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Technology and Investment, 2010, 1, 110-113
doi:10.4236/ti.2010.12013 Published Online May 2010 (http://www.SciRP.org/journal/ti)
Simple Differential Equations of A & H Stock Prices and
Application to Analysis of Equilibrium State
1 2
Tian-Quan Yun , Tao Yun
1
Department of Mechanics, School of Civil Engineering and Transportation, South China University of Technology,
Guangzhou, China
2
China Construction Bank Guangdong Branch, Guangzhou, China
E-mail: cttqyun@scut.edu.cn, samyun@126.com
Received December 13, 2009; revised December 18, 2009; accepted December 20, 2009
Abstract
Similar to the simplest differential equation of stock price, a set of simultaneous differential equations of
stock prices of the same share in both A and H stock markets have been established. This is a set of simulta-
neous nonlinear differential equations, which can be solved by iteration method via a proof by g-contraction
mapping theorem. Further more, the exact solution for equilibrium state and an example of checking the
price prediction of “China Petroleum” (601857) at a conference held in May 2008 are given.
Keywords: A & H Stock Markets, Differential Equation, G-Contraction Mapping, Equilibrium State, Stock
Value
1. Introduction 2. Simultaneous Differential Equations of
Stock Prices of A & H Stock Markets
The “A-stock market” in China is a new developing mar-
ket approaching to connect with the regulations of inter- Similar to the establish of simplest differential equation
national markets. The Hong Kong stock market (H-stock of stock price [9], we set up:
market) is a district international stock market. Many Ch- a) Equations of Amount of Purchasing and Selling:
inese companies issue their stocks in both A and H stock
1
markets. The difference between A and H stock market A ()tpx()tm[x()tey()t] (1)
pa
makes the same share have different prices in A and H A()tsx(t) m[x(t)ey()t]
stock markets, and thus speculating on different stock pr- sa (2)
ices happens frequently. The technique of evaluating the where t represents time; A , A represent the amount
value of a stock is most important for investors. Usually, p s
for the same share, a higher value, or PE ratio, is esti- of purchasing and selling respectively; x , y represent
mated in A-stock market than that for H-stock market the stock price (unit by RMB) of a share in A and H
due to many reasons. However, no paper concerned with stock market respectively; p , s , m , e are con-
quantitative analysis for stock price in A & H stock mar- a a
kets has been found. This paper establishes a set of si- stants. In which we assume that the amount of purchas-
multaneous differential equations for stock prices in A & ing A is inverse proportion to the stock price x , and
p
H stock markets based on a certain mathematic model proportion to the difference of [ey x]; the amount of
developed by a serious works of the author [1-11]. The selling A is proportion to the stock price x and mi-
simultaneous differential equations can be solved by it- s
eration method via the similar proof by g-contractive nus proportion to the difference of [ey x], since selling
mapping theorem [12]. Further more, an exact solution and purchasing are opposite action of trading.
Equation of the Changing Rate of Stock Price:
for a usefully special case is obtained. An example of ch- b)
ecking the predicting of stock price of “China Petroleum” We assume that the changing rate of stock price is pro-
(601857) at a conference held in May 2008 is given. portion to the difference between demand and supply,
Copyright © 2010 SciRes. TI
T. Q. YUN ET AL. 111
i.e., where e is the ratio of A and H stock prices at equilib-
rium
x()tdx/dtg[A(t)A()t] state.
(3)
ps There are a lot of argue on the value of e. How much
where g is a constant to make sure same dimension on should e be?
both sides. [ey x]
If = 0, then substituting (1), (2) into (3), we
c) Simultaneous Differential Aquations of Stock Pri-
ces in A & H Stock Markets have
1
x()tpx(t) sx()t (12)
Substituting (1), (2) into (3), we have aa
1 The solution of (12) is
yy()t T(x)axt()bx (t)cxt()
1 (4)
1/2
am1/(2 eg), bp/(2me), pp2
a aa (13)
()( )exp(2 )
xt x st
(5) 0 a
ss
cm(2 s)/(2me), aa
a
Similarly, for H stock market, we have where x x(0) is the stock price at the beginning of
0
1 the equilibrium state.
H ()tpy()tn[xey] (6)
ph Similarly, for [ey x] = 0, substituting (6), (7) into
H ()tsy()tn[xey], (7)
sh (8), we have
1
yt()g[H()t H()t], (8) y()tpy()tsy()t
ps (14)
hh
where Hp, Hs are the amount of purchasing and sell- The solution of (14) is
ing respectively; p , s , n are constants. Substituting 1/2
h h pp2
hh (15)
yt() ( y)exp(2st)
(6), (7) into (8), we have 0 h
ss
1 hh
(9)
x xt() T(y)ay()tby ()tcy()t
2 hhh where yy (0) is the stock price at the beginning of
an1/(2 g),bp/(2n), 0
hhh the equilibrium state.
