,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