![Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014
50
The Models: (3.4b) and (3.5a) suggest the optimal prediction of the expected value of assets
under fractal scaling exponentሺܥ − ܸሻ =
మ
ଶ
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 (3.4b), we observed that when ܽ = 0 , ߙ = 0,the
equation reduces to ܸሺ,ݏ ݐሻ = ܸ݁`௧
.This means that the expected value is being determined by
the interest rate ݎ and time .ݐ If ܽ = 4, ߙ = 2ݍ
ଶ
,(3.4b) reduces to
ܸሺ,ݏ ݐሻ = ܸ݁ݔ ቄ
ିସ
మ௧ௌషమ
ఙమ ± √2ݍቅ ݁௧
thisalso means that the growth rate depends on price,
time, and interest rate.
Considering (3.5a), we also observed that when ܽ = 0, the equation becomes ܸሺݏሻ = 0, this
signifies no signal. If ܽ = 4, (3.5a) becomes ܸሺݏሻ = ቀ
ଶ
మ
௦
ቁ
ఉ
ቊ݁ܣ
ఒభ
మ
మ
ೞ + ݁ܤఒమ
మ
మ
ೞ ቋ ,this implies
that there is signal. We now further look at it when ݍ = 1 to have ܸሺݏሻ = ቀ
ଶ
௦
ቁ
ఉ
൜݁ܣ
మഊభ
ೞ + ݁ܤ
మഊమ
ೞ ൠ.
Hence, if ߣଵ ܽ݊݀ ߣଶare negative, the equation decays exponentially. On the otherhand if
ߣଵܽ݊݀ ߣଶare positive , the equation grows exponentially.
References
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Jubilee of Benoit Mandelboirot M.L. Laipidus and Frankenhuijsen, Eds) 261-338 Ame. Mak. Soc.](https://image.slidesharecdn.com/1314mathsj03-201030124848/85/OPTIMAL-PREDICTION-OF-THE-EXPECTED-VALUE-OF-ASSETS-UNDER-FRACTAL-SCALING-EXPONENT-11-320.jpg)
OPTIMAL PREDICTION OF THE EXPECTED VALUE OF ASSETS UNDER FRACTAL SCALING EXPONENT
- 1. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014 41 OPTIMAL PREDICTION OF THE EXPECTED VALUE OF ASSETS UNDER FRACTAL SCALING EXPONENT Bright O. Osu1 and Joy I. Adindu-Dick 2 1 Department of Mathematics, Abia State University, P M B 2000, Uturu, Nigeria. 2 Department of Mathematics, Imo State University, Nigeria. ABSTRACT In this paper, the optimal prediction of the expected value of assets under the fractal scaling exponent is considered. We first obtain a fractal exponent, then derive a seemingly Black-Scholes parabolic equation. We further obtain its solutions under given conditions for the prediction of expected value of assets given the fractal exponent. Keywords: Fractal scaling exponent, Hausdorff dimension, Black-Scholesequation. 1. INTRODUCTION Financial economist always strive for better understanding of the market dynamics of financial prices and seek improvement in modeling them. Many studies have found that the multi-fractal is more reasonable to describe the financial system than the monofractal. 