






































Microsoft Word - 00_tresc.docx


 

 

DYNAMIC ECONOMETRIC MODELS 
Vol. 9 – Nicolaus Copernicus University – Toruń – 2009 

Anna Pajor  
Cracow University of Economics 

Bayesian Analysis of the Box-Cox Transformation  
in Stochastic Volatility Models† 

A b s t r a c t. In the paper, we consider the Box-Cox transformation of financial time series in 
Stochastic Volatility models. Bayesian approach is applied to make inference about the Box-Cox 
transformation parameter (λ). Using daily data (quotations of stock indices), we show that in the 
Stochastic Volatility models with fat tails and correlated errors (FCSV), the posterior distribution 
of parameter λ strongly depends on the prior assumption about this parameter. In the majority of 
cases the values of λ close to 0 are more probable a posteriori than the ones close to 1. 

K e y w o r d s: Box-Cox transformation, SV model, Bayesian inference.  

1. Introduction 
 The continuously compounded rates of return (or logarithmic returns) as 
well as the simple rates of return are commonly used in econometric analyses of 
financial data. These two types of data transformation are applied arbitrarily. In 
the derivatives pricing literature there is the tradition of using logarithmic re-
turns, but when the logarithmic return is modelled as a conditionally Student-t 
distributed random variable, the conditional expected simple rate of return is 
infinite. It violates the finite second moment condition for the asset payoff in 
call option pricing (see Duan, 1999). Duan (1999) uses the generalized error 
distribution (GED) for the logarithmic returns that also exhibits fat tails and 
includes the normal distribution as a special case. Other researchers build model 
with sample returns instead of log-returns and with the Student-t distribution 
(see e.g. Hafner, Harwartz, 1999; Härdle, Hafner, 2000; Bauwens, Lubrano, 
2002). However, both the logarithmic return and simple one are variants of the 
well-known Box-Cox transformation of the xt/xt-1 ratio (where xt denotes the 
asset price at time t) with parameter 0 and 1, respectively. In the paper, we con-
                                                 

† Research supported by a grant from Cracow University of Economics. The author would 
like to thank Janusz Jaworski for language verification of the manuscript. 



Anna Pajor 

 

 

82

sider the Box-Cox transformation of financial time series in Stochastic Volatili-
ty (SV) models. Bayesian approach is applied to make inference about the Box-
Cox transformation parameter (λ). As parameter λ is estimated along with other 
unknown parameters, information in the data is used to determine which trans-
formation is appropriate for the data.  
The structure of the article is as follows: section 2 consists of a short presenta-
tion of the Bayesian SV model with fat-tails correlated errors for the trans-
formed data, section 3 focuses on the empirical results, and finally, section 4 
incorporates the conclusions. 

2. Bayesian AR(1)-FCSV Model for the Transformed Data 
 Let xt denote the price of an asset at time t, t = 0, 1, ..., T. The Box-Cox 
transformation of the xt/xt-1 ratio is defined as:  

⎪⎩

⎪
⎨

⎧

=

>
−

=

−

−

−

0)/ln(

0
1)/(

),/(

1

1

1

λ

λ
λλ

λ

tt

tt

tt

xx

xx
xxB , t = 1, ..., T. 

For ),/( 1 λ−tt xxB  we use an autoregressive structure1: 

,]),/([),/( 121111 ttttt xxBxxB εδλρδλ +−=− −−−    t = 1, ..., T, (1) 

where {εt} is the stochastic volatility process with fat-tails and correlated errors 
(FCSV), introduced by Jacquier et. al., (2004). The discrete-time FCSV process 
can be written as: 

,/ tttt hu ωε =   (2) 

,lnln 1 thtt hh ησφγ ++= −   (3)  

,/)(~ 2 ννχωt   ωt ⊥ (ul , ηl),   t, l ∈ {1, …, T}, 

(ut , ηt)′ ~ ,
1

1
,0 ⎟⎟

⎠

⎞
⎜⎜
⎝

⎛
⎟⎟
⎠

⎞
⎜⎜
⎝

⎛
ρ

ρ
IN    t = 1, …, T.  

