Calculation of Changing of Holding Shares or Currencies by Instant Diffusion Equation of Price Changing with Multiple Sources

doi:10.4236/me.2012.35068

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Modern Economy, 2012, 3, 522-525

http://dx.doi.org/10.4236/me.2012.35068 Published Online September 2012 (http://www.SciRP.org/journal/me)

Calculation of Changing of Holding Shares or Currencies

by Instant Diffusion Equation of Price Changing with

Multiple Sources

Tianquan Yun

School of Civil Engineering and Transportation, South China University of Technology, Guangzhou, China

Email: [email protected]

Received February 20, 2012; revised March 20, 2012; accepted March 28, 2012

ABSTRACT

The purpose of this paper is application of the instant diffusion equation to the calculation of strategy on changing of

owning shares or currencies. The strategy of selling share(s) with maximum changing rate of price-ratio and purchasing

share(s) with minimum changing rate of price-ratio (SMaPMi) is calculated by instant diffusion equation with multiple

sources of stock-price changing. The spacial coordinate is defined by stock-price changing ratio at a time before diffu-

sion, which ensures that the changing of holding shares using SMaPMi has maximum profit at time next to diffusion

ending, based on the comparability of changing rates of stock-price. Summary, SMaPMi would be worked if operation

is proper. Calculating example of owning currencies using SMaPMi is given.

Keywords: Stock Market Strategy; Strategy of Changing Holding Shares or Currencies; Instant Diffusion Equation of

Price Changing

1. Introduction

Cyclic changing is a general phenomenon in nature and

social life. The so-called “cyclic changing of focus on

share(s) with high changing rate from share group to share

group” is often seen in Chinese stock markets. According

to this phenomenon, a strategy SMaPMi of changing of

holding share(s) is designed. There are many strategies

used in stock market, for example, the strategy of buying

winners and selling losers [1]; cross holding strategy to

increase control [2], etc. However, no literature on quan-

titatively analysis of strategy used in stock market based

on certain (not on probability) theory has been found. In

this paper, a strategy of selling share(s) with maximum

price-ratio changing rate and purchasing share(s) with

minimum price-ratio changing rate (simplify as “SMaPMi”)

is calculated by instant diffusion equation of stock-price

changing [3].

In Section 2, the diffusion equation of one source of

price changing shown in [3] is repeated and extended to

multiple sources of price changing. It also gives more

clear explanation on diffusion transmitted each other among

sources of price-changing. Further more, the price chang-

ing is replaced by the price changing ratio for accurate

description, and thus the diffusion equation and its solu-

tion of multiple sources of stock-price changing are re-

placed by its correspondences of stock-price changing

ratio.

In Section 3, a strategy of SMaPMi is described.

In Section 4, the spacial coordinate x of stock-price





,pxt is defined.

In Section 5, discussion on operation in details is made.

In Section 6, a calculating example for changing of cur-

rencies using SMaPMi is given. Finally, a conclusion is

made.

2. Instant Diffusion Equation Due to Price

Changing of Commodities

For convenience to the readers, let us repeat the instant

diffusion equation shown in [3]:





 (1)



where



,ppxt

jj represents the price of commodity

x at time t due to a raising price changing source at xj.

Equation (1) is explained by the Newton’s second law,

where j



 represents a “force” applied in an equi-

librium state; 22





,exp

jjjjj

pxtAkxxct

represents “transmitted accel-

eration”; D is a constants similar to the inertia of mass in

mechanic. By method of departing variables, the solution

of (1) is shown as follows.







(2)





T. Q. YUN 523





0,0

ppx



where unknowns

, kj and cj are determined by initial

boundary conditions.. These initial boundary conditions are:

1) ,

0t





px p



, by (2), we have

p (3)







jjj

pt px

 ,0

px t











,01 ,0j

px p



 j

x p





, by (2), we

have

cpp

 (4)

 



,00,0

jjj

tpxpxpx





xxp





 



，,

x, by (2), we have

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

Dck

(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

exp

,0 ex,0,0

,0,0 d

jjj

pxt

pppxx

pxpx pxx

pxpxt x

























jjj

ppt











(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 rr



(10)

In the following, we use , , and instead





,pxt p,



, and respectively for all the above

formulas. From (9), we have:





 









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

ssppp

npxf npxf  (12)

where subscripts S and P denote selling and purchasing

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





rt are

arranged in order, such that

 



 

1010

i i

rs t





rtrstrt (13)

Constructing a series:

x





(14)

And making a correspondence between these two se-

ries, e.g.,

rs t (15)

Then, (15) holds for different symbol of Si, i.e., (15)

holds for





rxt



Let x be an axis, such that all xi lies on it, then the

spacial coordinate x is defined by

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





px = 7.545 (RMB/share), by (10), we

have





7.545 7.310.333

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r



exp exp

Pp p

rxt rxt

rx txrx t





 





 



 



(17)





0ss

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

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,







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).

T. Q. YUN

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