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). In subsequent years, this new theory was developed and finally it brought a more general concept of multi-scaling. It allows one to study the global and local behavior of a singular measure or in other words, the mono-and multi-fractal properties of a system. In economy, multi-fractal is one of the well-known stylized facts which characterized non-trivial properties of financial time series (Eisler,2004). The multi-fractal model fundamentally differs from previous volatility models in its scaling properties. The emphasis on scaling originates in the work of Mandelbrot (1963), for extreme variations and Mandelbrot (1965),and Mandelbrot and Van ness (1968) for long memory. Multi- fractality is a form of generalized scaling that includes both extreme variations and long memory. Several studies have examined the cyclic long-term dependence property of financial prices, including stock prices (Aydogan and Booth, (1988);Greene and Fielitz, (1977)). These studies used the classical rescaled range (R/S) analysis, first proposed by Hurst (1951) and later refined
- 2. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014 42 by Mandelbrot and Wallis and Matalas (1970), among others. 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; Cheung, Lai, and Lai, 1994). 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 and Karasinski, 1991).Hull and White (1987) among others followed the traditional approach to pricing options on stocks with stochastic volatility which starts by specifying the joint process for the stock price and its volatility risk. Their models are typically calibrated to the prices of a few options or estimated from the time series of stock prices. Ugbebor et al (2001) considered a stochastic model of price changes at the floor of stock market. On the other hand, Osu and Adindu-Dick (2014) examined multi-fractal spectrum model for the measurement of random behavior of asset price returns. They investigated the rate of returns prior to market signals corresponding to the value for packing dimension in fractal dispersion of Hausdorff measure. They went a step further to give some conditions which determine the equilibrium price, the future market price and the optimal trading strategy. In this paper we present the optimal prediction of the expected value of assets under the fractal scaling exponent. We first obtain a fractal exponent, then derive a seemingly Black-Scholes parabolic equation. We further obtain its solutions undergiven conditions for the prediction of expected value of assets given the fractal exponent. 