Journal of Mathematical Finance, 2013, 3, 329-334
http://dx.doi.org/10.4236/jmf.2013.32033 Published Online May 2013 (http://www.scirp.org/journal/jmf)
An Empirical Study of Option Prices under the Hybrid
Brownian Motion Model
Hideki Iwaki1, Lei Luo2
1Faculty of Business Administration, Kyoto Sangyo University, Kyoto, Japan
2Finance Division, Citigroup Global Markets Japan, Tokyo, Japan
Email: iwaki@cc.kyoto-su.ac.jp, lei.luo@citi.com
Received February 20, 2013; revised April 3, 2013; accepted April 21, 2013
Copyright © 2013 Hideki Iwaki, Lei Luo. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In this paper, we mainly discuss an empirical study of option prices under the hybrid Brownian motion model devel-
oped by [1]. In a specific case of parameters, we have a simple transition probability density function that has a fat-
tailed feature as time passes. We show some empirical evidences that the feature of the model reflects the real market
price movements in Japanese stock market. Furthermore, we make a performance comparison between the hybrid
model and the BS model using Nikkei 225 call options. In general our results show that the hybrid model is slightly
better than the BS model.
Keywords: Hybrid Brownian Motions; Fat-Tailed Properties; Non-Normal Distribution; Empirical Studies; Nikkei 225
Index Options
1. Introduction
Measuring the risk of financial assets and pricing the
financial instruments such as options are very important
for financial institutions and other companies. In this area,
the normal distribution is commonly used as a distribu-
tion of stock return and the Black-Scholes (BS) model
based on the normal distribution is mostly used in prac-
tice. But fat-tailed and non-normal distributions of stock
return are shown by plenty of empirical evidences. As
early as 60s, [2,3] suggested the use of heavy tailed dis-
tributions to describe daily returns. Reference [4] com-
pared the stable distribution and t-distribution for stock
prices of United States. Reference [5] presented an em-
pirical evidence of the normal inverse Gaussian dis-
tribution and t-distribution in the European market. As to
Japanese market, [6] extensively studied on a Pearson
type distribution and made an empirical study in Japa-
nese stock market.
The financial crisis of 2007-2009 made it clear that the
extreme movements of stock return were much more
likely than normal distribution. After the financial crisis
the world financial market has become more and more
turbulent. European sovereign debt crisis is getting worse
and the governors of EU are trying to solve the problem.
The companies in Japan also suffered the loss from af-
termath of the earthquake in northern Japan in March
2011.
Based on the facts mentioned above, we need a new
approach which reflects the fat-tailed property to evalu-
ate financial assets and instruments. In the paper, we in-
vestigate validity of a stochastic differential equation
(SDE) that is a hybrid of arithmetic and geometric
Brownian motion introduced by [1]. They made an
analysis on a model of the market that involves two types
of market participants such as fundamental traders and
technical traders. Through the analysis they pointed out
that the log-return approximately follows such a hybrid
Brownian motion.
When we specify some parameter values, we can ex-
plicitly derive a transition probability density and the
variance of its distribution is increasing exponentially as
time passes. This special case might reflect the real mar-
ket movements in short term especially in some situa-
tions like financial crises. We have fitted the model to the
Japanese stock market. We show the distribution can
describe the market well compared with normal distribu-
tion if we focus on the short term or daily.
A well-known empirical bias associated with the BS
model is the volatility smile which is shown by [7] etc.
The volatility is assumed to be constant in the BS model,
but realized volatility of underlyings actually rise and fall
C
opyright © 2013 SciRes. JMF
H. IWAKI, L. LUO
330
over time. Corresponding to the observed volatility fluc-
tuation, [8] introduced a stochastic volatility model and
[9] gave a closed-form solution for option pricing under
the stochastic volatility model. Reference [10] has made
a thorough empirical study about the performance of the
stochastic volatility model. When volatility is merely a
function of the current asset level and time, we have a
local volatility model which is a useful simplification of
the stochastic volatility model. The local volatility model
is developed by [11]. The hybrid model is a kind of local
volatility models.
To show the performance of the hybrid model in op-
tion pricing empirically, we compare the performance
between the hybrid model and the BS model using Nik-
kei 225 index options. Since we cannot get a closed-form
option pricing formula under the hybrid model, we im-
plement the risk neutral option pricing by applying a tri-
nomial tree method. We compare two models in an in-
sample parameter estimation and an out-of-sample pric-
ing performance. Our empirical evidence indicates that
generally the hybrid model is slightly better than the BS
model.
The rest of the paper is organized as follows. Section 2
introduces the hybrid Brownian motion model and its
special case. We can use this special case to describe the
turbulent financial environments. Section 3 discusses fat-
tailed property of the model. If we focus on distributions
of daily returns, we can find that the model can fit the
market data well. Section 4 shows the results of our em-
pirical studies on comparison of option pricing under the
hybrid model with that under the BS model and discusses
some performance issues. Section 5 is our conclusion.
2. The Hybrid Brownian Motion
2.1. A General Model
We consider a market containing a stock with share price
at time . The log-return tt
S0t
X
is given by
0
log
tt
X
SS. Reference [1] introduced a hybrid SDE
mixing both arithmetic and geometric Brownian motions
to describe fluctuation of t
X
. They make a model of
market with two different types of agents. One set of
agents consists of fundamental traders acting independ-
ently of the current value of t
X
. The other set consists
of technical traders that trade based on the historical val-
ues
|0
s
X
st. They show that after some appro-
priate linearization and approximation, the aggregation of
those two type trader's impacts on the log-return of stock
prices leads to a general form of the hybrid SDE;

