1

The Forecasting Performance of Implied Volatility Models for TAIEX Options

by Chang-Wen Duan, Ken Hung, and Philip Russell

I. Introduction

Volatility forecasting has spawned a research industry and has been of interest to both academics

and practitioners becaucse of its role in option pricing, risk management, and portfolio

management. Several approaches have been used for forecasting realized volatility using both

historical and implied measures. We observe four main streams of time series volatility

forecasting models: historical data models, ARCH (auto regressive conditional hetroscedasticity)

models and varitions of ARCH such as GARCH, EGARCH, stochastic volatility models, and

implied volatilty models. Historical volatility tends to be effective when volatilty levels are stable.

The emphasis of late has been on implied volatility.

Implied volatility shows the market’s ex-ante forecast of the volatility of the underlying

asset. Traditionally, implied volatility has been calculated using the BSOPM. One can determine

the implied volatility by observing the market price of the option and then inverting the option

pricing formula. This will yield the volatility implied by the market price. Several studies have

examined the informational efficiency and predictive ability of implied volatility and the results

are mixed. In this paper, we further explore the relative performance of volatility forecasting

models. The reults are analyzed using monthly TAIEX Options (TXO) for the period 2001-2008

for a total of 1586 trading days. We use three different volatilty measures: Bayesian ARFIMA

volatility of ARFIMA volatility (ARFIMA-V), implicit price level with volatility (ATM-IV) and

endogenous volatility of volatility function (DVF-IV). We conduct both in-sample and out-of-

sample tests to compare the models and also consider the implications of the forecasting model

for investment performance. Empirical findings suggest that 3,6,9 and 15 minutes TXO interval

estimation of the true volatility is relatively stable. ARFIMA model provides the best forecast of

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volatility with higher explanatory power. Finally, DVF model's forecast performance of volatility

leads to best investment perormance.

Keywords: inclusive return, long memory, ARFIMA, implied volatility

The remainder of the paper is organized as follows. Section 2 provides a a brief review of

the literature followed by introduction of the data and models in section 3. Section 4 discusses

the results and section 5 concludes.

2. Literature review (not completed)

3. Data and Methodology

3.1 Data

Our sample consisted of monthly TAIEX Options (TXO) listed as of December 24, 2001 to May

20, 2008, giving us a total of 1586 trading days. Taiwan Futures Exchange launched European

style TXO on April 1, 2001. We used short-term contracts due to the illiquid nature of the longer

term contracts. We restricted our data to pre-May 20, 2008 as the Taiwan index plunged

significantly after the Presidential election. The data was retreived from Taiwan Economic

Journal (TEJ) database. To test the effectiveness of the predictive models, we used 100 days

outside the sample period from December 19, 2007 to May 20, 2008. Implied volatility estimates

are based on risk-free interest rate for Taiwan's first-year bank deposit rate. If the implied

volatility is negative or greater than 100%, it is eliminated. Also, if the contract expires within 7

days, it is eliminated. Thus the option contracts used in the sample have a maturity period of 8 to

30 days.

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3.2 Models

We test three diffferent models for forecasting, which seem to be more in line with the real world,

as introduced below.

(1) Inclusive return regression model

In order to explore the predictive ability of volatility models, the predictive power of

explanatory variables and selection of appropriate predictor variables is important. The

inclusive regression model is expressed as follows:

1 1 10 | | |( ) ( ) ( ) ( )e e e

t t t

IV ATM ATM DVF DVF RV RVt t t t tLn Ln Ln Ln

(1)

where. β and ε are the regression coefficients and error term, (e

t 1

RVt | )is implied volatility

predictive value (e

t 1

ATMt | ), (

e

t 1

DVFt | ) is function of endogenous fluctuations in volatility

forecast value and fit through ARFIMA volatility after the real predictive value.

(2) A single variable regression model

To compare the single predictor variables on the predictive power of implied volatility, we use

the folowing single variable regression model:

10 1 |( ) ( )t

IV forecastt t tLn Ln

(2)

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which for the first dayIVt can be expressed as a price level implied volatility forecast

1| t

forecastt

as a function of endogenous fluctuations and volatility forecasts ARFIMA model to fit different

range of the true volatility forecasts.

(3) The price of the classification

In much of previous research, price (Moneyness, M) is calculated using the Delta as proposed

classification criteria. The purpose of this paper is to find implied volatility of good predictive

value so if the Delta is based on the volatility that is one of the input variables, it contradicts the

purpose of this study. Bakshi, Cao, and Chen (1997) deal with the calculation of price

classification. Tai Zhiqi goods (TXF) is used to replace the daily closing price and is calculated as

follows:

F

MK

(3)

Where F is the Tai Zhiqi closing price of the goods in recent months, if no closing price on the

day the settlement is used, K is the strike price;

Estimates of Predictor Variables

In order to use the predicted value of the implied volatility forecasts, we must first construct

estimates of these predictor variables. We first have to estimate the price level implied volatility

