The Multi-fractal Spectrum Model for the Measurement of Random Behaviour of Asset Price Returns

To forecast the market risk, assessing the stock price indices is the foundation. Multi-fractal has lots of advantage when explaining the volatility of the stock prices. The asset price returns is a multi-period (multi-fractal dimension) market depending on market scenarios which are the measure points. This paper considers the multi-fractal spectrum model (MSM) to measure the random character of asset price returns, aimed at deriving the MSM version of the random behaviour of equity returns of the existing ones in literature. We investigate the rate of returns prior to market signals corresponding to the value for packing dimension in fractal dispersion of Hausdorff measure. Furthermore, we give some conditions which determine the equilibrium price, the future market price and the optimal trading strategy.


Introduction
The multifractal formalism was introduced in the context of fully-developed turbulence data analysis and modeling to account for the experimental observation of some deviation to Kolmogorov theory (K41) of homogenous and isotropic turbulence [1]. The predictions of various multiplicative cascade models, including the weighted curdling (binomial) model proposed by [2] were tested using box-counting (BC) estimates of the so-called f(α) singularity spectrum of the dissipation field [3]. Alternatively, the intermittent nature of the velocity fluctuations were investigated via the computation of the D(h) singularity spectrum using the structure function (SF) method. In financial economics, multi-fractal spectrum model (MSM) has been used to analyze the pricing implications of multi-frequency risk .The models have had some success in explaining the excess volatility of stock returns compared to fundamentals and the negative skewness of equity returns. They have also been used to generate multi-fractal jump-diffusion [4].

Original Research Article
The multifractal formalism of multi-affine functions amounts to compute the so-called singularity spectrum D(h) defined as the Hausdorff dimension of the set where the Hölder exponent is equal to h. Once one has the function, h(t),characterizing the Holder exponent , one can construct the singularity spectrum . It is believed that the singularity spectrum is a synonym to the multi-fractal spectrum.
The specifics of how to calculateh (t) and D(α) in practice vary from approximating them outright by their definition or by using wavelets to approximate the Holder exponent and then using a Legendre transform to approximate the multi-fractal spectrum.
MSM is a stochastic volatility model [5,6] with arbitrarily many frequencies. MSM builds on the convenience of regime -switching models, which were advanced in economics and finance [7]. MSM is closely related to the multi-fractal model of Asset Returns (MMAR) [8]. MSM improves on the MMAR's combinatorial construction by randomizing arrival times, guaranteeing a strictly stationary process. MSM provides a pure regime-switching formulation of multi-fractal measures, which were pioneered by [9,10,11].
Extensions of MSM to multiple assets provide reliable estimates of the value-at-risk in a portfolio of securities [12]. MSM often provides better volatility forecasts than some of the best traditional models both in and out of sample [13]. Report considerable gains in exchange rate volatility forecasts at horizons of 10 to 50 days as compared with GARCH (1,1), Markov-switching GARCH, [14,15] and Fractionally Integrated GARCH [16]. [17] obtained similar results using linear predictions. Building on the multifractal formalism presented earlier, MMAR represents a simple compound process that has a closed-form multifractal spectrum. MMAR produces volatility clustering, heavy tails and long memory in returns.
Our aim in this paper is to apply some tools of multifractal analysis to the measurement of random behaviour of the stock market price returns. The problem associated with random behaviour of stock exchange has been addressed extensively by many authors [18,19]. [20] 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. On the other hand, [21] considered astochastic model of price changes at the floor of stock market. Here the equilibrium price and the market growth rate of shares were determined. There have been some works with considerable extensions and constrains subsequently [22,23]. In this paper we present a dynamic stochastic multi-fractal spectrum model of variation of the stock returns aimed at deriving the MSM version of the random behaviour of equity returns of the existing ones in literature. The equilibrium price and growth rate of an asset, using a linear rate of return, ‫ݎ‬ ௧ are determined. We first developed a method of computing the so called singularity spectrum which is equivalent to computing the Holder exponent. Further more, the optimal trading strategy is determined.

