researching and devising mehods ha could aid hem and ohers in foreseeing an accurae sock marke oucome. Sock marke commodiies predicion is coninuously being aemped. Bu unforunaely unil now, here isn' a 100% accurae echnique creaed o do i ye. Sock marke is he erm given o he ac of rading company shares, opions, socks, and oher securiies and is derivaives. The sock opion has a number of players, which could be range from an individual sockholder o a very large corporae rader. These players can be anybody coming from any par of he world. Trading in he sock opion can be done privaely wih an aorney or wih a professional sock exchange dealer who have he power o execue he order. For he mos par, sock opion price is very volaile in naure so ha he price is very ough o predic. Tha's he reason why volailiy is sudied in his hesis. In he pas, people almos widely used he regression mehod, ime series mehods, and he neural nework mehods o predic sock price. Due o persisen sudies, he changes in he sock marke can now be calculaed in a relaively accepable precision. In his hesis, we use a differen kind of approach o predic he opion price. The performance here are he various effors carried ou by sock 5

16 marke expers o predic he marke's movemens. I depic he empirical procedure in Secion 3. and he applicabiliy of Modified RT (AMM-RT) model is also discussed. 6

17 CHAPTER Lieraures Review.1 Sock Opion Pricing Tradiionally, he approach of pricing he opion divides hree major secions. Secion One: Formula soluion (Closed soluion): Black and Scholes opion pricing model. Secion Two: Numerical Analysis soluion: Using numerical approach, like compuer simulaion, o calculae opion price. For example, ree algorihm, Mone-Carlo simulaion and finie differenial approach. Secion Three: Analyic approximae model: This approach combines he above wo mehods. For example, Barone-Adesi and Whaley (1987) deduce he analyic formula soluion of American opion. Mos researchers use risk free arbirage o deduce closed form soluion and find a parial differenial equaion and is soluion. However, he derivaion process is more complicaed and difficul since we couldn find is closed-soluion in many siuaions, especially he pah-dependency opion. Harrison and Kreps (1979) develop anoher kind of mehod o solve he pricing issue of he derivaive commodiy which is so-called maringale 7

18 pricing mehod. This mehod, comparing wih solving he parial differenial equaion, is easier o solve and involve wih fewer mahemaical echniques. Thus, recenly he maringale pricing mehod is used repeiiously. Alhough closed form formula is simple and compuing fas, no all he pricing of opions exis he closed form soluion. Besides, i is usually applicable o he pricing of European opion bu no o American opion and oher exoic opion. Moreover, we should adop he numerical approach o handle he opion pricing under disconneced ime. If we know he pah of our arge asse price, we can use he Mone Carlo approach o simulae arge asse price s possible pah repeaedly. Thus, we can ge he price of plain vanilla ype opion. Ye, his approach would cos a lo of processed ime and suffer from poor compuing efficiency. Cox, Ross, and Rubinsein (1979) develop binominal ree model (CRR model), which breakhrough he original BS model s assumpions and applicaive range. CRR model describe he arge asse price s behavior in discree ime saus. I also deduces he risk naure pricing model excep he arbirage opporuniy. I should be noed ha CRR model assume arge asse reurn s volailiy is consan when i is buil. Besides, he binominal ree model can add he segmenal ime seps on ree diagram o increase pricing accuracy, which also solve he issue of consuming a lo of ime of Mone Carlo mehod. Ye, when he pah-dependency issue exiss, he nodes of he ree diagram will increase exponenially due o he increase of segmenal ime seps. Thus we can conclude ha he radiional binominal ree model hardly o handle as soon as he opion becomes more complicaed. Boyle (1986) exends he binominal ree o rinomial ree. Furher, for binomial ree and rinomial ree, Tian (1993) proposes differen kind of esimaion mehods 8

19 of parameer. Tian also verifies and compares he pricing efficiency of he wo models. Even hough he rinomial ree s diffusive ended nodes are more han he binomial ree s, he segmenal ime sep will be smaller. Thus, he rinomial ree model can capure more complee price probabilisic disribuion funcion. Base on his advanage, we can find relaively accurae opion price and we can also verdic ha he pricing efficiency of he rinomial ree model is beer han which of he binomial ree model.. GARCH Opion Pricing Model Duan (1995) proposes he GARCH opion model in He develops he opion pricing model when sock opion follows he discree ime GARCH (p, q) process (proposed by Bollerslev, 1986). Following, we will describe he GARCH opion pricing model using he sandard discree ime GARCH (1, 1) specificaion. Because he simple GARCH (1, 1) wih normal disribuion assumpion is he mos commonly used, we will use GARCH (1, 1) as our esimaing model. Based on he LRNVR of Duan (1995), he esimaion of variance will no change wih he Measure siuaion, hus, we only need o apply simply GARCH (1, 1) o esimae he variance ou of sample. We le S be he arge asse price a daa, h be he condiional variance a inerval, 1 ha is one day. The behavior of arge asse price can be expressed as below: S S 1 ln( ) r h h, ~ N(0, h ) (1) f 1 h h 1 1 The oher corresponding definiions of symbol are illusraed as follows: () 9

