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© 2014 Nicolaus Copernicus University. All rights reserved.  
http://www.dem.umk.pl/dem 

D Y N A M I C  E C O N O M E T R I C  M O D E L S  
DOI: http://dx.doi.org/10.12775/DEM.2014.008  Vol. 14 (2014) 145−160 

Submitted October 20, 2014  ISSN 
Accepted December 23, 2014 1234-3862 

Joanna Górka*
 

Option Pricing under Sign RCA-GARCH Models 

A b s t r a c t. After Black and Scholes’s groundbreaking work, the literature concerning 
pricing options has become a very important area of research. Numerous option valuation 
methods have been developed. This paper shows how one can compute option prices using 
Sign RCA-GARCH models for the dynamics of the volatility. Option pricing obtained from 
Sign RCA-GARCH models, the Black and Scholes’s valuation and other selected GARCH 
option pricing models are compared with the market prices. This approach was illustrated by 
the valuation of the European call options on the WIG20 index. The empirical results indicat-
ed that RCA-GARCH and Sign RCA-GARCH models can be successfully used for pricing 
options. However none of the models can be indicated as the best one for the option valua-
tions for every period and every time to maturity of the options.  

K e y w o r d s: Sign RCA-GARCH models, option pricing, GARCH models.  

J E L Classification: G13. 

Introduction 

 Following the seminal work of Black and Scholes (1973) and Merton 
(1973), the option literature has developed into an important area of re-
search. The Black-Scholes formula (henceforth BS) assumes that stock price 
varies according to the geometric Brownian motion. The relationship be-
tween the geometric Brownian motion and the BS formula presents the fol-
lowing equivalence (Elliott and Kopp, 1999)1: 

                                                 
* Correspondence to: Joanna Górka, Nicolaus Copernicus University, Faculty of Econom-

ic Sciences and Management, 13A Gagarina Street, 87-100 Toruń, Poland, e-mail: joan-
na.gorka@umk.pl. 

1 This equivalence is obtained by applying Itô Lemma with function   tt SSf log .  



Joanna Górka 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 145–160 

146

,
2

2

0

tWt

ttttt eSSdWSdtSdS
 








 

  (1) 

where: 

tS  – a stock price, 

  – the drift rate, annualized expected value of St, 
t  – time, 

tW  – the Wiener process (Brownian motion),  

0  – the annualized volatility of St. 
 The BS formula assumes that the returns of the underlying asset (stock 
price) follow a normal distribution with constant volatility. Empirical evi-
dence has shown, however, that the model is in conflict with facts, especially 
for short-run returns2. The financial markets research indicated that financial 
series, such as stock returns, foreign exchange rates and others, exhibit lep-
tokurtosis and volatility varying in time. Hence the assumption of constant 
volatility is often strongly violated. Therefore, several option valuation mod-
els have been developed to incorporate stochastic volatility. One approach is 
to use continuous-time stochastic volatility models. Another approach is to 
use discrete-time generalized autoregressive conditionally heteroskedastic 
(GARCH) models (amongst others Engle,1982; Bollerslev,1986). 
 The choice of discrete-time GARCH models for this study was motivat-
ed by two facts, that: 
 the inclusion of linear autoregressive dynamics, AR(1), affects option 

prices (Hafner and Herwartz, 2001), 
 the random coefficient autoregressive models with the sign function 

(Sign RCA) are straightforward generalization of the constant coeffi-
cient autoregressive models (Thavaneswaran et al., 2006a). 

The random coefficient and the sign function have influenced the uncondi-
tional kurtosis. The value of unconditional kurtosis in the RCA-GARCH and 
the Sign RCA-GARCH models is bigger in comparison with ordinary AR-
GARCH. In addition, the sign function allows the modelling of asymmetry, 
such as response of returns on various information from the market.  
 The purpose of this work is to apply the Sign RCA-GARCH models to 
pricing European call options, and compare these results and results obtained 
from the Black-Scholes model and from other selected GARCH options 
pricing models with the market prices. Such use of Sign RCA-GARCH 

                                                 
2 More frequent than monthly. 



Option Pricing under Sign RCA-GARCH Models 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 145–160 

147

models as far as we know has not been applied in option pricing except the 
work by Górka (2012).  