1211 22
ddd
tttt
d,
t
X
Xt WXW
 
  (1)
where 1
, 2
, 1
and 2
are constants, and where
1t and 2t
W are standard Brownian motions. In Equa-
tion (1), the part of the arithmetic Brownian motion is
related to the fundamental trades. On the other, the part
of the geometric Brownian motion is related to the tech-
nical trades. We can of course reduce it to an SDE with a
single noise term. If
W
is the infinitesimal correlation
coefficient between the two Brownian motions, then we
can rewrite the SDE as

222 2
12121 2
dd 2
X
d
tt ttt
.XtXX W
 
 (2)
The model has five parameters 1
, 2
, 1
, 2
and
. W can explain them intuitively. 1
e
is t funda-
mental volatility related to the economy. 2
he
is t in-
vestor volatility related to the market confidence of in-
vestors. 1
he
is the drift related to the economy itself. 2
is the drift controlled by the investor sentiment.
represes the relationship between the two volatilities
above.
nt
2.2. A Special Case
Although there might be some positive or negative rela-
tionship between 1
W and 2
W, investors cannot get the
accurate information about the relationship because of
information asymmetry in the financial market. They
might just react randomly to the price of stock. So it
might be reasonable to assume 0
. 1
also has mi-
nor influence to analyze the model especially for a short
term. Thus we adopt such a special case that 10
.
Under the special case, [1] show the variance
t
VX
can be written in such a tractable form that





2
2
2
1
1
2
2
2
2
2
22
2 23
112
1e
~2,
t
t
VX
ttOt









where
2
2
2
2
1.

plays an essential dynamic role. Assuming the condi-
tion 0
, we can find that the variance grows expo-
nentially as time passes. It makes fatten the tails of dis-
tribution of t
X
. Under the assumption that 0
we
can simplify Equation (2) as
2
222
2
12
dd
tt d
2tt
.XtX W

 (3)
X
We observe the momentum property from Equation
(3). Since 2
in the general model is substituted by
2
22
, the positive (negative) return will try to bring a
positive (negative) return next day. This mean-reversion
condition in turn allows an equilibrium to establish. In
general, the transition probability density of t
X
is an
integral of a hyper-geometric function of complex pa-
rameters. However [1] show that under this special case
Copyright © 2013 SciRes. JMF
H. IWAKI, L. LUO 331
with 0
, the transition probability density
,
f
xt
of t
X
is explicitly given by



222
12
2
21
221
1
,fxt
2π
1
exp sinh.
2
tx
tx







(4)
We note that we do not have any closed form transi-
tion probability density except for this special case.
3. An Evidence of a Fat-Tailed and
Non-Normal Distribution
We show evidence such that logarithmic returns of Nik-
kei 225 index follow a fat-tailed and non-normal distri-
bution. We use daily closing prices of Nikkei 225 index
from January 04, 2008 to September 30, 2011 which
corresponds to 915 observations. Figure 1 shows normal
QQ-plots for the corresponding logarithmic returns. For
the Nikkei 225 index distribution, both tails are heavier
than the Gaussian.
Since our main interest is in the tails of the distribu-
tions, we use graphical logarithmic left and right tail tests
employed by [5] to examine the fit in the tails. The
graphical tests were performed as follows. Let
F
x
denote the estimated cumulative distribution function of
the fitted distribution, computed by numerical integration
of Equation (4), and
1,,
N
X
X
log
the order statistic of
the historical data. A plot of

t
F
X against t
X
superimposed onto a plot of

g 11N
lo against t
X
shows the left tail fit for the fitted distribution, and a plot
of against