Basic Tools and Preliminaries
Let ሺܲ ௧ ሻ ௧ஹ denote the price process of a security, in particular of a stock. To allow comparison of investments in different securities, it is necessary to investigate the rates of return defined by Most authors prefer these rates, which correspond to continuous compounding, to the alternative The reason for this is that the return over n periods, for example n days, is then just the sum; As a model for stock prices the natural candidate is now the multi-fractal process ൫ܺ ௧ ௗ, ൯ ௧ஹ This can also be written in the form ܻ݀ ௧ = ܻ ௧ ܼ݀ ௧ with ܼ ௧ = ܺ ௧ ௗ, . The solution of this equation is the Doleans-Dade exponential [24].
The MSM model can be specified in both discrete time and continuous time.

Discrete Time
Let ܲ ௧ denote the price of a financial assets, and let denote the returns over two consecutive periods. In MSM, returns are specified as Where µ and ߪ are constants and ሼߦ ௧ ሽ are independent standard Gaussians. Volatility is driven by the first-order latent Markov state vector: Given the volatility state ‫ܯ‬ ௧ the next-period multiplier ‫ܯ‬ ,௧ାଵ is drawn from a fixed distribution M with probability ߛ and is otherwise left unchanged ‫ܯ‬ ,௧ drawn from distribution M with probability ߛ . with probability 1 − ߛ The transition probability is specified by The sequence ߛ is approximately geometric with ߛ ≈ ߛ ଵ ೖషభ at low frequency. The marginal distribution ‫ܯ‬ has a unit mean, has a positive support, and is independent of ‫.ܭ‬ In empirical applications, the distribution M is often a discrete distribution that can take the values ݉ or 2 − ݉ with equal probability. The return process ‫ݎ‬ ௧ is then specified by the parameters ߠ = ሺ݉ , ߤ, ߪ ത, ܾ, ߛ ଵ ሻ, Note that the number of parameters is the same for all ݇ ത > 1.