ce(2 ns)/(2n), (10) Substituting (13), (15) into (11), we get
hh
2 1/2
(4), (9) are the simultaneous differential equations of
(/ps)[(/ps)x]exp(2st)
aa aa0 a
ee()t (16)
stock prices. Which are a cycling nonlinear differential 2
(/ps)[(p/s)y]exp(2st)
hh hh0 h
equations, and can be shown the existence of solutions
* * * * Once the coefficients p , s , p , s have been found
x ()t and yt(), i.e., the fixed points x ()t T [Tx( (t))], a a h h
2 1
** ([10] for the determination of coefficients), e can be
yt()T[T(y()t)]
and 12 , via g-contractive mapping theo-
[12]. We are not going to the details of the proof but calculated by (16).
rem However, how can we find an equilibrium state from
focus our attention to the more useful problem, i.e., the markets data?
where is the equilibrium point? The “equilibrium state” is an ideal concept, in which
no money moves on balance from either A-stock market
3. The Exact Solution of Simultaneous to H-stock market or from H-stock market to A-stock
Differential Equations for a Special market for speculating profits. But how can we know no
Case, the Equilibrium State[ey- x] = 0 money moving between both stock markets from markets
data? The market data only provide information of stock
[ey x]
We say that = 0 is an equilibrium state, at prices, turn-over volumes. The coefficients p ,s , p , s
which no profit can be made from speculating the dif- a a h h
can be calculated from the market data [10] of different
ference between the A and H stock prices, or no money time in a short time interval such that these coefficients
moves on balance from A-stock to H-stock (or from keep unchanged. However, it is more accurate to use the
H-stock to A-stock) for speculation. If [eyx] > 0, information at the same time. In the following, we con-
then, A-stock price x is chipper, and thus money sider a special case of an equilibrium state, i.e.,
x()t =
moves from H-stock market to A-stock market; If 0 and = 0 hold at the same time
yt() tt , i.e., both
[ey x] e
< 0, then money moves from A-stock market to stock prices x and y are in a “stationary point” (or
ock market.
H-st the so-called “Doji” in marketing term, or a “cross star”
[ey x]
In an equilibrium state = 0, we have in Chinese marketing term).
ee()t x()t/y()t (11)
xt() 0
, and yt()0, (17)
Copyright © 2010 SciRes. TI
112 T. Q. YUN ET AL.
Substituting (17) into (12) and (14), we have both “Petroleum China” (HK0857) and “China Petro-
x()tp ( /s) leum” (601857) become the focus of attention for A & H
(18)
eaa stock speculators and investigators. When the price of
yt() (p/s) “China Petroleum” went down from 48 (Yuan of RMB)
ehh
(19) to about 20, since Oct. 2007, many analysts revised their
Substituting (18), (19) into (11) we have price bottom estimations from 45, 42, 40, 35, 32, 30, 25,
20. At last, 16.7 (the issuing price) was considered as the
ep ()s/(ps) bottom line by the market. The author analyzed the mar-
ah ha (20) ket data according to the theory of equilibrium state
Since at a stationary point, x()t , yt() keep un- based on simultaneous differential equations of A, H
e e
changed, then x()tx(0) x, yt()y(0)y, i.e., stock prices and made a prediction that “16.7 is not the
e 0 e 0 bottom” published in a “Collecting papers” of “forum on
let the time te be the starting time t 0 of the equi- district economic cooperation and district development”
librium state, then (16) becomes held on May 23-25, 2008 [13].
ex / y From the markets data, we tried to find both “Doji”
(21)
00 appeared at the same time 2008-04-10, the opening price
(21) and (20) are equivalent, but (21) is easier to know 17.11, the closing price 17.35 (near “Doji”), turn-over
x and y from the market data. volume 51.887 (million Yuan, RMB) of “China Petro-
0 0
No leum” (601857); while opening price 10.20 (HKD),
w we have not used the information of turn-over
volumes. Notice that Substituting (3) and (8) into (17), Closing price 9.82 (HKD) (near “Doji”), turn-over vol-
together with (1), (2), (6), (7), we have ume 172.785 (million HKD) of “Petroleum China”
1 (HK0857). We considered that the time 2008-04-10 can
AAt() A()tpx()tsx()t, (22)
p esea0ea0e be viewed as the time t of equilibrium state and x =
e 0
1 17.35, y0 = 9.82 × 0.90 = 8.838 (0.90 is the ratio of
H Ht() H(t)py()tsy()t
, (23)
pe se h0e h0e RMB to HKD). By (21), we have
where A and H are the turn-over volumes of the ex / y= 1.963 (26)
stock s in A and H markets respectively. 100
Now we have three independent Equations (22) and By (25), we have
(23) and can determine three unknown coefficients pa eH/A172.785/(0.951.887)3.70
= p , s , s , (let p =p , i.e., the purchasing condi- 2 0, (27)
h a h a h
tion is the same for A and H stock markets) However, ee , which means that 2008-04-10 is
12
p = Ax = p = Hy , s = H / y , A/ x = s , not a strict equilibrium state, in which the turn-over
a 0 h 0 h 0 0 a volume in H-stock market is 3.7 times the turn-over
(24) volume in A-stock market. This means that the same
From (24) and (21), we have stock “Petroleum China” (HK0857) is much chipper than
eH / A that of “China Petroleum” (601857), therefore the specu-
, (25) lating money rushed into H-stock market, and made a
Substituting x , y , A , and H into (21) and (25), larger turn-over volume. This also showed that the price
0 0 of (601857) 17.35 had rooms for getting down, espe-
if both (21) and (25) are satisfied, i.e., cially for the tendency of (HK0857) was going down
x /yH /A, (26)
00 (opening price < closing price) at 2008-04-10. Again,
i.e., the turn-over volume is inverse proportion to the even if 2008-04-10 is an equilibrium state, then, 17.35 is
stock price, then, we can consider that such a state is in a mean value (or the value of stock) and is not the bot-