where the abbreviation ″IN″ denotes that the random vectors concerned are 
independent and normally distributed, ⊥ denotes stochastic independence.  
In the FCSV process, when ρ is equal to zero, ht is the inverse precision in the 
conditional distribution, p(εt|ht), that is, (v/v-2)ht (for v > 2) is the conditional 
variance. Thus, the FCSV model specifies a log-normal autoregressive process 
for the conditional variance factor (ht) with correlated innovations in the condi-
tional mean and conditional variance equations, i.e. in (2) and (3), respectively. 
                                                 

1 We use the autoregressive structure, because financial time series such as stock market in-
dices often present positive autocorrelation of order one of the returns (see Campbell et al., 1997). 



Bayesian Analysis of the Box-Cox Transformation in Stochastic Volatility Models 

 

 

83

One interpretation for the latent variable ht is that it represents the random, un-
even and autocorrelated flow of new information into financial markets (see 
Clark, 1973). The parameter φ is related to the volatility persistence, and σh is 
the volatility of the log-volatility. The above model captures the leverage effect 
when the correlation ρ is negative. In fact, if ρ is negative, then a negative inno-
vation ut is associated with higher contemporaneous and subsequent volatilities. 
On the other hand, a positive innovation ut is connected with a decrease in vola-
tility (see Jacquier et al., 2004).  
The Bayesian model is characterized by the joint probability density function of 
the untransformed xt/xt-1 ratios (i.e. y = (y1, ..., yT)′, where yt = xt/xt-1), the latent 
variables (i.e. h = (h1, ..., hT)′, ω = (ω1, ..., ωT)′), and of the parameter vector θ : 

)(),|,,()|,,,( )0()0( θyθωhyyθωhy ppp = , (4) 

where 

,)2(||

|),(|'
2
1exp)|(),|,,(

5.1

1

5.05.0*

1

1*
)0(

−

=

−−

=

−

∏

∑

×

×
⎭
⎬
⎫

⎩
⎨
⎧

⎟⎟
⎠

⎞
⎜⎜
⎝

⎛
−=

t

T

t
t

TT

T

t
tt

h

Jtrvpp

ωπ

λ

Σ

yrrΣωyθωhy
 

),(
2

exp
22

)|( ),0(
1

2
1

1
2

ttt

T

t
Ip ωωνωννν

ν
ν

+∞
−

=

−

⎟
⎠
⎞

⎜
⎝
⎛−⎟

⎠
⎞

⎜
⎝
⎛Γ⎟

⎠
⎞

⎜
⎝
⎛= ∏ω  

,
1

2
*

⎟⎟
⎠

⎞
⎜⎜
⎝

⎛
=

hh

h

σρσ
ρσ

Σ ,)',( thtt u ησ=r  ,)',,,,,,,( 2
11 λνρσφγρδ h=θ   

y(0) denotes initial values. The Jacobian J(λ, y) is .),(
1

1∏
=

−=
T

t
tyJ λλ y  

Our model specification gets completed by assuming the following prior struc-
ture: 

),()(),()()()()(),,,,,,,( 2
11

2
11 λνρσφγρδλνρσφγρδ pppppppp hh =  

where we use proper prior densities of the following distributions: 
δ1 ~ N(0, 1), ρ1 ~ U(-1,1) γ ~ N(0, 100), φ ~ N(0, 100) I(-1,1)(φ), v ~ Exp(0.1), 
τ ~ IG(1, 0.005), ψ|τ ~ N(0, τ /2), )1(, 22 ρστρσψ −== hh . 