(ATMt ), true volatility (

RVt ) function with endogenous fluctuations in volatility (

DVFt ) and

other three estimates. Further application of these estimates into a predicted value predict the

implied volatility (IVt ), the estimates for the price level implied volatility predictive value

(e

t 1

ATMt | ). ARFIMA fits the true volatility predictive value (

e

t 1

RVt | ) function with endogenous

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fluctuations in the volatility forecast (e

t 1

DVFt | ). These predictor variables on the concept of

implied volatility forecasts, can be expressed as follows:

1| t

forecastIV

tt

(4)

IVt refers to time t,

t-1

forecast

t | Ωσ , the implied volatility estimates for the time t-1 can be applied to

obtain all the information (Ω) under the estimate of the volatility of the predicted value,e

t 1

ATMt | ,

e

t 1

DVFt | and

e

t 1

RVt | . The information includes a t-1, t-5 t-20 days with volatility estimates, and

further takes the average for volatility forecasts. Because we use volatility forecast that includes

the information content of the early, rather than the general estimates as independent variables,

the predictive value is expected to have more explanatory power.

Investment performance simulation

We used foreign strategic investment portfolio simulation model to test whether volatility

forecasts can correctly predict. Simulation period includes data from December 19, 2007 to May

20, 2008, including 100 days of data outside the model period. Cross style portfolios are

constructed using recent months price of the option-based level, according to the concept of delta

neutral, and find the market price closest to theoretical K value of TXO contract to carry out

transactions.

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In the trading strategy portfolio simulation, we collected short-term and long-term portfolio

volatility prediction generated using implied volatility forecast models. The model can be

obtained through the BS-TXO's theoretical price, and the theoretical price compared with the

actual settlement price of the day. If the theoretical price is less than the daily settlement price, we

use short-term portfolio, otherwise we use long-term portfolio.

Volatility estimation

Implied volatility estimates and forecasts

To obtain estimates of implied volatility, we evaluate the model through the BS to solve the

inverse function. Newton - McPherson method (Newton-Raphson Method) to solve the future.

Newton - McPherson principle of law for the first initial of the volatility to make a guess, re-use

value for their continuous adjustment of Vega, is used to approximate the tangent, relative to the

dichotomy, the convergence speed is faster. Second, implied volatility on the price level

calculation predicted values, according to price our classification, the selected price level of

implied volatility estimates of the average, and further calculate t-1, t-5 and t-20 volatility

estimates on the average, in time t price level as implied volatility forecasts.

Dumas, Fleming, and Whaley (1998) adopt a different approach from the BS model, They note

that the strike price of the option implied volatility is influential, and the number of days from the

due date will affect the the time value of options, so one can build a set of variables to explain the

implied volatility. In addition, they also attempted to capture the implied volatility to explain the

use of non-linear part of the square term, and combined the strike price and expiration day

interaction of the remaining cross-terms. Dumas, Fleming, and, Whaley (1998) use these

variables to explain the evaluation model of the same period of the implied volatility, and are able

to get a good explanation. Because this model uses five options of the important variables, and

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model specification limit less than the BS model, therefore, we refer to Dumas, Fleming and,

Whaley (1998) set a variable way, change the value of the variable spot price of the futures price,

to describe the implied volatility smile with the price of mobile features, perform the following

formula:

2 250 1 2 3 4)( ( ) ( ) ( )IV

t t t t t t t t

F F FLn T T T

K K K (5)

Where α and ε are the regression coefficients and error term; to further minimize the use of

market prices and the BS pricing model estimated price between the theoretical MSE, the

following equation:

2

, , , , , ,1

1, , , , , ,

NIV

t i t i t t i t t i t i t t i ii

Min Mktpr BSpr F K r T F K TN

(6)

Through the above equation, we can get parameters to the value, the further use of these

estimated parameter values to estimate volatility as a forecast basis. In order to predict the same

way as the previous approach, the model's predictive value is also taken at t-1, t-5 and t-20 to

estimated the mean value of the predicted value at time t. Once a day (6) to get a daily value of

parameters to further the time t-1, t-5 and t-20 of the coefficient values and time t paired data

option contract, the time available t-1, t-5 and t-20 of the DVF-IV, finally, the three DVF-IV

estimates to be the average, will receive a time t of the DVF-IV predicted to form inclusive of

return of the predictor variables.

True volatility forecast

(1) the true volatility (RV) According to ABDL (2001) proposed the volatility of high frequency

information is calculated, the main square through the days of short term increases in the total

remuneration to as daily volatility of the index, as a cumulative volatility (integrated volatility),

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many literature data on the use of high frequency volatility estimation method to estimate this all

this:

2 2, ,i it RV t t

i

r (7)

Data for the days in which the price of a continuous time interval Δ return.

ABDL (2001) pointed out that when the sample the higher the frequency, time series, the

smaller the range can be split to avoid the measurement error when estimating the impact. But

from a statistical observation interval if the split is too small, you can not get the consistency of

volatility estimates; contrary, if the interval partition is too large, the market is susceptible to

noise results to estimate bias. Therefore, (8) estimated that the true volatility, the range of the

division determines the true volatility estimates are not biased and effectiveness.