Continuous Time
MSM is similarly defined in continuous time. The price process follows the diffusion: where is a standard Brownian motion, and ߤ and ߪ ത are constants.
When the number of components ݇ ഥ goes to infinity, continuous-time MSM converges to a multifractal diffusion , whose sample paths take a continuum of local Holder exponents on any finite time interval.
If the latest observation ܲ ௧ of the designated process at time ‫ݐ‬ > ݀ is conditioned on the information; available up to time ‫ݐ‬ − 1, then where (using (10)), and The function ℎሺ. ሻ with parameters ߙ = ሼߙ : ݇ = 0,1, … , ݀ሽ is defined as so that given the history (18) of the process up to time ‫ݐ‬ − 1, the conditional distribution of ܲ ௧ is normal with mean ߤ ௧ and variance Formally, if the price of an asset at time ‫ݐ‬ is ܲ ௧ for ‫ݐ‬ ∈ ሾ0, ܶሿ and ܶ < ∞, then the associated logprice process is given by The log-price process is then modeled as a compound process where ܹሺ‫ݐ‬ሻ is Brownian motion and ߠሺ‫ݐ‬ሻ is a stochastic process termed trading time (which is the cumulative distribution function (c.d.f.) of a random multifractal measure, and the processes ܹሺ‫ݐ‬ሻ and ߠሺ‫ݐ‬ሻ are assumed to be independent.
The fractal dimension is the basic notion for describing structures that have a scaling symmetry. Scaling symmetry means self-similarity of the considered object on varying scale of magnification.
[26] introduced the first notion of dimension, providing a measure for filling space which allows for the possibility of non-integral dimensions. Then D-dimensional Hausdorff measure on a set A is given by Define the optimal covering of this set using spheres of variable radius ‫ݎ‬ . The Hausdorffdimension ‫ܦ‬ ு is the value of ‫ܦ‬ at which ‫ܯ‬ jumps from ∞ to 0, while the dimension ‫ܦ‬ can be calculated as (ܰ is the number of small pieces that go into the larger one and ܵ is the scale to which the smaller pieces compare to the larger one). Equivalently for a given precision level ߝ > 0, ܰሺߝሻ satisfies a power law as ߝ → 0 so that In equation (23b), ‫ܦ‬ is a constant called the fractal dimension, which helps to analyze the structure of a fixed multifractal. For a large class multifractals, the dimension ‫ܦ‬ሺߙሻ coincides with the multifractal spectrum. For any ߙ ≥ 0, the set ܶሺߙሻ can be defined as the Hölder exponent ߙ with a fractal dimension‫ܦ‬ሺߙሻ satisfying 0 ≤ ‫ܦ‬ሺߙሻ ≤ 1.
If ‫ܣ‬ is any set inℝ , the Hausdorff measure of a subset ‫ܣ‬ of ℝ is defined by where the infimum extends over all countable covers of ‫ܣ‬ by sets ܿ of diameter ݀ሺܿ ሻ < ߜ, ℎ is monotone, right continuous and ℎሺ0 ା ሻ = 0 .
Let ߤ be a finite Borel measure in ℝ , consider the spherical density of ܷ at‫ݔ‬ given by where ‫,ݔ‪ሺ‬ܤ‬ ‫ݎ‬ሻ is a ball of radius ‫ݎ‬ and center ‫.ݔ‬ The behavior of this density on a set ‫ܧ‬ ⊂ ℝ as a function of ‫ݔ‬ connects ℎሺ‫ܧ‬ሻ with the size of ‫ܧ‬ in some sense. For example, in (25) then where ߣ is a constant [28,29].
In order to apply the density on random sets, we need to construct a suitable Borel finite measure ⋃ on ℝ with support in‫ܧ‬ using the sample path of a process .It is natural to try a measure which is uniformly spread over ‫,ܧ‬ so an obvious candidate for‫ܧ‬ሺ‫ݓ‬ሻ = ܺ ௧ ௗ, where ܺ ௧ ௗ, is a stochastic process in ℝ of index d , is the occupation measure defined by Equation (28) is relevant to Hausdorff measure.
Let ሺℝ , ߚሺℝ ሻሻ be a measurable space and let ݂: ߚሺℝ ሻ → ℝ be a measurable function. Let ߣ be a real valued function defined on ߚሺℝ ሻ. Then the multi-fractal spectrum with respect to the function ݂and ߣis defined by ‫ܦ‬ሺߙሻ = ߣሼ‫ߚ߳ݔ‬ሺℝ ሻ: ݂ሺ‫ݔ‬ሻ = ߙሽ.
The multifractal formalism of multi-affine functions amounts to compute the so-called singularity spectrum‫ܦ‬ሺߙሻ defined as the Hausdorff dimension of the set where the Hölder exponent is equal to ߙ [33].
To obtain the function ݂ሺ‫ݔ‬ሻ = lim sup ்ሺሻ ሺሻ in our case, we require the local asymptotic behavior of the sample path of the process.And what comes to mind is the subject of the law of iterated logarithm (LIL).
To this end we assume a double stochastic integrals by a direct adaptation of the case of the Brownian motion and set In what follows, we now state;

Theorem 1:
For ‫ݐ‬ > 0 and ℎሺ‫ݐ‬ሻ = ‫ݐ2‬ log ‫݈݃‬ ଵ ௧ , the so-called singularity spectrum ‫ܦ‬ሺߙሻ defined as the Hausdorff dimension of the set where the Hölder exponent is equal to ߙ is given by

Proof
Let ݀ be a predictable process valued in a bounded interval ሾߙ , ߙ ଵ ሿ for some real parameters 0 ≤ ߙ ≤ ߙ ଵ , and ܺ ௧ ௗ ≔ ‫‬ ‫‬ ܽ ܹ݀ ܹ݀ ௨ For the first inequality, we have by the law of the iterated logarithm for the Brownian motion.
where ݀ ሙ = ߜ ିଵ ሺߙ ത − ݀ሻ is the value inሾ−1,1ሿ. It then follows from the second inequality that; For the prove of the second inequality, we assume without loss of generality that ‖݀‖ ∞ ≤ 1.

