iBusiness, 2013, 5, 54-57
doi:10.4236/ib.2013.51b012 Published Online March 2013 (http://www.scirp.org/journal/ib)
Empirical Study on B-S Model Based Pricing of Warran t s
in China
——With Calculation of Standard Deviation by Modified EMA Model
Zhaoyuan Geng, Qi Ding, Junchi Zhang
Department of Applied Economics, Business School of Zhejiang University City College, Hangzhou, China.
Received 2013
ABSTRACT
This article valuated theoretical prices of covered warrants in china through fitting temporal series of target warrant
prices and market information of corresponding underlying securities. Furthermore, the author surveyed the deviation
between market price and theo retical price of warrant. In order to eliminate the inaccuracy caused by constant volatility
assumption of B-S Model, the author creatively used Modified Exponentially Moving Average (Modified EMA) Model
to calculate the historical volatility of market prices of warrants and fixed the best dilutio n factor through Grid Search
Technique. Also, validities o f calculatio n of historical vo latilit y by Modified EMA Mod el and original EMA Model are
compared. Original data came from market information during the period 2005-2008 supported by security trading
software and calculatio n was d one via Excel. At the same time when Chinese government was going to revive covered
warrant market, this article provided a more accurate method of pricing covered warrant, also known as modified B-S
Model.
Keywords: Warrant Pricing; Covered Warrant; B-S Model; EMA Model; Grid Search Technique
1. Introduction
Pricing warrant has been researched a lot in abroad,
which has perfect pricing theory system and with B-S
model for the main body of the effective measurement
method system [1]. Black-schoals (B-S) model over-
comes the limitations of the study of early scholars, pro-
viding the stock options and warrants pricing methods
reliably. While the study in China is still developing,
lacking correction of own assumptions of B-S model. In
the background that Chinese government was going to
revive covered warrant market, the author creatively puts
forward Mod ified Exp onentia lly Movi ng Avera ge (Mod-
ified EMA) Model to measure the volatility of the asset,
fixed B-S model’s constant volatility assumptions, and
get the more suitable model of pricing covered warrants
for China.
2. Model Constru ction
The article aims to construct an accurate pricing method
of covered warrant, which essentiall y is a Mo d ified
Black-Scholes (Mod ified B-S) Model.
The formul a of Original Black-Scholes (Original B-S)
Model as shown below:
12
(,)( )()
rt
e
CSTSNdK Nd
−
= −
(1)
2
1
11
ln (1)
2
S
dT
K
T
σ
σ

= ++


(2)
21
d dTt σ =−−
(3)
where C(S,T): the value of European call option;
S: present price of underlying assert;
T: Due date of warrant;
K: striking pric e of option;
r: he risk-free rate of rate of the inves tment due in T;
σ
: the annual volatility of the price of underlying a s-
sert;
N(x) means cumulative probability distribution func-
tion of sta nd a rd normal distribution variab le [2].
The pricing object in this article is the C hinese covered
warrant, which can be seen as a typical European call
option， so it is reasonable to use the original B-S Model.
However, the rigo rous limitati ons attached to the original
B-S Model on the financial market, underlying asserts
and the characteristics of the option have affected the
accuracy of the pricing to different degrees [3]. Therefore,
this article is to amend this model.
Apart from the Behavioral Financial, almost all the
pricing models in the spin-off pricing have included the
perfect market hypothesis, so the improvement in terms

