using neural network for forecasting txo price under different volatility models

8
Using neural network for forecasting TXO price under different volatility models Ching-Ping Wang a,1 , Shin-Hung Lin b,2 , Hung-Hsi Huang c,, Pei-Chen Wu d a Graduate Institute of Finance, Economics, and Business Decision, National Kaohsiung University of Applied Sciences, No. 415, Jiangong Rd., Sanmin District, Kaohsiung City 80778, Taiwan b Department of Finance, National Yunlin University of Science & Technology, No. 123, University Rd., Section 3, Douliou City 64002, Taiwan c Department of Banking and Finance, National Chiayi University, No. 580, Sinmin Rd., Chiayi City 60054, Taiwan d Graduate Institute of Finance, National Pingtung University of Science and Technology, No. 1, Hseuhfu Rd., Neipu, Pingtung 91201, Taiwan article info Keywords: Neural networks Black–Scholes model Grey theory GARCH DVF abstract This study applies backpropagation neural network for forecasting TXO price under different volatility models, including historical volatility, implied volatility, deterministic volatility function, GARCH and GM-GARCH models. The sample period runs from 2008 to 2009, and thus contains the global financial crisis stating in October 2008. Besides RMSE, MAE and MAPE, this study introduces the best forecasting performance ratio (BFPR) as a new performance measure for use in option pricing. The analytical result reveals that forecasting performances are related to the moneynesses, volatility models and number of neurons in the hidden layer, but are not significantly related to activation functions. Implied and deter- ministic volatility function models have the largest and second largest BFPR regardless of moneyness. Particularly, the forecasting performance in 2008 was significantly inferior to that in 2009, demonstrating that the global financial crisis during October 2008 may have strongly influenced option pricing performance. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction The Taiwan options market was established by the Taiwan Fu- tures Exchange (TAIFEX) on December 24, 2001. TAIEX options (TXO) were the only options traded in 2001. Besides, other options were subsequently introduced to date, including TEO, TFO, STO, MSO, XIO, GTO and TGO. 3 Trading volume in TXO contracts totaled 1.57 million during 2002, and rapidly increased approximately 96.9 million during 2006. Although a total of eight different options are currently traded on the TAIFEX, TXO options still comprise over 90% of trading volume. 4 Consequently, accurately evaluating TXO prices is important from both academic and practical perspectives. Black and Scholes (1973) developed a European option pricing for- mula (known as the BS formula), and several theoretical option pric- ing models have subsequently been suggested (Cox, Ross, & Rubinstein, 1979; Heston & Nandi, 2000; Rubinstein, 1994). The BS formula presents option price as depending on five factors, including the current price and return volatility of the underlying asset, strike price, risk-free interest rate and time to maturity. Volatility is the only one of these factors that is not directly observable. Conse- quently, the literature introduces various models for estimating volatility. Theoretical option pricing models generally assume that the underlying asset return follows a normal distribution and the mar- ket is frictionless. Since these assumptions deviate from reality, the above option pricing models have difficulty in efficiently evaluat- ing option prices. Consequently, it is desirable to consider an infor- mal mathematic formula for option pricing, such as the neutral network method. Consequently, this study aims to use neural net- work for forecasting TXO prices under different volatility models. The neural network structure used in this study is the backpropa- gation neutral network (BPNN), since this is the most popular structure. The input neurons in BPNN include the five factors of the BS formula. Since the volatility is not observable, this study provides five volatility models, including HV (historical volatility), IV (implied volatility), DVF (deterministic volatility function), GARCH and GM-GARCH (Grey Model-GARCH) models. Addition- ally, the futures index is substituted for the spot index, since the spot excludes cash dividends. 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.11.038 Corresponding author. Tel.: +886 5 2732831; fax: +886 5 2732889. E-mail addresses: [email protected] (C.-P. Wang), [email protected] (S.-H. Lin), [email protected] (H.-H. Huang), [email protected] (P.-C. Wu). 1 Fax: +886 7 3836380. 2 Fax: +886 5 5312079. 3 TEO, TFO, STO, MSO, XIO, GTO and TGO represent the Electronic Sector Index Options, Finance Sector Index Options, Equity Options, TAIFEX MSCI Taiwan Index Options, NonFinance NonElectronics Sub-Index Options, Gretai Securities Market Stock Index Options and Gold Options, respectively. All options in the Taiwan market are European-style. 4 During 2002 through 2009, annual trading volume of TXO contracts for each year totaled approximately 1.57, 21.7, 43.8, 80.1, 96.9, 92.6, 92.8 and 72.1 million, respectively. For more information please see http://www.taifex.com.tw. Expert Systems with Applications 39 (2012) 5025–5032 Contents lists available at SciVerse ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

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Expert Systems with Applications 39 (2012) 5025–5032

Contents lists available at SciVerse ScienceDirect

Expert Systems with Applications

journal homepage: www.elsevier .com/locate /eswa

Using neural network for forecasting TXO price under different volatility models

Ching-Ping Wang a,1, Shin-Hung Lin b,2, Hung-Hsi Huang c,⇑, Pei-Chen Wu d

a Graduate Institute of Finance, Economics, and Business Decision, National Kaohsiung University of Applied Sciences, No. 415, Jiangong Rd., Sanmin District,Kaohsiung City 80778, Taiwanb Department of Finance, National Yunlin University of Science & Technology, No. 123, University Rd., Section 3, Douliou City 64002, Taiwanc Department of Banking and Finance, National Chiayi University, No. 580, Sinmin Rd., Chiayi City 60054, Taiwand Graduate Institute of Finance, National Pingtung University of Science and Technology, No. 1, Hseuhfu Rd., Neipu, Pingtung 91201, Taiwan

a r t i c l e i n f o a b s t r a c t

Keywords:Neural networksBlack–Scholes modelGrey theoryGARCHDVF

0957-4174/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.eswa.2011.11.038

⇑ Corresponding author. Tel.: +886 5 2732831; fax:E-mail addresses: [email protected] (C.-P. W

(S.-H. Lin), [email protected] (H.-H. Huang), CoWu).