20 S : he arge asse price a period (day) h : he condiional variance of he arge asse price a period : he collecion of all informaion before period -1 : he sandard normal random variable a period, ha is, ~ N(0, h) 1 r f : risk free rae : uni risk premium The above GARCH (1,1) sysem follows he sandard GARCH parameer resricion. And his model follows he resricion as: 0, 0, < 1, ( ) 1 Based on Duan s model (1995), he asse price process under locally risk-neuralized pricing measure Q can be rewrien as: S S 1 1 ln( ) r h, ~ N(0, h ) (3) f 1 h h h ( ) (4) 1 Among hese equaions, is he sandard normal random variable in he corresponding risk-neuralized pricing measure Q. Under his modified model, he parameers waiing for esimaing are,,, respecively. Furhermore, he risk premium ampuaes from he equaion (3) under measure Q. In oher word, he equaion (3) is independen of he parameer. This propery indicaes ha ones can assume ha he invesors are risk-neural. In he risk-neural world, he presen value of any cash flow can be obained by discouning is expeced value a risk-free rae as well as he expeced reurn of any sock commodiy is he risk-free ineres rae. 10

21 .3 Richken and Trevor Tree Richken and Trevor (1999) develop he rinomial ree algorihm based on rinomial ree model o capure he pah of price and volailiy and advanced o evaluae he European opion and American opion. Since he sock opion volailiy is incompleely sandard disribuion, Richken and Trevor assume ha sock opion volailiy follows he GARCH model and he sochasic process. They develop he ree laice of he variances of sock yield rae, probabiliy, and sock price, ec. A. Consruc he ree laice of he variance and probabiliies: The key o an efficien implemenaion is o design an algorihm ha avoids an exponenially exploding number of sae, Toward his goal, we begin by approximaing he sequence of single period lognormal random variables in equaion (3) by a sequence of discree random variables. In paricular, we se ln( S ) y. Based on GARCH (1, 1) model under Q measure, he model in equaion (3) ~ (4) can be rewrien as : 1 y 1 y rf h, 1~ N(0, h ) (5) Thus, h h h ( ) (6) 1 E( y ) y r 1 h Var( y ) h 1 f 1 The GARCH (1,1) process can be approximaed by he following laice model, and he superscrip of each parameer a denoing approximaion : a a y 1 y j n, j 0, 1,,..., n (7) 11

22 h h h a a a a 1 ( ) (8) Symbol n deermine he segmenal number in each period (day). If n=, hen we segmen wo subinerval in each period. The symbol n also decides he branch of he ree diagram ha akes on n+1 value. When n=, we have five sae variables of each period (day) such ha here are wo values smaller he curren price, one value unchanged, and anoher wo value larger han he curren price. The symbol indicaes he jump parameer which allows he variance and mean of he nex period s logarihmic price o mach he rue momens as well as ensuring ha he (n+1) probabiliy values are valid in he inerval [0, 1]. By RT s heorem, is chosen such ha a h ( 1) (9) The gap beween wo neighbor logarihmic prices is decided by he spacing parameer n. By RT s heorem, is a fixed consan which follows he relaionship: n n (10) I would be noed ha he pah dependence issue will occur when we use GARCH model. In Figure 1, we should noe ha he number of variance may no be 1. Observe ha differen saes may pick differen saes η. Take node (,-1) for example, here will be hree possible pahs achieving node (,-1), which means here would be hree possible variances on node (,-1). We only reserve he maximum and he minimum variance on each node. Thus, he wo variances on node (,-1) will leads o wo possible η. For he 1

23 smaller variance, we jus need se η=1; however, we should se η= o saisfy equaion (10). Consequenly, if we don pu a limiaion o hese variances, we will have 5 pahs deriving from node (,-1). Fig.1 The ree laice of variance based on GARCH model (=3, n=1,). We assume K=, r f = 0, λ=0, β 0 = , β 1 =0.9, β 1 =0.04, c=0, S 0 =1000, h 0 = (h 0 is he iniial variance). The op (boom) 13