1. Theoretical Framework 

1.1. Option Pricing 

 As a consequence of the equation (1), the price of a European call option 
is given by equation: 

,)()(= 21 dNKedNSc r
tt

  (2) 

where: 

,
2

=

2

1 










 r

K

S
ln

d

t

 ,12  dd   

tS  – the stock price at time t, 
K  – the exercise price, 
r  – the risk-free interest rate, 
  – the time to maturity of the option,  
 N  – the cumulative normal density function,  

  – the volatility of rate of the return on the stock. 
 The valuation of derivative is about moving to the world free of risk, in 
which risky assets have the same return as the risk-free. The general idea of 
the valuation of derivatives is based on the following theorem. 
Theorem (Elliott and Kopp, 1999). 
If the process S satisfies the equation 

tttt dWSdtSdS  = , (3) 

then also satisfies the equation 

tttt WdSdtrSdS
~

=  , (4) 

where is r  the risk-free interest rate, t
r

WW tt 


=
~

 is a Wiener process. 

 Duan (1995) introduced the GARCH option pricing model by generaliz-
ing the traditional risk neutral valuation methodology to the case of condi-
tional heteroskedasticity.  

 Letting the conditional mean t  and conditional variance 2
t  be meas-

urable functions with respect to the information set (F), the general model 



Joanna Górka 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 145–160 

148

under the data generating probability measure P is given by (Hafner and 
Herwartz, 2001): 

 
 ,;;,

,1,0...~

,=

22 




tsf

Ndii

y

sst

t

tttt




 (5) 

where f  is a parametric function with parameter vector  . 
 In GARCH models, it is not possible to find a risk-neutralization proce-
dure that leaves unchanged the marginal variance of the process or the con-
ditional variance beyond one period. Therefore Duan (1995) introduced the 
local risk-neutral valuation relationship (LRNVR; equivalent martingale 
measure Q). The local risk-neutral valuation relationship is an essential fea-
ture of the equivalence of the conditional variances under the data generating 
probability measure P (historical measure) and the equivalent martingale Q.  
 Under the measure Q, the model is as follows (Hafner and Herwartz, 
2001): 

 
 

 

,

,

,;;,

,1,0...~

,=

22

t

t
t

ttt

sst

t

ttttt

r

tsf

Ndii

y


















 (6) 

where:  
r  is the risk-free interes rate,  

ry tt =]|[E 1FQ ,  

   11 |var|var   tttt yy FF QP . 
 This procedure leaves unchanged the one period ahead conditional vari-
ance and the conditional expected future return is equal to the risk-free inter-
est rate at each time t. Discounted asset price under the measure Q is a mar-
tingale.  

1.2. Option Pricing under Sign RCA-GARCH Models 

 The RCA-GARCH models were proposed by Thavaneswaran et al., 
(2006a). The RCA(1)-GARCH(1,1) model has the following form: 

  ,1 tttt yy         (7) 



Option Pricing under Sign RCA-GARCH Models 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 145–160 

149

,ttt    (8) 

,  2
1-t1

2
110

2
t   t  (9) 

where ,  ,0  ,1  1  are parameters of model,  ,,0...~ 2
 diit  

 ,,0...~ 2
 diit  ,00   ,01   .01   

Theoretical properties of this model can be found among others: Górka 
(2012), Thavaneswaran et al., (2006a, 2006b, 2008, 2009). It is worth noting, 
that the value of unconditional variance and kurtosis increases in comparison 
with ordinary AR(1)-GARCH(1,1). 
 We can define the RCA(1)-GARCH(1,1) option pricing model under the 
historical measure P: 

 
  ,
,1,0~

,)(=

2
11

2
110

2

1









ttt

t

ttttt

N

yy





 (10) 

and under the measure Q: 

 
  

 
,

,

,1,0~

,=

1

121
1

2
11

2
1110

2

1















t

tt
t

tttt

t

ttt

ry

N

ry









 (11) 

where 1r  is the one-day risk-free interest rate. 
 For the RCA(1)-GARCH(1,1) model, like for the AR(1)-GARCH(1,1) 
(Hafner and Herwartz, 2001), we can obtain the unconditional variance un-
der the measure Q. 
Proposition 1 (Górka 2012). Under the measure Q, the unconditional vari-
ance of yt under stationarity is finite if  

  11 1
22

1    , 

and 

    
  1

22
1

222
110

11

1
var












r

yt
Q . 