The prior distribution for δ1 is standardized normal, U(-1,1) denotes the uniform 
distribution over (-1,1). The prior distribution for φ is normal, truncated by the 
restriction that the absolute value of φ is less than one (I(-1, 1)(.) denotes the indi-
cator function of the interval (-1, 1), which is the region of stationarity of lnht). 
The symbol IG(v0, s0) denotes the inverse Gamma distribution with mean  



Anna Pajor 

 

 

84

s0/(v0-1) and variance )]2()1/[( 0
2

0
2
0 −− vvs (thus, when ρ = 0, the prior mean for 

2
hσ  does not exist, but the precision, 2−

hσ , has a Gamma prior with mean 200 
and standard deviation 200). The symbol Exp(a) denotes the exponential distri-
bution with mean 1/a (thus the prior mean for v is equal to 10 with the standard 
deviation equals 10). The prior distribution for (ψ, τ) induces a prior distribu-
tion for ),( 2

hσρ , which has the following form:   

,)1()()2()(),(
22

0
2

0

0
22

0

00 )1(2
)(

5.12)1(125.05.0
0

1
00

2 hh

p
v

s
v

h
v eepvsp σρ

ψρσ
σρ ρσπρσ −

−
−

−−−
−

+−−− −Γ=  

ν0 = 1, s0 = 0.005, ψ0 = 0, p0 = 2 (similar to Jacquier et al., 2004). 
As far as the prior distribution for λ, we assume that our prior information re-
garding this parameter can be represented by the following: 

a) a non-standard distribution on the interval [0, 1]: )1()( xxep −−∝ βλ , where 
β = 30. This prior distribution is symmetrical and U-shaped, as shown in 
Figure 1. 

b) the beta distribution with parameters 0.5 and 0.5; 
c) the uniform distribution on the interval [0, 1]; 
d) the exponential distribution with mean 1; 

 
Figure 1. Prior distributions for the Box-Cox transformation parameter (λ) 

 As regards the initial condition for ht, i.e. h0, we assume that it is equal to 1.  
The joint posterior distribution is then 

.|||),(|'
2
1exp

)|()(),|,,(

5.1

1

5.05.0*

1

1*

)0(

−

=

−

=

−
∏∑

⎭
⎬
⎫

⎩
⎨
⎧

⎟⎟
⎠

⎞
⎜⎜
⎝

⎛
−×

×∝

t

T

t
t

T
T

t
tt hJtr

vppp

ωλ ΣyrrΣ

ωθyyθωh

 

0
0.2
0.4
0.6
0.8

1
1.2
1.4
1.6
1.8

2

0

0.
05 0.
1

0.
15 0.
2

0.
25 0.
3

0.
35 0.
4

0.
45 0.
5

0.
55 0.
6

0.
65 0.
7

0.
75 0.
8

0.
85 0.
9

0.
95 1

a - non-standard b - beta c - uniform d - exponential



Bayesian Analysis of the Box-Cox Transformation in Stochastic Volatility Models 

 

 

85

The posterior probability density function is used to make inference about the 
parameters and latent variables.  

3. Empirical Results 
 We consider ten international stock market indices, namely the S&P 500, 
NASDAQ 100, DJIA (for the US), NIKKEI (for Japan), the CAC 40 
(for France), the DAX (for Germany), the FTSE 100 (for the UK), WIG 20 
(for Poland), HANG SENG (for China), SPTSE 60 (for Canada). 
The data set consists of the daily closing quotations of the stock market indices 
from January 2001 (or 2002) until February (or March) 2009 (see Table 1). 
Basic descriptive characteristics of the daily price ratios are presented in Table 
1. All series of xt/xt-1 ratio exhibit strong kurtosis, and they have highly non-
normal (truncated by zero) empirical distributions.  