equilibrium. tom line, so that 16.7 (the so-called “political bottom”,
“technical bottom” etc.) is not a real bottom of the A
4. An Example of Checking the Prediction stock market.
on Stock Price of “China Petroleum” According to the above analysis, the author made a
(601857) at a Conference Held on May prediction that “16.7 is not the bottom” at a conference
23-25, 2008 [13]. held in May 23-25, 2008 [13]. The history shows that the
prediction is correct, the stock price of “China Petrole-
um” went down and break the issuing price 16.7 since
“China Petroleum” (601857) has the largest weight on June-July 2008, until now its price is below 13.8 (lowest
Shanghai SSE Index and the tendency of the prices of 9.9).
Copyright © 2010 SciRes. TI
T. Q. YUN ET AL. 113
5. Conclusion Remarks 1999, pp. 721-728.
[5] T. Q. Yun, “Basic Equations, Theory and Principles For
Stock value estimation is an important evaluation to in- Computational Stock Markets (III)—Basic Theory,” Ap-
vestigators. Stock value is defined herein as the stock pr- plied Mathematics and Mechanics, Vol. 21, No. 8, 2000,
ice at the “equilibrium state”. The “equilibrium state” is pp. 861-868.
an ideal concept, in which no money moves on balance [6] T. Q. Yun, “Analysis of Financial Derivatives by Me-
chanical Method (I)—Basic Equation of Price Of Index
between A and H stock markets for gaining speculating Futures,” Applied Mathematics and Mechanics, Vol. 22,
profits from the difference of A and H markets. How to No. 1, 2001, pp. 118-125.
find the equilibrium state from the markets data? At first, [7] T. Q. Yun, “Analysis of Financial Derivatives By Me-
finding both “Doji” (i.e., the stationary point) at the same chanical Method (II)—Basic Equation of Market Price of
time t from the daily K-line of A and H stock markets; Option,” Applied Mathematics and Mechanics, 2001, Vol.
e 22, No. 9, pp. 1004-1011.
then, calculating the ratios e and e by (26) and (27),
if e e 1 2 [8] T. Q. Yun, “The Application of Game Theory to
1 = 2 , i.e., the turn-over volume is inversely pro- Stock/Option Trading,” Forecasting, in Chinese, Vol. 20,
portion to the stock price, then, te is the time at equilib- No. 5, 2001, pp. 36-38.
rium. Usually, it is hard is find a strict equilibrium state
from the markets data, and a near equilibrium state is [9] T. Q. Yun and G. L. Lei, “Simplest Differential Equation
excepted for rough estimation. of Stock Price, Its Solution and Relation to Assumptions
of Black-Scholes Model,” Applied Mathematics and Me-
chanics, Vol. 24, No. 6, 2003, pp. 654-658.
6. References [10] T. Q. Yun and J. S Yu, “Mathematical Analysis for Op-
erators—Gaps, Optimum Pushing Up, Tactics of Run
[1] T. Q. Yun, “A Basic Integral-Differential Equation of Away, and Reliability Calculation,” Research on Finan-
Changing Rate of Stock Price,” Journal of South China cial and Economic Issues (additional copy), in Chinese,
University of Technology, in Chinese,Vol. 24, No. 6, 1996, Vol. 5, May 2005, pp. 32-35.
pp. 35-39. [11] J. S. Yu, T. Q. Yun and Z. M. Gao, Theory of Computa-
[2] T. Q. Yun, “A Short-Term Prediction of Stock Price for tional Securities, Scientific Publishers, Beijing, 2008.
the Normal Case,” Journal of South China University of [12] T. Q. Yun, “Fixed Point Theorem of Composition G-
Technology, in Chinese, Vol. 25, No. 5, 1997, pp. 47-51. Contractive Mapping and its Applications,” Applied
[3] T. Q. Yun, “Basic Equations, Theory and Principles for Mathematics and Mechanics, Vol. 22, No. 10, 2001, pp.
Computational Stock Markets (I)—Basic Equations,” Ap- 1132-1139.
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[4] T. Q. Yun, “Basic Equations, Theory and Principles for and District Economic Development, Journal of Eco-
Computational Stock Markets (II)—Basic Principles,” nomic Research, University of Guangzhou, Guangzhou,
Applied Mathematics and Mechanics, Vol. 20, No. 7, 23-25 May 2008, pp. 599-604.
Copyright © 2010 SciRes. TI
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