Empirical Study on B-S Model Based Pricing of Warrants in China
opyright © 2013 SciRes. IB
55
of the prerequisite hypothesis of the financial market is
impractical.
Then the solely necessary and feasible amendment lies
on the prerequisite hypothesis of the underlying assert.
Evidences have accumulated to show that the deviation
caused by the share profits prepayment hypothesis is not
significant, and furthermore the covered warrant itself
does not produce the shareholding dilution effect, so the
amendment of the shareholding dilution effect can be
excluded. Upon this, the article has finally chosen the
constant volatility rates hypothesis as the breach to
amend the original B-S Model.
In order to amend the constant volatility hypothesis of
B-S Model, the author creatively used Modified EMA
Model to calculate the volatility of yield of underlying
warrants. EMA (Exponentially Moving Average) is a
dynamic s ta tistic，which esse ntially is a predictive value
by adding observations in the past and giving longer
value a lower weightings coefficients [4]. Original EM A
Model supposes the yield of underlying assert obeying
Gaussian distribution—t
R
～
2
(0, )N
σ
, the formula as
sho wn blow
2
0
()
1
R
n
µ
σ
−
=−
∑
(4)
2 22
11 1
(1 )()
ttt t
R σλµ µσ
−− −
=− −+
(5)
where t σ : the volatility of the t;
λ
: dilution factor (0＜
λ
＜1);
µ
: average yield rate in sample periods;
1t
R−: the yield r a te in the t-1 [5].
SIAH Continuous Information Connecting Hypothesis
indicates that Volatility has a positive correlation with
volume. Based on this hypothesis, Rong He amended
EMA Model in 2009. The formula as shown bl ow:
2 22
11 1
(1 )()
t Lttt
VR σ γλµµσ
−− −
= +−−+
(6)
where
γ
is constant coefficient;
L
V
is the long-period volatility of the yield rate of
warrant.
According to the two theory models above, the author
has deduced a more accurate pricing method of covered
warrant used in this article, the modified B-S Model.
Using the modified EMA Model to calculate t σ
, and
supposing n = 60, which mea ns usi ng the sta ndar d d evia-
tion 60 trading days before the opening day of the sample
term as the original volatility 0 σ
, one can get the final
σ
used in the modified B-S Model.
3. Analysis of Chinese’s Data Warrants
Based on Empirical
3.1. Data’s Selec tions
This evidence in the article is used in Time-series data,
including Angang JTC1, Wuliang YGC1 and Yage
QCB1 three Covered Warrants’ price data from the
opening to the delisting and the corresponding underly-
ing stock price data.
In addition to warrants and the price of the underlying
asset, B-S Model also requires Risk-free Interest Rate
over the same period. This article uses the 7-Day Bond
Repurchase Rate instead of the Risk-free Interest Rate,
the data logger for details sees appendix A.
3.2. Real Diagnosis Examination
1) Determination of Attenuation factor
λ
.
In order to determine the best attenuatio n factor value,
this article uses the Grid Search Technique,
λ
value is
0.1, 0.5, 0.8 and 0.95 carry on the pilot calculation sepa-
rately. According to the definition, the best attenuation
factor must make the theoretical price and the market
price cumulative departure is smallest, also namely rea-
lizes
{ }
2
1
min ()
T
i ii
CP
=
∑−
,
i
P
is the market price, i
C
is the theoretical price. The result compiles see Table 1:
λ
= 0.95
2) Modified EMA Model validity examination
In order to check the Modified EMA Model’s validity,
this article has compared three sign stock income fluctu-
ation rate separately under the Modified EMA Model and
the Original EMA Model measurement cumulative de-
parture. It compiles the result to reference Tables 1 and
2: The result demonstrates, in all four pricing of warrants
estimate, uses the Modified EMA Model to measure the
sign stoc k income fluctuation rate forms price accumula-
tion deviation is smaller than EMA Model. Obviously,
the Modified EMA Model has achieved the anticipated
effect, this has a lso guara nteed this article validit y, which
revises to the Modified B-S Model.
Table 1. Modified EMA Model accumulation price devia-
tion tabl e.
λ
=0.1
λ
=0.5
λ
=0.8
λ
=0.95
Angang JTC 1 97.942 95.965 95.985 95.013*
Wulian g YGC1 1595.977 1596.124 1595.823 1595.299*
Yage QCB1 362.917 350.266 344.466 327.194*
Note: *is the accumulation deviation the most minor term.
Table 2 . Ori ginal EM A Model accumulati on price devi ation
table.
λ
=0.1
λ
=0.5
λ
=0.8
λ
=0.95
Angang JTC 1 369.924* 370.948 372.923 375.016
Wulian g YGC1 1791.567 1796.074 1784.153* 1809.885
Yage QCB1 553.957* 576.433 603.248 625.416
Note: *is the accumulation deviation the most minor term.