Table 1. Sample characteristics for the data sets used 
time series 

(xt/xt-1 ratio of:) average std. dev. kurtosis period 
from: - to:  # obs. T 

WIG 20 1.0000 0.0162 4.9800 02.01.2001 – 13.02.2009  2035 
S&P 500 0.9998 0.0139 13.3286 03.01.2002 – 06.03.2009 1805 

NIKKEI 225 0.9999 0.0163 11.3553 07.01.2002 – 06.03.2009 1760 
FTSE 100 0.9999 0.0137 10.9021 03.01.2002 – 06.03.2009 1813 

DAX 1.0000 0.0169 8.6642 03.01.2002 – 06.03.2009 1825 
NASDAQ 100 1.0000 0.0178 7.8639 03.01.2002 – 06.03.2009 1808 

CAC 40 0.9998 0.0159 9.7031 03.01.2002 – 06.03.2009 1838 
SPTSE 60 1.0001 0.0132 14.2830 03.01.2002 – 06.03.2009 1798 

HANG SENG 1.0002 0.0164 15.0382 03.01.2002 – 06.03.2009 1789 
DJIA 0.9999 0.0130 12.5726 02.01.2001 – 13.02.2009 2039 

Note: The data were downloaded from the website http://finance.yahoo.com. 

 In Table 2 we present the posterior means and standard deviations (in paren-
thesis) of the parameters, in the case of the AR(1)-FCSV model with the uni-
form prior for λ on [0, 1]. Our posterior results are obtained in Gauss 9.0 using 
MCMC methods: Metropolis-Hastings within the Gibbs sampler (see, e.g. Pajor 
2003 and Jacquier et al., 2004 for detail).2 First, for more series the autoregres-
sive parameters seem to be insignificantly different from zero. The posterior 
distributions of δ and ρ1 are located close to zero. Second, all indices have per-
sistent volatility as shown by φ - the lowest posterior mean is 0.927 (for the 
WIG20 index), the highest one is 0.97 (for NASDAQ). It means that the half-
life of shock to volatility, HL = ln(0.5)/ln(φ), is equal to about 9 days for the 
WIG20 index and 20 days for the NASDAQ index. We observed that the 
NASDAQ index exhibits a lower variability of volatility as shown by the preci-
sion, σh

-2. As regards the leverage effect parameter, ρ, the posterior means of ρ 
are negative, from -0.15 for the WIG20 index to -0.62 for the CAC40 index. 
                                                 

2 The results are obtained using 100 000 burnt-in and 1000 000 final Gibbs passes. 



Anna Pajor 

 

 

86

The parameter ρ is estimated precisely with a standard deviation around 0.068. 
Almost all the posterior mass of ρ is in the negative region. Thus, the leverage 
effect is strong for all indices excluding the WIG20 index, for which it is signif-
icantly lower. The posterior means of the degrees of freedom are between 16 
(for the HANG SENG index) and 39 (for the FTSE 100 index). The HANG 
SENG index has the lowest posterior mean of degrees of freedom of the Stu-
dent-t distribution. For the remaining indices the posterior mean of v is above 
23, indicating that the normal conditional distribution would not be strongly 
rejected by the data.  

Table 2. Posterior means and standard deviations (in parenthesis) of the parameters of 
the AR(1)-FCSV model, in the case of λ ~ U[0, 1] 

parameter WIG 
20 

S&P 
500 

NIKKEI  
225 

FTSE 
100 

DAX NASDAQ 
100 

CAC 
40 

SPTSE 60 HANG 
SENG 

DJIA 

δ1∗104 4.74 4.74 6.52 5.43 9.70 5.77 6.59 8.68 7.20 4.14 
 (3.15) (1.65) (2.56) (1.63) (2.15) (2.63) (1.99) (1.78) (2.37) (1.60) 

ρ1 0.027 -0.095 -0.034 -0.092 -0.059 -0.070 -0.080 -0.060 0.005 -0.074 
 (0.023) (0.023) (0.024) (0.024) (0.023) (0.024) (0.023) (0.024) (0.023) (0.022) 
γ -0.622 -0.332 -0.429 -0.357 -0.328 -0.297 -0.335 -0.493 -0.408 -0.348 
 (0.119) (0.048) (0.068) (0.052) (0.049) (0.048) (0.047) (0.074) (0.069) (0.051) 

φ 0.928 0.965 0.951 0.962 0.963 0.966 0.963 0.948 0.955 0.963 
 (0.014) (0.005) (0.008) (0.006) (0.006) (0.006) (0.005) (0.008) (0.008) (0.005) 