2. THE MODEL Consider the average fractal dimension which is the optimal extraction part to be ݂ሺݔሻ = ଵ ∆ఈ ݂ሺߙሻ݀ߙ ఈೌೣ ఈ . (2.1) Here, ߙ is the singularity strength or the holder exponent, while ݂ሺߙሻ is the dimension of the subset of series characterized by ߙ and ݂ሺݔሻ is the average fractal dimension of all subsets. ∆݂ሺߙሻ = ݂ሺߙ௫ሻ − ݂ሺߙሻ ∆ߙ = ሺߙ௫ − ߙሻ. If the process follows the Hausdorff multi-fractal process we have ݂ሺݔሻ = ଵ ∆ఈ ݂ሺߙሻ ݀ߙ ఈೌೣ ఈ = ݈݅݉ݑݏ ఓሺሺ௫,ሻሻ ഀሺ//ሻ = ܷሺሻ = ∅ݎா ݀ߤሺݍሻ. (2.2) Let (ܴ , ߚሺܴ ሻሻ be a measurable space and ݂: ߚሺܴ ሻ → ܴ be a measurable functionL et ߣ be a real valued function on ߚሺܴ ሻ , then the multi-fractal spectrum with respect to the functions ݂ܽ݊݀ λis given by
- 3. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014 43 ܦሺܽሻ = ߣሼߚ߳ݔሺܴ :݂ሺݔሻ = ܽሽ (2.3) whereλ is taken to be the Hausdorff dimension. Xiao (2004), defined ݂ሺݔሻ = ݈݅݉ݑݏ ఓሺሺ௫,ሻሻ ഀሺ//ሻ = ܸఓ ∅ ሺܧሻ (2.4) from which Uzoma (2006), derive another gauge function to be ݂ሺݔሻఒ = ݈݅݉ݑݏ ఓ൫ሺ௫,ሻ൯ ഀሺ//ሻഊ = ܥ∅ ሺܧሻ. (2.5) Letܦሺܽሻbe multi-fractal thick points of ߤ and ܺ௧ be Brownian motion in ܴ if ݊ > 3 then for all 0 ≤ ܽ ≤ ସ మ, Xiao (2004) showed that ݀݅݉ ቄܴ߳ݔ : lim→ ݑݏ = ఓሺሺ௫,ሻሻ ഀሺ//ሻ = ܽቅ= 2 − మ ଶ (2.6) withݍ > 0 a Bessel function given as ଶିଶሺ௫ሻ , ߤ൫ܤሺ,ݔ ݎሻ൯ is the sojourn time,ܽ the singularity strength andݎradius of the ball. We assume that ݂ሺݔሻ − ݂ሺݔሻఒ ⇒ ܥ∅ሺܧሻ − ܸఓ ∅ሺܧሻ, (2.7) where λ = distortion parameter defined on ߚሺܴሻ, ݀݅݀݁ݐݑܾ݅ݎݐݏ ܽݏ ߤ dynamics and is governed by the useful techniques for Hausdorff dimension. For (2.7) is not equal to zero, we obtain in the sequel its value of which we shall call the fractal exponent ߙ. 2.1.1 ESTIMATION OFࢻ Given a real function ܳ௧, which is continuous and monotonic decreasing for ݐ > 0 with lim௧→ ܳ௧ = +∞ Frostman (1935), defined capacity with respect to ܳ௧ ∶ suppose E is bounded borel set in ܧܽ݊݀ ܳ௧ then ߤ is a measureable distribution function defined for Borel subsets of E such that ߤሺܧሻ = 1 ܷሺሻ = න ∅ݎ ா ݀ߤሺݍሻ ⇒ ݂ሺݔሻ = 1 ∆ߙ න ݂ሺߙሻ ݀ߙ ఈೌೣ ఈ Where ݎ denotes the distance between p and q, exists for ߳ ܧ and is finite or +∞ . U(p) is ∅- potential with respect to the distribution ߤ. Define ∅ − ܿܽݕݐ݅ܿܽ of E denotedܥ∅ ሺܧሻ by (i) if ܸ∅ሺܧሻ = ∞ ݐℎ݁݊ ܥ∅ሺܧሻ = 0 (ii) if ܸ∅ሺܧሻ < ∞ ݐℎ݁݊ ܥ∅ሺܧሻ − ܸ∅ሺܧሻ ≠ 0. Given ܴ௧ as the closure of ,ݔ it is clear that if x is not in ܴ௧ then
- 4. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014 44 ܦఓሺݔሻ = lim→ ݑݏ ఓሺሺ௫,ሻሻ ሺଶሻ = 0 (2.8) Thus if ܧ = ቄܴ߳ݔ :lim→ ݑݏ ఓ൫ሺ௫,ሻ൯ ሺଶሻ = 0ቅ (2.9) Then ܧ ∩ ܴ௧ = ∅ Applying (2.3 ) and (2.8) we see that ܦሺ0ሻ = ߣሼ߳ݔ ܴ :݂ሺݔሻ = 0ሽ (2.10) gives the Hausdorff dimension of ,ܧ ܧ ∩ ܴ௧ = ∅. From the gauge function ܥ∅ሺܧሻ = ܦఓ (x) ℎሺݎሻ = ݎଶ ሺ݈݃ ଵ ሻఒ , ߣ > 1 (2.11) is the correct gauge function such that ݂ሺݔሻ = 