Addition to considering the range of split, the interval length of construction is also subject to the

true volatility of Taiwan stock market, matching each time. Although ABDL (2001) shows days

five minutes interval is a measure of the true volatility of the best frequency, but the study of the

subject matter is foreign exchange market with this study is different. Taiwan stock market is

about every 20-45 seconds match time, reference ABDL (2001), ABDL (2003) and Pong,

Shackleton, Talor, and Xu (2004) recommendations, we set the minimum interval length is 3

minutes, up to 30 minutes interval in multiples of 45 seconds to estimate the true volatility of

simulation. Second, in order to avoid fluctuations in the opening and closing time zone cluster,

the beginning and end of each trading day interval are to be deleted.

Simulation results shown in Figure 1, we find that volatility is relatively stable range of 27-30

minutes were 6,9-12,14-15,17-23 and other, more in line with 45 seconds have multiple

relationships were 6,9, 15 and 30 minutes, so we will adopt the four intervals formed by the

follow-up to do the real volatility of the true volatility estimate.

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(2) The long memory of the ARFIMA model related

Volatility of financial returns on the verification, the discovery of many documents that are

consistent, the real behavior of volatility has long memory of the phenomenon (Andersen,

Bollerslev, Diebold, and Labys (2001, 2003) and Andersen, Bollerslev, Diebold, and Ebens

(2001)). They believe that the true volatility is no single one (unit root) case, but the series has

presented scores of integration, there are a few very slow response phenomenon. Thus, it can

describe the long memory nature through the process of ARFIMA time series models to

characterize. Therefore, in order to avoid measurement error, we will solve by ARFIMA model to

these concerns.

ARFIMA presents the basic model, we take the natural volatility of the real number was set to

ARFIMA time series models in the stability of the series, the ARFIMA (1, d, 0) model is the

volatility of the following formula:

0 1 1(1 ) ( ) ( )d RV RVt t tL Ln Ln e (8)

Where L is the gap factor, for the true volatility, ω is the coefficient, e is the error term.

In the ARFIMA (p, d, q) model specification process, p and q values must take precedence to

determine the estimated value of d, the reference number of for volatility fit ARMA (p, q) process

are that, over a high level of p / q are the model coefficients will lead to biased estimates is too

large, many papers often p = 1 / q = 1 or p = 1 / q = 0 to be a good fit can be the result. In order to

avoid a high level of p and q produce bias, so we ARFIMA (1, d, 0) to set the initial estimate of

way, then follow the smooth analog d value, to determine the d value. Finally, according to the

best ARFIMA (p, d, q) model is derived to estimate the true volatility, we also adopt the same

estimate of the prediction methods, the time t-1, t-5 and t-20 to the estimated value The average

forecast for time t the value.

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According to Black (1976), Christie (1982), Schwert (1990) and Cheung and Ng (1992) pointed

out that the changes and fluctuations in the same period a negative relationship between returns,

this relationship through the leverage effect (leverage effect) to explain; This theory concluded

that fluctuations in stock price movements will cause a change, and fluctuations in the share price

will not be back to, in other words, price changes are due, is a result of changes in volatility.

Therefore, we believe that volatility in the forecast, the index returns also contain information

about future volatility of the content, so we further set the dummy variable (I), to characterize the

leverage effect, which is currently a rate of return (rt-1) small to zero, a dummy variable is 1, and

zero otherwise, add ARFIMA (1, d, 0) model, the formation of ARFIMA (1, d, 0) + Leverage

effect model, denoted by ARFIMA-L model:

10 1 1 2 1 0(1 ) ( ) ( ) ( )t

d RV RV RVt t t r tL Ln Ln Ln I e

(9)

1 1( ) /t t t tr Index Index Index = One day on the Taiwan Stock Index return rate.

Offenders are usually found in the literature, the message event occurs when the jump, will

increase the frequency of sampling, will result in unstable parameter estimates. Anderson,

Bollerslev, Diebold, and Labys (2003) demonstrate the use of non-parametric methods will jump

from the days of data were composed of isolated, they mainly use the theory of change with a

second high-frequency data to measure the jump variables, such non-parametric estimation is

defined as the second energy variation measurements (bi-power variation measure, BV), BV is

calculated by, get a non-negative value of jump variables.

References Andersen, Bollerslev and Diebold (2003) method, first we define the BV is calculated:

1

21 1 , ( -1) ,

2

( )t t j t jj

BV r r

(10)

Among them, the style and subject, as the size of the daily sampling frequency. According to the

concept, we can reflect that the size of Jump, Jump model is:

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1 1 1[ ( ) ( ) , 0 ]RVt t tJ Max BV (11)

We further variables into the ARFIMA model of J or ARFIMA-L mode available ARFIMA (1, d,

0) + Jump (ARFIMA-J) mode or ARFIMA (1, d, 0) + Leverage effect + Jump (ARFIMA- LJ)

model:

ARFIMA-J model:

0 1 1 1(1 ) ( ) ( )d RV RV Jumpt t t tL Ln Ln J e (12)

ARFIMA-LJ model:

10 1 1 2 1 0 1(1 ) ( ) ( ) ( )t

d RV RV RV Jumpt t t r t tL Ln Ln Ln I J e

(13)

Which βJump to jump coefficient. Therefore, the true volatility related to ARFIMA model fitting,

the total execution ARFIMA, ARFIMA-L, ARFIMA-J with the ARFIMA-LJ four kinds of model

ARFIMA related.