24 number is he maximum (minimum) variance (muliplied by 10 5 ). Due o he pah dependence issue, he variance of nex ime will be influenced by he variance of he previous ime. Wih he ime increase, he number of variance of each node will no be only one. In order o overcome his problem, we mus le he number of variance of each node fixed as well as express he all possible variance one node may possessed. We uilize inerpolaion mehod o ge he oher K- variances using he maximum and minimum variance, hus we can keep he number of variance of each node fixed. max h and min h represen he possible maximum and he minimum condiional variance which come from all pahs of he laice. RT model divide inerval beween max h and a h ino K pars. Le h ( i, k) be he k h min condiional variance of node (, i ): a min max min h ( i) h ( i) h ( i, k) h.( k 1), k 1,,..., K (11) K 1 Thus, we finish he frame of RT laice model. When we consruc he variance of he RT laice model, meanwhile, we can obain he following probabiliies: p u h ( r h / ) 1/ n a a f n n (1) p m a h 1 (13) n 14

25 p d h ( r h / ) 1/ n a a f n n (14) a h : he variance of arge sock s rae of reurn (uni: year) : jump parameer n : sock price sep beween before and afer dae, a n and h0 n r f : risk neural ineres rae n: decide he segmenal number of each period (1 day). If n=, hen we segmen n subinervals of one day. B. Consruc he ree laice of sock price Afer seing he variances and probabiliies of he RT laice model, we nex begin o consruc he sock price of he ree. Le y ln S and y y j, j 0, 1,,..., n. n Following we define he node (, i) represens he h dae and he i h price, hus, y ( i) y(0) i, i M ( ),...0,... M ( ), M () and M () sand d u for he maximum and minimum unis of price-ascending and price-descending, respecive. For example, y (1) y(0) or y (1) y(0) 3. d u C. Pricing he sock opion price In order o esablish he laice model, in he above discussion, RT model se he logarihmic price a dae is a y and he condiional variance is 15

26 a h. In he Sep C, afer he underlying asse price and volailiy laice are seing, sock opion price (no variance) can be evaluaed on he ree using a he backward inducion. Le C ( i, k) is opion price of node (, i) relaed o he k h condiional variance. A his ime, he corresponding sock price is S () i y () i e and k=1,,, K. Regardless of he corresponding variance, when he opion srikes, he reurn should be C ( i,1) C ( i,)... C ( i, K) Max 0, S ( i) X (15) T T T T where X is he srike price and S () i is he sock price a dae T. We will T show he corresponding relaion beween variance and sock price a mauriy day (=T) in Figure. Fig. The corresponding relaion beween variance and sock price a mauriy day (=T). If he k h condiional variance of node (, i) is h (i,k), we can calculae he rue condiional variance of node (+1, i+jη) a nex period: 16

27 1 ( j ( r h ( i, k)) a n f nex ( ) [ ] (16) a h ( i, k) h j h h where j 0, 1,,..., n However, a node (+1, i+jη), we have sored opions for only K differen variance level. When (n+1) is larger han K, here may no be a nex variance enry corresponding exacly o h ( j ). Following we use inerpolaed mehod o decide he sock opion price which corresponds o nex h ( j ). We assume h ( j) locaes beween he L h condiional variance nex and (L+1) h condiional variance of node ( 1, i j ). (one node can nex possess K variance) Thus, he corresponding sock opion price of h ( j ) is: in erp a a C j q j C i j L q j C i j L ( ) ( ) (, ) (1 ( )) (, 1) (17) 1 1 where, h ( i j, L 1) h ( j) a nex 1 ( ) (18) a a h 1( i j, L 1) h 1( i j, L) q j nex We show he relaion of h ( j ) in Figure 3. In his way, eiher node (+1, nex i+jη) conains a variance enry (or, opion price) ha maches h ( j ), or he relevan informaion is inerpolaed from he close wo enries, will have is corresponding opion price. 17

28 nex Fig. 3 The illusrucion of he locaion of h ( j ) As a resul, we can ge he sock opion price corresponding o k h condiional variance a node (, i): n a rf (, ) ( ) in erp ( ), 0, 1,,..., (19) j n C i k e P j C j j n We can use equaion (16) ~ (19) wih he backward recursion o ge he opion price a day 0. We also describe equaion (19) using following Figure 4. a Fig. 4 The illusraion of evaluaing sock opion price C ( i, k ) The above menioned probabiliy disribuion (1)-(14) can be expressed as n ju jm jd p( j) pu pm pd ju, jm, j j d u jm j, wih j,, 0 u jm jd and ju jm jd n. d 18