1 Fax: +886 7 3836380.2 Fax: +886 5 5312079.3 TEO, TFO, STO, MSO, XIO, GTO and TGO represen

Options, Finance Sector Index Options, Equity OptionOptions, NonFinance NonElectronics Sub-Index OptioStock Index Options and Gold Options, respectively. Allare European-style.

4 During 2002 through 2009, annual trading volumetotaled approximately 1.57, 21.7, 43.8, 80.1, 96.9,respectively. For more information please see http://w

This study applies backpropagation neural network for forecasting TXO price under different volatilitymodels, including historical volatility, implied volatility, deterministic volatility function, GARCH andGM-GARCH models. The sample period runs from 2008 to 2009, and thus contains the global financialcrisis stating in October 2008. Besides RMSE, MAE and MAPE, this study introduces the best forecastingperformance ratio (BFPR) as a new performance measure for use in option pricing. The analytical resultreveals that forecasting performances are related to the moneynesses, volatility models and number ofneurons in the hidden layer, but are not significantly related to activation functions. Implied and deter-ministic volatility function models have the largest and second largest BFPR regardless of moneyness.Particularly, the forecasting performance in 2008 was significantly inferior to that in 2009, demonstratingthat the global financial crisis during October 2008 may have strongly influenced option pricingperformance.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The Taiwan options market was established by the Taiwan Fu-tures Exchange (TAIFEX) on December 24, 2001. TAIEX options(TXO) were the only options traded in 2001. Besides, other optionswere subsequently introduced to date, including TEO, TFO, STO,MSO, XIO, GTO and TGO.3 Trading volume in TXO contracts totaled1.57 million during 2002, and rapidly increased approximately 96.9million during 2006. Although a total of eight different options arecurrently traded on the TAIFEX, TXO options still comprise over90% of trading volume.4 Consequently, accurately evaluating TXOprices is important from both academic and practical perspectives.Black and Scholes (1973) developed a European option pricing for-

ll rights reserved.

+886 5 2732889.ang), [email protected]

[email protected] (P.-C.

t the Electronic Sector Indexs, TAIFEX MSCI Taiwan Indexns, Gretai Securities Marketoptions in the Taiwan market

of TXO contracts for each year92.6, 92.8 and 72.1 million,ww.taifex.com.tw.

mula (known as the BS formula), and several theoretical option pric-ing models have subsequently been suggested (Cox, Ross, &Rubinstein, 1979; Heston & Nandi, 2000; Rubinstein, 1994). The BSformula presents option price as depending on five factors, includingthe current price and return volatility of the underlying asset, strikeprice, risk-free interest rate and time to maturity. Volatility is theonly one of these factors that is not directly observable. Conse-quently, the literature introduces various models for estimatingvolatility.

Theoretical option pricing models generally assume that theunderlying asset return follows a normal distribution and the mar-ket is frictionless. Since these assumptions deviate from reality, theabove option pricing models have difficulty in efficiently evaluat-ing option prices. Consequently, it is desirable to consider an infor-mal mathematic formula for option pricing, such as the neutralnetwork method. Consequently, this study aims to use neural net-work for forecasting TXO prices under different volatility models.The neural network structure used in this study is the backpropa-gation neutral network (BPNN), since this is the most popularstructure. The input neurons in BPNN include the five factors ofthe BS formula. Since the volatility is not observable, this studyprovides five volatility models, including HV (historical volatility),IV (implied volatility), DVF (deterministic volatility function),GARCH and GM-GARCH (Grey Model-GARCH) models. Addition-ally, the futures index is substituted for the spot index, since thespot excludes cash dividends.

The input layer

F

The output layer

The hidden layer

K T r σ 1

c

Fig. 1. The neural network structure.

5 The constant 1 represents the threshold value. Since the volatility r cannot beobserved, Section 3 of this study introduces five models for volatility estimation.

6 Section 4 of this study compares the prediction performances of the adopted 2, 3or 4 neurons.

7 The logistic function and hyperbolic function are quoted from Chapter 4 of Haykin(1999).

5026 C.-P. Wang et al. / Expert Systems with Applications 39 (2012) 5025–5032

Historical volatility models are relatively simple and widelyused, and estimate volatility using the standard deviation of histor-ical asset returns. The BS formula is used to extract the implied vol-atility from market option prices (Schmalensee & Trippi, 1978).Gemmill (1986) and Fleming (1998) verified that the implied vol-atility model outperforms the historical volatility model in fore-casting option prices. To capture the nonconstant relationbetween volatility and strike prices, Dumas, Fleming, and Whaley(1998) developed a deterministic volatility function model thatperformed well in predicting option prices. Furthermore, theGARCH (generalized autoregressive conditional heteroskedasticity)model devised by Bollerslev (1986) has been verified to have goodexplanatory power for financial asset returns volatility. Akgiray(1989) demonstrated that the GARCH(1,1) model outperformsalternatives in predicting stock return volatilities. Successive stud-ies have applied the GARCH model to forecast stock or index returnvolatilities (Claessen & Mittnik, 2002; Corredor & Santamaría,2004; Lamoureux & Lastrapes, 1993). Recently, Chen, Hsin, & Wu,2010 addressed that the GARCH model outperforms alternativein forecasting exchange rate volatility. Additionally, Kayacan, Ulu-tas, and Kaynak (2010) employed the grey model to predict ex-change rates.