 The Sign RCA(1)-GARCH(1,1) models proposed by Thavaneswaran, et 
al., (2006a) have the following form: 



Joanna Górka 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 145–160 

150

,)(= 11 ttttt ysy     (12) 

,= ttt   (13) 

,= 2
11

2
110

2
  ttt   (14) 

 where  ,  , 0 , 1 , 1  are parameters of model,  ,,0...~ 2
 diit  

 ,,0...~ 2
 diit  ,00   ,01   ,01   









 0<for1

0=for0

0>for1

=

t

t

t

t

y

y

y

s .  

The sign function  ts  has the interpretation: if  t , the negative 

value of   means that the negative (positive) observation values at time 
1t  correspond to a decrease (increase) of observation values at time t . In 

the case of stock returns it would suggest (for returns) that after a decrease of 
stock returns, the higher decrease of stock returns occurs than expected, and 
in the case of the increase of stock returns the lower increase in stock returns 
occurs than expected.  
 Theoretical properties of this model can be found among others: Górka 
(2012), Thavaneswaran et al., (2006a, 2006b, 2008, 2009). It is worth noting, 
that the adding the sign function has influence the increase of unconditional 
variance and kurtosis in comparison with RCA(1)-GARCH(1,1), and there-
fore also with ordinary AR(1)-GARCH(1,1). 
 Under the historical measure P the Sign RCA(1)-GARCH(1,1) option 
pricing model can be defined:  

 
  ,
,1,0~

,)(=

2
11

2
110

2

11









ttt

t

tttttt

N

ysy





 (15) 

 Under the martingale measure Q the Sign RCA(1)-GARCH(1,1) option 
pricing model takes the form: 

 
  

 
,

,

,1,0~

,=

1

1221
1

2
11

2
1110

2

1















t

ttt
t

tttt

t

ttt

rys

N

ry









 (16) 



Option Pricing under Sign RCA-GARCH Models 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 145–160 

151

For the Sign RCA(1)-GARCH(1,1) model, like for the RCA(1)-
GARCH(1,1), we can obtain unconditional variance under the measure Q. 
Proposition 2 (Górka 2012). Under the measure Q, the unconditional vari-
ance of yt under stationarity is finite if 

  11 1
222

1    , 

and 

    
  1

222
1

2222
110

11

1
var












r

yt
Q . 

1.3. Monte Carlo Simulations 

 GARCH models are very popular and effective for modeling the volatili-
ty dynamics in many asset markets. Unfortunately, existing GARCH models 
do not have closed-form solutions for option prices. These models are typi-
cally solved by simulation. The Monte Carlo simulation procedure for option 
pricing can be described in following steps (Duan, 1995; Hafner and Her-
wartz, 2001; Lehar et al., 2002; Piontek, 2002, 2004): 

1. Parameter estimation under the empirical measure P. 
2. Simulation of sample paths for the underlying asset price under the 

equivalent martingale measure Q (50000 paths), i.e. 


 




 
n

s
stisti

n

s
stinr

tni eSS 1
0,0,

1

2
0,1

0

5.0

,


, 

where: i  – i-th path, 
sti 0,  – the current values of the innovation,  

2
, 0 sti 

  – the current values of the variance. 

3. Correction to the standard Monte Carlo simulation procedure (empirical 
martingale simulation, Duan and Simonato, 1998), i.e. 

,

1
,

1

,*
,

1

0





m

i
nim

ninr
tni

S

S
eSS  

where: m  – number of paths. 
4. Discounting the expected payoffs to yield of the Monte Carlo price of  

option, i.e. 