(3) ARFIMA model estimation

ARFIMA model since its inception, the model has been widely used to analyze the data with a

long memory time modeling, parameter estimation and forecasting. Hosking (1981) on the long

memory parameter d values may range from the routine nature and the reversibility study found

that when 0 <d ≦ 0.5, the number of columns, has the constancy, reversibility, and long memory

features; Geweke and Porter-Hudak (GPH, 1983) is further proposed to the least squares method

to estimate the number of d bands. Focus of these studies are nothing more than to explore the

long memory parameter can present the content of the series, without further wish to estimate the

other parameters discussed.

To observe the ARFIMA model to study the literature shows that the parameter estimation

method much more expensive than two-stage estimation method, which estimates the value of d

first, and then to estimate the d value of the input variables, further estimates that all other model

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parameters. However, the two-stage parameter estimation method, in order to want to estimate the

value of d as input variables, and then further estimated model parameter values, within the

endogenous variables as model input variables seem to have committed a logical error. Koop, Lay,

Osiewalski and Steel (1997) estimates of these errors is divided into three types, there are

maximum likelihood (maximum likelihood), approximate maximum likelihood (approximate

maximum likelihood) and two-stage process (two- step procedure). According to Koop et al

(1997) The validation results show that the Bayesian method is compared to these estimates have

many advantages, particularly that it can provide more accurate estimates.

GPH compared to the traditional way, using the Bayesian method has many advantages, and the

weights in advance, collection, estimation method can solve the other problems that can not be

ruled out, so we use Bayesian methods to estimate the ARFIMA (1, d , 0) model to determine the

long memory characteristics of the part of the differential value of parameter d, and in order to

obtain more stable estimates, We have also Bayesian estimation method to simulate the d value.

4. Discussion of empirical results (to be completed)

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Table 1、價性與到期日分類下之隱含波動率選擇權契約剩餘到期日為 8-120 日之契約，並刪除隱含波動率大於 1、

負值與未收斂者，契約期間為 2001 年 12 月 24 日至 2007 年 12 月 18 日

之每日交易資料。價性 (F/K)採以近月份期貨價格配適，深度價內

(DITM)、價內 (ITM) 、價平 (ATM)、價外 (OTM) 與深度價外 (DOTM)

分別為價性大於 1.06、介於 1.03~1.06、介於 0.97~1.03、介於 0.94~0.97

與小於 0.94；其中交易量為該價性之樣本內期間買權契約的總交易口數。

價性 到期日買權 賣權

交易量 平均數 中位數 交易量 平均數 中位數

DITM

all 444157 0.2547 0.2451 460103 0.2895 0.27901-30 355534 0.3042 0.2886 368703 0.3407 0.3144

31-60 67472 0.2444 0.2386 80750 0.2820 0.277361-90 14333 0.2332 0.2257 7577 0.2623 0.2580

91-120 6818 0.2351 0.2338 3073 0.2710 0.2628

ITM

all 2170138 0.2303 0.2241 1534528 0.2466 0.23221-30 1912303 0.2500 0.2386 1308041 0.2679 0.2548

31-60 228703 0.2249 0.2145 210707 0.2425 0.230661-90 22427 0.2233 0.2145 10935 0.2354 0.2193

91-120 6705 0.2255 0.2225 4845 0.2450 0.2306

ATM

all8564135

30.2174 0.2080

56906857

0.2380 0.2225

1-307624680

30.2176 0.2128

50272932

0.2397 0.2241

31-60 9101520 0.2169 0.2064 6354311 0.2391 0.224161-90 241011 0.2174 0.2048 237454 0.2344 0.2193

91-120 52019 0.2185 0.2080 42160 0.2422 0.2257

OTM

all3769527

80.2184 0.2064

31278561

0.2426 0.2241

1-303027089

60.2224 0.2145

25598356

0.2507 0.2306

31-60 7059260 0.2167 0.2016 5386392 0.2414 0.224161-90 322966 0.2166 0.2016 265692 0.2364 0.2177

91-120 42156 0.2199 0.2080 28121 0.2454 0.2290

DOTM

all1701169

50.2459 0.2451

21488202

0.2565 0.2403

1-301269708

00.2711 0.2693

16402746

0.2859 0.2741

31-60 4031752 0.2382 0.2386 4630448 0.2481 0.233861-90 216984 0.2322 0.2306 364934 0.2389 0.2209

91-120 65879 0.2347 0.2306 90074 0.2455 0.2290

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Table 2. IV估計值、ATM-IV預測值與DVF-IV預測值