Although the BS formula is historic and familiar both to aca-demics and industry insiders interested in option pricing, it fre-quently has inferior out-of-sample performance than the neuralnetwork method (Amilon, 2003; Anders, Korn, & Schmitt, 1998).Several investigations have verified that the backpropagationmethod outperforms the traditional option pricing model in termsof forecasting performance (Lachtermacher & Fuller, 1995; Malli-aris & Salchenberger, 1996; Yao, Li, & Tan, 2000). Furthermore,Wang (2009a, 2009b) and Lin and Yeh (2009) applied the back-propagation neutral network to price TXO call options and com-pared forecasting performances among various volatility modelsand moneynesses using RMSE, MAE and MAPE as the performancecriteria. The sample periods used by Wang (2009a) and Lin and Yeh(2009) run from 2003 to 2004, while that used by Wang (2009b)runs from 2005 to 2006. Lin and Yeh (2009) found that the GARCHmodel achieves better forecasting performance than the HV, IV,GARCH(1,1) and GM(1,1) volatility models. Wang (2009a) foundthat GM(1,1)-GARCH(1,1) outperforms the HV, IV, GARCH(1,1)and GM(1,1)-GARCH(1,1) volatility models for in-the-money andat-the-money options. However, for the out-of-the-money anddeep-out-of-the-money options, the GARCH model has the bestperformance. In a study comparing the GARCH, GJR-GARCH andGrey-GJR-GARCH volatility models, Wang (2009b) found that theGrey-GJR-GARCH volatility model yields the best forecastingperformance.

Similar to Wang (2009a, 2009b) and Lin and Yeh (2009), thisstudy uses backpropagation neural network with one hidden layerto forecast TXO price under different volatility models, using RMSE,MAE and MAPE as the performance criterion. However, this studydiffers from previous works in several ways. First, besides HV, IV,GARCH and GM-GARCH, this study further investigates the perfor-mance of the DVF volatility model, which can capture the volatilitysmile phenomenon (Dumas et al., 1998; Rubinstein, 1994; Zhang &Xiang, 2008). Second, this study examines both call and put op-tions. This study thus can compare the forecasting results of calland put option prices. Third, the spot index is replaced by the fu-ture index, thus avoiding the paid cash dividend and thus satisfy-ing the asset return assumption in the BS formula. Fourth, thesample horizon runs from 2008 to 2009, an interval that includesthe global financial crisis in October 2008. Consequently, this studytests whether forecasting performances remain unchanged duringthe financial crisis period. Fifth, besides moneynesses and volatilitymodels, this study further investigates forecasting performancesusing the impacts of activation functions, number of neurons in

the hidden layer, and different sample periods. Finally, since themeasures of RMSE, MAE and MAPE are not additive measures, thisstudy introduces the best forecasting performance ratio (BFPR) as anew performance measure for option pricing.

The remainder of this paper is organized as follows. Section 2introduces the methodology of backpropagation neural networkand the forecasting performance criteria of RMSE, MAE and MAPE.Next, Section 3 describes the volatility models, including HV, IV,DVF, GARCH and GM-GARCH models. Section 4 then demonstratesthe empirical results, including data description, the best forecast-ing performance ratios for various classifications, and the interac-tion of volatility models and moneynesses. Finally, Section 5presents conclusions.

2. Backpropagation neural network

The backpropagation neural network (simply BPNN) comprisesan input layer, one or more hidden layers, and an output layer. Fol-lowing Haykin (1999), this study employs BPNN with one hiddenlayer for pricing options. Since according to the BS formula optionprice is determined by the five factors, including current price ofthe underlying asset S0, strike price K, time to maturity T, risk-freerate r, and volatility r, several studies use these five factors as theneurons in the input layer; for example, Lin and Yeh (2009) andWang (2009a, 2009b). However, since the spot index in Taiwanstock market does not reflect cash dividends, the futures index Fis being substituted for S0 in this study.

Fig. 1 illustrates the neutral network structure with one hid-den layer. The input layer contains the six neurons of F, K, T, r,r, and 1.5 The hidden layer in this study contains 2, 3, or 4 neu-rons.6 The output layer contains only one neuron, c, which repre-sents the estimated or predicted option price. For generality, theneutrons in the input and hidden layers are denoted by{I1, I2, I3, . . .} and {m1, m2, . . .}, respectively. The activation functionconnecting the input layer and hidden layers is the logistic functionor hyperbolic tangent function.7 Accordingly, the kth neuron valuein the hidden layer is

mk ¼1

1þe�netkfor the logistic function

enetk�e�netk

enetkþe�netkfor the hyperbolic function

(ð1Þ

C.-P. Wang et al. / Expert Systems with Applications 39 (2012) 5025–5032 5027

where netk ¼P

iWkiIi � bk, Wki denotes the weight connecting theith neuron in the input layer and the kth neuron in hidden layer,and bk denotes a threshold. The activation function connecting thehidden and output layers is linear, such that the estimated or pre-dicted option price is calculated by

c ¼X

k

Wkvk � b ð2Þ

where Wk denotes the weight connecting to the kth neuron in thehidden layer, and b is a threshold. The estimation or prediction per-formance for the BPNN is measured by the following error function:

E ¼ 12

Xj

ðcj � cjÞ2 ð3Þ

where cj and cj represent the actual and estimated values for the jthobserved option price. The Levenberg–Marquarft algorithm is usedto minimize the error function and find these values of cj.8

Using the backpropagation neural network, this study respec-tively forecasts the call and put option prices. Market data are par-titioned into two parts, training and testing sets. Similar to Wang(2009a, 2009b) and Lin and Yeh (2009), the training part containsthe earlier 70% of the data and the testing part contains theremaining 30%. This study respectively adopts RMSE, MAE andMAPE, defined below to measure pricing errors, following to An-ders et al. (1998), Lin and Yeh (2009) and Wang (2009a, 2009b).9

RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1n

Pnj¼1ðcj � cjÞ2

s

MAE ¼ 1n

Pnj¼1jcj � cjj

MAPE ¼ 1n

Pnj¼1j cj�cj

cjj

8>>>>>>>>><>>>>>>>>>:

ð4Þ

where n is the number of observations.