0. Note that the occupation measure associated with Brownian motion in ݊ ≥ 3 has a simple meaning for it becomes lim→ ݑݏ ்ሺሻ ሺଶሻ , (2.12) Where ܶሺݎሻ = ܫሺ௫,ሻ ଵ ܺሺݏሻ݀ݐ (2.13) is the total time spent in ܤሺ,ݔ ݎሻ up to time 1. THEOREM 1 Let ܸ∅ሺܧሻbe as in (2.4) and define ܥ∅ ሺܧሻ capacity of ܧ to be ܥ∅ሺܧሻ as in (2.5). If ܸ∅ሺܧሻ = ∞ then ܥ∅ ሺܧሻ = 0 and given ܸ∅ሺܧሻ < ∞, then the dimension capacity ∅ (equivalent to our fractal exponent) is given by ܥ∅ ሺܧሻ − ܸ∅ ሺܧሻ ≠ 0 ⟹ మ ଶ .
- 5. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014 45 Proof We shall proof this in two parts; the value of ܥ∅ሺܧሻand ܸ∅ ሺܧሻ. For ܥ∅ ሺܧሻ, Let there be a Brownian motion in ܴ , ݊ ≥ 2 , then there exist a positive constant c such that for ℤ ≥ ℤ > 0, ሼܶሺݎሻ ≥ ℤݎଶሽ ≤ exp ሺ−ܿݖሻ (Taylor, 1967) .Let ܺ௧ be a Brownian motion in ܴ , ݊ ≥ 3. Suppose ℎሺݎሻ = ݎଶ ሺ݈݃ ଵ ሻఒ , ߣ > 1. Then following Uzoma (2006), we have Lim→ ݑݏ ்ሺሻ ሺଶሻ = 0. (2.14) For a fixed ߳ > 0 ܽ݊݀ ܽ → 0 ܽݏ ݇ → ∞defineܧఒ = ቄܶሺܽሻ ≥ ߳ ܽ ଶ ሺ݈݃ ଵ ೖ ሻఒ ቅ by ܲሺܧఒሻ ≤ ݁ݔ ቄ−ܿሺ݈݃ ଵ ೖ ሻఒ ቅ ≤ ݁ݔ ቄ−ܿሺ݈݃ ଵ ೖ ሻఒఢ ቅ = ሺ݈݃ ଵ ೖ ሻିఒఢ , (2.15) hence ∑ ܲሺܧఒሻ < ∞, ݂݅ ߣ > ଵ ఢ > 1. Thus by Borel Cantelli lemma, we have ܲሺܧఒ , ݅. 0ሻ = 0 therefore there exist ܽ such that ቄܶሺܽሻ < ߳ ܽ ଶ ሺ݈݃ ଵ ೖ ሻఒ , ݅. 0ቅ for some ܽ ≤ ܽ so that lim ೖ→బ ݑݏ ܶሺܽሻ ܽ ଶ ሺ݈݃ ଵ ೖ ሻఒ ≤ ߳ ݂ݎ ߣ > 1 Allowing ߳ → 0, ݏℎݏݓ ݐℎܽݐ ܲ lim ೖ→బ ݑݏ ܶሺܽሻ ܽ ଶ ሺ݈݃ ଵ ೖ ሻఒ = 0 > 0, ߣ > 1 By the Blumenthal zero- one law, we have ܲ ቈlimೖ→బ ݑݏ ்ሺೖሻ ೖ మ ሺ భ ೌೖ ሻഊ = 0 = 1, ߣ > 1 (2.16) Hence, by monotonicity of T and h, we have
- 6. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014 46 lim → ݑݏ ܶሺݎሻ ℎሺݎሻ ≤ ሺܫ + ߳ሻ lim →∞ ݑݏ ܶሺܽሻ ሺܽ ଶ ݈݃ ଵ ೖ ሻఒ ′ ߣ > 1 and the result is established Thus if λ> 1 and ܧ = ቄܴ߳ݔ : lim୰→ sup µሺሺ୶,୰ሻሻ ୰మ/୪୭୰/λ = 0ቅ then from (2.6) dim ܧ = ቄܴ߳ݔ :lim୰→ sup µሺሺ୶,୰ሻሻ ୰మ/୪୭୰/λ = 0ቅ = 2 when ݊ > 3 a.s (2.17) For the second part, it has been shown thatܸ∅ሺܧሻ = 2 − మ ଶ (Xiao, 2004). If ܸ∅ሺܧሻ = ∞ then ܥ∅ ሺܧሻ = 0 and given ܸ∅ሺܧሻ < ∞, then the dimension capacity ∅ is given by ܥ∅ ሺܧሻ − ܸ∅ ሺܧሻ . Putܸఓ ∅ሺܧሻ = ఓ൫ሺ௫,ሻ൯ ഀሺ||ሻ = ݂݊ܫఓܸఓ ∅ሺܧሻ where ߤis a measure with respect to ∅-capacity of ܧ on the function ܸ∅ ሺܧሻand ܥఓ ∅ ሺܧሻ = ఓ൫ሺ௫,ሻ൯ ഀሺ୪୭ ሻഊ = ܵݑఓܥఓ ∅ ሺܧሻ, then ܥఓ ∅ሺܧሻ − ܸఓ ∅ሺܧሻ = ܵݑఓܥఓ ∅ሺܧሻ − ݂݊ܫఓܸఓ ∅ሺܧሻ = 2 − ቀ2 − మ ଶ ቁ = మ ଶ , (2.18) as required. 3. Optimal expected value of assets under fractal scaling exponent 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 ℎܵ − ܸ, (3.1) measured by the fractal index ܥ∅ሺܧሻ − ܸ∅ ሺܧሻ ≠ 0. After an elapse of time ∆ݐ the value of the portfolio will change by the rate ℎሺ∆ܵ + ݐ∆ܦሻ − ∆ܸin view of the dividend received on h units held. By Ito’s lemma this equals ℎሺߤݐ∆ݏ + ߪܵ∆ݖ + ݐ∆ܦሻ − ቆ ߲ݒ ߲ݐ + ߲ݒ ߲ݏ ߤܵ + 1 2 ߲ଶ ܸ ߲ܵଶ ߪଶ ܵଶ ቇ ∆ݐ + ߲ݒ ߲ܵ ߪܵ∆ݖሻ