  .|0,maxE
0

1

0

*
,

Q
tni

nr
t KSec F   



Joanna Górka 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 145–160 

152

where: 
0tc  – the corresponding call price obtained by Monte Carlo simu-

lation, QE  – the risk-neutral conditional expectation operator. 
The risk-free interest rate ( 1r ) was approximated on the basis of the interest 
rate of the WIBID3 and the WIBOR4. 
 To quantify the deviation of theoretical option prices from the prices 
observed at the market the statistical error measures were applied (Lehar et 
al., 2002): 
 the relative pricing error 

t

tt

c

cc 


ˆ
RPE , 

 the absolute relative pricing error 

t

tt

c

cc 


ˆ
ARPE , 

where tc  and tĉ  denote the observed price and the model price, respectively. 
 The RPE is a measure of the bias of the pricing model. A non-zero RPE 
may therefore indicate the existence of systematic errors. The ARPE 
measures both the bias and the efficiency of pricing (Lehar et al., 2002). 

2. An Empirical Analysis 

 The data used in the empirical study were the WIG20 index and prices of 
the European call options on WIG20 index on the Warsaw Stock Exchange 
(WSE). The sample period for the WIG20 runs from 19-th of November 
2003 to 21-th of February 2011. Evolution of the WIG20 index was dis-
played in Figure 1. The WIG20 index went up during the first 4 years, after 
that it rapidly went down and since 2009 started to increase again.  

                                                 
3 Warsaw Interbank Offered Rate. 
4 Warsaw Interbank Bid Rate. 



Option Pricing under Sign RCA-GARCH Models 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 145–160 

153

 

Figure 1.  The WIG20 index (November 19, 2003 – February 21, 2011) 

 Two periods were chosen to calculate option prices. The first one was at 
the turn of 2007–2008 (it is about 3 months long) and the second one was at 
the turn of 2010–2011. First day of the option pricing at the turn of year 
2007–2008 was made on 22-th of November 2007 (valuation on the Novem-
ber 23, 2007) and 16-th of November 2010 at the turn of 2010–2011 (valua-
tion on the November 17, 2010). 
 Option pricing were made using: 
 the standard Black-Scholes (BS),  
 Monte Carlo simulation (MCS; Duan's method). 

 For comparison, the valuation of options were also made using the 
AR(1)-GARCH(1,1) model and the AR(1)-GJR-GARCH(1,1) model. For all 
model specifications, parameter values were obtained from the MLE using 
WIG20 index daily logarithmic returns. The sample sizes on which models 
were estimated are as follows: 
 252 observations (~ year), 
 504 observations (~ 2 years), 
 1008 observations (~ 4 years). 

All computation were made using authors codes written in GAUSS 6.0. 
 Example results of the valuation of European call options for WIG20 
stock index on a particular day (November 17, 2010), on three month to 
maturity and different sample sizes, and market prices of these options (the 
closing price) are shown in Table 1.  



Joanna Górka 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 145–160 

154

Table 1.  European call option prices for WIG20 stock index on three months to 
maturity and market prices of these options 

Strike (K) BS RCA-GARCH Sign RCA-GARCH AR-GARCH AR-GJR-GARCH 
Market price 
17.12.2010 

252 observations 
2300 476.72 491.21 493.56 524.49 494.79 483.50 
2400 377.64 399.56 403.62 433.41 405.20 388.50 
2500 278.57 314.31 320.34 348.30 322.02 294.90 
2600 179.50 238.20 246.04 271.50 247.50 221.05 
2700 80.43 173.31 182.39 204.68 183.29 133.00 
2800 1.31 120.84 130.42 149.05 130.64 78.00 
2900 0.00 80.65 89.98 104.83 89.51 45.10 
3000 0.00 51.66 59.92 71.13 58.89 23.00 
3100 0.00 31.79 38.66 46.57 37.26 9.10 

504 observations 
2300 476.72 487.08 489.88 486.94 487.81 483.50 
2400 377.64 390.36 392.63 389.99 391.45 388.50 
2500 278.57 297.02 298.39 296.19 298.16 294.90 
2600 179.50 211.25 211.15 209.70 211.28 221.05 
2700 80.43 138.46 136.63 136.25 135.72 133.00 
2800 1.31 82.89 79.92 80.40 76.51 78.00 
2900 0.00 45.48 42.27 43.25 36.75 45.10 
3000 0.00 23.27 20.53 21.61 14.78 23.00 
3100 0.00 11.24 9.37 10.18 4.87 9.10 