3. Volatility models

3.1. Historical volatility and implied volatility

Let St denote the stock index at date t, where the daily returnRt = St/St�1 � 1. Referring to Anders et al. (1998), Yao et al. (2000)and Amilon (2003), the daily historical volatility is

rt ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

29

X30

d¼1

ðRt�d � RÞ2vuut ; R ¼ 1

30

X30

d¼1

Rt�d ð5Þ

The average number of annual trading days is 250 during 2008 and2009.10 Accordingly, the annual historical volatility

rH;t ¼ rt

ffiffiffiffiffiffiffiffiffi250p

ð6Þ

The spot index S0 is substituted by the futures price F, and the BSformula for the call option price c and the put option price p is asfollows:

c ¼ e�rT ½FNðd1Þ � KNðd2Þ�p ¼ e�rT ½KNð�d2Þ � FNð�d1Þ�

d1 ¼lnðF=KÞþ1

2r2T

rffiffiTp ; d2 ¼ d1 � r

ffiffiffiTp

8>><>>: ð7Þ

8 The algorithm, developed by Levenberg (1944) and Marquardt (1963), efficientlysolves the least-squares estimations of nonlinear parameters.

9 RMSE is a common measurement of pricing error used in the literature; forexample, by Amilon (2003), Christoffersen and Jacobs (2004), and Gemmill (1986).

10 Trading data are collected from the beginning of 2008 to the end of 2009, and thesamples for each year comprise 249 and 251 trading days, respectively.

where N( � ) denotes the cumulative distribution function of thestandard normal variable. Besides r, the other parameters in Eq.(7) can be observed from the option market. Consequently, the im-plied volatility can be obtained from Eq. (7). Since each option con-tract can correspond with an implied volatility, the estimatedimplied volatility for date t in this study is equally averaged bythese implied volatilities extracted from all the traded option con-tracts at date t � 1.

3.2. The deterministic volatility function

Rubinstein (1994) illustrated that the BS implied volatility mayvary with different moneynesses. Moreover, Dumas et al. (1998)examined the prediction and hedging performances on a determin-istic volatility function. Furthermore, Peña, Rubio, and Serna(1999) suggested that a nonconstant relation exists between thestrike price and the implied volatility. These phenomena aretermed the volatility ‘‘smile’’ or ‘‘smirk’’ (Zhang & Xiang, 2008).Accordingly, this study assumes that the volatility satisfies a deter-ministic function

rðK; FÞ ¼ a0 þ a1ðK=FÞ þ a2ðK=FÞ2 ð8Þ

where K represents the strike price. For the traded options at datet � 1, r(K, F) represents the BS implied volatility. Subsequently,using all the traded options at date t � 1 as the sample, parametersa0, a1 and a2 are estimated by the OLS regression coefficients in Eq.(8). Next, the estimated volatility at date t is

rt ¼ a0;t þ a1;tðK=FÞ þ a2;tðK=FÞ2 ð9Þ

where a0;t , a1;t and a2;t denote the estimates of a0, a1 and a2,respectively.

3.3. The GARCH(1,1) model

The GARCH(1,1) model is frequently applied to forecast stockindex return volatility; for instance, Akgiray (1989) and Claessenand Mittnik (2002). Additionally, RiskMetrics by Morgan (1995)adopts GARCH(1,1) for estimating stock return volatility.11 Mathe-matically, the GARCH(1,1) model can be presented as follows:

Rt ¼ Rþ et

r2t ¼ aþ be2

t�1 þ cr2t�1

(ð10Þ

where the error term et � Nð0; r2t Þ, Rt and R are daily and average

returns, respectively, and a, b and c are constant parameters.

3.4. GM-GARCH model

Deng (1982) first suggested using the grey system theory tomanage a stochastic control system problem. Subsequently, greysystem theory has been widely applied in both academic researchand practical contexts. (Deng, 1989; Liu & Forrest, 2007; Wang &Liu, 2009) The grey model (GM) transfers disordered data to a reg-ular series without the need to conduct numerous observations.First-order grey model with one variable (GM(1,1)) is the basictype of grey system. Recently, GM(1,1) has been applied to im-prove stock index forecasting performance (Chang & Tsai, 2008;Chen, Hardle, & Jeong, 2010) and option pricing (Lin & Yeh, 2009;Wang, 2009a, 2009b). Since GARCH(1,1) has good power to explain

11 For details regarding the application of GARCH(1,1), please see Jorion (2007) andthe RiskMetrics Technical Manual (1995).

5028 C.-P. Wang et al. / Expert Systems with Applications 39 (2012) 5025–5032

stock returns, as demonstrated by Wang (2009a), this study com-bines GM(1,1) and GARCH(1,1) create a GM-GARCH model. Thatis, the original error term sequence in Eq. (10) is replaced by theGM(1,1) forecasting error term sequence, calculated as followsaccording to Wang (2009a, 2009b).

Let e(0) denote the original error term sequence with tobservations,

eð0Þ ¼ eð0Þð1Þ; . . . ; eð0ÞðtÞ� �

ð11Þ

This study uses daily observations and t equals 6 days. The nonneg-ative sequence x(0)(i) is defined as

xð0Þ ¼ xð0Þð1Þ; . . . ; xð0ÞðtÞ� �

xð0ÞðiÞ ¼ eð0ÞðiÞ �min16i6t

eð0ÞðiÞ� �(

ð12Þ

Moreover, the first-order cumulative sum sequence x(1) is definedby

xð1Þ ¼ fxð1Þð1Þ; . . . ; xð1ÞðtÞg

xð1ÞðiÞ ¼Pi

j¼1xð0ÞðjÞ

8><>: ð13Þ

The background value is defined by

zð1Þðiþ 1Þ ¼ 0:5½xð1Þðiþ 1Þ þ xð1ÞðiÞ� for i ¼ 1; . . . ; t � 1: ð14Þ

The GM(1,1) forecasting error term at date t + 1 is

eð0Þðt þ 1Þ ¼ 1� eað Þ xð0Þð1Þ � ba

� �e�at þmin

16i6tfeð0ÞðiÞg ð15Þ

where a and b are respectively estimated by a and b, as follows:

a

b

� �¼ ZTZ �1

ZTX; X ¼xð0Þð2Þ

..