選擇權契約剩餘到期日為 8-30 日之契約，並刪除隱含波動率大於 1、負值與未

收斂者，契約期間為 2001 年 12 月 24 日至 2007 年 12 月 18 日之每日交易資料。

t 日預測值為時間 t-1、t-5 和 t-20 日之估計值予以平均後而得之。

統計量 IV ATM forecastIV

DVF forecastIV

ALL Call Put ALL Call Put ALL Call Put平均數 0.2797 0.2637 0.2894 0.2481 0.2551 0.2703 0.2729 0.2592 0.2851

標準差 0.0740 0.0756 0.0836 0.0612 0.0625 0.0649 0.0618 0.0698 0.0720

偏態係數 0.5021 0.6116 0.7191 0.2979 0.2903 0.2494 0.2932 0.4412 0.3486

峰態係數-

0.10500.4077 0.8725

-0.8039

-0.2347

-0.4302

-0.5549

-0.0789

-0.2155

最大值 0.5473 0.5917 0.7189 0.4391 0.4929 0.4889 0.4680 0.5115 0.5175

最小值 0.1350 0.0997 0.1115 0.1305 0.0988 0.1305 0.1532 0.0913 0.1188

樣本數 1486 1486 1486 1466 1466 1466 1466 1466 1466

Table 3、ARFIMA(1, d, 0) 相關模型估計真實波動率預測值

區間為估計真實波動率之報酬率計算區間；Leverage 與

Jump 分別表示為槓桿效果與跳躍過程。t 日預測值為時間

t-1、t-5 和 t-20 日之估計值予以平均後而得之。

區間 統計量 ARFIMAARFIMA

+LeverageARFIMA+Jump

ARFIMA+L+J

6Min

平均數 0.8766 0.8774 0.8766 0.8766

標準差 0.0248 0.0315 0.0250 0.0137

最大值 0.9411 0.9970 0.9415 0.9612

最小值 0.8164 0.8016 0.8167 0.8097

樣本個數 1465 1465 1465 1464

9Min

平均數 0.8817 0.8825 0.8818 0.8815

標準差 0.0250 0.0323 0.0253 0.0133

最大值 0.9503 1.0152 0.9506 0.9635

最小值 0.8184 0.8085 0.8155 0.8125

樣本個數 1465 1465 1465 1464

15Min

平均數 0.8259 0.8268 0.8259 0.8260

標準差 0.0218 0.0304 0.0219 0.0119

最大值 0.8819 0.9418 0.8822 0.8764

最小值 0.7697 0.7584 0.7705 0.7836

樣本個數 1465 1465 1465 1464

30 平均數 0.5871 0.5878 0.5872 0.5877

17

Min 標準差 0.0082 0.0174 0.0094 0.0136

最大值 0.6104 0.6454 0.6108 0.6191

最小值 0.5650 0.5536 0.5367 0.5159

樣本個數 1465 1465 1465 1464

Table 4. 單一變數包含式迴歸：ARFIMA模型

迴歸模型為：e

t 1

IV Model RVt 0 t | tLn( ) Ln( )

，其中e

t 1

RVt | 分別為 ARFIMA、

ARFIMA+Jump、ARFIMA+Leverage 與 ARFIMA+Leverage+Jump 等模型估計之真實

波動率；估計之真實波動率區間並分為 6、9、15 與 30 分鐘四個區間； IVt 則為 B-

S 隱含波動率估計值。

Model6 Min 9 Min 15 Min 30 Min 6 Min 9 Min 15 Min 30 Min

ARFIMA ARFIMA+Leverage

β00.9322 *** 1.1054 *** 1.0624 *** 4.7104 ***

-0.6806 *** -0.8519 ***

-0.4623 *** 0.1383

(11.34) (12.34) (11.88) (19.65)(-

13.52)(-

18.11) (-7.74) (1.07)

βModel 0.432 *** 0.4785 *** 0.5053 *** 1.4414 *** 0.1182 *** 0.088 *** 0.1777 *** 0.3457 ***(27.38) (27.06) (26.61) (25.13) (12.67) (9.91) (14.33) (11.25)

F 749.9 *** 732.3 *** 708.3 *** 631.6 *** 160.6 *** 98.2 *** 205.3 *** 126.7 ***

R2 0.3389 0.3336 0.3262 0.3015 0.0989 0.0629 0.1231 0.0797Model ARFIMA+Jump ARFIMA+Jump+Leverage

β00.9242 ***

-0.1142 1.0637 *** 2.3596 ***

-0.6597 *** -0.8013 ***

-0.4681 *** -0.122

(11.15) (-1.51) (11.88) (11.64)(-

12.81) (-16.6) (-7.85) (-1.02)

βModel 0.4312 *** 0.237 *** 0.5056 *** 0.8781 *** 0.1222 *** 0.0986 *** 0.1765 *** 0.2834 ***(27.03) (15.91) (26.6) (18.13) (12.76) (10.74) (14.26) (10.01)