29 The variance number a each node in his ree laice is differen; indeed, one node maybe has more han one variance (i.e. K variances). The pricing resul shows ha he convergence velociy will be influenced by he laice branches and he number of variance a each node. When he number of each node is fixed as well as he mulinomial ree s branch exends, hen, he convergence velociy increases. On he oher hand, when he mulinomial ree s branch is fixed as well as he number of each node increases, hen, he price would converge closely o is rue price. Cakici and Topyan (000) modify he RT model, which is so-called RTCT model. In heir poin of view, RT s model is no meaningful due o he nex dae s possible variance produced by inerpolaion mehod. This sep could be reserved unil he opion price is calculaed in he backward inducion. In he forward sep which consrucs he ree laice, we only need o reserve he maximum and minimum variances and calculae hese wo facors influence on nex period. Thus, Cakici and Topyan find ha he model s volailiy would be more close o rue asse s volailiy when he difference of he variance in each node is equal. Furhermore, when he accuracy increases, he calculaing ime would decrease, convergence velociy be beer, and he price be more unbiased. Even he GARCH model s parameers change; he resul will sill be he same..4 Adapive Mesh Model When we use he laice model o price sock opion, here are essenially wo kinds of approximaion errors in any pricing echniques of 19

30 laice framework, which are disribuion error and nonlineariy error, respecively. 1. Disribuion Error: When we use laice model o price sock opion, we use a finie se of nodes wih probabiliies (i.e. binominal or rinomial) o approximaes he rue asse price disribuion wih coninuous lognormal densiy. Alhough he mean and variance of he discree disribuion of laice model are mached by he coninuous disribuion of laice model, he discrepancy beween hem sill leads o disribuion error in sock opion value. If we increase he ime sep number of he laice model (i.e. increase n), he discree disribuion of laice model will approach o he coninuous disribuion of laice model. Wih he ime sep number of laice model increasing, he disribuion error decreases.. Nonlineariy Error: If he opion payoff funcion is highly nonlinear, pricing his nonlinear region wih only one or several nodes (i.e. binominal or rinomial) would give a poor approximaion o he average value over he whole inerval. For example, when sock price pass hrough hese regions: around he srike price, he sock opion price mees he crossroad, and barrier opion approaches o he barrier price, hen he sock price s biy perurbaion will lead o he large change of he sock opion price(i.e. jumps or mees he crossroad). The above siuaions (he sock opion price jumps or mees he crossroad) are called nonlineariy error. The nonlineariy error can be reduced by increasing he ime sep number of he laice model. Even hough he nonlineariy error can be reduced, i also will occur while he ime sep number increases o some hreshold number. Thus, we apply Adapive Mesh model in pricing 0

31 sock opion o minimized nonlineariy error wih only sligh compuaion increase. Originaing from rinomial laice model, S. Figlewski and B. Gao (1999) propose he Adapive Mesh Model (AMM) which adds he mesh poin densiy parially o modify he inefficiency and calculaing error of he rinomial laice model. AMM is a kind of rinomial ree laices ha applying higher resoluion fine mesh o where nonlineariy error occurs. AMM model use ln( S ) o subsiue he original arge asse price S as he variable for calculaing node s price. This is also he major difference beween AMM and he original rinomial ree model proposed by Boyle (1986). The AMM adops he characerisic of numerical analysis mehod. I can adjus is seing and limiaion of is parameers based on differen warran conrac. In he following secion, we discuss AMM applying in European opion and American opion, respecively. (I) European opion: AMM follows he serious assumpion of BS model. In he risk neural pricing environmen, he arge asse S obeys he generalized Wiener process and saisfies he logarihm normal disribuion. The arge asse in AMM can be expressed as: ds S d dz We assume X * (0) * ln( S), hen dx d dz. Among his equaion, r q ( is he expeced rae of reurn; q is he insananeously dividend paymen rae; is he insananeously volailiy rae; dz expresses 1

32 he Wiener process). Thus we can rewrie equaion (9) as: d ln( S) ( r q ) d dz (1) The advanage of his ransform is leing he all asse price o change regularly a a fixed quaniy d and dz. Therefore, we can use he idea of he finie difference o handle he price fronier. Thus, we go on o increase he densiy of mesh poin on he price fronier locally and ensure he price can converge rapidly o increase he calculaing accuracy. Trinomial ree model assumes he arge asse price will have hree kinds of changes unil he nex period comes: ascending, unchanging, and descending. The AMM also reains his characerisic and hypohesize he occurring probabiliies are P u, P m and P d, respecively. If he probabiliy and he range ha price changes are symmerical, he range of price change h (h is so-called price sep) should saisfies dz ~ N(0, d) under he Geomeric Brownian Moion. For his reason, in he coninuous diffusion process, he hypohesis of model is composed of he summaion of occurring probabiliies, firs degree parial differenial equaion (1 s PDE), second degree parial differenial equaion ( nd PDE), and forh degree parial differenial equaion (4 h PDE)o form he following simulaneous equaion: 1 P P P () u m d E[ X ( ) X ( )] 0 P h P 0 P ( h) (3) u m d E X X P h P P h [( ( ) ( )) ] u m 0 d ( ) (4) E X X P h P P h [( ( ) ( )) ] 3 u m 0 d ( ) (5) Wih some algebraic effors, we obain he following equaion:

33 P 1/ 6, P / 3, P 1/ 6, h 3 (6) u m d The above deducion is he rinomial process for appropriaing he arge asse price disribuion, in oher words: * * X X h, wih P u 1/ 6 0, wih P /3 m h, wih P 1/6 According o he above derivaion, S. Figlewski and B. Gao (1999) use AMM o find he single node s price a ime wihou considering he fine mesh srucure. As he above-menioned, in he logarihm normal disribuion, if he sep beween each node of he ree is consan (h and Δ ), we can make use of he explici finie difference o develop he fine mesh srucure. Thus he approach would decrease he lineariy error. Because he conrac of he European opion is succinc, he model wih symmery will increase he convergence speed while compuing. Figlewski and Gao (1999) sugges o replace he original logarihmic asse price X* by he average mean-adjused logarihmic asse price ( X X * () ). In oher d words, he mean of * X will be zero a any ime. This also implies he early process for he original X is: X X h, wih Pu 1/ 6, wih P /3 (7) h, wih P 1/6 Therefore, in he condiion ha he asse price x and mauriy dae T, he general formula of sock opion price can be wrien as: m d 3

34 r C( X, ) e ( P ( h, ) C( X h, ) P ( h, ) C( X, ) u P ( h, ) C( X h, )) d X C( X, T) ( e X ), X (8) In he above equaions, he boundary condiion () of he mauriy day means he value in he bracke is posiive or zero, which is he same wih he siuaion of general laice model. Noe ha Eq. (8) allows he probabiliies (i.e. P u, P m, P d ) would vary wih h and, whereas hey are fixed in he curren case of Eq. (7). Following, we will describe he applicaion of AMM o European Opion (i.e. Plain Vanilla Opion). We use Figure 5 o illusrae he fine mesh srucure of one-level AMM around srike price a mauriy day. We will consruc he one-level fine mesh beween dae T and dae T-Δ. In Figure 5, he coarse laice is he original rinomial ree wih price and ime seps h and Δ, is denoed by heavy lines. The ligh lines represen he fine mesh wih price sep size h/4 and ime sep size Δ/4. The fine mesh covers all he node of he coarse mesh a ime sae T-Δ. The saring nodes of he fine mesh m include A, A 3, A 4, and A 5. In he fine mesh branching from node A, X is he highes ou-of-he-money node while X is he lowes in-he-money node. Since all branches saring from nodes below A 1 all end up in-he-money and all branches saring from nodes above A 6 are all expired ou-of-he-money. So here is no need o fine he mesh below node A 1 and above node A 6. When he laice model used o evaluae sock opion, he nonlineariy error would occur in he dae closing o he mauriy day. Thus, in he Wiener process, price sep h is direcly relaed o he variaion duraion (i.e. 4

35 h ). For ha reason, while we apply he one-level fine mesh srucure for pricing our arge asse, he price volailiy h and duraion lengh would conver o h /and /4, respecively. Besides, he one-level fine mesh will consruc beween ime T and ime T-Δ. For wo-level mesh, he price volailiy h and duraion lengh would conver o h /4and /16, respecively and i will be consruc beween ime T and ime T-Δ/4. Consequenly, if we ake M-level fine mesh srucure, he corresponding parameers will change o hm h/ M and /4 M, respecively. And i will be consruced beween ime T and ime T. If we increase one ( 1) 4 M level o he laice, he number of node will increase 5. Even hough he CRR model and rinomial ree model could achieve o convergence by increasing he segmenal number of period n comprehensively, hese approaches are no effecive enough lime AMM. For CRR model or rinomial ree model, here are (N +1) nodes of price compuaion in oal, where N is he number of price sep. Therefore, while cuing he price sep in half o reduce he nonlinear error, i would lead N become quadrupled (h is direcly relaed o ) which implies 16 imes compuaion amoun han before. On he oher hand, we compare hem wih M AMM. For example, we see he 1-level AMM in Figure 5 and find ha we only need o add 40 nodes of price compuaion o he criical region. (The oal number of node of 1-level AMM: 5; The coarse mesh region of 1-level AMM: 1; he fine mesh region of 1-level AMM including he overlap region, hen we only need o increase 5-1=40 nodes) On he oher hand, -level AMM wih only 5 ime seps, which is much more accurae han a sandard rinomial ree wih 50 ime seps, and only a lile less accurae 5