.

xð0ÞðtÞ

2664

3775; Z ¼

�zð1Þð2Þ 1

..

. ...

�zð1ÞðtÞ 1

2664

3775 ð16Þ

12 For example, on January 22, 2008, the expiration months for TXO are February,March, April, June and September; while the delivery months for TX are February,March, June, September and December. The futures price for April delivery = Thefutures price for March delivery � 2/3 + the futures price for June delivery inJune � 1/3.

4. Empirical results

4.1. Data description

Intraday call and put option prices data for TXO over 500 trad-ing days are extracted from the InfoWinner database during Janu-ary 1, 2008 to December 31, 2009. Notably, the sample horizonincludes the global financial crisis in October 2008. To comparethe results during different periods and during the financial crisisperiod, this study separately considers seven periods. Periods Aand B represent the first and second halves of 2008; periods Dand E represent the first and second halves of 2009; periods Cand F represent the whole years of 2008 and 2009; while periodG represents the two-year sample horizon of 2008 to 2009, respec-tively. Referring to Anders et al. (1998), this study excludes optionswith zero prices and expirations of less than 7 days. The risk-freerate is represented by the one-year term deposit interest rate of-fered by the Bank of Taiwan.

The expiration periods of TXO options have spot month, the nexttwo months, and the next two quarterly calendar months. The deliv-ery months of TAIEX futures (TX) contracts have spot month, thenext month, and the next three quarterly months. Since the TXOexpiration months do not always match the TX delivery months, thisstudy adjusts the futures price based on the weighted average of the

near two futures prices.12 Referring to Rubinstein (1985), Heston andNandi (2000) and Andreou, Charalambous, and Martzoukos (2008),this study classifies moneyness into four levels: in-the-money(M > 1.05), at-the-money (0.95 < M 6 1.05), out-of-the-money(0.85 < M 6 0.95), and deep out-of-the-money (M 6 0.85), wheremoneyness M = F/K for call options and M = K/F for put options.

Table 1 lists the number of traded option contracts for variousmoneyness and time periods. The sample contains 30,904 call con-tracts and 28,525 put contracts in the sample. Meanwhile, 17,868call contracts were traded in 2008 and 13,036 in 2009, while12478 put contracts were traded in 2008 and 16,047 in 2009.Hence, call options are generally more attractive than puts in theTXO market. Particularly, the call and put options weremore attractive before and after the financial crisis periods,respectively.

This study performs a total of 84 experiments and separatelycalculates the training and test errors of RMSE, MAE, and MAPE,corresponding to two option styles (call and put), two activationfunctions (logistic and hyperbolic tangent), three differentneurons in the hidden layer (2, 3 and 4), and seven time periods.For convenience, the logistic and hyperbolic tangent functionsare simplified by the logsig and tansig functions. Each experi-ment contains 60 different training and testing errors, corre-sponding to five volatility models, four moneynesses, and threedifferent numbers of neurons (2, 3 and 4) in the hidden layer.For simplicity, this study only presents the three experimentsduring the whole sample period (2008/01/01–2009/12/31), listedin Tables 2–4. The three tables respectively display the RMSE,MAE and MAPE as calculated using the logsig activation functionfor call options.

Table 2 shows that the in-the-money call options have largertraining and testing errors (RMSE) than the others for various vol-atility models and numbers of neurons, except in the historicalvolatility and four neurons cases. This result is consistent withWang (2009b) but is inconsistent with Wang (2009a) and Linand Yeh (2009), where the in-the-money call options have small-est RMSE in most cases. Table 3 shows that the in-the-money calloptions have larger training and testing errors (MAE) than theothers for most cases, consistent with Wang (2009a, 2009b) andLin and Yeh (2009). Table 4 illustrates that the training andtesting errors (MAPE) in the deep-out-of-the-money call optionssignificantly exceed those in other moneyness options. This resultis consistent with Wang (2009a) and Lin and Yeh (2009) but isinconsistent Wang (2009b), in which the MAPE in out-of-the-money call options frequently exceeds that in deep-out-of-the-money call options. The results listed in Tables 2–4 can beintuitively explained as follows. The in-the-money options arepriced highest relative to others, and the pricing errors of RMSEand MAE are positively related to the magnitudes of optionprices. Additionally, the MAPE is extremely volatile when theoption prices are small. This property directly leads to MAPEbeing largest for deep-out-of-the-money options.

4.2. Best forecasting performance ratio

The empirical results include 1680 testing errors (RMSE, MAEand PAPE), corresponding to two option styles, two activation func-tions, three different numbers of neurons, five volatility models,

Table 1Option observations for various moneynesses and time periods.

Period Call options Put options

In At Out DOut Total In At Out DOut Total

A 785 2481 3269 1624 8159 1058 2533 2453 913 6957B 698 1843 2488 4680 9709 2180 1877 1233 231 5521C 1483 4324 5757 6304 17868 3238 4410 3686 1144 12478D 1763 1887 1819 1019 6488 989 1802 1903 2404 7098E 1755 2780 2006 7 6548 939 2663 2684 2663 8949F 3518 4667 3825 1026 13036 1928 4465 4587 5067 16047G 5001 8991 9582 7330 30904 5166 8875 8273 6211 28525

Note: the moneyness M = F/K for call options and M = K/F for put options. In, At, Out and DOut respectively represent in-the-money (M > 1.05), at-the-money(0.95 < M 6 1.05), out-of-the-money (0.85 < M 6 0.95), and deep out-of-the-money (M 6 0.85).