Empirical Study on B-S Model Based Pricing of Warrants in China
56
3) Modified B-S Model estimate rational examination
In rational e xamination part, t his article rec ognizes the
information of three warrant stocks and sign stock price
to Modified B-S Model, and obtains the theory price of
each warrant B-S model. In or de r to guar a nt ee t he integr-
ity of the demonstration, the author retained four war-
rants in the estimate process separately from opening to
the drawing all trading day data.
The key points of this part are the quota examination
revises, the valid ity of Modified B-S Model in our coun-
try warrant exchange market. Chart 1-3 visually show
bias and fit of the three warrants market price and theo-
retical price of the B-S Model:
0
1
2
3
4
5
6
7
8
9
2005/12/5
2005/12/22
2006/1/13 2006/2/10
2006/3/1
2006/3/20
2006/4/6
2006/4/25 2006/5/19
2006/6/7
2006/6/27 2006/7/14
2006/8/2
2006/8/21
2006/10/13
2006/11/1
2006/11/20
权证收盘P 权证理论价格Ci
Figure 1 . Angang JTC1- M arket price an d theoretical pri ce
comparison chart.
0
10
20
30
40
50
60
2006/4/3
2006/5/18 2006/6/28
2006/8/4
2006/9/12
2006/10/26
2006/12/4 2007/1/15
2007/3/2
2007/4/12 2007/5/30
2007/7/9
2007/8/15 2007/9/24 2007/11/7
2007/12/14
2008/1/24 2008/3/10
权证收盘P 权证理论价格Ci
Figure 2. Wuliang YGC1-Market price and theoretical
price compa rison cha rt .
0
5
10
15
20
25
30
2006/5/22
2006/6/9
2006/6/29 2006/7/19
2006/8/8
2006/8/28 2006/9/15
2006/10/12
2006/11/1
2006/11/21 2006/12/11 2006/12/29
2007/1/23 2007/2/12
2007/3/9
2007/3/29 2007/4/18
权证收盘P 权证理论价格Ci
Figure 3. Yage QCB1-Market price and theoretical price
comparison chart.
In order to verify the validity of pricing, t his article has
calculated in the entire sample period various warrant
market prices and between the B-S model theory price
deviation, and establishes the reasonable deviation range
is ±5%. The concrete computational method is: Dif =
( )/
ii i
PC C−
,
Dif
is a deviation, i
P is the market price,
i
C is the theoretica l price. T he statistical resul t sees Ta-
ble 3.
The result showed that three warrant price overall
deviation besides “Yage QCB1” in ±5%, indicates the
market price close theoretical price. The sample whole's
average degree of deviation is 7.24%. Certainly, “Yage
QCB1” surpasses 20% degree of deviation is not allo wed
to neglect. Its positive number's large-scale deviation
indicated that its market value is higher than the B-S
Model theory value obviously.
4) The empirical analysis result
The outcome shows there are some partial differences
between the modified B-S Model and market price of
covered warrants in China, in addition, market price is
alwa ys higher than theoretical price. The author considers
the major reason why warrant price is estimated is that
excess liquidity and excessive speculation in warrant
market.
The article used domestic warrant market compares
with internal stock market and foreign warrant market,
the fluidity of our domestic warrant market is obviously
too high. In Table 4 2006 years of data, for example,
have at le ast the number of i s sued to the mainl and market
created all trade authority card of authority card is the
largest t urno ver. T his high liquidity is pr obably author ity
card products are the main cause of market overestimate.
And Xiao Gang in the 2007 study also confirmed the
warra nt and its u nd er l yi ng s to ck war r ants s ize and l iquid-
ity will have a significant impact on p ricing.
As China's warrants market, one of the rare financial
derivatives, along with the reform and open of authority
card is investors ' attention. I n addition to rational facto rs,
investors' “Herding Effect” can also explain the overes-
timation of the phenomenon. In China, small investors,
individual investors accounted for the majority of war-
rants. Their limited financial kno wledge to simply be the
expected future market ups and downs as the main fac-
tors affecting the price of warrants, and this study marks
the 2005-2008 bull market in the do mestic stock market,
market expectations for the future generally optimistic.
This optimism is expected to bring the warrants will re-
sult in heat in this part of the study warrants be overesti-
mated.
Table 3. M arket price and theoretical price deviatio n table.
Covered
Warrant Angang JTC1 Wuliang YGC1 Yage QCB1 Average
Dif 4.85% -4.35% 21.23% 7.24%

Empirical Study on B-S Model Based Pricing of Warrants in China
opyright © 2013 SciRes. IB
57
Table 4. The world's major market trading information
summary warrants.
Exchange
The end of the total
num ber of covered
warrants in circulat ion Turnover
（
USD
）
2005 2006 2007 2005 2006 2007
Australian Stock
Exchange 2,447 3,019 4,028 4,985 7,311 17,428
Malaysia Stock
Exchange 12 33 120 277 934 3,843
Hong Kong Stock
Exchange 1,304 1,959 4,614 110,168
23,411 610,380
Korea Stock
Exchange 72 1,387 1,646 41 43,689 73,039
Singapore Stoc k
Exchange 455 521 883 6,521 9,156 19,594
Taiwan Stock
Exchange 540 694 2,085 4,424 5,388 7,717
China S tock
Exchange 4 26 — 21,548 243,900
—
Source: World Federation of Exchange Annual Report (2005-2008)
4. Conclusion
The paper is based on B-S model for warrants and mea-
surement asset price volatility after adjust ing the EMA
Model to build a pricing covered warrant which is k nown
as modified B-S Model. Evidence can be seen through
the entire process, use Modified EMA Model to the
market price of the asset of mark of volatility estimates
that a good results. However, Modified B-S Model used
in Chinese covered warrants on the theory of measure-
ment to get the price and the actual market prices deviate
fr om the r emain s. But s uch case s b y t he ma rke t struc tur e,
a specific period of optimism explained. Most of aut hor-
ity card effective pricing made by Modified B-S Model
will be an optimization sc heme in China.
This article argues that there should continue to study
the following two questions: First, the duration of war-
rants priced deviations in different periods of comparison.
Second, the market liquidity will affect the warra nt prices.
So such resear ch will lead to the warrants pricin g system
in China more perfect.
5. Acknowledgment
This paper is supported by the construct program of the
key laborator y in Hangzhou.
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