F 730.7 *** 253.3 *** 707.6 *** 328.7 *** 162.9 *** 115.3 *** 203.3 *** 100.1 ***

R2 0.3344 0.149 0.326 0.1835 0.1007 0.0742 0.122 0.0641括號內為 t 值，並依 1%、5%和 10%顯著水準分別標示為***、**和*。

18

Table 5. 分買賣權之單一變數包含式迴歸：ARFIMA 模型

迴歸模型為：e

t 1

IV Model RVt 0 t | tLn( ) Ln( )

，其中e

t 1

RVt | 分別為 ARFIMA、

ARFIMA+Jump、ARFIMA+Leverage 與 ARFIMA+Leverage/Jump 等模型估計之真

實波動率；估計之真實波動率區間並分為 6、9、15 與 30 分鐘； IVt 則為 B-S 隱

含波動率估計值。

Option Call Put

Min 6 Min 9 Min 15 Min 30 Min 6 Min 9 Min 15 Min 30 Min

Model ARFIMA

β0

1.2687 *** 1.4437 *** 1.3306 *** 5.3699 *** 0.7915 *** 0.9689 *** 0.9722 *** 4.5195 ***(15.00

) (15.54) (14.13)(21.02

) (8.31) (9.37) (9.49)(16.62

)

βModel0.5093 *** 0.5583 *** 0.5762 *** 1.6148 *** 0.3996 *** 0.446 *** 0.4801 *** 1.389 ***(31.36

) (30.44) (28.82)(26.42

) (21.85) (21.84) (22.08)(21.36

)

F 983.7 *** 926.3 *** 830.4 *** 698.3 *** 477.4 *** 477.1 *** 487.7 *** 456.1 ***

R2 0.40 0.39 0.36 0.32 0.25 0.25 0.25 0.24

Model ARFIMA+Jump

β0

1.2552 *** 0.0848 1.3233 *** 2.6115 *** 0.7872 *** -0.1930 ** 0.9798 *** 2.3599 ***(14.71

) (1.07) (14.01) (0.22) (8.20) (-2.29) (9.57)(-

10.53)

βModel0.5075 *** 0.2893 *** 0.5747 *** 0.9539 *** 0.3993 *** 0.2158 *** 0.4818 *** 0.8715 ***(30.92

) (18.46) (28.66) (0.05) (21.63) (13.00) (22.16)(16.28

)

F 955.8 *** 340.8 *** 821.3 *** 331.7 *** 468 *** 169 *** 491 *** 265.1 ***

R2 0.40 0.19 0.36 0.18 0.24 0.10 0.25 0.15

Model ARFIMA+Leverage

β0

-0.6456 ***

-0.8477 ***

-0.4088 *** 0.2535 * -0.6865 *** -0.845 ***

-0.4709 *** 0.1236

(-11.94)

(-16.72) (-6.37) (1.82) (-12.4)

(-16.43) (-7.15) (0.87)

βModel0.1369 *** 0.1012 *** 0.2025 *** 0.3887 *** 0.1119 *** 0.0839 *** 0.1701 *** 0.3356 ***(13.67

) (10.58) (15.20)(11.73

) (10.91) (8.65) (12.44) (9.98)

F 187 *** 111.9 *** 231.1 *** 137.6 *** 119 *** 74.8 *** 154.8 *** 99.6 ***

R2 0.11 0.07 0.14 0.09 0.08 0.05 0.10 0.06

Model ARFIMA+Jump+Leverage

β0-0.6207 ***

-0.7946 *** -0.421 ***

-0.0688 -0.6666 *** -0.7959 ***

-0.4724 ***

-0.0994

(-11.24)

(-15.31) (-6.57) (-0.53)

(-11.77)

(-15.05) (-7.19) (-0.76)

βModel0.1417 *** 0.1124 *** 0.1999 *** 0.3116 *** 0.1157 *** 0.0942 *** 0.1698 *** 0.2821 ***(13.81

) (11.39) (15.02)(10.17

) (10.98) (9.36) (12.45) (9.13)

F 190.6 *** 129.7 *** 225.7 *** 103.5 *** 120.6 *** 87.7 *** 155 *** 83.3 ***

R2 0.12 0.08 0.13 0.07 0.08 0.06 0.10 0.05表中括號內為 t 值，並依 1%、5%和 10%顯著水準分別標示為***、**和*。

19

Table 6. 單一變數包含式迴歸：價平選擇權與內生性波動函數

迴歸模型為：e

t 1

IV Model Modelt 0 t | tLn( ) Ln( )

，其中e

t 1

Modelt | 分別表示為 ATM-IV

預測值(e

t 1

ATMt | )與 DVF-IV 預測值(

e

t-1

DVFt | )； IV

t 則為 B-S 隱含波動率估計值。

Model ATM DVFOption Call Put ALL Call Put ALL

β0

-0.3128 ***

-0.2012 ***

-0.1772 ***

-0.2005

-0.1056 ***

-0.0024

(-10.04) (-6.79) (-6.72) (-8.86) (-4.93) (-0.12)