Table 2RMSE for call options adopted logsig function during 2008–2009.

Moneyness HV IV DVF GARCH GM-GARCH

TrE TeE TrE TeE TrE TeE TrE TeE TrE TeE

Panel A: 2 neurons in the hidden layerIn 58.96 104.56 296.19 423.95 64.44 102.56 61.50 108.07 505.71 630.87At 56.38 50.56 109.28 115.30 51.27 48.21 59.10 48.96 161.16 158.01Out 46.47 46.13 92.71 115.46 43.63 39.05 48.88 43.09 124.44 192.72Dout 25.28 21.03 92.76 30.04 23.96 20.13 26.76 20.56 144.08 30.42

Panel B: 3 neurons in the hidden layerIn 105.16 159.41 48.35 76.61 62.59 80.96 58.49 68.39 59.51 108.25At 75.43 70.50 49.80 55.17 44.53 47.49 55.93 47.06 57.85 58.00Out 77.18 55.85 35.52 47.88 34.70 33.79 41.75 34.92 47.54 49.42Dout 40.16 29.27 18.33 16.57 19.91 19.15 22.76 18.17 24.90 21.44

Panel C: 4 neurons in the hidden layerIn 51.07 81.87 47.70 82.36 60.88 90.24 58.15 127.42 57.65 61.96At 51.88 48.67 45.02 50.02 44.25 49.86 58.28 48.75 71.06 54.40Out 40.05 37.42 33.55 31.31 35.54 38.27 46.33 41.52 56.16 43.55Dout 22.83 20.11 22.19 18.80 20.42 19.73 25.42 19.95 26.53 24.36

Notes: TrE and TeE represent training errors and testing errors, respectively. In, At, Out and DOut represent in-the-money, at-the-money, out-of-the-money and deep out-of-the-money, respectively.

Table 3MAE for call options adopted logsig function during 2008–2009.

Moneyness HV IV DVF GARCH GM-GARCH

TrE TeE TrE TeE TrE TeE TrE TeE TrE TeE

Panel A: 2 neurons in the hidden layerIn 36.92 49.61 209.78 270.14 43.52 61.66 39.51 56.30 429.53 483.97At 42.93 37.01 88.67 91.66 37.19 35.98 45.51 35.19 124.89 127.27Out 34.53 35.03 70.97 92.79 31.79 29.49 36.29 32.70 104.57 165.19Dout 14.90 13.18 70.68 68.45 14.36 12.72 15.35 13.13 112.43 124.66

Panel B: 3 neurons in the hidden layerIn 73.14 98.55 31.72 43.49 42.96 56.84 38.67 47.64 37.41 55.12At 58.10 55.74 35.36 43.60 32.01 35.39 43.37 34.55 42.96 43.61Out 56.61 46.80 23.52 37.60 25.54 25.11 31.31 25.91 35.12 37.17Dout 25.07 21.72 10.67 10.23 12.39 12.43 14.49 12.53 14.98 13.47

Panel C: 4 neurons in the hidden layerIn 34.83 48.02 31.54 50.72 40.57 54.66 37.47 60.67 42.37 42.46At 40.13 37.70 31.29 37.97 32.05 37.05 44.65 35.20 59.00 42.82Out 29.52 28.73 22.89 22.71 25.24 29.34 33.59 30.63 46.16 33.21Dout 15.63 13.87 14.75 12.70 13.73 13.17 16.26 13.48 18.53 16.48

Notes: TrE and TeE represent training errors and testing errors, respectively. In, At, Out and DOut represent in-the-money, at-the-money, out-of-the-money and deep out-of-the-money, respectively.

13 Logsig and tansig activation functions are widely applied to neural networks. Thelogsig function effectively imitates real numbers while the tansig function learnsfaster (Haykin, 1999). Amilon (2003) employed the logsig function while Anders et al.(1998) and Yao et al. (2000) applied the tansig function to training data.

C.-P. Wang et al. / Expert Systems with Applications 39 (2012) 5025–5032 5029

four moneynesses, and seven time periods. Since the measures ofRMSE, MAE and MAPE are not additive measures, this studyintroduces the best forecasting performance ratio (BFPR) as anew performance measure. BFPR is defined as the ratio of theoccurrence frequencies of the smallest testing errors for a particu-lar category divided by the total occurrence frequencies. Table 5summarizes the empirical results related to BFPR.

Panel A of Table 5 compares the BFPR of the logsig and tansigfunctions.13 Panel A contains 1680/4 = 420 testing errors, where

Table 4MAPE for call options adopted logsig function during 2008–2009.

Moneyness HV IV DVF GARCH GM-GARCH

TrE TeE TrE TeE TrE TeE TrE TeE TrE TeE

Panel A: 2 neurons in the hidden layerIn 0.049 0.052 0.245 0.260 0.057 0.072 0.052 0.060 0.537 0.512At 0.194 0.183 0.336 0.396 0.162 0.151 0.198 0.167 0.459 0.652Out 1.748 1.361 3.626 3.356 1.627 1.203 1.698 1.279 6.845 7.961Dout 3.398 3.973 65.860 78.237 3.289 4.003 3.591 4.138 102.893 126.413

Panel B: 3 neurons in the hidden layerIn 0.091 0.108 0.044 0.049 0.057 0.071 0.052 0.060 0.050 0.058At 0.252 0.281 0.133 0.181 0.131 0.133 0.174 0.129 0.193 0.211Out 2.839 2.294 0.816 1.116 0.922 0.684 1.042 0.594 1.663 1.395Dout 9.442 10.954 2.769 3.349 6.484 8.526 7.083 8.394 3.523 3.970

Panel C: 4 neurons in the hidden layerIn 0.049 0.055 0.044 0.062 0.053 0.063 0.050 0.062 0.063 0.054At 0.164 0.165 0.118 0.137 0.133 0.146 0.187 0.161 0.254 0.175Out 0.852 0.791 0.870 0.684 1.192 0.873 1.069 0.887 1.801 0.755Dout 4.376 4.752 3.524 3.933 8.000 8.522 6.425 6.230 4.913 5.386

Notes: TrE and TeE represent training errors and testing errors, respectively. In, At, Out and DOut represent in-the-money, at-the-money, out-of-the-money and deep out-of-the-money, respectively.