σh
-2 22.233 16.972 17.711 13.728 15.277 25.790 15.973 13.672 15.958 17.871 
 (5.675) (2.809) (3.395) (2.123) (2.592) (5.279) (2.491) (2.447) (3.156) (2.996) 

ρ -0.153 -0.607 -0.55 -0.578 -0.612 -0.492 -0.62 -0.484 -0.372 -0.55 
 (0.081) (0.063) (0.065) (0.061) (0.059) (0.081) (0.063) (0.067) (0.075) (0.064) 

ν 23.04 27.20 38.47 39.75 31.77 30.34 31.33 37.85 16.04 26.99 
 (9.96) (11.45) (14.52) (14.61) (12.76) (12.06) (12.55) (14.55) (7.22) (11.17) 

λ 0.397 0.472 0.504 0.448 0.405 0.387 0.399 0.497 0.401 0.433 
 (0.255) (0.265) (0.266) (0.263) (0.256) (0.253) (0.255) (0.266) (0.256) (0.262) 

Table 3. Posterior means and standard deviations (in parenthesis) of λ, in the case of the 
exponential prior distribution for λ (d) 

parameter WIG 
20 

S&P 
500 

NIKKEI  
225 

FTSE 
100 

DAX NAS-DAQ 
100 

CAC 
40 

SPTSE 60 HANG 
SENG 

DJIA 

λ 0.431 0.666 0.801 0.579 0.45 0.412 0.44 0.798 0.437 0.531 
 (0.371) (0.556) (0.624) (0.497) (0.382) (0.35) (0.377) (0.652) (0.376) (0.453) 

Finally, we consider the posterior evidence regarding the Box-Cox transforma-
tion parameter. Figure 2 shows the prior and posterior distributions for λ in the 
case of the WIG20 index. We see from the graphs that the prior distribution for 
λ strongly affects the posterior distribution for this parameter, e.g., a U-shaped 
prior distribution implies the U-shaped posterior distribution. In the case of the 
uniform prior for λ on the interval [0; 1], for most stock indices (considered 
here) the posterior mean is smaller than the prior mean, but the dispersion of 
posterior distribution is close to that of the prior distribution (in the case of c, 



Bayesian Analysis of the Box-Cox Transformation in Stochastic Volatility Models 

 

 

87

the prior mean is equal to 0.5, the prior standard deviation is equal to 0.288). 
Even though the prior distribution is symmetrical, in the majority of cases the 
posterior distributions are asymmetrical. The values of λ from the interval  
[0, 0.5] are more probable a posterior than those from [0.5, 1] (see the quantiles 
of the posterior distributions of the Box-Cox transformation parameter  
in Table 4). 

a – non-standard 

 

b – beta distribution 

 
c – uniform distribution on the interval [0, 1] 

 

d – exponential sistribution 

 
Figure 2. Prior (solid line) and posterior (bars) distributions for λ (the WIG20 index) 

 In the case of the non-standard prior distribution for λ considered in (a), 
except for the NIKKEI and SPTSE 60 indices, the posterior medians are below 
0.1, but the probability that λ is less than 0.9 is not zero. In Table 5 we present 
the posterior probabilities that λ is in the interval [0, 0.01] and in the interval 
[0.99, 1]. Except for the NIKKEI index, the values of λ from the interval 
[0, 0.01] are more probable a posterior than those from the interval [0.99, 1]. 
Thus the data transformations which are close to the log-return are more proba-
ble a posterior than those which lead to the simple return.  