βModel

0.7628 *** 0.8103 *** 0.7974 *** 0.8492 *** 0.9162 *** 0.9896 ***

(34.74) (37.22) (43.72) (53.07)(56.20

)(64.17

)F 1206.8 *** 1385.1 *** 1911.7 *** 2816.1 *** 3158.9 *** 4118.1 ***

R2 0.45 0.49 0.57 0.66 0.68 0.74括號內為 t 值，並依 1%、5%和 10%顯著水準分別標示為***、**和*

Table 7. 預測TXO隱含波動率之包含式迴歸：6分區間

模型:e e e

t 1 t 1 t 1

IV Int ATM ATM DVF DVF RV RVt t | t | t | tLn( ) Ln( ) Ln( ) Ln( )

其中 σIV 為 TXO 隱含波動率估計值，e

t 1

ATMt | 、

e

t 1

DVFt | 與

e

t 1

RVt | 分別為 ATM-

IV 預測值、DVF-IV 預測值與 ARFIMA 模型配適之真實波動率預測值；所

有估計值皆為透過契約剩餘到期日 8-30 天計算而得。

Model σIV β0 βATM βDVF βRV F R2

ARFIMA

ALL0.1178 *

-0.0286 0.9838 *** 0.0325 ** 1375.

5***

0.74

(1.92) (-0.82)(30.98

) (2.06)

Call0.0949 0.0708 ** 0.7329 *** 0.0689 ***

963.4 ***0.66

(1.31) (2.45)(25.92

) (3.77)

Put0.1399 ** 0.0927 *** 0.8113 *** 0.0494 ***

1089.5

***0.69

(2.11) (2.92)(31.01

) (3.11)

ARFIMA+Jump

ALL0.1202 *

-0.0353 0.9893 *** 0.0334 ** 1365.

1***

0.74

(1.95) (-1.00)(31.00

) (2.10)

Call0.0807 0.0693 *** 0.7381 *** 0.0652 ***

955.2 ***0.66

(1.11) (2.4)(26.04

) (3.56)

20

Put0.1429 ** 0.0907 *** 0.8107 *** 0.0506 ***

1076.0

***0.69

(2.14) (2.85)(30.87

) (3.16)

ARFIMA+Leverage

ALL

-0.0108 0.0162 0.9776 *** -0.0029 1370.

4***

0.74

(-0.37) (0.53)(30.73

) (-0.51)

Call

-0.1508 *** 0.0983 *** 0.7748 *** 0.0029

949.6 ***0.66

(-4.21) (3.49)(29.39

) (0.43)

Put

-0.0505 0.1415 *** 0.8095 *** 0.0006 1079.

1***

0.69

(-1.47) (4.92)(30.83

) (0.09)

ARFIMA+Jump+Leverage

ALL

-0.0113 0.0105 0.9824 *** -0.0027 1360.

8***

0.74

(-0.38) (0.34)(30.76

) (-0.47)

Call

-0.1489 *** 0.0943 *** 0.7781 *** 0.0034

945.9 ***0.66

(-4.11) (3.35)(29.42

) (0.49)

Put-0.052 0.1406 *** 0.8085 *** 0.0007

1066.1

***0.69

(-1.49) (4.87)(30.69

) (0.11)表中括號內為t值，並依1%、5%和10%顯著水準分別標示為***、**和*。

21

Table 8. 預測TXO隱含波動率之包含式迴歸：15分區間

模型:e e e

t 1 t 1 t 1

IV Int ATM ATM DVF DVF RV RVt t | t | t | tLn( ) Ln( ) Ln( ) Ln( )

其中 σIV 為 TXO 隱含波動率估計值，e

t 1

ATMt | 、

e

t 1

DVFt | 與

e

t 1

RVt | 分別為 ATM-IV

預測值、DVF-IV 預測值與 ARFIMA 模型配適之真實波動率預測值；所有估計

值皆為透過契約剩餘到期日 8~30 天計算而得。

Model σIV β0 βATM βDVF βRV F R2

ARFIMA

ALL0.1404 **

-0.0275 0.9817 *** 0.0409 ** 1376.

9*** 0.74

(2.17) (-0.82)(30.99

) (2.30)

Call0.0916 0.0784 *** 0.7387 *** 0.0714 ***

961.4 *** 0.66(1.20) (2.75) (26.4) (3.50)

Put0.1928 *** 0.0907 *** 0.8076 *** 0.0674 ***

1093.5

*** 0.69(2.72) (2.95)

(30.91) (3.67)

ARFIMA+Jump

ALL0.1472 **

-0.0293 0.9819 *** 0.0429 ** 1377.

5*** 0.74

(2.27) (-.87)(31.00

) (2.42)

Call0.0885 0.0788 *** 0.7394 *** 0.0705 ***

961.2 *** 0.66(1.16) (2.76)

(26.45) (3.46)

Put0.2045 *** 0.0885 *** 0.8075 *** 0.0705 ***

1095.0

*** 0.69(2.89) (2.88)

(30.92) (3.85)

ARFIMA+Leverage

ALL

-0.0144 0.0167 0.9777 ***

-0.0042 1370.

