Optimal Prediction of Expected Value of Assets Under Fractal Scaling Exponent Using Seemingly Black-Scholes Parabolic Equation
Bright O. Osu^{1}, Joy Ijeoma Adindu-Dick^{2}
^{1}Department of Mathematics, College of Physical and Applied Sciencs, Michael Okpara University of Agriculture, Umudike, Nigeria
^{2}Department of Mathematics, Faculty of Physical and Biological Sciences, Imo State University, Owerri, Imo State, Nigeria
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Bright O. Osu, Joy Ijeoma Adindu-Dick. Optimal Prediction of Expected Value of Assets Under Fractal Scaling Exponent Using Seemingly Black-Scholes Parabolic Equation. Mathematics Letters. Vol. 2, No. 2, 2016, pp. 19-24. doi: 10.11648/j.ml.20160202.11
Received: July 7, 2016; Accepted: July 27, 2016; Published: October 11, 2016
Abstract: Assessing the stock price indices is the foundation of forecasting the market risk. In this paper, we derived a seemingly Black-Scholes parabolic equation. We then solved this equation under given conditions for the optimal prediction of the expected value of assets.
Keywords: Fractal Scaling Exponent, Black-Scholes Equation, Assets Price Return, Optimal Value, Parabolic Equation
1. Introduction
The problem associated with random behavior of stock exchange has been addressed extensively by many authors (see for example, Black and Scholes, 1973; and Black, et al., 1991). The concept of "fractal world" was proposed by Mandelbrot in 1980’s and was based on scale-invariant statistics with power law correlation (Mandelbrot, 1982). Fang et al., (1994) examined the relevance of fractal dynamics in major currency futures market. Fractal dynamics are forms of dynamics characterized by irregular cyclical fluctuations and long term dependence. They estimated directly the fractal structure in currency futures prices based on a time series model of fractional processes. Based on the self-similarity property of fractal, Tokinaga and Moriyasu, (1997) forecasted the time series by the fractal dimension which was obtained via the wavelet transform. Xiong, (2002) also applied the wavelet to measure the fractal dimension of Chinese stock market. Muzy, et al., (2000) estimated the statistical self-similarity exponents from the data and made a quadratic fit for some low order moments. Several studies have examined the cyclic long-term dependence property of financial prices, including stock prices (Greene and Fielitz, (1977); Aydogan and Booth, (1988)). These studies used the classical rescaled range (R/S) analysis, first proposed by Hurst (1951) and later refined by Mandelbrot and Wallis, (1969) and Wallis and Matalas, (1970), among others. Using R/S analysis, Greene and Fielitz, (1977) studied 200 daily stock returns of securities listed on the New York stock exchange and they found significant long range dependence. A problem with the classical R/S analysis is that the distribution of its regression-based test statistics is not well defined. As a result, Lo (1991) proposed the use of a modified R/S procedure with improved robustness. The modified R/S procedure has been applied to study dynamic behavior of stock prices (Lo, 1991; and Cheung, et al., 1994). Teverovsky et al., (1999) and Willinger et al., (1999) identified a number of problems associated with Lo’s method. In particular, they showed that Lo’s method has a strong preference for accepting the null hypothesis of no long range dependence. This happens even with long-range dependent synthetic data. To account for the long-range dependence observed in financial data, Cutland et al., (1995) proposed to replace Brownian motion with fractional Brownian motion as the building block of stochastic models for asset prices. An account of the historical development of these ideas can be traced from Cutland et al., (1995), Mandelbrot, (1997) and Shiryaev, (1999). In this paper, we will derive a seemingly Black-Scholes parabolic equation. This equation is being solved under given conditions for the optimal prediction of the expected value of assets.
2. The Model
Consider a portfolio comprising h unit of assets in long position and one unit of the option in short position. At time, T the value of the portfolio is
(1)
measured by the fractal index .
After an elapse of time, , the value of the portfolio will change by the rate in view of the dividend received on h units held. By Ito’s lemma this equals
OR
If we take
(2)
the uncertainty term disappears, thus the portfolio in this case is temporarily riskless. It should therefore grow in value by the riskless rate in force i.e.