Table 5The best forecasting performance ratio (%).

Call options Put options

RMSE MAE MAPE RMSE MAE MAPE

Panel A: activation functionslogsig 48.99 49.22 47.86 50.12 50.69 50.44tansig 51.01 50.78 52.14 49.88 49.31 49.56

Panel B: hidden neurons2 26.51 26.17 28.00 24.40 22.34 23.633 35.91 34.23 35.33 34.36 34.02 31.164 37.58 39.60 36.67 41.24 43.64 45.21

Panel C: time periodsA 4.13 7.50 9.09 0.83 0.83 9.09B 10.74 17.50 5.79 1.67 1.67 24.79C 0.00 0.00 0.00 0.00 0.00 0.00D 0.00 0.83 9.09 3.33 4.17 13.22E 50.41 41.67 52.89 85.00 85.00 42.15F 4.96 3.33 11.57 5.00 4.17 5.79G 29.75 29.17 11.57 4.17 4.17 4.96

Panel D: volatility modelsHV 5.33 6.55 13.37 7.74 7.10 8.33IV 47.34 46.43 40.12 50.00 53.25 48.21DVF 33.73 34.52 29.07 32.74 28.99 32.14GARCH 5.92 5.95 9.88 4.76 5.92 5.95GM-GARCH 7.69 6.55 7.56 4.76 4.73 5.36

Note: periods A and B represent the first half and second half of 2008; periods D andE represent the first half and second half of 2009; periods C and F represent thewhole years of 2008 and 2009; and period G represents the two-year samplehorizon of 2008 to 2009, respectively. HV, IV, DVF, GARCH and GM-GARCH repre-sent the historical volatility, implied volatility, deterministic volatility function,GARCH(1,1) and GM(1,1)-GARCH(1,1) models.

5030 C.-P. Wang et al. / Expert Systems with Applications 39 (2012) 5025–5032

the logsig and tansig functions respectively have 48.99% and51.01% cases of the smallest RMSE. Since the interval of BFPRfor the logsig function ranges 48.99% to 50.69%, the difference inforecasting performance between the two activations isinsignificant.

Panel B of Table 5 compares the BFPR among different numbersof neurons, and contains 1680/6 = 280 testing errors.14 For both calland put options, the BFPR is monotonously and significantly increas-

14 Consider the best neurons for only one hidden layer in the neural network.Basically, more hidden neurons may overfit the data (Amilon, 2003). Haykin (1999)and Amilon (2003) indicated that the optimal neurons depend on differentexperimentations. Yao et al. (2000) chose two to six neurons, while Wang (2009a),Wang (2009b) and Lin and Yeh (2009) adopted only two.

ing in the number of neurons. On average, the BFBR for four neuronsis approximately 40.66%, while that BFBR for two neurons is only25.18%. This result indicates that future works should add the neu-rons to the hidden layer. For example, Amilon (2003) found the opti-mal numbers of neurons may be 10, 12 and 14, using theperformance measure of RMSE. Moreover, Hsiao, Lin, and Chiang(2003) calculated RMSE with from 1 to 15 neurons in the hiddenlayer.

Panel C of Table 5 compares the BFPR among different sampleperiods.15 Panel C contains 1680/14 = 120 testing errors. Further-more, period E (2009/07/01–2009/12/31) markedly outperformsthe other periods with the BFPR of RMSE (50.41%), MAE (41.67%)and MAPE (52.89%) for the call options and of RMSE (85.00%), MAE(85.00%) and MAPE (42.15%) for the put options. Notably, all theBFPR in Period C (2008/01/01–2008/12/31) are zero, revealing thatthe data characteristics before the global financial crisis are inconsis-tent with those after the crisis (October 2008). Restated, the 70%training set in Period C over January through September 2008 doesnot occur during the financial crisis, while the 30% testing set overOctober through December 2008 does occur during the financial cri-sis. Additionally, the global financial crisis caused to enormous mar-ket volatility up until the first half of 2009. Accordingly, the datacharacteristics are stable throughout 2009, causing the strong fore-casting performance.

Panel D of Table 5 compares the BFPR among various volatilitymodels. Panel D contains 1680/10 = 168 testing errors. Wang(2009a) examined the forecasting performance of volatility mod-els, including HV, IV, GH, and GM-GARCH. Besides the four volatil-ity models, this study further considers the DVF volatility model.Overall, the implied volatility model significantly outperformsthe others, with BFPR ranging from 40.12% to 53.25%. Meanwhile,the DVF model exhibits the second best performance with BFPRranging from 28.99% to 34.52%. Furthermore, the other three vola-tility models do not exhibit a consistent order of BFPR. Since thevolatilities in the IV and DVF models are extracted from the optionprices while the others (HV, GARCH and GM-GARCH) are extractedfrom the historical stock index returns, this result indicates thatthe informational extent of implied volatility significantly exceedsthe historical volatility. Excluding the IV and DVF models, the re-

15 Yao et al. (2000) divided the data into training, validation and testing sets,according to time sequence, time to maturity, and random selection. Amilon (2003)tested the data for eight non-overlapping periods but did not compare forecastsamong different periods.

Table 6Interaction of volatility model and moneyness on BFPR (%).