1008 observations 
2300 476.72 495.32 495.34 495.31 519.97 483.50 
2400 377.64 405.43 405.45 405.42 440.70 388.50 
2500 278.57 322.00 322.00 321.97 368.18 294.90 
2600 179.50 247.43 247.40 247.40 303.11 221.05 
2700 80.43 183.66 183.60 183.63 245.94 133.00 
2800 1.31 131.79 131.69 131.76 196.75 78.00 
2900 0.00 91.65 91.51 91.62 155.33 45.10 
3000 0.00 62.22 62.06 62.19 120.91 23.00 
3100 0.00 41.44 41.29 41.41 93.08 9.10 

Note:  The bold number denotes the theoretical option prices the closest to the market price. 

 Option prices calculated by the BS formula were underestimated, while 
option prices calculated by other models were overestimated for the sample 
size of 252 observations and 1008 observations. For the sample size of 504 
observations the option prices were overestimated for some strikes, but for 
other – underestimated. It depends of the type of option and of the model on 
which theoretical option prices were calculated. For the out-of-the-money 
options differences between market prices and theoretical option prices cal-
culated by models on the simple size of 504 observations were small, while 
for others simple sizes these differences were greater. In this study, for the 



Option Pricing under Sign RCA-GARCH Models 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 145–160 

155

sample size of 504 observations the theoretical option prices were closest to 
the market prices. It holds for all model specifications.  

 
Figure 2.  The annualized implied volatility of the RCA-GARCH option pricing 

model (504 observations; day of the option pricing – November 16, 2010) 

 
Figure 3.  The annualized implied volatility of the Sign RCA-GARCH option pricing 

model (504 observations; day of the option pricing – November  16, 2010) 

 Figures 2 and 3 show the relationship between implied volatility, the 
exercise price and the time to maturity of the option. This shape resembles a 
smile and is called the volatility smile. It is often observed in financial mar-
kets (Lehar et al., 2002; Piontek, 2002). When the time to maturity increases, 
the smile tends to become flatter. With the increase of the time to maturity of 



Joanna Górka 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 145–160 

156

the option the increase (Figure 3) or decrease (Figure 2) of volatility for op-
tions with the same moneyness is often observed. This result is similar to the 
result obtained from other GARCH models (Piontek, 2002, 2004; Hafner 
and Herwartz, 2001; Duan, 1995). 
 On the basis of the valuation of options for a day, it is difficult to draw 
more general conclusions on the usefulness of RCA-GARCH and Sign 
RCA-GARCH models. Therefore, the valuation of options in the two periods 
were made using different models as a result of progressive estimation mod-
els. In the first period (November 23, 2007 – January 25, 2008) the values of 
OW20C85 and OW20F8 options with different exercise prices for the next 
63 days were determined. In the second period (November 17, 2010 – Feb-
ruary 21, 2011) the valuation of  OW20C1 and OW20F1 options with differ-
ent exercise prices for the next 67 days was made. 

Table 2. The mean option pricing errors  180>  

Error Options 
RCA-GARCH Sign RCA-GARCH AR-GARCH AR-GJR-GARCH 

252 observations 

RPE 
ITM 0.8079 0.1860 0.1988 0.2022 
ATM 1.8958 0.4736 0.5088 0.4358 
OTM 5.8016 1.1948 1.2816 1.0673 

ARPE 
ITM 0.8079 0.1873 0.1988 0.2067 
ATM 1.8958 0.4736 0.5088 0.4408 
OTM 5.8016 1.1948 1.2816 1.2015 

  
504 observations 

RPE 
ITM 0.0764 0.0911 0.2228 0.2022 
ATM 0.1517 0.1830 0.5339 0.4358 
OTM 0.3341 0.4280 1.4065 1.0673 

ARPE 
ITM 0.0812 0.0996 0.2228 0.2067 
ATM 0.1596 0.1979 0.5339 0.4408 
OTM 0.3589 0.4911 1.4198 1.2015 

  
1008 observations 

RPE 
ITM 0.3268 0.3114 0.3268 0.3628 
ATM 0.8260 0.7756 0.8260 0.8607 
OTM 2.2219 2.0916 2.2217 2.1698 

ARPE 
ITM 0.3268 0.3176 0.3268 0.3628 
ATM 0.8260 0.7884 0.8260 0.8607 
OTM 2.2219 2.1179 2.2217 2.1698 

Note:  ATM – at-the-money, ITM – in-the-money, OTM – out-of-the-money. The bold number indicate 
minimum of absolute error. 