0

0.5

1

1.5

2

2.5

3

3.5

4

0

0.
09

0.
18

0.
27

0.
36

0.
45

0.
54

0.
63

0.
72

0.
81 0.
9

0.
99

0

0.5

1

1.5

2

2.5

3

3.5

4

0

0.
09

0.
18

0.
27

0.
36

0.
45

0.
54

0.
63

0.
72

0.
81 0.
9

0.
99

0

0.5

1

1.5

2

2.5

3

3.5

4

0

0.
09

0.
18

0.
27

0.
36

0.
45

0.
54

0.
63

0.
72

0.
81 0.
9

0.
99

0

0.5

1

1.5

2

2.5

3

3.5

4

0

0.
18

0.
36

0.
54

0.
72 0.
9

1.
08

1.
26

1.
44

1.
62 1.
8

1.
98



Anna Pajor 

 

 

88

Table 4. Posterior quantiles for λ 
 

quantile of order 
WIG 
20 

S&P 
500 

NIKKEI  
225 

FTSE 
100 

DAX NASDAQ 
100 

CAC 
40 

SPTSE 
60 

HANG  
SENG 

DJIA 

 0.05 0.003 0.004 0.005 0.004 0.003 0.003 0.003 0.005 0.003 0.004 
 0.25 0.016 0.023 0.033 0.021 0.016 0.015 0.016 0.031 0.016 0.019 
a 0.5 0.042 0.082 0.824 0.062 0.044 0.039 0.042 0.665 0.042 0.055 
 0.75 0.135 0.957 0.972 0.937 0.161 0.111 0.144 0.97 0.133 0.914 
 0.95 0.988 0.995 0.996 0.993 0.989 0.985 0.988 0.996 0.988 0.992 
 0.05 0.005 0.011 0.016 0.008 0.005 0.005 0.004 0.013 0.004 0.007 
 0.25 0.088 0.163 0.217 0.131 0.095 0.081 0.088 0.194 0.087 0.116 
b 0.5 0.277 0.433 0.516 0.375 0.294 0.26 0.28 0.489 0.278 0.343 
 0.75 0.573 0.746 0.806 0.693 0.597 0.545 0.575 0.787 0.576 0.658 
 0.95 0.939 0.981 0.987 0.973 0.945 0.926 0.939 0.985 0.938 0.963 
 0.05 0.042 0.063 0.076 0.055 0.044 0.04 0.042 0.072 0.043 0.051 
 0.25 0.185 0.252 0.286 0.229 0.192 0.178 0.186 0.278 0.188 0.215 
c 0.5 0.362 0.462 0.506 0.429 0.374 0.35 0.366 0.496 0.369 0.409 
 0.75 0.585 0.686 0.723 0.656 0.597 0.571 0.589 0.716 0.591 0.636 
 0.95 0.864 0.914 0.929 0.901 0.871 0.855 0.865 0.926 0.867 0.891 
 0.05 0.034 0.058 0.077 0.047 0.036 0.032 0.035 0.071 0.034 0.044 
 0.25 0.158 0.252 0.322 0.213 0.167 0.153 0.161 0.307 0.16 0.196 
d 0.5 0.332 0.52 0.652 0.445 0.351 0.32 0.341 0.63 0.337 0.411 
 0.75 0.598 0.932 1.13 0.805 0.629 0.574 0.615 1.122 0.61 0.74 
 0.95 1.172 1.771 2.037 1.577 1.208 1.11 1.186 2.095 1.186 1.435 

Note: Prior distributions for λ: a – non-standard, U-shaped on the interval [0, 1], b – beta distribution,  
c – uniform distribution on [0, 1], d – exponential distribution. 

Table 5. Posterior results for λ 

case index: WIG 
20 

S&P 
500 

NIKKEI 
225 

FTSE 
100 DAX NASDAQ 

100 
CAC 
40 

SPTSE 
60 

HANG 
SENG DJIA 

 
u=Pr(λ<0.01|y) 0.173 0.127 0.102 0.142 0.172 0.178 0.172 0.105 0.174 0.149 

a v=Pr(λ>0.99|y) 0.041 0.083 0.109 0.068 0.044 0.038 0.042 0.100 0.040 0.060 
 u/v 4.238 1.531 0.934 2.088 3.942 4.693 4.111 1.052 4.370 2.492 
 u=Pr(λ<0.01|y) 0.077 0.050 0.040 0.060 0.073 0.081 0.079 0.045 0.081 0.064 