t
FX
log 1t
X
superimposed onto a
plot of


log 1Nt 1N, the right tail fit.
Figure 1. QQ-plots for log returns of Nikkei 225 index.
Figure 2 shows the plots. The upper panel in the fig-
ure shows the left tail fit, and the lower panel the right
tail fit. The circles correspond to the empirical data, the
line plotted around the circles corresponds to the distri-
bution Equation (4), and the line deviating from the cir-
cles corresponds to the normal distribution. We can find
the distribution Equation (4) fits the index daily return
quite well.
4. Option Pricing and Empirical Tests
4.1. An Numerical Approach for Option Pricing
As we discussed in the previous sections, the hybrid
model has some good features to describe the financial
market. In some situations the pricing of options based
on this model might be better than the traditional pricing
based on the BS model. Since we cannot find a closed
form option pricing formula under the hybrid model, we
use a trinomial tree pricing method to derive the prices of
European call options.
We assume that there exists a constant instantaneous
risk-free rate and there does not exist any arbitrage
opportunities. Let us define
r
e, 0,
rt
tt
YStT
.
By Ito’s lemma, we have
00
de e,d,0,;
rt rt
ttt
YYtWtTY

,S
where



2
2
10
,lnSt SSS
 
 2
,
and where t
W denotes the Wiener process under the
risk neutral probability measure.
We construct a trinomial tree as follows. Parting pe-
riod
0,T into N parts and tTN . For every node
Figure 2. Left and right tail plots for Nikkei 225 index.
Copyright © 2013 SciRes. JMF
H. IWAKI, L. LUO
332

,ni , , the value of 0,1,,;0, 1, 2,nNi
Yt
is given by

0
,
y
niSiy
with a constant
y
. We set

e, 3
rT
yStS
t

to
satisfy such constraints that every transition probability
must be positive and that .
;,pnij

Yt

0yN 
jumps from
,ni to three nodes;


 
1,,,1, ,1,,.nijnininijni 
Here the width of jump is given by
,jni
 

ee ,,
,
rtrnt yni nt
jni t
y



1
,
where
a means the largest integer less than or equal
to . At each node (, , the risk-neutral transition
probability is given by
a)ni
(;,)pni j






 
22
2
2
22
2
2
;,
ee,,
1if
,
ee,, if ,,
,
0oth
rtrnt
rtrn t
pnij
yni nttji
jni y
yni nttjijni
jni y





,
erwise.
Since the option price at the maturity is given by
, where K denotes the
strike price, and the risk-neutral transition probability has
been calculated at every node, the current option price
can be derived by cal- culating backwards from
the final nodes to the initial node .
 
,max ,e ,0
rT
CNjyNj K

0, 0C
0,
0
4.2. Data Description
We use Nikkei 225 index call option prices for the em-
pirical study. Options written on this index are the most
actively traded European type contracts in Japan. We use
daily closing prices of the premiums during October
2011. We also adopt three month TIBOR as the risk-free
rate. For the simplicity we don’t adjust for dividends. We
divide the option data into several categories according
to either the moneyness or the term to expiration. As to
the moneyness, we classify an option contract as at-the-
money (ATM) if 0.97 1.03SK, out-of-the-money
(OTM) if 0.97SK, in-the-money (ITM) otherwise.
As to the term to expiration, we classify an option con-
tract as short-term if days, medium-term if
, long-term otherwise. Since the long term
option contracts are traded scarcely, we exclude them
from the test.
60T
60 180T
4.3. Parameter Estimation and In-Sample Fit
In this section, we make some performance tests by
comparing option pricing under the hybrid model with
that under the BS model. We follow the approach taken
by [10]. In order to apply the hybrid model for option
pricing, we need to estimate the model parameters. Strike
price K and time to maturity T are specified in the con-
tract. We also know spot price S and interest rate r
from the market. However, since structural parameters
1
and 2
are unobservable, we estimate them as fol-
lows. First collect N prices of Nikkei 225 index call op-
tions on a given day in October 2011. For each
1, ,nN
, let n and n
T
K
be respectively the time to
maturity and the strike price of the n-th option. Let
K
ˆ,,
nn
CtT n
denote the observed market price and
K,,
nn
CtT n
be the model price. For each n, we define
a function of 1
and 2
by

12 ˆ
,,,,,
nnnnnn
CtTKCtTK


.
n
Then we estimate the parameters so as to minimize the
sum of squared errors;