36 han a 1000 ime seps binomial ree which require 50 imes greaer execuion ime. Alhough he binomial ree runs disincly faser, i is only abou half as accurae as he sandard rinomial ree and much less accurae han he AMM. Furhermore, he 1-level AMM is abou four imes as accurae as he sandard rinomial ree. The -level AMM, wih "finer mesh, is even abou four imes as accurae as he1-level AMM. These descripions also indicae AMM can reduce he nonlineariy error wihou sacrificing is efficiency. If we increase more level number M, we will obain more accuracy. When we increase one level o he laice, he number of node will only increase 5. I won add oo much compuing ime o he whole model. 6

37 h/ Fig. 5 A one-level AMM for a pu opion of Plain Vanilla Opion around srike price a mauriy day. (II) American opion: For American opion, he nonlineariy error is also largely accouned for he error in he las ime sep. Besides, here is also an approximaion error wih regard o where he early exercise occurs. While we use AMM o evaluae American opion, we should se up he 7

38 fine mesh srucure around he las several periods execuing prices, using he calculaing mehod of he previous AMM for European opion. We use he AMM laice in Figure 6 o illusrae. In he coarse mesh, we se he srike price X as he cener poin and selec he wo neighbor asse price X (node A 11 ) and X (node A 1 ) as he criical region of he fine mesh srucure. In order o achieve he accurae resul, Figlewski and Gao (1999) believe ha he calculaing pah of he fine mesh srucure should covers he region of in-he-money and ou-of-he-money. Hence he calculaing range of he coarse mesh node which connecs he fine mesh should exend from ( X, X ) o( X h, X h). In oher word, in he mauriy day, we exend he original criical region from ( A11, A1) o ( A10, A 13). For he dae T-Δ, he nodes A and A 5 have he same asse price wih nodes A 10 and A 13 a mauriy dae. From( A, A 5 ), we also spread heir calculaing range o ( X i h, X i h), ha is, ( A8, A 15). Thus, he whole fine mesh srucure is surrounded by he rapezoid composed of nodes A, A 5, A 15 and A 8. When we calculae he American opion, we mus handle he fine mesh srucure firs. Is process is similar o he general laice model. Take subscripion o warran for example, he fine mesh node B in Figure 6, whose warran price is formed by A 15, B 1, and A 14 : r f e /4 ( P ( h/, / 4) C P ( h/, / 4) C P ( h/, / 4) C B u A m B u A P ( h/, / 4) C ) (9) d A 14 8

39 Fig. 6 AMM mesh srucure char. 9

40 CHAPTER 3 Mehodology 3.1 Empirical Procedure In his secion, we discuss he corresponding assumpion, limiaion and he operaion mehod of Modified RT model (AMM-RT). Originaed form Adapive Mesh Model proposed in 1999 by S. Figlewski and B. Gao., we also apply fine mesh srucure during he period of (T-1, T). T is he mauriy day here. In he RT model, he ime sep during (T-1, T) is 1, ha is n=1. The laice srucure of AMM-RT is no only based on RT model, bu also wih he idea of AMM. We cu he period of (T-1, T) ino m subinerval (we call he segmenal level of he las rading day m, i.e. we use m=, 3, 5 in he hesis). The approach in (T-1, T) possesses he essence of AMM. In he following menions, we inroduce he empirical procedure of Modified RT model under he sock opion price predicion. A. Using Original RT ree laice before period T-1 In his secion, we simplify GARCH (1, 1) model and do parameer 30

41 esimaion in he firs. We use he assumpion in RT model and K=3 (hree variance in each node). The risk free rae r f is.5%. We use he arge sock s rae of reurn and GARCH model wih ou of sample esimaion o esimae he parameers of our esimaing period ω, α, β, λ and he iniial variance h 0. For each day of our evaluaing period, each day will has iself GARCH parameers (ω, α, β, λ, h 0 ). Then, for each evaluaing day, we se ω, α, β and he variance of rae of reurns of asse h 0 as he beginning value o consruc RT model. The pricing empirical procedure before dae T-Δ is shown sep by sep as follows: Sep1: Le n=1 o consruc he rinomial ree and j=1, 0, -1. Sep: Calculae h0 and n. Since n=1, hus n. n Sep3: Using he inequaliy 1 o find he value. h Sep4: Subsiue he variance h (he iniial value of h is h 0 and h 0 is known) of his period (day) ino formula of h +1 (he variance of he nex period under pas period variances have known) h 1 ( h ) h o find he variance of nex period (i.e. h +1 ). Sep5: Due o he pah dependence issue, he variance of nex day will be influenced by he variance of he previous day. Wih he ime increase, he number of pah arriving a each node will increase. Thus, he number of variance of each node will increase oo. Then, here will be more han one variance in each node. Thus, when we proceed o he nex dae, we should compare he value of variance of each node 31