b v=Pr(λ>0.99|y) 0.018 0.035 0.044 0.030 0.019 0.016 0.018 0.039 0.017 0.024 
 u/v 4.294 1.418 0.913 1.999 3.829 4.924 4.402 1.137 4.769 2.656 
 u=Pr(λ<0.01|y) 0.011 0.007 0.006 0.008 0.011 0.012 0.011 0.006 0.011 0.009 
c v=Pr(λ>0.99|y) 0.003 0.005 0.006 0.004 0.003 0.002 0.003 0.006 0.003 0.004 
 u/v 4.485 1.546 0.933 2.084 3.733 5.293 4.222 1.027 4.184 2.604 

Note: Prior distributions for λ: a – non-standard, U-shaped on the interval [0, 1], b – beta distribution,  
c – uniform distribution on [0, 1]. 

It is important to stress that even though the prior distribution of λ has a strong 
effect on the posterior distribution of λ, it does not affect the posterior distribu-
tion of the remaining parameters. Thus in Table 3 we present the posterior cha-
racteristics only of λ, obtained in the AR(1)-FCSV model with the exponential 



Bayesian Analysis of the Box-Cox Transformation in Stochastic Volatility Models 

 

 

89

distribution for the Box-Cox transformation parameter. Although the prior mean 
is equal to 1, for all series the posterior mean is less than 1. 
Finally, in Table 6 we present the results of the formal Bayesian model compar-
ison. We consider three AR(1)-FCSV models: with, respectively, λ = 0 (M1),  
λ = 1 (M2), and λ ~ U(0, 1) (M3). If λ = 1, the relation (1) is linear in the simple 
returns. If λ = 0, it is linear in the logarithmic returns. To obtain the marginal 
data densities we use the Newton and Raftery method (see Newton and Raftery 
1994). The Newton and Raftery estimator is quite stable for all our models. The 
drawback of this method in the FCSV models is that the models differ from one 
another by quite a few orders of magnitude. 
For all series, assuming equal prior model probabilities, the AR(1)-FCSV model 
with λ = 0 (log-returns) is more probable a posterior than with λ = 1 (simple 
returns). Only for the DJIA index, the AR(1)-FCSV model with the uniform 
prior distribution of λ is quite a few orders of magnitude better than that  
with λ = 0.  

Table 6. Posterior probabilities (under equal prior model probabilities) and marginal 
data densities of the observation vector y in Mi model (based on the Newton – 
Raftery method) 

 
 

Index 
M1: λ = 0 M2: λ = 1 M3: 0 < λ < 1* p(y|M1) p(y|M2) p(y|M3)* 

WIG 20 0.9754 0.0000 0.0246 2.4⋅10-170 1.8⋅10-176 6.0⋅10-172 
S&P 500 0.9995 0.0000 0.0005 2.8⋅10-176 2.0⋅10-186 1.4⋅10-179 

NIKKEI 225 0.9997 0.0000 0.0003 1.5⋅10-188 7.5⋅10-196 4.2⋅10-192 
FTST 100 0.9931 0.0069 0.0000 4.1⋅10-161 2.8⋅10-163 4.3⋅10-177 

DAX 1.0000 0.0000 0.0000 3.4⋅10-122 2.4⋅10-141 1.3⋅10-129 
NASDAQ 100 1.0000 0.0000 0.0000 9.3⋅10-103 1.6⋅10-108 1.7⋅10-113 

CAC 40 1.0000 0.0000 0.0000 5.5⋅10-192 2.4⋅10-202 3.3⋅10-197 
SPTSE 60 1.0000 0.0000 0.0000 3.3⋅10-45 2.5⋅10-53 1.1⋅10-55 

HANG SENG 1.0000 0.0000 0.0000 3.3⋅10-50 5.2⋅10-55 2.9⋅10-55 
DJIA 0.0000 0.0000 1.0000 2.3⋅10-202 7.8⋅10-204 2.6⋅10-195 

Note: *The results are obtained in the AR(1)-FCSV model with the uniform prior for λ on the interval (0, 1). 