Volatility Call options Put options

RMSE MAE MAPE RMSE MAE MAPE

Panel A: in-the-moneyHV 7.14 11.90 11.11 7.14 4.65 4.76IV 47.62 52.38 51.11 47.62 46.51 54.76DVF 33.33 30.95 33.33 38.10 41.86 40.48GARCH 2.38 0.00 2.22 7.14 6.98 0.00GM-GARCH 9.52 4.76 2.22 0.00 0.00 0.00

Panel B: at-the-moneyHV 2.38 0.00 2.38 4.76 4.76 4.76IV 45.24 40.48 47.62 73.81 78.57 80.95DVF 45.24 45.24 42.86 21.43 16.67 14.29GARCH 4.76 11.90 4.76 0.00 0.00 0.00GM-GARCH 2.38 2.38 2.38 0.00 0.00 0.00

Panel C: out-of-the-moneyHV 2.38 4.76 23.26 7.14 7.14 4.76IV 52.38 45.24 34.88 42.86 50.00 42.86DVF 33.33 38.10 18.60 42.86 40.48 47.62GARCH 7.14 7.14 20.93 4.76 2.38 2.38GM-GARCH 4.76 4.76 2.33 2.38 0.00 2.38

Panel D: deep-out-of-the-moneyHV 9.30 9.52 16.67 11.90 11.90 19.05IV 44.19 47.62 26.19 35.71 38.10 14.29DVF 23.26 23.81 21.43 28.57 16.67 26.19GARCH 9.30 4.76 11.90 7.14 14.29 21.43GM-GARCH 13.95 14.29 23.81 16.67 19.05 19.05

Note: HV, IV, DVF, GARCH and GM-GARCH represent the historical volatility,implied volatility, deterministic volatility function, GARCH(1,1) and GM(1,1)-GARCH(1,1) models.

C.-P. Wang et al. / Expert Systems with Applications 39 (2012) 5025–5032 5031

sult shows that the complicated volatility models (GARCH and GM-GARCH) need not outperform the simple ones (HV). The result inPanel D is clearly inconsistent with those of Wang (2009a,2009b) and Lin and Yeh (2009). Meanwhile, Wang (2009a) foundthat GM-GARCH and GARCH models outperform the other modelsin terms of their forecasting ability; Wang (2009b) identified theGrey-GJR-GARCH volatility approach as having the best forecastingperformance; and Lin and Yeh (2009) identified the GARCH modelas the best performer. The results listed in Panel D differ fromthose of Wang (2009a, 2009b) and Lin and Yeh (2009), possibly be-cause of the differences between this study and them. First, thesample horizon in this study runs from 2008 to 2009, and includesthe global financial crisis period during October 2008. Next, thisstudy can select different activation functions (logsig andtansig) and different numbers of neurons in the hidden layer(2, 3 and 4).

4.3. Interaction of the volatility model and moneyness

Similar to Wang (2009a, 2009b) and Lin and Yeh (2009), thisstudy examined whether the volatility models and moneynessesinteract on the BFPR, as shown in Table 6. Table 6 demonstratesthat the IV and DVF models have larger BFPR than the others forin-the-money, at-the-money, and out-of-the-money options. Par-ticularly, the GARCH and GM-GARCH models have zero BFPR forat-the-money put options. IV is best for deep-out-of-the-moneyoptions, with the exception of the MAPE of put options. These re-sults obviously differ from those of Wang (2009a) and Lin andYeh (2009).

On average, IV and DVF have the largest and second largestBFPR regardless of moneyness. Besides IV and DVF, HV andGM-GARCH have larger average BFPR for in-the-money anddeep-out-of-money options, respectively. Overall, forecasting per-formances are related to moneyness in HV, GARCH and GM-GARCHbut not in IV and DVF.

5. Conclusions

This study applied backpropagation neural network to forecastTXO price under different volatility models, including the HV, IV,DVF, GARCH and GM-GARCH models. The TXO intraday data con-tain call and put option prices over 500 trading days from 2008through 2009. The sample horizon includes the global financial cri-sis in October 2008, and thus allows the comparison of results fordifferent periods, including the financial crisis period. Since themeasures of RMSE, MAE and MAPE are not additive measures invarious classifications, this study introduces the best forecastingperformance ratio (BFPR) as a new performance measure for optionpricing. The key findings are as follows.

First, since the pricing errors of RMSE and MAE are positively re-lated to the magnitudes of option prices, the in-the-money optionsare priced highest relative to the others. Additionally, becauseMAPE is extremely volatile when option prices are very low, MAPEis largest for deep-out-of-the-money options. Second, the forecast-ing performance difference between the logsig and tansig activa-tions is insignificant. Third, BFPR is monotonously andsignificantly increasing in the number of neurons. Consequently,future studies should add the neurons to the hidden layer. Fourth,the forecasting performance during 2008 was significantly inferiorto that during 2009. Restated, the global financial crisis could sub-stantially affect the option pricing performance. The data charac-teristics before the global financial crisis thus are clearlyinconsistent with those after the crisis period (October 2008).Additionally, the data characteristics are quite stable throughoutthe whole year in 2009, contributing to the strong forecasting per-formance. Fifth, overall, the IV model significantly performs theother models, while the DVF model performs second best. Thisranking indicates that the informational extent of implied volatilitysignificantly exceeds the historical volatility in option pricing. Fur-thermore, the complicated volatility models (GARCH and GM-GARCH) need not outperform the simple ones (HV). Two explana-tions may exist for this study obtaining different results to Wang(2009a, 2009b) and Lin and Yeh (2009): first, the sample horizoncontains the global financial crisis period; second, this study allowsfor flexibility in selecting different activation functions and num-bers of neurons in the hidden layer. Finally, on average, IV andDVF have the largest and second largest BFPR regardless of money-ness. Overall, the forecasting performances are related to money-ness in HV, GARCH and GM-GARCH but not in IV and DVF.

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