42 in he dae. Nex, we reserve he maximum and he minimum variance o calculae he η of nex dae. (Tha is, we subsiue h ino h 1 o calculae he η of nex dae). We REPEAT Sep4 o Sep5 unil day T-1. Sep6: Afer Sep5, we already consruc he ree laice of variance before dae T-1. A he same ime, we also consruc he ree laice of probabiliy. p p p u m d h ( r h/ ) f n h 1 n h ( r h/ ) f n n n B. Using AMM-RT Model in he las (T-1, T) period Afer using he RT laice srucure before T-1 period (his is so-called coarse mesh srucure in our model), we will apply our modified mehod o consruc he fine mesh srucure during las (T-1, T) period. In he original RT model, n=1 is used during las (T-1, T) period. In he following menion, we will use differen value of m (m is he segmenal level in he las rading day) in he las period. The modified RT (AMM-RT) model wih m=, 3, 5 will be discussed and compared. For convenience o describe, we show Figure 7 o explain he laice in he las period. A day is cu ino m = 3 3

43 periods, and he jump size urn o m. For one inermediae node 3 a day T-1, m + 1 saes a day T follow each sae a day T-1. ( y, h ) T 1 T 1 m=3 1 day Fig. 7 For node a period T-1, when a period (day) is cu ino m = 3 periods, and he jump size is m 3 In our AMM-RT model, we only need o add moderae node in he las during las (T-1, T) period. This will no cos much compuing amoun as well as increase pricing accuracy. Alhough AMM also increase is node in he las ime sep (for 1-level AMM, we increase he mesh densiy in he las (T-1, T) period), AMM can capure more complee price probabilisic disribuion funcion and he condiional variance. We have inroduced AMM algorihm in Chaper. The probabiliy disribuion of AMM ree laice is fixed and he price sep and ime sep are also fixed oo. Even hough he fine mesh in he las ime sep increase he accuracy of pricing, i seems no o be 33

44 enough efficienly. For our AMM-RT model, he probabilisic funcions are no only non-fixed bu also he variance of each ime sep updae wih ime. This will capure more informaion of arge asse han AMM and achieve more accuracy a he same ime. The AMM-RT model cu he las period (day) ino m subinervals (i.e. increase m), and hus, he discree disribuion of laice model will more approach o he coninuous disribuion of laice model. Wih m increasing, he disribuion error decreases. Besides, he nonlineariy error will occur when RT model applying in some exoic opion. For example, when barrier opion approaches o he barrier price, he nonlineariy error occurs. I sugges our AMM-RT model wih he same essence as AMM will be able o price his ype opion. The procedure in he (T-1, T) will be shown as following Sep7~Sep14. We assume m=, 3, 5 in he (T-1, T). Sep7: In he Sep 7, we increase m o add he mesh densiy. Le m=, 3, 5, respecively o consruc he rinomial ree and j 0, 1,,..., m Sep8: Calculae h0 and m. m=, 3, 5. m Sep9: Using he inequaliy h T 1 1 o find he value of las period (day) T. Sep10: Subsiue he variance h T-1 ino formula of h +1 h 1 ( h ) h o find he variance of nex period (i.e. h T ). Sep11 Afer Sep 10, we already consruc he variance ree laice. A he 34

45 same ime, we also consruc he ree laice of probabiliy of las period (day). p p p u m d h ( r h / ) 1/ m f n h 1 n h ( r h / ) 1/ m f n n n Sep1: We consruc he ree laice of sock price. For period (0, T-1): y 1 y j, j 0, 1,,..., n For period (T-1, T): yt yt 1 j m, j 0, 1,,..., m Sep13: Calculae he sock opion price; his price a every node should be he same. C ( i) C ( i) Max 0, S ( i)-x max min T T T Sep14: Afer he Sep13, we apply he backward recursion and discoun, and hen we can ge he sock opion price a day =0. Using he equaions recursively as follow: 1 ( j ( r h ( i, k)) h j h h j n m a n f nex ( ) [ ] ; a h ( i, k) 0, 1,,..., (or ) in erp a a C j q j C i j L q j C i j L ( ) ( ) (, ) (1 ( )) (, 1) 1 1 n a rf (, ) ( ) in erp ( ), 0, 1,,..., (or ) j n C i k e P j C j j n m 35

46 If we increase m, he disribuion error will decrease. Furhermore, when he laice model used o evaluae sock opion, he nonlineariy error would occur in he dae closing o he mauriy day. Thus we only cu he ime sep in he las period o rack he asse price and reduce he nonlineariy. If he segmenal level m is larger in he period (T-1, T), we can obain more accuracy. Besides, i won add oo much compuing ime o he whole model. 36