 The obtained values of option prices were subsequently split according 
to the time to maturity of the options (in days), i.e. 
                                                 

5 Type of call options. 



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DYNAMIC ECONOMETRIC MODELS 14 (2014) 145–160 

157

 short maturity ( 60 ), 
 medium maturity ( 180<60  ), 
 long maturity ( 180> ). 

 Then, in the first place the option at-the-money (ATM) was determined, 
and then statistical measures of errors for the four options in-the-money 
(ITM) and four options out-of-the-money (OTM) were calculated6. The re-
sults for the second period  were shown in Tables 2–4. 

Table 3. The mean option pricing errors  180<60   

Error Options 
RCA-GARCH Sign RCA-GARCH AR-GARCH AR-GJR-GARCH 

252 observations 

RPE 
ITM 0.2549 0.1153 0.1117 0.1435 
ATM 0.6812 0.3362 0.3339 0.4123 
OTM 4.5922 1.6938 1.6740 1.8805 

ARPE 
ITM 0.2585 0.1209 0.1147 0.1441 
ATM 0.6966 0.3676 0.3392 0.4123 
OTM 4.6231 1.7395 1.6768 1.8805 

  
504 observations 

RPE 
ITM –0.0002 0.0426 0.1000 0.1532 
ATM –0.1188 0.0651 0.2646 0.4006 
OTM 0.1759 0.5314 1.5774 2.2177 

ARPE 
ITM 0.0913 0.0685 0.1166 0.1676 
ATM 0.3081 0.1690 0.3375 0.4734 
OTM 0.7708 0.6223 1.6448 2.4192 

  
1008 observations 

RPE 
ITM 0.1702 0.1901 0.1702 0.2003 
ATM 0.5001 0.5692 0.5001 0.5282 
OTM 2.5682 3.2605 2.5682 2.2497 

ARPE 
ITM 0.1715 0.2031 0.1715 0.2035 
ATM 0.5010 0.6349 0.5010 0.5448 
OTM 2.5682 3.3311 2.5682 2.2857 

Note:  ATM – at-the-money, ITM – in-the-money, OTM – out-of-the-money. The bold number indicate 
minimum of absolute error. 

 Obtained results depend mainly on the time to maturity and size of sam-
ple. However, the smallest absolute error values were received for the sam-
ple of 504 observations regardless of the choice of the model for the theoret-
ical option prices (see Table 2–4). For each time to maturity and size of 

                                                 
6 Firstly, 4 options in-the-money and out-of-the-money were right next to the option at-

the-money. Second, for each day of option pricing the error measures had had 3 values, one of 
each type of option. For the whole of the period (for example, a option with medium maturi-
ty), the result was the average of the results for the option of this period (for example, for the 
option of 180<60  ). 



Joanna Górka 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 145–160 

158

sample different conclusions may be drawn. For example, for long time to 
maturity (Table 2) and the sample of 504 observations the smallest values of 
the mean pricing error (absolute and relative) were obtained for RCA-
GARCH models, while for the short time to maturity (Table 4) – for Sign 
RCA-GARCH models. This holds for each type of options. In this study, 
regardless of the sample size, the out-of-the-money options for the short time 
to maturity were substantially overestimated. The similar results for the long 
time to maturity for the first period (November 23, 2007 – January  25, 
2008) were found. 