4. Conclusions 
 The paper presents the stochastic volatility models with the Box-Cox trans-
formation of financial time series. The widely used logarithmic and simple re-
turns are nested into the Box-Cox transformation by setting λ = 0 and λ = 1, 
respectively. Using daily data, we show that in the stochastic volatility model 
with fat tails and correlated errors, the posterior distribution of the Box-Cox 
transformation parameter strongly depends on the prior assumption about this 
parameter. Our empirical results show that in the majority of cases the values of 



Anna Pajor 

 

 

90

λ close to 0 are more probable a posteriori than the ones close to 1. The formal 
Bayesian model comparison indicates that the Box-Cox transformation with 
λ = 0 (log-return) is preferred by the data in the FCSV model. However, the 
posterior distributions of λ show that the simple returns are not completely in-
appropriate.  

References  
Bauwens, L., Lubrano, M. (2002), Bayesian Option Pricing Using Asymmetric GARCH Models, 

Journal of Empirical Finance, 9, 321–342. 
Campbell, J.Y., Lo, A.W., MacKinlay, A.C. (1997), The Econometrics of Financial Markets, 

Princeton University Press, Chichester 1997. 
Clark, P.K. (1973) A Subordinated Stochastic Process Model with Finite Variance for Speculative 

Prices, Econometrica, 41, 135–155. 
Duan, J.-C. (1999), Conditionally Fat-Tailed Distributions and the Volatility Smile in Options, 

Working Paper, http://www.bm.ust.hk/~jeduan. 
Hafner, C.M., Harwartz, H. (1999), Option Pricing under Linear Autoregressive Dynamics, Hete-

roskedasticity, and Conditional Leptokurtosis, Journal of Empirical Finance, 8(1), 1–34. 
Härdle, W., Hafner, C.M. (2000), Discrete Time Option Pricing with Flexible Volatility Estima-

tion, Finance and Stochastics, 4(2), 189-207. 
Jacquier, E., Polson, N., Rossi, P. (2004), Bayesian Analysis of Stochastic Volatility Models with 

Fat-tails and Correlated Errors, Journal of Econometrics, 122, 185–212. 
Newton, M.A., Raftery, A.E. (1994), Approximate Bayesian inference by the weighted likelihood 

bootstrap (with discussion), Journal of the Royal Statistical Society B 56, 3–48. 
Pajor, A. (2003), Procesy zmienności stochastycznej SV w bayesowskiej analizie finansowych 

szeregów czasowych, czasowych (Stochastic Volatility Processes in Bayesian Analysis of 
Financial Time Series), doctoral dissertation published by Cracow University of Econom-
ics, Kraków.  

Zellner, A. (1971), An Introduction to Bayesian Inference in Econometrics, J. Wiley, New York. 
 

Bayesowska analiza transformacji Boxa i Coxa dla w modelach  
o zmienności stochastycznej 

Z a r y s  t r e ś c i. Celem artykułu jest statystyczna analiza transformacji Boxa i Coxa ilorazu cen 
instrumentów finansowych w modelach FCSV. Stosowane jest podejście bayesowskie, które 
pozwala zbadać, w jakim stopniu dane modyfikują wstępne przekonanie o parametrze transfor-
macji. Wyniki empiryczne pokazują, że założenia o rozkładzie a priori parametru transformacji 
ma istotny wpływ na kształt brzegowego rozkładu a posteriori tego parametru. Jednak w większo-
ści rozważanych przypadków rozkłady te, w porównaniu z rozkładami a priori, są przesunięte w 
kierunku zera. Zatem transformacje ilorazu cen dające wartości bliskie logarytmicznej stopie 
zwrotu są bardziej prawdopodobne a posteriori niż transformacje prowadzące do prostej stopy 
zwrotu.  

S ł o w a  k l u c z o w e: transformacja Boxa i Coxa, model SV, wnioskowanie bayesowskie.  