47 CHAPTER 4 Numerical Illusraion 4.1 Daa Analysis To examine he empirical performance of he GARCH opion pricing model, we applied he model o daily closing prices of he Taiwan Sock Exchange Capializaion Weighed Sock Index (TAIEX) and is corresponding TAIEX opions. For simpliciy, we will jus consider he call opions here. We use he index and is corresponding opions based on he following consideraion. The firs reason is ha he index and he opion daa are freely available on he websies. Furhermore, he TAIEX index opion is he mos acively raded European-syle opion in Taiwan. Thus, he TAIEX opion marke is chosen o es he empirical performance of he Black-Scholes model, RT model and AMM-RT model. In nex secion, we will focus o esimae he call opion price in Sepember 007 (007/9/3~ 007/9/31, 18 rading days). We use he TAIEX index wih he sample period from Sepember o December 007 (pas 5 years) o esablish he GARCH volailiy dynamic. There are 139 observaions. 37

48 4. Numerical Analysis In his hesis, we will apply our AMM-RT rinomial laice model o price he sock opion price. Firs, we should esimae he parameers of GARCH model under P measure. We use TAIEX as our approximaing samples. Here, we choose TAIEX index wih he sample period from Sepember, 00 o Augus 31, 007 as esimaive period of GARCH model. For example, he call opion price of 007/9/3 will be esimaed under he esimaive period 5 years prior o his day (i.e. 00/9/~007/8/31). Following, we use rolling sample mehod o esimae he subsequen parameers of GARCH model. Fig. 8 shows he daily observaions of TAIEX during 00/9/ 007/1/31. Based on Bakshi, Cao, and Chen (1997), Duan and Zhang (001), we define a call opion is said o be a-he-money if he moneyness is beween (1.00, 1.03), in-he-money if he moneyness is beween (1.03, 1.06), ou-of-he-money if he moneyness is beween (0.94, 0.97) and deep in-he-money if he moneyness is greaer han1.06 and deep ou-of-he-money if he moneyness is less han We ampuae he daa whose moneyness greaer han 1.1 or smaller han 0.9, because he volume if rade of hem are small. Table 1 provides he average and sandard deviaion of call opion prices repored for each moneyness caegory, and also shows he numbers of observaions in hese caegories for he period from Sepember 1, 007 o Sepember 31, 007 in Figure 8. 38

49 Table 1 Summary Saisics for TAIEX Call Opions (Sepember)* Moneyness (S/K) DOTM <0.94 OTM ATM ATM ITM DITM >1.06 Average Sd. Dev Number Sum 388 *The summary saisics of TAIEX call opion near closing prices are repored for each moneyness caegory. Moneyness is defined as S/K, where S denoes he closing value of he TAIEX and K denoes he exercise price of he opion. The sample period is from Sepember 1, 007 o Sepember 31, 007 wih a oal of 559 call opions. For he selecion of opion daa, we ampuae he rading days which are less han 7 days (since he volailiy is large) and more han 40 days (since he volume of rade is small) away from he esimaed rading day. 10,000 9,000 8,000 7,000 6,000 5,000 4,000 3, Figure 8. TAIEX during 00/9/ 007/1/31, 131 daily observaions. From Figure 8, we find ha he TAIEX rend appears buoyancy during 00~007. The index rises from (00/9/) o (007/8/31) 39

50 and subsequenly has inense vibraion. The index is a 007/1/31. We also show he rae of reurn (log reurn) of TAIEX during 00/9/ ~ 007/1/31 in Figure Fig. 9 Rae of reurn (log reurn) of TAIEX during 00/9/ 007/1/31 wih 130 daily observaions. I is noed ha he observaion will lessen 1 afer selecing he log reurn. Volailiy clusering is also observed in he Figure 9, a large value ends o follow by anoher large value. This is known as he condiional heeroscedasiciy. Thus his daa is suiable o be analyzed by GARCH opion pricing model. 40

51 We also show he relaive saisics of TAIEX in Table. Table The elemenal saisic of TAIEX during 00/9/ 007/1/31 wih 130 daily observaions. Saisics Mean Median Maximum Minimum Sd. Dev Skewness Kurosis From Table, he average rae of reurn is posiive, which also indicaes he rend beween 00/9/ 007/1/31 appears buoyancy. The rae of reurn appears o shif o lef (he skewness is negaive) and possesses fa ail, which also accords wih he characerisic of he rae of reurn of Index. 41

52 We also show he esimaed parameers of he GARCH model under P measure in Table 3. Table 3 he esimaion of he GARCH model under P Measure (00/9/ 007/8/31, 139 observaions) S S 1 ln( ) r h h, ~ N(0, h ) f 1 h h 1 1 Esimaed parameer (-.74) (3.166) (8.93) (80.194) *The value in he bracke is he value, which used o evaluae he opion price of 007/9/3. We sill need o esimae he parameers of he GARCH model again using rolling sample mehod when we evaluae he forhcoming days opion price (007/9/4~007/9/31). 4