m0 x59 h22 y81 ff3 fs12 fc0 sc0 ls0">0
,00,0
j
jjj
tpxpxpx


jj
xxp
 

,
j
x
x, by (2), we have
00
=
j
jj
kpp
(5)
Substituting (3), (4), and (5) into (2), we have


00jjj
pp t

000
,exp
jjjj
pxtpp pxx
(6)
Substituting (2) into (1), we have
2
j
j
Dck
t
(7)
If the price changing is falling down, i.e., cj < 0, then,
the last term +cjt in solution (2) is replaced by –cjt, such
that the constant D is always positive.
The above is repeated from [3]. Now, we make two
improvements on [3]:
2.1. Diffusion Equation with Multiple Sources of
Price Changing
The diffusion equation for one source of price changing
is extended to diffusion equation with multiple sources of
price changing.
Suppose that there are n sources of price changing ap-
pearing consequently at time 0
, if 0

, .nΛ
0,0


,
then, we can consider that the n sources
of price changing is appearing at the same time interval

. The diffusion equation due to n sources of price
changing is the sum of (1). The solution of diffusion equa-
tion due to all n sources of price changing is therefore:
1, 2,j


 


000
1
,
exp
,0 ex,0,0
,0,0 d
n
jjj
j
jjj
jjj
pxt
pppxx
pxpx pxx
pxpxt x



00
jjj
j
ppt
x
(8)
where replaces , since x (include xj) is assumed to be
continuous. According to the theorem of middle value of
integral, we have



00mm
pptn
000
,
exp
mmmm
pxt
pppxx

(9)
where
p p
is the average stock price of all
stocks; 0m and 0m are its partial derivatives respect
to x and t respectively. Equation (9) shows that the solu-
tion of multiple sources of price changing can be ex-
pressed by one source of price changing at x = xm.
It should be noticed that the diffusion process is a
process of transmitted each other among all sources and
is completed instantly. The diffusion with beginning at
time 0 and end at the time 0+ changes an old equilibrium
state to a new equilibrium state. Otherwise, the diffusion
process is still continuous, until a new equilibrium state
is formed. Since (9) shows that the solution of n sources
can be represented by one source at x = xm, so that we
don’t need the details of initial and boundary conditions
for each source, but just need the initial and boundary
conditions for one source at x = xm.
2.2. Diffusion Equation of Price Changing Ratio
with Multiple Sources
To measure the degree of a changing state, using the rate
of price-ratio changing
,rxtt
is better than using
price changing rate
,pxt t
. Where the price-ratio
changing
,rxt is defined by

000
,,,,,,r xtp xtp xtp xttt




,rxt rr
(10)
In the following, we use , , and instead
of
,pxt p,
, and respectively for all the above
formulas. From (9), we have:
p
 
00 00
,exp ,
mmommmm
rxtrrrxxrr tn



(11)
(11) is the solution of diffusion equation of multiple sources
of price-ratio changing. Our calculation is based on (11).
3. Strategy of Selling Shares with Maximum
Price Changing Rate and Purchasing
Shares with Minimum Price Changing
Rate (SMaPMi)
The strategy SMaPMi is one of strategies used by the
author for changing of holding shares. SMaPMi intends
to gain maximum difference of stock-price changing from
selling and purchasing. The goal of many speculators take
part in speculation is just for obtaining different price from
purchasing and selling. They do not care much on good
or bad of earning ability of company of a share, but just
care on how to get maximum profit from trading. So that
SMaPMi is a typical strategy of speculation. In the fol-
lowing, the SMaPMi is calculated.
According to the condition of equal price changing of
the holding shares, we change the holding shares by

,0 1,0 1
s
ssppp
npxf npxf  (12)
where subscripts S and P denote selling and purchasing
Copyright © 2012 SciRes. ME
T. Q. YUN
524
respectively; nS and nP are the numbers of selling and
purchasing shares respectively; f is dealing fee per share.
s is the price of selling shares, which, for SMaP-
Mii, has a maximum changing rate at time t = 0; while
p is the price of purchasing shares, which, for
SMaPMi, has a minimum changing rate at time t = 0.

,0px

,0px

,pxt

,pxt
4. Definition of Spacial Coordinate x of
Stock-Price
The spacial coordinate x of price of commodity
is defined by the price relation degree among commodi-
ties in [3]. Share is a kind of commodity, what is the price
relation degree in shares? A stock-price reflects the com-
bination of affecting factors: e.g., earning ability of the
company, economic environment (relation between sup-
ply and demand; interest rate; exchanging rates; policy;
etc.) and the strategy of buyers and sellers, etc. Stock-
prices connect each other by the connection of compa-
nies (e.g., trading connections; guarantee relationship;
and loan contract, shares-holding contract, etc.); and the
comparability of stock-prices. Comparing to commodity,
shares is easier or conveniently to speculate and its price
is determined more on the strategy of buyers and sellers.
The key of description of diffusion phenomena of stock-
price changing is the definition of spacial coordinate. Def-
erent definitions lead to different phenomena of diffusion
description.
For the diffusion phenomenon of going up (or falling
down) of stock-price from a share(s) to other share(s) is
called “recycling changing of shares group” in Chinese
marketing term, the definition of spacial coordinate is
defined by the following process:
According to the market data at time t0, all
0
rt are
arranged in order, such that
 