The particular case
When the mean interest rate of some stocks does not depend on other stocks in the market, we consider again the following stochastic differential equation (SDE); Equation (35) is our so called multifractal spectrum model defined in continuous time. The most probable path ߮ሺ‫ݐ‬ሻ associated with this equation satisfies Equation (36) means that an investor has invested his money in a stock with a linear mean return ߤ ௧ and volatility ߪሺ‫ܯ‬ ௧ ሻ and his real return rate is most likely to be given by Instead of the usual ߤ.
The Ito's Formula on (35) leads to the model equation where ܹ = ܹሺܲ ௧ , ‫ܭ‬ ௧ ሻ is the investment worth and ‫ܭ‬ ௧ is the investment over period ‫.ݐ‬ Following the assumption in [22] and the references therein, we have (40) reduced to an ordinary differential equation of the form The solution of (39) by the method of change of variable using Euler's substitution is; with where is the positive characteristics root of (41).
We have assumed here that ܹሺܲሻ is twice differentiable such that Using equations (7), (15), (19) and (21), we have (40) as with (using 45b) Equation (46) is the worth growth rate of an investor under MSM.
Under equilibrium condition, the discounted profit from a unit capacity at ܲ must be equal to the expected unit cost ܲ ത of the risky stock option. This therefore implies that by (47), we have Solving for ‫ܣ‬ in (47) and (48) and equating the results gives;

The general case
In the real world in general, markets are neither ideal nor complete. Therefore the model equation (40) can hardly be seen as real world behaviour of a stock market price.

∆௧
ቁ is the rate of stock price changes at time ‫.ݐ‬

Theorem 2
Let ܹሺܲ, ‫ݐ‬ሻ be the investment output, ‫ݎ‬ the linear discount rate (as in equation (7)) and ܲ the stock prices. The PDE (58) Proof: To remove ‫ݎ‬ from (54), set and so that (54) becomes; By the method of separation we obtain [34], we obtain a special solution of the form; Using equations (7), (18) and (19) (7), (18) and (19)). We therefore have the future price as In general we have the growth rate of the investor's portfolio as (using equations (59) and (56))
Our objective here is to find the optimal value function such that

Conclusion
The multi-fractal spectrum model has some successes in explaining excess volatility of stock returns compared to fundamentals and negative Skewness of equity returns as well as generating multi-fractal spectrum. By (50) there is no market signal as it tends to zero, meaning that the market is likely to crash at such point, signifying insolvency in asset returns. Equation (23a) can be represented by a renormalized probability distribution of local Holder exponents, called the multifractal spectrum by the form Denoting by ܰሺߙ, ∆‫ݐ‬ሻ the number of intervals ሾ‫,ݐ‬ ‫ݐ‬ + ∆‫ݐ‬ሿ required to cover ߬ሺߙሻ we can write (23b) ܰሺߙ, ∆‫ݐ‬ሻ~ሺ∆‫ݐ‬ሻ ିሺఈሻ .
The market prices correspond to the values of ߙ between ߙ and ߙሺ݂ ௫ ሻ [36].
Furthermore, equation (71) can be solved for the value function ‫ܪ‬ and replaced in (72) in order to obtain the optimal investment strategies.
We have only consider herein, the continuous case. An on-going work considers an extension where we will derive the MSM with the discrete version and establish its comparative effectiveness with the continuous case. An empirical illustration to compare both the MSM for equity returns and those in existing literature will be shown.