- 7. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014 47 or ሺℎߤܵ + ℎܦሻ − ቆ ߲ݒ ߲ݐ + ߲ݒ ߲ݏ ߤܵ + 1 2 ߲ଶ ܸ ߲ܵଶ ߪଶ ܵଶ ቇ ∆ݐ + ሺℎߪܵ − ߲ݒ ߲ܵ ߪܵሻ∆ݖ If we take ℎ = డ௩ డௌ (3.2) 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. ሺℎߤܵ + ℎܦሻ − ቆ ߲ݒ ߲ݐ + ߲ݒ ߲ݏ ߤܵ + 1 2 ߲ଶ ܸ ߲ܵଶ ߪଶ ܵଶ ቇ ∆ݐ = ሺℎܵ − ܸሻݐ∆ݎ Thus ܦ ߲ܸ ߲ܵ − ቆ ߲ݒ ߲ݐ + 1 2 ߲ଶ ܸ ߲ܵଶ ߪଶ ܵଶ ቇ = ሺ ߲ݒ ߲ܵ ܵ − ܸሻݎ So that డ డ௧ + ሺݏݎ − ܦሻ డ డௌ + ଵ ଶ డమ௩ డௌమ ߪଶ ܵ = ܸݎ . (3.3) Proposition 1: Let ܦ = 0 (where D is the market price of risk), then the solution of (3.3) which coincides with the solution of డ௩ డ௧ + ଵ ଶ డమ డௌమ ߪଶ ܵଶ = 0 (3.4a) is given by VሺS, tሻ = Vexp ቄ ିଶα୲ୗതషమ σమ + λSതቅ ݁௧ . (3.4b) For proof see (Osu and Adindu –Dick, 2014). Proposition 2:For ܦ ≠ 0, the solution of (3.3) is given as: ܸሺݏሻ = ቀ మ ଶ௦ ቁ ఉ ቊ݁ܣ ఒభ ೌ మ మೞ + ݁ܤఒమ ೌ మ మೞ ቋ, (3.5a)
- 8. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014 48 Where ߣଵ = − ଶ ௭ ± ට ସ ௭మ + ଼ ௭మఙమand ߣଶ = ± ଵ ௭ ට4 + ଼ ఙమ (3.5b). Proof We take ܼ = ఈ ௌ ; ܸሺݏሻ = ܼఉ ܹሺܼሻ. (3.6) Thus ݀ݖ ݀ݏ = − ߙ ܵଶ = − 1 ߙ ܼଶ ݀ݒ ݀ݏ = ܸ݀ ܼ݀ . ܼ݀ ݀ܵ = − 1 ߙ ܼଶ ሺߚܼఉିଵ ܹ + ܼఉ ܹ݀ ܼ݀ ሻ = − ଵ ఈ ሺߚܼఉାଵ ܹ + ܼఉାଶ ௗௐ ௗ ሻ. Hence ௗమ ௗௌమ = ௗ ௗௌ ሺ ௗ ௗ ). ௗ ௗௌ = − ଵ ఈ ܼଶ ሺߚሺߚ + 1ሻܼఉ ܹ + ߚܼఉାଵ ௗௐ ௗ + ሺߚ + 2ሻܼఉାଵ ௗௐ ௗ + ܼఉାଶ ௗమௐ ௗమ ሻ. In this case V is not dependent on .ݎ Substituting into the given differential equation we have ܼݎఉ ܹ = ߪଶ 2 ሺߚሺߚ + 1ሻܼఉ ܹ + ߚܼఉାଵ ܹ݀ ܼ݀ + ሺߚ + 2ሻܼఉାଵ ܹ݀ ܼ݀ + ܼఉାଶ ݀ଶ ܹ ܼ݀ଶ ሻ +ቀ ఈ − ܦቁ ቀ ିଵ ఈ ቁ ሺߚܼఉାଵ ܹ + ܼఉାଶ ௗௐ ௗ ሻ Cancelling byܼఉ and collecting like terms we have 0 = ߪଶ 2 ܼଶ ݀ଶ ܹ ܼ݀ଶ + ܹ݀ ܼ݀ ൬ߪଶሺߚ + 1ሻܼ − ܼݎ + ܦ ߙ ܼଶ ൰ + ܹ ቆ ߪଶ 2 ߚሺߚ + 1ሻ − ߚݎ + ߚ ܦ ߙ ܼቇ − ݓݎ = ߪଶ 2 ܼଶ ݀ଶ ܹ ܼ݀ଶ + ܹ݀ ܼ݀ ܼ ൬ߪଶሺߚ + 1ሻ − ݎ + ܦ ߙ ܼ൰ + ܹ ቆ ߪଶ 2 ߚሺߚ + 1ሻ − ݎሺߚ + 1ሻ + ߚ ܦ ߙ ܼቇ Let ߚ = 0.ݎ = ఈ ܼ (3.7)
- 9. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014 49 We obtain ܼଶ ௗమௐ ௗ௭మ + 2ܼ ௗௐ ௗ௭ − ଶௐ ఙమ = 0. (3.8) Let ߣଵ and ߣଶ be the roots of the equation, then ߣଵ + ߣଶ = − 2 ݖ ߣଵߣଶ = − 2ݎ ܼଶߪଶ Now, ݀ଶ ܹ ݀ݖଶ − ሺߣଵ + ߣଶሻ ܹ݀ ݀ݏ − ߣଵߣଶܹ = 0 or ݀ ݀ݖ ൬ ܹ݀ ܼ݀ − ߣଶܹ൰ = ߣଵ ൬ ܹ݀ ݀ݏ − ߣଶܹ൰ Then ܹ݀ ܼ݀ = ܻ, ܻ = ൬ ܹ݀ ݀ݏ − ߣଶܹ൰ Which gives ܻ = ݁ܥఒమ௭ with solution ݁ିఒభ௭ ܹ = ܥ ݁ሺఒభିఒమሻ௭ ݀ݖ + ܤ (3.9) (Where C and B are arbitrary constants). Hence ܹሺݖሻ = ݁ܣఒభ௭ + ݁ܤఒమ௭ (3.10) ܸሺݏሻ = ܼఉ ܹሺܼሻ = ቀ ߙ ݏ ቁ ఉ ቄ ݁ܣఒభ ഀ ೞ + ݁ܤఒమ ഀ ೞ ቅ = ቀ మ ଶ௦ ቁ ఉ ቊ ݁ܣఒభ ೌ మ మೞ + ݁ܤఒమ ೌ మ మೞ ቋ (3.11) 4. Conclusion