5*** 0.74

(-0.43) (0.54)(30.76

) (-0.53)

Call

-0.1401 *** 0.0977 *** 0.7731 *** 0.0062

949.8 *** 0.66

(-3.39) (3.47)(29.14

) (0.66)

Put

-0.0423 0.1394 *** 0.8095 *** 0.003 1079.

2*** 0.69

(-1.07) (4.83)(30.85

) (0.33)

ARFIMA+Jump+Leverage

ALL

-0.0136 0.0163 0.9778 ***

-0.0039 1370.

4*** 0.74

(-0.40) (0.53)(30.76

) (-0.50)

Call-

0.1404 *** 0.0977 *** 0.7732 *** 0.0061949.8 *** 0.66

22

(-3.40) (3.47)(29.15

) (0.65)

Put

-0.0404 0.1388 *** 0.8095 *** 0.0035 1079.

2*** 0.69

(-1.02) (4.81)(30.85

) (0.39)表中括號內為t值，並依1%、5%和10%顯著水準分別標示為***、**和*。

Table 9. 預測誤差表

A. 單一變數預測模型

ModelMSE MAE MAPE

CALL ALL CALL ALL CALL ALL

樣

本

內

ATM 0.0450 0.0304 0.1651 0.1369 0.1273 0.1111DVF 0.0281 0.0184* 0.1309 0.1035 0.1008 0.0842

ARFIMA6min 0.0491 0.0463 0.1745 0.1725 0.1369 0.1401ARFIMA15min 0.0523 0.0472 0.1819 0.1747 0.1428 0.1424

ARFIMA6min+Jump 0.0493 0.0465 0.1748 0.1729 0.1374 0.1406ARFIMA15min+Jump 0.0526 0.0472 0.1822 0.1746 0.1431 0.1423

樣

本

外

ATM 0.1030* 0.1031 0.1497 0.1457 0.1468 0.1385DVF 0.1087 0.1127 0.1715 0.1763 0.1634 0.1638

ARFIMA6min 0.1838 0.1388 0.3783 0.3106 0.3177 0.2645ARFIMA15min 0.1526 0.1224 0.3679 0.3181 0.3143 0.2775

ARFIMA6min+Jump 0.1790 0.1358 0.3722 0.3061 0.3123 0.2604ARFIMA15min+Jump 0.1526 0.1219 0.3543 0.3073 0.3022 0.2681

B. 包含式迴歸預測模型

內ARFIMA15min 0.0276 0.0183* 0.1297 0.1036 0.1297 0.1036

ARFIMA15min+Jump 0.0276 0.0183* 0.1297 0.1036 0.10000.0843

*外 ARFIMA15min 0.1106* 0.1186 0.1719 0.1918 0.1597 0.1864

ARFIMA15min+Jump 0.1108 0.1140 0.1721 0.1808 0.1599 0.1676

Table 10、以包含式迴歸預測模型之投資績效模擬結果模擬交易期間為樣本外之 2007 年 12 月 19 日至 2008 年 5 月 20 日，共 100 個

樣本外交易日，投資模擬以近月份的價平 TXO 契約為投資標的。ATM 表單一

ATM 模型預測而得之波動率，透過 B-S 模型求得理論價格，再經由交易策略所得

之投資績效；DVF 為以單一內生性波動函數預測而得之波動率。ADF 表以 ATM、

DVF 和 ARFIMA 真實波動率組合的包含式迴歸模型。

單位：% 不考慮交易成本 考慮交易成本

Model ATM DVF ADF ATM DVF ADF

持

有

一

天

正報酬平均 21.82 21.73 21.43 26.44 25.63 28.32

負報酬平均 -7.95 -8.12 -7.97 -10.20 -10.21 -9.88

總平均 5.99 *** 6.25 *** 5.81 *** 1.08 *** 1.32 *** 0.89 ***

T 值 11.15 11.88 10.67 2.59 2.82 2.45

23

持

有

一

週

正報酬平均 64.80 64.23 67.12 63.37 62.69 66.21

負報酬平均 -24.43 -24.49 -24.05 -26.86 -26.94 -26.56

總平均 8.88 ** 9.81 *** 8.77 ** 3.57 4.49 3.48

T 值 2.12 2.35 2.08 -0.21 -0.15 -0.23投資績效皆以百分比呈現，檢定統計量參考 Johnson(1978)的調整後 t 檢定，並依 1%、5%和 10％

顯著水準標示***、**和*。

Figure 1、不同樣本區間模擬之真實波動率平均值

6%

9%

12%

15%

18%

3M 6M 9M 12M 15M 18M 21M 24M 27M 30M

24

6分鐘 9分鐘

5分鐘 30分鐘

Figure 2、真實波動率在不同樣本頻率下之 ACF 圖

Figure 3、模擬 ARFIMA(1, d, 0) 模型之各單位期間之 d 值

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6 7 8 9 10 11

6 Min 9 Min 15 Min 30 Min