.
Thus
So
(3)
Proposition 1: Let (where is the market price of risk), then the solution of equation (3), which coincides with the solution of
(4)
with
, (5a)
(5b)
and is assumed constant, is given by
.(using equation (5a)) (6)
with
(using equation (5b)). (7)
Where is the investment output, the discount rate, and the variance of the stock market price.
Proof: Let (where is the market price of risk), then equation (3) becomes
(8)
In order to remove the effect of the discount rate () from equation (8), we let and set
(9a)
and
. (9b)
Hence equation (8) becomes
(10)
By the method of separation of variables, we assume a solution of the form
Hence
, (11a)
and
. (11b)
Substituting equations (11a) and (11b) in equation (10) gives
(12)
or
(13a)
and
(13b)
These are ordinary differential equations for with
(using equation (13a)) and having solution
(14)
Also
(using equation (13b))
So that and with solution
(15)
Hence, we obtain a special solution of the form
(16)
But
(as in equation (7)).
Solving for in the above equation gives
,
and
.
Equating this result to equation (9b) gives
and
(17)
Equating equation (9a) to equation (16) gives
(18)
Proposition 2: Let (where is the market price of risk), then the solution of equation (3) where is not a constant, coincides with the solution of
(19)
with
(20a)
and
(20b)
is given by
(21)
with
(using equation (20b)). (22)
Where is the investment output, the discount rate, the variance of the stock market price, and are arbitrary constants.
Proof: From equation (8), to equation (9b) we have equation (3) reduced to
(23)
By the method of separation of variables, let the solution of equation (23) be Hence, and . Equation (23) becomes . Therefore
(24)
By separation of variables equation (24) becomes
(25)
From equation (25) we have
That is,
(26)
We then solve equation (26) using Euler’s substitution method.
Let , then
and . (27)
Also
(28)
,
and . (29)
Equation (26) becomes
(30)
Let be the solution of equation (30), hence ; . Equation (30) becomes
Our auxiliary equation becomes
.
Therefore
(31)
and
(32)
From equation (27) we have , but , hence
Our general solution becomes
(33)
From equation (25) we have
(34)
The solution of equation (34) becomes
(35)
But from equations (33) and (35) we have
(36)
But
(as in equation (22)).
Solving for in the above equation gives
and
.
Equating this result to equation (9b) gives
and
(37)
Equating equation (9a) to equation (36) gives
(38)
where and are arbitrary constants; and are as defined in equations (31) and (32).
Proposition 3: For , the solution of equation (3) is given as:
(39)
where
and . (40)
Proof
We take
(41)
Thus
.
Hence
).
.
In this case V is not dependent on. Substituting into the given differential equation we have
(42)
Cancelling by and collecting like terms we have
Or
.
Let
and . (43)
We obtain
(44)
Let and be the roots of the equation, then
..
Now,
or
Then
Which gives with solution
(45)
(Where C and B are arbitrary constants).
Hence
(46)
(47)
3. Conclusion
The Models: equations (6), (21) and (39) suggest the optimal prediction of the expected value of assets under fractal scaling exponent which we obtained. We derived a seemingly Black Scholes parabolic equation and its solution under given conditions for the prediction of assets values given the fractal exponent. Considering equation (6), we observed that when , the equation reduces to .This means that the expected value is being determined by the interest rate and time . If , equation (6) reduces to , this also means that the growth rate depends on price, time, and interest rate.
Considering equation (21), we observed that when our singularity strength, , our fractal exponent, , equation (21) becomes If our singularity strength, , our fractal exponent, , equation (21) reduces to . When , we have . This means that the expected value depends on stock price, interest rate, and time. If and are positive, the stock price increases, hence the investment output increases. On the other hand, if and are negative, the stock price decreases and this leads to decrease in investment output. Considering equation (39), we also observed that when , the equation becomes , this signifies no signal. If equation (39) becomes , this implies that there is signal. We now further look at it when to have . Hence, if are negative, the equation decays exponentially. On the other hand, if are positive, the equation grows exponentially.
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