12
2
,1
1
min, .
N
n
n


2
(5)
We employed the function optim in the R software to
implement the minimization in Equation (5). Daily aver-
ages of the estimated parameters are reported in Table 1
followed by the standard errors in the parentheses.
denotes the total sum of squared errors and
SSE
SSE de-
notes an average of in the sample period.
SSE
In general we can find the hybrid model improves the
goodness of fit. In most of the trading days, the values of
1
are similar in two models and those of 2
are very
small near to zero. It might indicate that the option prices
are mostly determined by the BS model. This is not sur-
prising because the most of the financial institutions
quote the prices based on the BS model. We can find that
the standard errors of 2
are significantly greater than
those of 1
. This may be explained as follows. In dif-
ferent days there are different market conditions not re-
flected in the BS model but described by 2
. The values
of 2
in the short term are larger than those in the long
term. This is consistent with that the price feedback in
volatility becomes larger and larger as time passes.
of the hybrid model in the short term is better than that in
the long term because option prices with the short term
are affected heavily by the volatility fluctuations and the
hybrid model can consider this kind of effect in option
pricing. The performance among the out-of-the-money
options is not as good as predicted, regardless to the time
to maturities. However, we note that most of out-of-the-
SSE
Copyright © 2013 SciRes. JMF
H. IWAKI, L. LUO
Copyright © 2013 SciRes. JMF
333
Table 1. Estimated parameters and in-sample fit.
All Options Short-term Medium-term
Parameters
BS Hybrid BS Hybrid BS Hybrid
0.238 0.252 0.232
a
(0.031) (0.038) (0.028)
0.237 0.250 0.231
1
(0.031) (0.039) (0.029)
0.081 0.453 0.096
2
(0.254) (0.700) (0.297)
SSE 47,557 47,196 17,138 16,926 26,588 26,328
SSE 1056 1048 613 605 1499 1485
OTM ATM ITM
Parameters
BS Hybrid BS Hybrid BS Hybrid
0.216 0.247 0.298
(0.026) (0.033) (0.053)
0.215 0.243 0.251
1
(0.027) (0.036) (0.049)
0.089 0.534 2.459
2
(0.277) (0.942) (1.479)
SSE 1672 1600 2263 2168 14,406 8307
SSE 59 56 275 264 1379 836
a
denotes the volatility of the BS model.
money options are relatively long maturities.
4.4. Out-of-Sample Performance
We have shown that the in-sample fit of the hybrid
model is generally better than that of the BS model.
However we could think that the improvement of the
performance might be due to increase of number of the
parameters. To deal with this concern, we next examine
pricing performance of each model in the out-of-sample.
In each day in October 2010, we first estimate the re-
quired parameters using the previous day’s option prices
by the same method used in the in-sample fit, and then
we use them as inputs to compute current day’s model-
based option prices. We measure the performance of
each model by calculating the residual sum of squares.
Our result is shown in Table 2. Generally the hybrid
model is slightly better than the BS model. Among others,
the hybrid model of options in-the-money outperforms
the BS model obviously. This might be due to the same
reason as the results of the in-sample test since almost all
the options in-the-money belong to the short-term cate-
gory. However, since 2
changes significantly in day-
to-day trading, the estimations may over-fit the prices of
next day. Especially, both prices of the short-term op-
tions and those of at-the-money are affected by the most
recent market changes. This may cause the performance
of the hybrid model not to be better than expected.
5. Conclusions
In this paper we have made an empirical test of option
pricing under the hybrid Brownian motion model intro-
duced by [1]. In a specific case of parameters of the
model, the variance grows exponentially and shows a
fat-tailed property as time passes. These features of the
model reflect the real market movements especially in
some situations like financial crisis or some event driven
by fluctuation. We have fitted the model to Japanese
stock market. We have also implemented the perform-
ance comparison between the hybrid model and the BS
model in two aspects as in-sample parameter estimation
and out-of-sample pricing prediction.
Our empirical evidence indicates that generally the
hybrid model is slightly better than the BS model. Since
the actual pricing of options in the market is based on the
H. IWAKI, L. LUO
334
Table 2. Out-of-sample pricing errors.
All Options Short-term Medium-term
BS Hybrid BS Hybrid BS Hybrid
SSE 58,292 58,246 20,565 20,583 33,870 33,774
SSE 1297 1296 733 733 1935 1932
OTM ATM ITM
BS Hybrid BS Hybrid BS Hybrid
SSE 3646 3638 4928 4962 32,995 25,793
SSE 128 127 627 632 3082 2468
BS model, it is reasonable that the BS model fits the
market data moderately. So it might be difficult to domi-
nate the BS model remarkably. But if we can improve the
algorithm of pricing method under the hybrid model, it
might be possible to get better results. Furthermore, al-
though we have made an empirical test using a special
case of the hybrid model explained in Section 2.2 for the
computational convenience, we might try to make similar
tests using the general model of the hybrid Brownian
motion in the near future.
6. Acknowledgements
Hideki Iwaki is supported in part by Grant-in-Aid for
Scientific Research (C) No. 22530310, Japan Society for
the Promotion of Science.
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