 
1010
,
i i
rs t

1,
ii
00
,
ii
rtrstrt (13)
Constructing a series:
x
x

0
,
ii
(14)
And making a correspondence between these two se-
ries, e.g.,
x
rs t (15)
Then, (15) holds for different symbol of Si, i.e., (15)
holds for
0
,
ii
x
rxt

0
,
Let x be an axis, such that all xi lies on it, then the
spacial coordinate x is defined by
x
rxt (16)
(16) is the definition of spacial coordinate x of
5. Discussion on Details
,pxt
and
,rxt

,pxt
.
At first, we choose a t0 or 0 and a t or 0+. We can choose
any time to be t0 and t. For example, we choose 2011-
12-15 be t0 or 0, from the market data, the average price
of Shenzhen stock market 0m = 7.53 (RMB/
share). We choose 2011-12-18 be t or 0+, then, from the
data we get
,0
m
px = 7.545 (RMB/share), by (10), we
have
7.545 7.310.333
m
r r
0. However, the 0m
and 0m of (11) are not easily to find out, since the aver-
age price at xm is not a real share and in fact the stocks is
not continuous so that its partial derivatives 0m and 0m
can only be found approximately. Nevertheless we can
avoid this difficult by the following calculation by (11):
r
&
rr
&




0
0
,,
exp exp
,,
ss
s
Pp p
rxt rxt
x
rx txrx t


 


 


 

(17)
0ss
x
where rt

0pp
, the selling share, must be in the pool
of holding shares, since in Chinese stock markets the
regulation of selling a share must be a holding share. While
the purchasing share
x
rt may be in or not in the
pool of holding shares.
(17) shows that if xS = xmax and xP = xmin, then,
maxmin becomes maximum. That is, SMaPMi
has maximum profit at time t > 0+, if no new breaking
factor on equilibrium state appearing after the end of
diffusion 0+.
,,rxtrxt
Discussion on some details:
1) How to determine the time interval of beginning
and end of the diffusion
0
It depends on the judge, or the wish of the operator.
For example, if the operator wishes the price-ratio of the
maximum changing rate of the selling share(s) is 10%,
then, he or she shall wait the time t until the condition
10% is reached, i.e.,
,tt 0,
0


or?
 

maxmax 0
max
max 0
,,
,10%
,
px tpx t
rx tpx t

,px t
max and
where
max 0
,pxt are known at time t
and t0 respectively.
2) Is the time of selling share(s) and purchasing
share(s) should be the same?
It depends on the judge of the operator. If the operator
can not judge the price of the purchasing share is going
up or down, then, it is best to purchase the share at the same
time of selling the share with maximum changing rate.
3) How to determine the volumes of selling and pur-
chasing shares?
It depends on the judge of the operator. In the case of
lacking in purchasing money, it is best to purchase the
volume according to (12).
Copyright © 2012 SciRes. ME
T. Q. YUN
Copyright © 2012 SciRes. ME
525
6. Example of Changing of Owning
Currencies Using SMaPMi
The following data are listed from the turnover record of
the ICBC account of the author.
(2012-02-16) 2674.44 USD 2050.36 EUR (due to
debt crisis of Greek, USD had maximum increasing and
EUR had minimum increasing, EUR/USD = 1.3044. Ac-
cording to SMaPMi, sold USD and buy EUR)
(2012-02-20) 2050.36 EUR 2707.09 USD (EUR
rebounded but should be temporal, EUR/USD = 1.3203)
(2012-03-12) 2707.09 USD 2066.17 EUR (due to
better employment record in USA, USD had a maximum
increasing and EUR had a minimum increasing, EUR/-
USD = 1.312). According to SMaPMi, sold USD and
buy EUD.
7. Conclusions
1) SMaPMi is suited for short term speculation, if op-
erator is proper.
2) The process of diffusion is a process from the be-
ginning of a breaking of an old equilibrium state to the end
of a new equilibrium state due to inertia. The calculation
of strategy of SMaPMi based on diffusion equation of
multiple sources, i.e. (17), is suited for time t 0
+ (the
end of the new equilibrium state) if no new breaking
factor appearing. SMaPMi is also suited for changing of
currencies.
REFERENCES
[1] N. Jegadesh and S. Titman, “Returns to Buying Winners
and Selling Losers: Implications for Stock Market Effi-
ciency,” Journal of Finance, Vol. 48, No. 1, 1993, pp. 65-
91. doi:10.1111/j.1540-6261.1993.tb04702.x
[2] R. K. Kakani and T. Joshi, “Cross Holding Strategy to
Increase Control: Case of the Tata Group,” XLRT Work-
ing Paper, 06-03, 2006, pp. 1-24.
[3] T. Q. Yun, “Instant Diffusion of Price Changing and
‘Time-Space Exchange’ Description,” Technology and
Investment, Vol. 2, No. 2, 2011, pp. 124-128.
doi:10.4236/ti.2011.22012