- 10. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014 50 The Models: (3.4b) and (3.5a) suggest the optimal prediction of the expected value of assets under fractal scaling exponentሺܥ − ܸሻ = మ ଶ 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 (3.4b), we observed that when ܽ = 0 , ߙ = 0,the equation reduces to ܸሺ,ݏ ݐሻ = ܸ݁`௧ .This means that the expected value is being determined by the interest rate ݎ and time .ݐ If ܽ = 4, ߙ = 2ݍ ଶ ,(3.4b) reduces to ܸሺ,ݏ ݐሻ = ܸ݁ݔ ቄ ିସ మ௧ௌషమ ఙమ ± √2ݍቅ ݁௧ thisalso means that the growth rate depends on price, time, and interest rate. Considering (3.5a), we also observed that when ܽ = 0, the equation becomes ܸሺݏሻ = 0, this signifies no signal. If ܽ = 4, (3.5a) becomes ܸሺݏሻ = ቀ ଶ మ ௦ ቁ ఉ ቊ݁ܣ ఒభ మ మ ೞ + ݁ܤఒమ మ మ ೞ ቋ ,this implies that there is signal. We now further look at it when ݍ = 1 to have ܸሺݏሻ = ቀ ଶ ௦ ቁ ఉ ൜݁ܣ మഊభ ೞ + ݁ܤ మഊమ ೞ ൠ. Hence, if ߣଵ ܽ݊݀ ߣଶare negative, the equation decays exponentially. On the otherhand if ߣଵܽ݊݀ ߣଶare positive , the equation grows exponentially. References [1] Aydogan, K.,Booth, G.G. (1988). “Are there long cycles in common stock returns?,” southern economic journal , 55:141-149. [2] Black F. Karasinski P. (1991). “Bond and options pricing with short rate and lognormal. Finan Analysis J. 47(4): 52-59. [3] Black F. Scholes M. (1973). “The valuation of options and corporate liabilities”, J Pol Econ. 81:637- 654. [4] Cheung,Y.W. ,Lai, K.S. Lai, M. (1994). “Are there long cycles in foreign stock returns?” Journal of international financial markets, institutions and money, forthcoming. [5] Greene, M. T. Fielitz B.D. (1997). “Long term dependence in common stock returns”, Journal of financial economics, 5:339-349. [6] Eisler Z. Kertesz J. (2004). Physica A. 343:603. [7] Frostman, O. (1935). “Potential d’equilibra et capacite des ensembles avec quelques applications a la theorie des fonctions”. Meddelanden fran lunds universitets mathematiska seminarium 3:1-18 [8] Hull J. White A.(1987). “The pricing of options on assets with stochastic volatilities. J. Finan 42:271- 301. [9] Hurst, H.E. (1951). “Long term storage capacity of reservoir”, Transactions of the American society of civil engineers , 116:770-799. [10] Lo, A.W.(1991). “Long term memory in stock market prices”, Econometrica, 59:1279-1313. [11] Mandelbrot B. B. (1963). The variation of certain speculative prices, Journal of business. 36:394- 419.,comptes rendus de l’academie des sciences de paris 260:3274-3277. Reprinted in Mandelbrot (2000). [12] Mandelbrot B.B. (1975). Une classe de processus stochastiques homothetiques a soi [13] Mandelbrot B.B. (1982). The fractal geometry of nature. Freeman. New york. [14] Mandelbrot B.B, Ness J. W. (1968). Fractional Brownian motion: fractional noises and application SIAM Review 10:422-437.
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