Table 4. The mean option pricing errors  60  

Error Options 
RCA-GARCH Sign RCAGARCH AR-GARCH AR-GJR-GARCH 

252 observations 

RPE 
ITM 0.0266 0.0287 0.0248 0.0517 
ATM 0.1834 0.1548 0.1891 0.2992 
OTM 1.7632 1.7607 1.8024 1.8375 

ARPE 
ITM 0.0442 0.0452 0.0474 0.0622 
ATM 0.2039 0.2035 0.2277 0.3092 
OTM 1.7765 1.9044 1.8163 1.8760 

  
504 observations 

RPE 
ITM 0.0250 0.0211 0.0479 0.0520 
ATM 0.1704 0.0942 0.3498 0.2910 
OTM 2.0401 1.1652 4.3350 3.5110 

ARPE 
ITM 0.0503 0.0471 0.0725 0.0751 
ATM 0.2180 0.1922 0.4135 0.3458 
OTM 2.1327 1.3513 4.4305 3.6669 

  
1008 observations 

RPE 
ITM 0.0413 0.0434 0.0413 0.0542 
ATM 0.2634 0.2738 0.2636 0.2903 
OTM 2.8569 2.9869 2.8585 1.6641 

ARPE 
ITM 0.0544 0.0565 0.0544 0.0651 
ATM 0.2747 0.2851 0.2748 0.3048 
OTM 2.8630 2.9928 2.8645 1.7021 

Note:  ATM – at-the-money, ITM – in-the-money, OTM – out-of-the-money. The bold number indicate 
minimum of absolute error. 

 Comparing the RPEs for the four different models (Table 2–4), one can 
see systematic overpricing across all models (except the RCA-GARCH 
model for the ITM and ATM options for 180<60   and the sample of 504 
observations – Table 3). In other words, the volatility of the underlying asset 
price was systematically overestimated.  
 In some cases the differences between the mean option pricing errors 
were small (e.g. for the sample of 1008 observations in Table 2, for the sam-
ple of 252 observations in Table 4), while in other cases these differences 



Option Pricing under Sign RCA-GARCH Models 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 145–160 

159

were substantial (e.g. for the sample of 504 observations in Table 2 or 4). It 
is worth noting, that Sign RCA-GARCH models outperform the selected 
GARCH models, because the absolute error were substantially lower (e.g. 
for the sample of 504 or 1008 observations in Table 2, for the sample of 504 
or 1008 observations in Table 4). However, the better performance of Sign 
RCA-GARCH models was not well established and depends on the time to 
maturity, size of sample or the period of the data. 

Conclusions 

 This paper has applied  Sign RCA-GARCH models to compute theoreti-
cal option prices. This approach was illustrated by the valuation of the Euro-
pean call options on the WIG20 index, together with a comparison of their 
values obtained on the selected GARCH models. It is difficult to make gen-
eral remarks, nevertheless the empirical results showed that: 
 the Black-Scholes model cannot explain the prices of out-of-the-money 

options, 
 RCA-GARCH and Sign RCA-GARCH models can be successfully ap-

plied in pricing options, 
 none of the models can be indicate as the best one for the option valua-

tions for every period and every time to maturity of the options, 
 the choice of a sample size for estimating the option pricing model has 

a significant impact on the option pricing, 
 the choice of the volatility model is important for achieving a satisfying 

pricing performance.  

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Modele Sign RCA-GARCH w wycenie opcji 

Z a r y s  t r e ś c i. Po ukazaniu się przełomowej pracy Blacka i Scholesa literatura dotycząca 
wyceny opcji stała się bardzo ważnym obszarem w badaniach. Zostały opracowane liczne 
metody wyceny opcji. W artykule tym pokazano, jak można obliczyć ceny opcji wykorzystu-
jąc model Sign RCA-GARCH do opisu dynamiki zmienności. Wyceny opcji uzyskane przed-
stawioną metodą oraz wyceny opcji uzyskanych z wykorzystaniem modelu Blacka-Scholesa 
i wybranych modeli GARCH zostały porównane z ceną rynkową. Podejście to zostało zilu-
strowane wyceną europejskich opcji kupna na indeks WIG20. Empiryczne wyniki wskazują, 
że modele RCA GARCH i Sign RCA GARCH mogą być z powodzeniem stosowane do 
wyceny opcji. Jednak żadnego z przedstawionych modeli nie można wskazać jako najlepsze-
go do wyceny opcji dla dowolnej wielkości próby czy dowolnego czasu pozostającego do 
wygaśnięcia opcji. 

S ł o w a  k l u c z o w e: model Sign RCA-GARCH, wycena opcji, modele GARCH. 


